(x) dx
for any function (p(x) G Go°(G). Consequently, the function v(x) is the weak derivative D^u(x) of u(x] in G. Corollary. Let u(x], v(x) G L\OC(G}. We assume that there exists a sequence {um(x)} G L\OC(G) such that each function um(x) has the weak derivative D"um(x] in G. We assume that um(x}^u(x)
in
Li oc (G),
D%um(x)->v(x)
in
L\OC(G] (7.3)
as 77i —> oo. Then the function u(x) has the weak derivative D"u(x) in G and D%u(x) = v ( x ) . PROOF. The assertion follows from the theorem. Indeed, by (7.3), for
32
1. Preliminaries
any function
—>•
/f
u(x}(p(x}dx,
G
! um(x}D«v(x}dx= (-1)H I D^um(x}^(x}dx-^
(-1)H f
v(x)(p(x}dx
G
as ra —>• oo.
D
Theorem 7.2. Lei a function u(x) have the weak derivative D%u(x) of class Lp(G), I ^ p < oo, i n a domain G. Then in any interior subdomam G' C G the following convergence takes place: (x} - D?u(x), LP(G')\\ -> 0,
h-*Q.
(7.4)
PROOF. For the sake of simplicity, we consider the average Uh(x] with the kernel (2.2) and a - (1, . . . , 1) in (2.3). We first establish that for 0 < h < p(dG, dG'} the following identity holds: ( D g n ) h ( x ) = D£uh(x),
x£G',
(7.5)
i.e., in any interior subdomain, the weak differentiation operators commute with the averaging operators provided that h is sufficiently small. On the basis of the properties of averages, we have
I
(7.6)
J
If 0 < h < p(dG, dG'), then for any x 6 G1 the function K((x - y ) / h ) (regarded as a function of y) belongs to the class C^°(G). By the definition of weak derivatives in (7.6), we have
•Since x G G' is arbitrary, we obtain (7.5). Taking into account (7.5) and Theorem 2.2, we obtain (7.4). D Based on the above theorems, it is easy to establish properties 5-7 of weak derivatives formulated at the beginning of the section. PROOF OF PROPERTY 5. Let G' be a bounded interior subdomain of G. By (7.5), for 0 < h < p(dG}dG'} we have D X ] u h ( x ] - 0, j = 1, . . . , n, x e G'. Since uh(x] 6 C°°(G), we have
u h ( x ) = c(h),
x&G1,
(7.7)
§ 8. Sobolev Spaces
33
where the constant c(h) depends on h. By properties of averages, we also have ||u/,(z) — u(x), Li(G')|| —>• 0 as h —> 0. Therefore, (u/ l (x)} is a Cauchy sequence, i.e., Jhlh3 — \\uhi(x) — Uh3(x), Li(G')\\ -> 0 as hi, /i2 ->• 0. On the other hand, from (7.7) it follows that J/u/i 2 — l c (^i) — c(h2)\fJ>(G') —)• 0 as hi,h% —> 0. This means that (c(/i)} is a Cauchy sequence. Consequently, there exists a constant c such that c — limc(A). By the uniqueness of limit, u ( x ) = c for almost h—¥Q
all x £ G'. Since G' is arbitrary, the function u ( x ) is constant almost everywhere. D PROOF OF PROPERTY 6. From the properties of averages as h -> 0 it follows that Uh(x)ijj(x) —> u(x)ij)(x) in L\OC(G) and Uh(x}DXi^(x] —>• u(x)DXiif>(x) in L\OC(G). By Theorem 7.2, we have DXi(uh(x))i{j(x) —>• in L\OC(G) as /i —>• 0. Therefore,
in L\OC(G) as h —>• 0. By the corollary to Theorem 7.1, the function u(x)t{)(x) has the weak derivatives in G; moreover, DXt(u(x)^(x}) = D X t ( u ( x ) ) i ^ ( x ) + u(x}D ii>(x). D PROOF OF PROPERTY 7. Consider the functions u h ( y ] = uh(a~l(y)) and Vih(y) = DXiUh\x=a-T.^, h > 0. Since y = a(x) is a diffeomorphism of class C1, we conclude that Uh(y) G Cl(Gy). From the properties of averages and Theorem 7.2 we conclude that Uh(y) —> u(y) in L\oc(Gy) and Vih(y) —* ^'(a" 1 ^)) in Lioc(Gy) as h —t 0. Applying the classical chain rule, we find n i=l
in L\oc(Gy) as A —>• 0. By the corollary to Theorem 7.1, the weak derivatives of u(y) exist in the domain Gy\ moreover,
1=1 The proof is complete.
D
§ 8. Sobolev Spaces In this section, we introduce the Sobolev spaces Wlp(G] and recall wellknown results concerning Sobolev spaces. In the isotropic case, these spaces
34
1. Preliminaries
were introduced by S. L. Sobolev [1-3]. The theory of distributional functions, the theory of function space of different! able functions and their applications to partial differential equations have been constructed on the basis of the results obtained in the cited works. Let / = ( / i , ... ,ln) be a vector with natural components, 1
we obtain a linear normed space called the Sobolev space and denoted by Wlp(G}. We will denote by \\u(x], Wlp(G}\ the norm (8.1). If not all components of the vector / are the same, then the spaces Wp(G) are often referred to as amsotropic Sobolev spaces. In this case, functions u(x) £ Wp(G) can have different properties along different directions. Theorem 8.1. The Sobolev space Wp(G) is complete. PROOF. Consider a Cauchy sequence { u r n ( x } } £ Wp(G), i.e., ||w l ( x ) ~~ u ( x ) , Wp(G) | —>• 0 as z, k —>• oo. By the definition of the norm (8.1), the sequences {um(x}}, {Dxjum(x}}, j — 1, . . . ,n, are Cauchy sequences in the space Lp(G) and have limits u(x), Vj(x), j = 1, . . . , n, respectively since Lp(G] is a complete space. By the corollary to Theorem 7.1, u(x) possesses all the weak derivatives DX}U(X] — Vj(x] in G. Therefore, k
as m —> oo.
D
The detailed study of Sobolev spaces and further references can be found in O. V. Besov, V. P. Il'in, and S. M. Nikol'skii [1], V. G. Maz'ya [1], S. M. Nikol'skii [1], S. L. Sobolev [3, 10], E. M. Stein [1], H. Triebel [1], S. V. Uspenskii, G. V. Demidenko, and V. G. Perepelkin [1]. Properties of the spaces Wp(G) depending on the domain G are also considered there. In particular, a large class of domains G such that the properties of the spaces Wp(G) are similar to those of the spaces Wp(Rn) has been studied. In this section, we consider the Sobolev spaces Wp(G) for such domains G. Definition 8.1. Let Q.J — l / l j , 0 < ctj < bj, j = 1 , . . . , n. A domain of the form n (V/ ,)< J/ ) =
a I\ JI {*•&y G K 'n : aJ? < yJh >< 6.-. &Ji J ' jJ = I ,' . . . ,n} ' J
(8.2) V /
is called an l-horn and the domain x + H(l,8) is called an l-horn with vertex x.
§ 8. Sobolev Spaces
35
Definition 8.2. We say that a subdomain G of a domain G satisfies the l-horn condition relative to G if there exist a = ( a i , . . . ,an), b — (61, . . . ,& n ), and S > 0 such that for any x 6 G the /-horn x + !!(/,<£) is contained in G. Definition 8.3. A change of coordinates y = p ( x ) , x 6 Gx, y G Gy, is said to be invariant relative to the class Wp if the following conditions are satisfied: (a) the mapping Gy is a diffeomorphism of class Ck , k = max /,-. l^i^n
(b) for any function u(x) G Wp(Gx) Cl ||tt(y),
Wlp(Gy}\\ <: \\u(x), Wlp(Gx}\\ <: c 2 p(y), Wlp(Gy)\\t
where u(y) = u((p~l(y}} and the constants c2 > GI > 0 are independent of u(x). We note that the transformation yj — ( — \ } k j x j , j — 1, . . . , n, is invariant relative to the class Wp for any / = (/i, . . . ,ln). Definition 8.4. We say that a domain G satisfies the l-horn condition if there exist finitely many subdomains GI, . . . , GN such that N
G=\jGk,
(8.3)
k =l
and each subdomain Gk (possibly, after an invariant transformation relative to the class Wp) satisfies the /-horn condition relative to the domain. Definition 8.5. We say that a domain G satisfies the strong l-horn condition if G satisfies the /-horn condition and there exists d > 0 such that, together with the representation (8.3), the following relations hold: N
G=\J G M>
Gk,d = { x E G k : p(x, G\Gk) > d}.
In the isotropic case l\ — . . . — ln, the domain (8.2) is a cone. In this case, domains satisfying the /-horn condition are referred to as domains satisfying the cone condition. Domains satisfying the /-horn condition were first introduced by 0. V. Besov and V. P. II 'in [1] who considered properties of anisotropic Sobolev spaces. Examples of domains satisfying the strong /-horn condition for any / are the entire space M n , the subspace R£, a parallelepiped with faces parallel to the coordinate planes. However, the disk {x\ -f- x\ < 1} satisfies the strong /-horn condition only if l\ = /2- Many examples of such domains are contained in 0. V. Besov, V. P. Il'in, and S. M. Nikol'skii [I],
36
1. Preliminaries
We formulate a number of important theorems concerning the Sobolev spaces Wp(G) for domains G satisfying the /-horn condition. We introduce the notation a — (QI, . . . ,an), QJ = l / l j , j — 1, . . . ,n, n a a
\= E ij=i
Theorem 8.2. Let (3 = (fa , . . . , /?„), /?,- ^ 0 6e integers, j = 1, . . . , n. If (3 a ^ 1 — |a|(l/p — l/^), 1 < p ^ 9 < cxo; then any function u(x] G Wp(G) has the weak derivative D%u(x) G Lq(G) and \\D^u(x),Lg(G)\\ ^ c \ \ u ( x ) , Wp(G)\\, where the constant c > 0 is independent of u ( x } . Theorem 8.3. Let u(x) G Wlp(G), 1 ^ p < oo, /? = (A, • • • , /?n), /% £ 0 6e integers, j = 1 , . . . , n . If /3a < 1 — |a|/p, £/ien f/ze wea& derivative D@u(x) can be changed in a set of measure zero in such a way that D%u(x) G C(G); moreover, sup \D%u(x)\ ^ c||u(x), iy'(G)||, wAere i/ie r£G
constant c > 0 z's independent of u ( x ) . Theorems 8.2 and 8.3 generalize the embedding theorems due to S. L. Sobolev [2, 3] to the isotropic case. By Theorem 8.2, for domains G satisfying the /-horn condition the norm (8.1) is equivalent to the norm
Theorem 8.4 (on extension). Let a domain G satisfy the strong l-horn condition. Then there exists a linear continuous extension operator IT : , Kp 0 be integers, j = 1, . . . ,n. If 0 = 1 — (3 a — \ a \ ( l / p — l / q ) > 0, 1 < p ^ q < oo, then any function u(x) G Wp(G) has the weak derivative D%u(x) G H^(G], A = (Ai, . . . , X n ) , Xj — IjO for Ij9 < 1 and Xj < 1 for Ij9 ^ 1, and the following estimate holds:
where the constant c > 0 is independent of u ( x ) . Theorem 8.6. Let u(x) G Wp(G), 1 < p < oo; and let /3 = (A, . . . ,/?„), /3j ^> 0, j = 1, . . . , n, be integers. If 0 = 1 - /3a - \a\/p > 0, then the
§ 8. Sobolev Spaces
37
weak derivative D%u(x) can be changed on a set of measure zero in such a way that D%u(x) £ CX(G], A = ( A i , . . . , A n ), \j - Ij9 for Ij9 < I and Xj < I for Ij0 ^ 1; moreover, \\D%u(x), C*(G)\\ ^ c\\u(x), WJ,(G)\\, where the constant c > 0 is independent of u ( x ) . Theorems 8.2, 8.3, 8.5, 8.6 are usually referred to as embedding theorems in different metrics. To conclude the section, we formulate the embedding theorem in different metrics and dimensions or the trace theorem. We first formulate the following assertion. Theorem 8.7. Let (3 = ( / ? ! , . . . , # » ) , fa ^ 0, j ( « ! , . . . ,am), m < n. If 9 = I -pa - \a\/p+ \a'\/q > 0,
l , . . . , n , a1 =
1 < p ^ q < oo,
(8.4)
then for any function u ( x ) £ C^°(Mn) and plane Em = {x — (x',x") 6 M n ) x' = (xi,... ,xm), x" — (xm+i,... ,xn) = const} the following estimate holds: \\DPxu(x',x"), Lq(Rm)\\ <: c \ \ u ( x ) , Wj(M n )||,
(8.5)
where the constant c > 0 is independent of u ( x ) and x". By this theorem, functions u(x] £ Wp(1$Ln), as well the weak derivatives D%u(x), can be defined on any plane Em with the help of the notion of the trace u\ D u\ Let u(x] £ W'(R n ). Consider a sequence {uk(x)} 6 C£°(Rn) such that fe ||u (x) — u ( x ) , Wp(Mn)|| —> 0 as k -> oo. If a multi-index 0 satisfies (8.4), then from the estimate (8.5) for any k\ and k% we obtain the inequality 3 k2 l M D/XU ,/! (X (T' , Xr"\) — —U nPn (r' (W m)l\ \\\ ^ < C\\U r\\iikl (X) IT} — U i/k2(r\ Ml \\L> (X , Xr"\J , LTq(M. (X), W VVp(]\$ (\&n)\\> XU
where the constant c > 0 is independent of k± and ^2- Consequently, [D!xuk(xl, x")} is a Cauchy sequence in the space L 9 (lR m ). Since the space -L q (M m ) is complete, there exists a function vp(x'} 6 L g (M m ) such that \\D%uk(x', x") - v p ( x ' } , L g (M m )|| -^ 0 as k -> oo. Definition 8.6. A function VQ(X') is called the trace of u(x] G Wp(Rn) on the plane Em and the function vp(x'}, \f3\ > 0, is called the trace of the derivative D^u(x) on Em. We use the notation vp(x'} = D^.u\Em- The trace D%u\Em is well defined, i.e., the function vp(x') is independent of the choice of an approximate sequence {uk(x)}, and [(^(a?'), L g (M m )|| $C c||u(x), W'(M n )||, where the constant c > 0 is independent of u ( x ) . Consider an arbitrary function u(x) 6 WL(G}, where the domain G C M n satisfies the strong /-horn condition. Repeating the above arguments, we can define the trace of a function u(x) and its derivatives on the plane crosssections Gm = G fl Em. Similar estimates hold: \\D%U\G , Lq(Gm}\\ ^ c\\u(x)tWl(G)\\.
38
1. Preliminaries
Stronger estimates for traces can be obtained on the basis of the Besov spaces Brq(Gm} (cf. O. V. Besov [1]). Let r = ( T I , . . . , r m ) , TJ = TJ + Xj, TJ ^ 0 be integer, 0 < Xj ^ 1, j = I , . . . , m, 1 < q < oo. Definition 8.7. By a Besov space Bq(Gm) we mean the linear normed space of functions v(y] £ Lq(Gm} with the norm
sup where 0 < 6 < oo, k = 1 for 0 < Xj < 1 and k — 2 for Xj — 1, j — 1, . . . , m. Theorem 8.8. Let u(x) G Wlp(G], let a domain G satisfy the strong l-horn condition, and let Gm — G fl Em, m < n. If a multi-index /3 = (j3\ , . . . ,/? n ) satisfies (8.4), then the trace D^.u\Gm = vp(x'} belongs to the space Brq(Gm}, where r = ( n , . . . ,r m ), TJ = IjO, j = 1 , . . . , m; moreover, \ \ v p ( x ' ) , B^(Gm)\\ ^ c||w(or), Wp(G)||, where the constant c > 0 is independent o f u ( x ) . The proof of Theorems 8.2-8.8 can be found in O. V. Besov, V. P. Il'in, and S. M. Nikol'skii [1], S. M. Nikol'skii [1], S. V. Uspenskii, G. V. Demidenko, and V. G. Perepelkin [1]
§ 9. Boundary-Value Problems for Ordinary Differential Equations on the Half-Axis Consider the differential operator with constant coefficients d\
dm
dm~l
We assume that the characteristic polynomial
satisfies the following condition: the equation L m (A) = 0 has no purely imaginary roots. We denote by A^~, . . . , A~ the roots lying in the left halfplane and by A^", ... , A+, v = m — p., the roots lying in the right half-plane. Let 1
k=l
9=1
§ 9. Boundary- Valued Problems for Equations
39
For n = m we set L+(X) = I . In this section, we consider boundary- value problems of the form t>0, .7i-
sup \u(t)\ < oo, t>0
where bj I — j are differential operators of order rrij ^ m — 1. We emphasize \ at J that the number of the boundary conditions at t = 0 is equal to the number of roots of the equation Lm(X) = 0 lying in the left half-plane. We consider the characteristic polynomials bj(X) of the boundary operators in the form
where qj(X) and 0j(A) are polynomials and the degree of 0j(A) is at most H-l. In other words, bj(X) = /%(A) (modL~(A)). Let /3j(X) = 0ji+0j 2 A + , j = 1 , . . . , fi. Introduce the matrix • • • ~T Pi u. •
D
021
02/j
022
We assume that the boundary operators satisfy the Lopatinskii condi-
tion (9.3)
Remark 9.1. By the definition of the matrix B, we can say that the Lopatinskii condition means the linear independence of the polynomials bj(X) by modulo L~(X}. Theorem 9.1. Let f ( t ) = 0. The boundary-value problem (9.1) has a unique solution for any data y>j, j = 1 , . . . , / / , if and only if the Lopatinskii condition (9.3) holds. PROOF. We note that if u(t) is a solution to the equation Lm(£)u = Q \at)
'
(9.4)
and \u(t)\ < oo for t ^ 0, then u(t] is also a solution to the equation - 0.
(9.5)
40
1. Preliminaries
Indeed, since the symbol Lm(\) is represented as the product of two mutually prime polynomials L~ (A) and L + (A), we can write u ( t ) in the form u ( t ) = ( c l U i ( t ) + ... + CnU-(t)) + ( d l u f ( t ) + ... + dl,u+(t)) = u-(t) + u+(t), where i/ + // = m and {u^(t}, . . . , u~(t}} is the fundamental system of solutions to equation (9.5), whereas {u*(t),... ,u+(t)} is the fundamental system of solutions to the equation L+ ( — 1 u — 0 (u+(t] = 0 for // = m}.
\dtj
For the fundamental system of solutions to equation (9.5) we can take the system {(ex:>t,tex:>t,...,tm)-1ex^t), j — l,...,r}, where rrij is the multiplicity of the root AJ , mj + . . . + mr — JJL. Similarly, for /j, < m the system of functions {(e A * f ,texk i } , . . ytnk~lexk £ ) ( k = 1, . . . ,p}, where n^ is the multiplicity of the root Ajj", n\ + . . . + np = v, form the fundamental system of solutions to the equation L"1" I — I u = 0.
dt
Taking into account that Re A^~ < 0, we conclude that | u ~ ( < ) | < oo for t ^ 0 for any constants GI, . . . , c^. On the other hand,, since Re A " > 0, we have |u + (/)| —> oo as t —> -foo if at least one constant of d\} . . . } d ^ differs from zero. Therefore, if u(t] is a bounded solution to equation (9.4) for t ^ 0, then d\ = . . . = d^ — 0, i.e., u ( t ) = u~(t}. Consequently, u ( t ) satisfies equation (9.5). We note that if u ( t ) is a solution to equation (9.5), then it is a bounded solution to equation (9.4) for t ^ 0. From the representation of the symbol bj(X) in the form (9.2) it follows that
By the above arguments, we conclude that the solution to the boundaryvalue problem (9.1) for f ( t ) = 0 is equivalently reduced to the solution of the boundary-value problem
,/i.
t > 0, This problem is equivalent to the problem
du — = Au, at
t > 0,
Bu
(9.6)
where
A-
0
0
...
1
— a~
— a~_,
. ..
—a i .
and the entries a • , j — 1, ... , /^, are coefficients of the polynomial
M L-(A) = JJ(A - A-) = A^ + a^-1 + ... + a~.
(9.7)
§ 9. Boundary- Valued Problems for Equations
41
Thus, for f(t) = 0 the boundary-value problem (9.1) is equivalent to the problem (9.6); moreover, if u(t) is a solution to the problem (9.6), then its first component is a solution to the problem (9.1) and u(t) takes the form The solution to the problem (9.6) can be written in the form u ( t ) = etAc\ moreover, the vector c should be found from the system Be = <J>. Therefore, the unique solvability of the problem (9.1) for any v?i, . . . ,f>n is equivalent to the condition det 5 ^ 0 . D Theorem 9. 2. If the Lopatinskii condition (9.3) holds, then the boundary-value problem (9.1) is uniquely solvable for any bounded function f(t) 6 C[0,oo), (pj, j = 1,. . . ,(A; moreover, sup |/(a)| +
|p
(9.8)
where the constant c > 0 is independent of f ( t ) , v ? i , . . . , (p^. PROOF. By Theorem 9.1, it suffices to construct a special bounded solution uo(t) to the nonhomogeneous equation « = /(*).
(9.9)
Then the solution to the boundary-value problem (9.1) can be written in the form u(t) — uo(t) + v(t), where v(t] is the solution to the problem d_ ~dt,
_
fd\ \"V
t=o
sup \v(t)\ < oo. t>0
The Lopatinskii condition guarantees the uniqueness of a solution. To find a bounded solution to equation (9.9), we proceed as follows. Consider the boundary-value problem on the axis teEl
'
SU
P
N<)l<°o-
(9-10)
-CX3
Since the equation Lm(X) — 0 has no imaginary roots, this problem is uniquely solvable for any bounded continuous right-hand side f ( t ) , and the solution can be represented in the form oo
w(t)=
f g(t-s)J(s}ds,
(9.11)
1. Preliminaries
42
where g(t) is the Green function. But we need to define a special bounded solution to equation (9.9) only on the half-axis t > 0. Therefore, it suffices to write a formula similar to (9.11) =
(9.12)
g(t-s)f(s)dS.
This function is a bounded solution to equation (9.9) for t > 0. To derive the estimate (9.8), we write integral formulas for the solution to the boundary-value problem (9.1). We explicitly write the Green function g(t). By definition, it is uniquely determined by the following conditions: d_ •km TJt (b)
sup \g(t}\ < oo, — oo
We denote by F~ a contour in the complex plane that surrounds all the roots A^" , . . . , A~ and by F + a contour surrounding all the roots A]*" , . . . , A+. It is easy to verify that the Green function g(t) takes the form I
t > o,
dX, r-
(9.13)
MA)
d\,
t < 0.
r+ Indeed, condition (a) follows from the Cauchy theorem, condition (b) is obvious by the choice of the contours F + and T~. Since for any contour F surrounding all the roots of the equation Lm(X) — 0 we have yk 1 1
X
Lm(X) condition (c) is satisfied. Taking into account formulas (9.12) and (9.13), we can write a special bounded solution to equation (9.9) in the form 1
r-
r+
§ 10. Boundary- Value Problems For Systems
43
We write integral formulas for the solution to the boundary- value problem (9.1) for f ( t ) = 0. Let 6 fcj be entries of the inverse Lopatinskii matrix B~[, and let a~ be the same as in (9.7). We introduce the polynomials
q=Q
Then the solution to the boundary- value problem (9.1) for f ( t ) = 0 can be written in the form
-
r-
Indeed, by the Cauchy theorem, we have (0 = 5>, Z7 ,•_•, j-i
r-
It is easy to verify the boundary condition since
rFrom (9.14) and (9.15) we obtain the following integral formula for the solution to the boundary- value problem (9.1): t
00
„
00
I f e^~s'^ f 1 : / 7TT~ d\ /(s) ds — I : J 2iri / 2ni J Lm(\) o p t r+ I"
/ *—^ 2m~ JI J- 1 r-
T / \ \ L~(\)
I ^P1 "•
\
J/ r»2m' o r+
Estimating the contour integrals, we obtain the estimate (9.8).
D
§ 10. Boundary-Value Problems for Systems of Differential Equations on the Half-Axis In this section, we consider boundary-value problems for systems of ordinary differential equations of the form
dt
._-=(f>, '-u
sup\u(t)\o
(10.1)
44
1. Preliminaries
where A is an (m x m)-matrix with constant entries dij. We indicate conditions on the boundary matrix B under which the boundary-value problem is uniquely solvable for any bounded functions f ( t ) G C[0,oo) and vector (p. These conditions are called the Lopatinskii conditions. We will assume that the matrix A has no purely imaginary eigenvalues. We denote by A^ , . . . , A~ the eigenvalues of A lying in the left half-plane and by A^", ... , A+, v — m — fj,, the eigenvalues lying in the right half-plane. We assume that 1
t > 0,
(10.2) sup \v(t)\ < oo
(if) = (p — B u o ( 0 ) ) .
t>0
We note that the solution uo(t) can be written in the integral form oo
UQ(t) = f G ( t - s ) f ( s ) d s ,
(10.3)
J
0
tA
where G(t) = e P_ for t > 0 and G(t) = -etA(I-P-} for t < 0; P_ is the projection on the invariant subspace of the matrix A corresponding to the eigenvalues Aj~, . . . , A~ AP- — P-A. To verify formula (10.3), it suffices to note that the matrix G(i) is the Green matrix of the boundary-value problem on the axis — u = Au + g ( t ) ,
sup
— oo < t < oo,
\u(t)\ < oo.
— oo
Let E~ be the invariant subspace of the matrix A corresponding to the eigenvalues A~ , j = 1, . . . , / / . It is obvious that dimE~ = \JL. av Lemma 10.1. The solution v(t) to the system — = Av is bounded for
t ^ 0 if and only if v(0) E E~ . PROOF. The solution to the homogeneous system can be written in the form v(t) - etAP-v(Q) + etA(I - P-)v(0) = V l ( t ) + v 2 ( t ) . There exists a constant 6 > 0 such that | V l ( t ) \ \ ^ \\etAP_\ \v(Q)\\ ^ ce~St\\v(Q)\\ for t > 0. Consequently, if sup ||n(i)| < oo, then the norm of the second term t>o
§ 10. Boundary- Value Problems For Systems
45
\\v2(t)\\ is bounded for t ^ 0. Since (/ - P_) 2 = I - P-, (I - P.)etA = etA(I -P-), e~tA(I - P-}v2(t) = (I- P-)v(O), we have ||(7 - P_)u(0)|| ^ \\e~tA(I - P_)|| \\v2(t)\\. Since \\e~tA(I - P-)\\ ->• 0 as t -> +00, we have (/ — P_)i>(0) = 0, i.e., v(0) G E~ . The converse assertion is obvious. Q By Lemma 10.1, the general formula for the solutions to the homogeneous system that are bounded in the half-axis {t > 0} takes the form (10.4) where Vj , . . . , i^ is a basis for the space E~ . Hence the necessary condition for the unconditional unique solvability of the boundary- value problem (10.2) is that the number of the boundary conditions is equal to fj., i.e., B should be a p. x m-matrix. In this case, the constants Ci, . . . , C M in (10.4) are a solution to the system of linear equations B(v\ . . . v^)c = ip. Therefore, the condition of the unique solvability of the problem (10.2) takes the form detJ3(t;i . . . i ^ ) ^ 0.
(10.5)
This condition is independent of the choice of the basis vi, . . . ,Vp. Theorem 10.1. The boundary-value problem (10.1) has a unique solution for any bounded function f ( t ) 6 C[0,oo) and vector
Chapter 2 The Cauchy Problem for Equations not Solved Relative to the Higher-Order Derivative In this chapter, we begin to study partial differential equations that are not solved relative to the higher-order time-derivative LQ(Dx}D[u +
Li-k(Dx)Dtu = f ( t , x),
x = ( X l , . . . , *„)
(0.1)
k=0
where Lo(Dx) is a quasielliptic operator. As was mentioned in Introduction, such equations appear in applications. In Section 1, we present some examples of equations of the form (0.1) describing physical processes in practice. It is natural to start with general properties of evolution equation by considering the Cauchy problem. There are many works devoted to the construction of the theory of Cauchy problem for equations of the form (0.1) (cf., for example, M. I. Vishik [1], A. I. Vol'pert and S. I. Hudyaev [1], S. A. Galpern [1-3], G. V. Demidenko [4, 5], Yu. A. Dubinskii [2], A. G. Kostyuchenko, and G. I. Eskin [1], A. A. Lokshin [1], A. L. Pavlov [2, 3], S. V. Uspenskii, G. V. Demidenko, and V. G. Perepelkin [1], G. I. Eskin [1], V. R. Gopala Rao and T. W. Ting [1], T. V. Gramchev [1], R. W. Carroll and R. E. Showalter [1], H. A. Levine [1, 2], and others). As is established, if the symbol Lo(i£) of the operator Lo(Dx) is not degenerate for £ G E n , then for certain classes of equations (0.1) it is possible to construct the theory of the Cauchy problem that is similar to the theory for hyperbolic and parabolic equations (cf., for example, L. Hormander [2]). If the symbol LO(Z£) can vanish at a point £ £ M n , no such a theory can be created. This fact was discovered by S. A. Galpern [1-3] who constructed the ./^-theory of the Cauchy problem. In particular, he obtained conditions on the data under which the Cauchy problem is solvable in the Sobolev space W^. The 47
48
2. Equations not Solved Relative to Higher-Order Derivative
solvability conditions for the Cauchy problem in the spaces Wp, I < p < oo, were established by G. V. Demidenko [4]. In this chapter, we consider the Cauchy problem for three classes of equations of the form (0.1): simple Sobolev-type equations, pseudoparabolic equations, and pseudohyperbolic equations (cf. the definitions in Section 2). In Section 3, we study the equations under the assumption that the symbol LO(^) of the operator LQ(DX] is not degenerate at £ £ M n , i.e., we consider equations with invertible operators at the higher-order derivative. Using the Fourier-Laplace transform and theorems about multipliers, it is easy to establish the well-posedness of such equations in the Sobolev spaces Wp like in the case of classical equations. In Section 4, we present the solvability theorems for the Cauchy problem for simple Sobolev-type equations and pseudoparabolic equations in the case where LQ(DX] is a homogeneous quasielliptic operator. This case is essentially different from the previous one by formulation and proof. The reason is that the symbol L/o(i£) is degenerate at £ = 0 and, consequently, it is impossible, generally speaking, to apply standard methods based on the Fourier-Laplace transform. Instead of this, a special regularization is suggested. It is based on some integral representation of summable functions (cf. Section 6). The corresponding techniques and the proof of the statements in Section 4 are explained in the following three sections. In Section 8, we consider equations with variable coefficients and integrodifferential terms. The influence of lower-order terms in equations (0.1) on the well-posedness of the Cauchy problem is also discussed there. In Section 9, we study the Cauchy problem for pseudohyperbohc equations. In Section 10, we give an interesting example of a hyperbolic system in hydrodynamics that is not solved relative to the higher-order time-derivative.
§ 1. Problems Leading to Sobolev-Type Equations Equations that are not solved relative to the higher-order time- derivative appear in a number of applied problems. Poincare was one of the first scientists who considered equations of the form (0.1) (cf. H. Poincare [1]). Later, such equations often attracted an attention of mathematicians and mechanicians. We give some examples. Example 1.1 (the Rossby wave equation). In 1939, Rossby [1] considered the equation AA« + /?Ar 2 u = 0,
n = 2.
(1.1)
This equation appears in the study of some waves in thin layers of a fluid on the surface of a rotating globe. In oceanology, equation (1.1) is known
§ 1. Problems Leading to Sobolev-type equations
49
as the Rossby wave equation. (Hereinafter, A is the Laplace operator with respect to the variables x — (xi, . . . , xn).) Example 1.2 (the Sobolev equation). In the 1940's, S. L. Sobolev studied small oscillations of a rotating ideal fluid. In particular, he derived the equation AD2u + uJ2D23u = f ( t , x ) ,
n = 3,
(1.2)
where u/2 is the angular velocity. S. L. Sobolev [7-9] studied the Cauchy problem, the first and second boundary- value problems for this equation in a cylindrical domain and formulated a number of new problems in mathematical physics. His study was the first deep investigation of equations that are not solved relative to the higher-order time-derivative. In the literature, equation (1.2) is called the Sobolev equation, whereas equations of the form (0.1) are often referred to as Sobolev-type equations. Example 1.3 (small-amplitude oscillations of a rotating viscous fluid). In the study of small- amplitude oscillations of a rotating viscous fluid, the following equation appears: v2A3u + uj2D23u = f ( t , x ) ,
(1.3)
where v > 0 is the viscosity coefficient. Example 1.4 (the gravity-gyroscopic wave equation and the internal wave equation). The study of small- amplitude oscillations in an incompressible rotating ideal exponentially stratificated fluid in the field of gravity force leads to the equation (A - {32} D2u + N2(D2Xi + D22)u + ^Dlu - u2p2u = 0,
n = 3, (1.4)
where /? is the stratification parameter, TV2 is the square of the VaisalaBrunt frequency, ui/2 is the angular velocity. This equation is called the gravity-gyroscopic wave equation or (in the case u = 0) the internal wave equation. We write the gravity-gyroscopic wave equation AD2 u + N2(D2XI + D22)u + u2D23u = 0
(1.5)
and the internal wave equation in the Boussinesq approximation 2
(D2Xi + D2y}u = Q.
(1.6)
50
2. Equations not Solved Relative to Higher-Order Derivative
Example 1.5 (the Boussinesq equation). The following equation of the form (1.4) (0- 2 A-l)D t 2 u + 7 2 A w r = 0
(1.7)
is called the Boussinesq equation. This equation appears in the description of longitudinal waves in rods, in the theory of long waves in water and waves in plasma (cf., for example, H. Ikezi [1] and G. B. Whitham [1]). Example 1.6 (the Barenblatt-Zheltov-Kochina equation). In 1960, the following equation of the form (0.1) was obtained by G. I. Barenblatt, Yu. P. Zheltov, and I. N. Kochina [1]:
(1.8) This equation appears in the study of the fluid filtration in fissured rocks. Equations (1. !)-(!. 6) are discussed in L. M. Brekhovskikh and V. V. Goncharov [1], S. A. Gabov and A. G. Sveshnikov [1, 2], R. K. Dodd, J. C. Eilbeck, J. D Gibbon, and H. C. Morris [1], J. Lighthill [1], P. H. LeBlond and L. A. Mysak [1], Yu. Z. Miropol'skii [1].
§ 2. Classes of Equations not Solved Relative to Higher-Order Derivative In this section, we describe the classes of equations that are not solved relative to the higher-order time-derivatives. We will consider the Cauchy problem and mixed boundary-value problems. We consider the differential operator with constant coefficients l
L(Dt,Dx] = L0(Dx)D t
/-i + Y^Li-K(D*}Dt>
x
= (*i» • • • > * « ) .
(2.1)
fc = 0
We suppose that the following conditions hold. Assumption 1. The symbol L(in, it;), £ = ( £ 1 , . . . , £ n )> of the operator L(Dt, Dx) is homogeneous relative to a =• (QQ, a) = (QQ, &\,. . . , an), where QO ^ 0 and I/a,- are natural numbers, i.e., L(ca°in,cai^) — cL(ir],i£), c>0. Assumption 2. The operator Lo(Dx) is quasielliptic, i.e., Lo(i£) = 0, f 6 M n , if and only if £ = 0.
§ 2. Classes of Equations
51
From Assumption 1 it follows that the symbol Lo(i£) is homogeneous relative to the vector a — (a.\, . . . , a n ) with homogeneity exponent 1 — /ao, i.e., Lo(cai£) = cl~la°Lo(i£), c > 0. Then Assumption 2 implies the twosided estimate
n
_ ,
where (£) 2 — ^ £/.
, £ €i M n and 0 < c\ ^ c2 are constants.
k-\
Definition 2.1. The differential equation L(Dt,Dx)u = f ( t , x )
(2.2)
is called a simple Sobolev-type equation if the operator L(Dt, Dx] satisfies Assumptions 1, 2 and ao = 0. Definition 2.2. Equation (2.2) is said to be pseudoparabolic if the operator L(Dt,Dx) satisfies Assumptions 1 and 2, ao > 0, and the following relation holds:
Definition 2.3. Equation (2.2) is said to be pseudohyperbolic if the operator L(Dt,Dx) satisfies Assumptions 1 and 2, ao > 0, and the following relation holds:
k=0
v
^'
has only real roots relative to r\. If the roots 7/i(£)) • • • > ^(0 are different for £ E Mn\{0}, then equation (2.2) is said to be strictly pseudohyperbolic. We will also consider Sobolev-type equations with lower-order terms
1-1 k=0
i =f(t,x),
(2.3)
i.e., the operator £(Dt, Dx) defining this equation takes the form £(Dt, Dx) — L(Dt, Dx] -f L'(Dt, Dx), where the leading part L(Dt,Dx) is an operator
52
2. Equations not Solved Relative to Higher-Order Derivative
satisfying Assumptions 1 and 2, /-i k=0
is the part containing lower-order terms; moreover, the symbols L ' l _ k ( i ^ ) of "lower-order" operators L'l_k(Dx}^ k = 0, . . . , / , satisfy the estimates
where £ 6 M n , 1 - a 0 A; ^ e > 0, (£) 2 = ^ £tand c > 0 is a constant. t=i We say that equation (2.3) is a simple Sobolev-type equation, pseudoparabolic or pseudohyperbohc if the equation without lower-order terms (2.2) has the corresponding type. We also can define "pseudoparabolic" and "pseudohyperbolic" equations in a similar way as in the case of classical partial differential equations. We note that simple Sobolev-type equations can be regarded as equations that are reducible to ordinary differential equations with bounded operator coefficients in Banach spaces. Indeed, let, for example, the symbol Lo(i^) of quasielliptic operator Lo(Dx) be real. Without loss of generality, we can assume that LQ(I£) > 0, £ G M n \{0}. Then the operator R : Lp(Mn) ->• W£(Rn), r = (1/cn, . . . , l/an), 1 < p < oo, defined by the formula
Rf(x)
= (27T)-/ 2
is a regularizer of the operator LQ(DX), i.e., RoLo(Dx) — I + T, where T is a smoothing operator. Since QQ — 0, the operators Ro(Li-h(Dx}-\-L'l_k(Dx}} : W f ( R n ) ->• W£(l&n), k = 0, 1, . . . , / , are bounded. Consequently, applying the operator /? to equation (2.3), we obtain the ordinary differential equation (/ + T0)Dltu +
7]_fc Dktu = R f ( t , x) k=Q
with bounded operators T^, A: = 0 , . . . , / , in the space 14^"(ffin). Examples of simple Sobolev-type equations without lower-order terms are the Sobolev equation (1.2), the gravity-gyroscopic wave equation (1.5), and the internal wave equation (1.6) in the Boussinesq approximation. The corresponding operators satisfy Assumptions 1 and 2 for LQ(DX) = A, a = (0,1/2,1/2,1/2). The Rossby wave equation (1.1), the gravity-gyroscopic
§ 2. Classes of Equations
53
wave equation (1.4), and the Barenblatt-Zheltov-Kochina equation (1.8) are simple Sobolev-type equations with lower-order terms. Equation (1.3) describing small- amplitude oscillations of a rotating viscous fluid is a pseudoparabolic equation. For this equation £(Dt,Dx) = L(Dt , Dx] + L'(Dt,Dx), where L(Dt,Dx) = ADt2 -2i/A 2 A + ^ 2 A 3 , L'(Dt, Dx) = u; 2 D 2 3 , L0(DX] = A, and a = (1/3,1/6,1/6,1/6). In addition to the differential equations (2.3), the following equations are of interest: /-i )u = (LQ(DX] + L'Q(Dx))Dltu k =0
t
-f- Ll_k(Dx})DfU + 2_] I (t — s}qLi+q+i(Dx)u(s, x}ds q=o{ = /(*,*).
(2.4)
We denote by £)t~1 an integral operator of the form
t D^ u(t,x) = I u(s,x)ds. l
0
Then the operator defining this equation takes the form 1-1 A;=0
We give an example of such an equation appearing in the study of dynamics of a viscous incompressible exponentially stratificated fluid. In the two-dimensional case, the study of small-amplitude oscillations of such a fluid in the field of gravity force leads to the equation
(cf., for example, S. A. Gabov and G. Yu. Malyshev [1]). Making the change up(t, x] — e~PX2u(t, x ) , we write this equation in the form
54
2. Equations not Solved Relative to Higher-Order Derivative
Integrating with respect to t and taking into account the initial conditions, we obtain the equation
The leading part of this equation is defined by the pseudoparabolic operator L(Dt,Dx) = A A - "A2; moreover, a = (1/2,1/4,1/4). We note that a similar equation in the Boussinesq approximation takes the form
ADtu - vk?u + N2D^1D2 u — f(x).
§ 3. Equations with Invertible Operator at the Higher-Order Derivative In this section, we consider the Cauchy problem for equations with lowerorder terms (2.3) under the assumption that the symbol of the operator LQ(DX] + L'0(DX) satisfies the two-sided estimate ci(l + (O1-"*0) ^ IMiO + 4>(''OI ^ c 2 (l + (O 1 -'" 0 ), n
where (<^) 2 = ^ £k
(3-1)
.
> £ ^ ^«
an<
^ <^2 > ci > 0 are constants.
k =l
By the estimate (3.1), a linear continuous operator Lo(Dx) -f L'Q(DX) : W™(Rn) -)• L p (E n ), m = ((1 - laQ)/ai, ... ,(l- Ia0}/an), I < p < oo, possesses the bounded inverse operator (L/o(Dx) + L'Q(DX)}~1 : L p —> Furthermore, , t
We consider the Cauchy problem ' t = 0~
l X
'
r\
i
U)
« -
r\
'
^> ' l
X& -i
"'
(3.2) ^
'
U, . . . , t - 1,
for simple Sobolev-type equations and pseudoparabolic equations. We look for a solution to the problem (3.2) for simple Sobolev-type equations in the function space V K ' > r ( ( 0 , T ) x M n ), r — ( 1 / a i , . . . ,l/an), I < p < oo, under the assumption that the weak derivative Dltu(t, x) belongs
§ 3. Equations with Invertible Operator
55
to Wp°' r ((0,T) x M n ) . Then all the terms (Li-k(Dx] + L\_k(Dx}} D*u(t, x) on the left-hand side of equation (2.3) belong to L P ((0,T) x M n ). By the definition of a solution to the Cauchy problem for simple Sobolevtype equations, the problem (3.2) can be written in the following equivalent form: Dltu + ^Ti-kDktu = (L0(DX) + L ' Q ( D x } ) - l f ( t , x } } k=0
where T/_ fc = (L0(DX) + L'0(DX))-1 Li-k(Dx), k = 0 , 1 , . . . , / - ! , are bounded linear operators in W£(M.n). The following assertion follows from the general theory of ordinary differential equations in Banach spaces. Theorem 3.1. Let f ( t , x ) £ L p ((0,T) x Rn). Then the Cauchy problem (3.2) for a simple Sobolev-type equation is uniquely solvable in the space Wp>r((Q,T} x M n ) and the following estimate holds: IK*, *), W£ r ((0, T) x M n )|| + \\D\u(t, x)yW°>r((Q, T) x M n )||
where the constant c(T) > 0 depends on T. We denote by ^(Mj+1), 1% 0, r = ( n , . . . ,rn), I < p < oo, 7 > 0, the Sobolev space of functions with exponential weight e '*, i.e., u ( t , x ] 6 ) ife-Vu(t,x) G V^' r (M+ +1 ). We introduce the norm
We formulate an assertion about the solvability of the Cauchy problem (3.2) in the half-space M^"+1. By a solution we mean a function u ( t , x ) £ ^(K+ +1 ), r = (1/ai, . . . , l/a n ), such that Dltu(t,x) € W Theorem 3.2. There exists 70 > 0 such that the Cauchy problem (3.2) for a simple Sobolev-type equation is uniquely solvable in the space PVp'^(R* +1 ) ; 7 ^ 70, for any right-hand side f ( t , x ) £ L pi7 (lR^ +1 ) anrf the following estimate holds:
where the constant c > 0 z's independent of f ( t , x } .
56
2. Equations not Solved Relative to Higher-Order Derivative
To illustrate Theorem 3.1, we consider the Cauchy problem tLi-k(Dx)Dtu
= f(ttx),
t > 0,
k=Q
If £ ciijtitj > 0, £ E Mn\{0}, a 0 < 0, and the order of Li-k(Dx) is not larger than 2, then for any function f ( t , x ) E L P ( ( Q , T ) x K n ) this problem has exactly one solution in the space Wp'2((0,T) x M n ). We proceed by considering the Cauchy problem for pseudoparabolic equations. We look for a solution in the class of functions W p 1 i7 (B&^_ ( _ 1 ), r — (1/ai, . . . , l/a n ), 1 < p < oo, under the assumption that the weak derivatives D$u(t,x) belong to Wp;^~ka°)r(^+i), k = 0, . . . , / . Theorem 3.3. There exists 70 > 0 such that the Cauchy problem (3.2) for a pseudoparabolic equation is uniquely solvable in the space W / p ' )7 (ffi^ +1 ) ; 7 ^ 70, for any right-hand side f ( t , x ) £ -Zy pi7 (JR^ +1 ) ; and the following estimate holds:
i /
^ II
t
V '
/ '
P )*Y
. )||,
*•
M-y-l/ll
(3.3)
where the constant c > 0 is independent of f ( t , x } . PROOF. We establish the existence of a solution to the problem (3.2) by the method of Fourier transform. We first assume that f ( t , x) is compactly supported with respect to x. As in the case of the Cauchy problem for hyperbolic and parabolic equations, we consider the Cauchy problem for the ordinary differential equation with parameter £ E M n :
This problem is obtained by the formal application of the Fourier operator with respect to x to the original problem. Taking into account the estimate (3.1), it is easy to show that there exists 70 > 0 such that the Cauchy problem (3.4) has a unique solution v ( t , £ ) E M^ 7 (Mj"), 7 ^ 70, for any f ( t , £ ) E L P)7 (MJ"). It is obvious that
57
§ 3. Equations with Invertible Operator the solution can be written in the form
t (3.5)
(3.6)
r(0 where F(£) is a contour in the complex plane surrounding all the roots of the equation £(ir),i£) = 0 with respect to 77. It is easy to verify this fact using the following assertion. Lemma 3.1. For £ £ M n the contour integral J(t,£) is a solution to the Cauchy problem ,tOv = 0, * > 0 , t_0=0,
j = <),...,/-2,
(3.7)
PROOF. By the definition (3.6) and the Cauchy theorem, we have ,0 = — / eitXdX = Q.
r(0 We verify the initial conditions:
=
(i\)j
f
1
„
*=° 27 / c(ix~iF}dX> £t i\
J
Afc/ I fr/\ j fcL I
r(0
+
where £(iA,tO - (I 0 («) + L'Q(i
+ H_ k=0
Since the contour F(^) surrounds all the roots A of the equation £(z'A,z£) = 0, the value of the integral Ij(£) remains unchanged if this contour is replaced with a circle CR with sufficiently large radius R that surrounds . Consequently, for j ^ / — 2 and any fixed £ £ E n we have
14(01 -
/
-dX
max
0
58
2. Equations not Solved Relative to Higher-Order Derivative
as R —> co, i.e., /j(£) = 0. For j = I — I we have
CR
I
Hence the contour integral (3.6) satisfies (3.7).
D
In what follows, we will essentially use Lemmas 3.2-3.5. Lemma 3.2. There exists a constant 70 > 0 such that the symbol of a pseudoparabolic operator satisfies the two-sided estimate
ci(i + <0 1 "' a o )(H'H-(0' a o )^l^,» I OI ^ c 2 ( l + {0 1 ~' ao )(H' + <0' a o ),
£eMn,
Rer>7o,
(3.8)
where c-i > c\ > 0 are constants. PROOF. The estimate (3.8) from above immediately follows from the homogeneity condition relative to the vector a of the symbol of the principal part of the operator jC(Dt,Dx}. To establish an estimate from below, we write the symbol £(r, i£) in the following form:
1-1
fc=0
We denote by m(r, z'^) the expression in the square brackets and show that for sufficiently large Re r > 0 and any £ E K n the following estimate holds: |m(r,»Ol^l/2.
(3.10)
59
§ 3. Equations with Invertible Operator For £ = 0 this estimate is obvious. Let £ 6 M n \{0}. Then
1-1 k=0
1-1
l-l
-i
1-1
l-l
l-l k=0
fc=0
(-1
-i
k=0 /-l
-1
1-
k=0 l-l
l-l
k=0
k=0
By the homogeneity with respect to the vector a of the symbol L/_fc(z'£), k — 0, . . - , / , of the quasielliptic operator LQ(DX) and the inequality (3.1), we have 1-1 k=0
A;=0
l-l
where a i , a 2 , £ , £ A : > 0 are constants. Since the principal part
1-1 k=0
satisfies the estimate (3.11)
60
2. Equations not Solved Relative to Higher-Order Derivative
where Re r ^> 0, £ G M n , and the constant as is independent of parameters, there exists 70 > 0 such that for any Re r ^> 70, £ € M n \{0) the inequality (3.10) holds. Taking into account (3.1), (3.10), and (3.11), from (3.9) we obtain the estimate (3.8) from below. D Lemma 3.3. Let 70 > 0 be the constant in Lemma 3.2. functions
^ (£('&> + 7, 'O)" 1 ,
ka0 + pa^l,
k^l,
Then the
(3.12)
for 7 ^ 70 are multipliers on L p (M n + i). PROOF. Using Lemma 3.2, it is easy to obtain the uniform estimates ^(^o + 7, t'O)" 1 )! ^ c < oo,
(fr,0 € Mn + i,
7 2 70,
for any (cro,0~), where cr; = 0 or 0 be the constant in Lemma 3.2. 7 ^ 70, £ 6 M n
Then for
CO
1
'
1,l^r
.
(3-13)
PROOF. Let l > °- Dtv t-t=o" n= °> A: = 0 , . . . , / - 1 . By (3.5), we have t V(t,t)=
Jj(t-T,t)
We apply the Laplace transform with respect to t to this integral. By Lemma 3.2, the integral
0
for 7 ^ 70 is defined in view of the properties of the Laplace transform
§ 3. Equations with Invertible Operator
61
On the other hand, applying the integral Laplace operator with respect to t to the differential equation and taking into account the zero initial conditions, we find ,*0 f e-^+^tv(t,^)dt = f eSince (p(i] is arbitrary, from these relations we obtain (3.13).
D
Lemma 3.5. Let 70 > 0 be the constant in Lemma 3.2. Then for
(3.14)
(^ +
^ (3.15)
PROOF. We consider the integral
0
k = 1, . . . , / — 1. Integrating by parts, we find /* ("7 + 7, »0 - -Di-lJ(t, 0 | t=0 +fo + 7)/*-i (ii? + 7, «0By Lemma 3.1, Ik(irj + -y,i^} = (^ + 7)7/5-1(^ + 7,^). Integrating k times and using Lemmas 3.1 and 3.4, we obtain (3.14). The identity (3.15) follows from (3.14) in view of Lemma 3.1: /-i - 0. D
By Lemma 3.2, we can apply the inverse Fourier transform with respect to x to the integral (3.5) and thereby obtain a formal solution to the Cauchy problem (3.2) in the form u(t,x) = (27r)-n/2 f eixtv(t,t)d£.
(3.16)
62
2. Equations not Solved Relative to Higher-Order Derivative
We establish estimates for the integral (3.16) which imply the existence of a solution to the problem (3.2) in the Sobolev space Wp'^(M+ +1 ), 7 ^ 7o, for any right-hand side f ( t , x ) £ L P)7 (]R^" +1 ). In what follows, we extend the function f ( t , x ) by zero for t < 0 and preserve the same notation for the extended function. We denote by /7 (77, £) the Fourier transform with respect to ( t , x ) of the function e~~1tf(t,x] and by f-y(t,^) the Fourier transform with respect to x of the same function. We estimate u ( t , x ] in three lemmas. Lemma 3.6. L e t f ( t , x ) € L P>7 (1R+ +1 ) ; 7 ^ To- Then for (3 = (/?i , . . . ,/?„), f3a <; I , the following estimate holds: \\D?u(t, x), L P)7 (R+ +1 )|| ^ c \ \ f ( t , x), Lp^(R++1}\\,
(3.17)
where the constant c > 0 is independent of f ( t , x). PROOF. It suffices to consider the case f ( t , x ) E C^°(M^ +1 ). By the definition (3.16), we have
We multiply this equality by e '*. Using the above notation and the Heaviside function d(t), we find
By the formula of the Fourier transform of convolution, this expression can be written in the form
X 0
By Lemma 3.4, we obtain the representation
§ 3. Equations with Invertible Operator
63
By Lemma 3.3, we obtain the inequality (3.17).
CD
Lemma 3.7. Let f ( t , x ) E L P|7 (1R^ +1 ), 7 ^ 70- Then ||X>} U (f,*),L p i 7 (R+ + 1 )|| ^ c\\f(t,x),LPn(R++1)\\,
(3.18)
where the constant c > 0 is independent of f ( t , x ) . PROOF. It suffices to consider the case f ( t , x ) £ Co°(IR+ +1 ). Using Lemma 3.1, we can write the derivative D\u(t, x) in the form
Dlt u(t, x] = (27T)-/ 2 / j* r—±—-f(tt 7 LO(Z^) + L 0 (z^J
0 ffin
By the inequality (3.1) and the Lizorkin theorem, we have \\u\t, x), L P|7 (M+ +1 )|| ^ where the constant c > 0 is independent of f ( t , x ) . Consider u2(t, x ) . Multiplying by e~~1t and using the Heaviside function 6 ( t ) , we find c ' •Uj / 2 f(I, /
/I
/I e*
By the formula of the Fourier transform of convolution, this expression can be written as follows: 7r)-(" +1 )/ 2 / /c'X'eo Ki E n oo
x
fe-^+^DltJ(r, \ •/ o
By Lemma 3.5, 2 U
(i, ar) - ( 2 7 r ) - n + 1 2
e°+((i^Q + 7)
64
2. Equations not Solved Relative to Higher-Order Derivative
Using Lemma 3.3, we find
where the constant c > 0 is independent of f ( t , x } . Taking into account a similar estimate for ul(t,x), we obtain (3.18). D Arguing as above, it is easy to establish the following assertion. Lemma 3.8. Let f ( t , x ) E L p , 7 (IRj +1 ), 7 ^ 70- Then for k ^ I and 0 = (0i, . . . ,0n), such that ka0 + 0a ^ I , the following estimate holds:
where the constant c > 0 is independent of f ( t , x ) . By Lemmas 3.6-3.8 , for any function f ( t , x ) G L pi7 (M^ +1 ) having compact support in x the integral (3.16) is a solution to the Cauchy problem (3.2) of class ^(R+ +1 ); moreover, Dktu(t,x] 6 Wp^~ka°}r (Mj +1 ), k = 0, . . . , / , and the estimate (3.3) is valid. Since the set of compactly supported functions is dense in L P|7 (K+ +1 ), this assertion is also valid for any / ( * , * ) € £ P , 7 (M+ +1 ). To prove the uniqueness of the solution to the problem (3.2), we establish the following assertion. Lemma 3.9. I f u ( t , x ) £ W_^(IR^ +1 ) is a solution to the Cauchy problem (3.2) with f ( t , x ) = Q, then u ( t , x ) - 0. PROOF. We first assume that the solution u ( t , x ) has compact support with respect to x. Then the Fourier transform of u with respect to x is a solution to the Cauchy problem for the following ordinary differential equation with parameter £ € M n : £(Dt,i£)v = Qt
00,
^Nt=0=0,
k = 0, . . . , / - 1.
By (3.1), the coefficient at the higher-order derivative Dltv differs from zero. Since the solution to the Cauchy problem is unique, we have v ( t , £ ) — 0. But, in this case, u(t,x) is the zero solution. By the above arguments, a compactly supported in x solution to the problem (3.2) is unique. By Lemmas 3.6-3.8, any w(t,x) 6 PFp'^(IR^"+1) with compact support in x D3tw ( _ 0 — 0, j = 0, . . . , / — 1, having the weak derivatives D^D^w(t,x) 6 L P ] 7 (M+ + 1 ), ka0 + 0a ^ I , Q ^ k ^ I, satisfies the inequality liiid T} W YV p I UJ{1 ,
n \ ft
\ T
/TO)"T" ^11
/o io^
§ 3. Equations with Invertible Operator
65
where the constant c > 0 is independent of w(t, x). Consider the general case. Let u(t,x) be a solution to the homogenous Cauchy problem (3.2) of class W^(M^ +1 ). For any e > 0 we construct a compactly supported with respect to x function ue(t,x) £ Wp'^(M^ +1 ), D ue
t \t=o= °> .7 = 0, . . . , J - 1 such that D^D^u£(t,x) € L P|7 (M+ +1 ), ka0 + f3a ^ I , k = 0,... ,1, and
<: 6.
By (3.19) the following estimate holds:
where the constant c > 0 is independent of £ and u£(t,x). Since £(Dt, Dx}u(t, x} = 0, we find
c\\jC(Dt,Dx)u£(t, x),
Taking into account the form of the operator £(Dt,Dx) and the quasihomogeneity condition for the leading part L(Dt, Dx}, we find
^ \\(Li-k(D k=0 I
E
E
k=0
6a^.l-ka0
I
k=0 8
Since e is arbitrary, we conclude that l l w ^ , ^ ) , Wp'^(M^ +1 )|| — 0.
D
By Lemma 3.9, a solution to the Cauchy problem (3.2) is unique in the class W p '£(IR+ +1 ). Theorem 3.3 is proved. D
2. Equations not Solved Relative to Higher-Order Derivative To illustrate Theorem 3.3, we give the following example: iijDXiX . + GO I Dtu + / ^ bpDxit = f ( t , x ) ,
t > 0,
|/3|^4
It
= 0.
If X^
ai
'J^^J
>
°>
XI ^^
<
°'
^
€
Mn\{0),
GO < 0,
then the operator
is pseudoparabolic, the symbol of the principal part
] + y ^ 0/3^ ,
V^
L
C0
\/3\=4
satisfies Assumptions 1, 2 in Section 2 with a = (1/2, 1/4,. . . , 1/4), and the symbol n
satisfies the estimate (3.1). Consequently, by Theorem 3.3, for any function f ( t , x ) £ L p i 7 (M^ + 1 ), 7 ^ 70, this Cauchy problem has a unique solution in the space W^^(R^+l}.
§ 4. Sobolev-Type Equations without Lower-Order Terms In this section, we consider the Cauchy problem L(Dt,Dx)u = f ( t , x ) ,
t>0, x 6
(4.1)
for simple Sobolev-type equations and pseudoparabolic equations without lower-order terms. We note that the study of the Cauchy problem for simple
§ 4. Sobolev-Type Equations without Lower-Order Terms
67
Sobolev-type equations or pseudoparabolic equations with nonzero initial conditions D^u t_Q= k(x), k = 0 , . . . , / — 1, is reduced to the study of problems of the form (4.1) by the change
We look for a solution to the problem (4.1) in the same classes of functions as in Section 3. However, as we will see below, the formulations of the well-posedness theorems significantly differ from those in Section 3. In particular, the problem (4.1) is not necessarily solvable in all Sobolev spaces of the scale Wp'^(M* +1 ) and for the proof of the existence theorem it is required that the right-hand side to be orthogonal to some polynomials. The number of these conditions is finite and depends on a and p. This essential difference from the results of Section 3 is caused by the fact that the operator L0(DX) : W*(Rn) -> Lp(Rn), s = ((I - /a 0 )A*i, • • • , (1 - l<*o)/an), 1 < p < oo, is not continuously invertible. We introduce the notation p' = p/(p — 1)> r — ( l / ^ i , . . . , l/an), \a\ = n
n
.
-\ i \1 v~* 27/ a , * i , . . . ,an}, (xy - 5J x,- .
Let us formulate some results for the problem (4.1) for simple Sobolevtype equations. Theorem 4.1. Let \a\/p' > 1. Then there exists 70 > 0 such that the Cauchy problem (4.1) for a simple Sobolev-type equation is uniquely solvable in the space Wp'^(M.^+1), 7 > 70, for any right-hand side f ( t , x ) 6 L P?7 (R^ +1 ) H Lpt-y(R^; Li(K n )) ; and the following estimate holds:
<: c(\\f(t, x), L pi7 (M+ +1 )|| + || ||/(*, x], L!^))), where the constant c > 0 is independent of f ( t , x ) . Theorem 4.2. Let \a\/p' ^ I , and let N be a natural number such that \a\/p' + Nam-in > 1 ^ \<*\/p' + (N — l)a m in- Then there exists 70 > 0 such that the Cauchy problem (4.1) for a simple Sobolev-type equation is uniquely solvable in the space W^(M+ +1 ) ; 7 > 70, for any function f ( t , x ) E LP|7(1R^+1) such that
(4.2)
68
2. Equations not Solved Relative to Higher-Order Derivative
The solution satisfies the estimate
where the constant c> 0 is independent of f ( t , x ) . We formulate similar assertions for pseudoparabolic equations. Theorem 4.3. Suppose that \a\/p' + lao > I , f ( t , x ] 6 L p?7 (M^ +1 ) fl Z/ P|7 (]R] I ~; Li(M n )) ; 7 > 0. Then there exists a unique solution u ( t , x ] £ Wp'^(M.n+i) t° the Cauchy problem (4.1) for a pseudoparabolic equation, and the following estimate holds: l>r
(R+
i }\\ -f + V^ / ^
]7^n_|_1J||
^ c(\\f(t, x), L P , 7 (M+ +1 )|| + where the constant c > 0 z's independent of f ( t , x ) . Theorem 4.4. Let \a\/p' + laQ ^ 1, and /e£ N be a natural number such that \a\/p'+laQ + Nam-m > 1 ^ \a\/p'+la0-\-(N— l)a m i n - Then the Cauchy problem (4.1) for a pseudoparabolic equation is uniquely solvable in the space W^(R++1), 7 > 0, for any function f ( t , x ) e LP^(R++1), (1 + ( x ) } N \ a \ f ( t , x) G Lp ] 7 (IRj";Li(]R n )) ; satisfying the orthogonality condition (4.2) and the following estimate holds:
i Ml -i- V^ II nkn(i
x), L P)7 (R+ +1 )|| + II ||(1 + ( x ) } a f ( t , x), Li(M n )||, where the constant c > 0 is independent of f ( t , x ) . These assertions follow from the general theorems formulated at the end of this section. From Theorems 4.1 and 4.3 it follows that the Cauchy problem (4.1) is well defined in the space Wp'^(S&n+l) if |a|/y 4- ^o > 1- In the case \ct\lp1 + IctQ ^ 1, from Theorems 4.2 and 4.4 it follows that the Cauchy problem is solvable in the space W'pj^(]R^ +1 ) if the right-hand side of the equation satisfies the additional orthogonality conditions (4.2). The following natural question arises: Are the conditions (4.2) essential for the solvability of the Cauchy problem (4.1)? The following theorem partially answers this question.
§ 4. Sobolev-Type Equations without Lower-Order Terms
69
Theorem 4.5. Let \a\Jp'+ laQ ^ 1, a 0 ^ 0, <*i = . . . = an, p ^ 2. TAen £Ae orthogonality conditions (4.2) are necessary for the solvability of the Cauchy problem (4.1) in the space Wp'^(M^ +1 ), 7 > 70PROOF. Assume the contrary. Let |a|/p' + QI + ICCQ > 1 ^ |c*|/p' -f IctQ. We assume that there exists a solution u(t, x) E Wp'^(M^"+1) to the problem (4.1) but f ( t , x} E QTO^n+i) does not satisfy the condition (4.2) for N = 1, i.e., F
fl
(1 x) T\ Hr (i, ax ^L ^ u.
(4. ^} (1.6)
Applying the Fourier operator and the Laplace operator to (4.1), we find L(r,i£)u(r,£) = f ( r , £ ) ,
r = irj + a, a > 70, £ E M n .
(4.4)
By assumption, the following estimate holds: ||w(f, x), V^^(]R^+1)|| ^ c(f) < oo, p -3O. / i [ ||«(i»7
i
f
\ »/
\
/ti
*
CO.
£>0 J
— 00
For f ^ 0 from (4.4) it follows that u(r,£) = f(r,£}/L(r,i£}. assumptions on the operator L(Dt, Dx) imply the inequality
Since the
where 0 < GI ^ 02, on any interval (a, 6), the following estimate holds: °-lf(r^},Lp,(e
< (0 < iWdrj
where c > 0 is a constant. We represent the function f(r, £) in the form n
f ( T , t ) = f(r,W + '£,tj / J=i o
}
Daif(Ti8)\t_^d\.
Using the Minkowski inequality, from the last estimate we find
6
c J + c'CfllKO" 14 -"*'- 1 , Lp,(e < (0 <
70
2. Equations not Solved Relative to Higher-Order Derivative
By the assumptions J < oo and |a p' + a\ + la® > 1, the right-hand side of the last inequality is finite for e 0. On the other hand, by (4.3), there exists an interval (a, 6) such that
Since 1 ^ M/V + lao, the left-hand side unboundedly increases as £ —> 0. We arrive at a contradiction. We treat the case JV > 1 in a similar way. D Remark 4.1. From the proof of Theorem 4.5 we see that the orthogonality conditions (4.2) are necessary for the solvability in Wp'^(M* +1 ), p
L(Dt,Dx)u+
%u = f ( t , x ) ,
t > 0,
We show the validity of Theorems 4.3 and 4.4 for this problem below. We illustrate the above arguments by an example of the Cauchy problem for the equation of small-amplitude oscillations of a rotating fluid (1.2), (1.3): l3u = f ( t , x),
t>0,
(4.5)
We recall that this equation is a simple Sobolev-type equation a = (0, 1/2,1/2, 1/2) for v — 0 and pseudoparabolic equation a = (1/3, 1/6, 1/6, 1/6) for v > 0. Therefore, r = (2, 2, 2) for v = 0 and r = (6, 6, 6) for v > 0. By Theorems 4.1 and 4.3, the problem (4.5) is uniquely solvable in the space Wp2 ( l R j ) for p > 3. Theorems 4.2 and 4.4 imply that for 3/2 < p ^ 3 this problem is solvable in the space if f f ( t , x ) d x = Q,
(4.6)
J
la
whereas for 1 < p <J 3/2 the following conditions are required: f
f
I f ( t > x ) d x = I X i f ( t , x ) d x = 0,
i = 1,2,3.
§ 4. Sobolev-Type Equations without Lower-Order Terms
71
By Theorem 4.5, these conditions are also necessary for the solvability of the problem (4.5) in the space Wp 2 ^(IR^) for 1 < p ^ 2. It is easy to establish the necessity of the condition (4.6) in the case 2 < p 0, x <= M 3 ,
« t = 0 = Dtu\t=Q= 0.
Let f ( t , x ] = %(/)• oo takes the form
We estimate the integral
\x-y\ in the L p -norm. Taking g(s) £ CQ°(]RI), 0 ^ g(s)
/
||$(x),
K3
\x-y\
g(\x - y\)(y) . _ ,_ x-y\ dy, L p (M 3 )
By the Young inequality, we have -,Li(M 3 ) (4.7)
Consequently, if p > 3, then
72
2. Equations not Solved Relative to Higher-Order Derivative
If p ^ 3, then the estimate (4.7) does not imply ||$(z), L p (E 3 )|| ^ c(p) < oo. We show that for the validity of this inequality the following condition is necessary: (:r) for |x| ^> 1p in the form I
I
I
— y| 7/1
Since for \y\
2p
\y\
1
we have $i(z) E inequality, we find
> 2p) for any p > 3/2. Using the Minkowski
1-1>Lp(|x|>2p)
\ \ $ ( x ) , Lp(\x\
, L p (|x| > 2p)\\ < oo.
However, for p
|x| > 2p), and
ip(x] dx ^ 0.
Consequently, we again see that the Cauchy problem in the space Wp2',£(IRj) is unconditionally solvable for p > 3 and is solvable under the orthogonality conditions on f ( t , x) for p oo with a certain rate. Furthermore, the decay rate depends on the vector a = (a 0 , «i, • • • , <*n) and increases if the right-hand side f ( t , x ) has the zero moments with respect to x. This conjecture helps us to understand why the number of additional
§ 4. Sobolev-Type Equations without Lower-Order Terms
73
solvability conditions for the Cauchy problem (4.1) in the space Wp can decrease as the exponent p increases. Therefore, it is reasonable to study the solvability of the problem (4.1) in larger function spaces, for example, in Sobolev spaces with power weights with respect to x. Furthermore, it is reasonable to expect that the problem is solvable under less restrictive conditions on the right-hand side f ( t , x ] . We will see that this fact takes place for special weight Sobolev spaces Wp'^ j(T (IR* +1 ). We define the weight Sobolev space Wlp^ >a(l&n+i)> 1 < P < °°, 7 > 0, >£>!«(* > x), L P (M+ +1 )||
From the definition of the spaces W j ; ^ i < y ( K + 1 ) and W j £ ( M + 1 ) it follows that <,;,o(M++1) - W&(R+ +1 ) and W&(R++l) -> W^a(R++1). We will denote by Li i 5 (M n ) the space of summable functions
Definition 4.1. By a solution to the Cauchy problem (4.1) we mean a function u(t,x) G PVp'^ a (M^ +1 ) having the weak derivatives D ^ D f U ( t , x) G Lp i7 (M^ +1 ) for kao + /3a = 1, k = 1 , . . . , / , and satisfying (4.1) almost everywhere. Theorem 4.6. Let \a\ > 1, and let \a\/p > a > 1 — |a|/p'. Then there exists 70 > 0 such that the Cauchy problem (4.1) for a simple Sobolevtype equations is uniquely solvable in the space W^ a (K^ +1 ), 7 ^ 70, o - ^ l , /or /(<,x) 6 I P)7 (IRJ +1 ) n L p i 7 ( M f ; L i i _ f f ( M n ) ) , anrf f/ie following estimate holds:
IK*, x), ;7 ^ c(||/(<, x), where the constant c > 0 z's independent of f ( t , x ) . Theorem 4. 7. Lei |a|+/a:o > 1, |a|/p > l-/a 0 -|a|/p / ; and f ( t , x ) eLp^(^+l)r}Lpi^t\Llia(iao.1}(Rn)), 7 > 0, o-^ 1. Then the
74
2. Equations not Solved Relative to Higher-Order Derivative
Cauchy problem (4.1) for a pseudoparabolic equation is uniquely solvable in the space Wlp^ CT(M^"+1) and the following estimate holds:
\\u(t, x), ^ : ; iff (K+ +1 )ii +
i
+ II 11(1 + (*)r(1-lao}f(t, x), Li(R where the constant c > 0 zs independent of f ( t , x ) . We return to the Cauchy problem (4.5) and illustrate the formulated theorems. As was already mentioned, the unconditional solvability of this problem in the Sobolev spaces W_ 2 '^(R* +1 ) holds for p > 3, whereas, in the case p 0. By Theorems 4.6 and 4.7, the Cauchy problem (4.5) is unconditionally solvable in the weight Sobolev spaces W^'^ ff(R^} for any 1 < p < oo, 1 — I&Q — \a\/p' < 0 such that for any right-hand side of the form f(t,x)=
52
^Fft(t}x)eLp^(R++l)t
/3a = l
Fp(t,x)€Lpn(R++1),
7>7o,
the problem (4.1) for a simple Sobolev-type equations has a unique solution u ( t , x ) E W'^IR^+J, 7 > 7o; and the following estimate holds: \ \ u ( t , x},W^(R++l) | + \\D\u(t, x), W^(R++l}\\
where the constant c> 0 is independent of f ( t , x ) . Theorem 4.9. Let f(t,x)=
52
DZF(,(t,x)eLP
f3a = l-la0
),
7>0.
§ 5. Approximate Solutions
75
Then the Cauchy problem (4.1) for a pseudoparabolic equation is uniquely solvable in Wp' 7 (E+ +1 ) ; 7 > 0, and the following estimate holds: u(t, x), Wj;;(M+ + OH + £ \\Dkt u(t, x), W P p °,l{ 1 - fcao)r (K++1 A:=0
where the constant c > 0 zs independent of f ( t , x ) . The above theorems are proved in the following three sections.
§ 5. Approximate Solutions to the Cauchy Problem for Equations without Lower-Order Terms In this section, we sketch the proof of the theorems formulated in Section 4. To prove the solvability of the Cauchy problem (4.1)
Dx}u = /(i, x),
t > 0, x E M n ,
we construct approximate solutions by a method developed by G. V. Demidenko [2, 4]. To fix an idea, we assume that a function f ( t , x) is compactly supported with respect to x. We consider the Cauchy problem for the ordinary differential equation with real parameter £ which is obtained by the formal application of the Fourier operator with respect to x to the problem (4.1):
, 0
A"t=o= «
,
(5.1)
*= <),...,/-I.
By the conditions on the operator L(Dt, Dx] indicated in Section 2, the co/-i efficient Lo(i£) in the equation L(Dt,i£)v = Lo(i^)Dltv+ ^ Li-k(i^)D^v = k=0 ^ f ( t , £ ) is degenerate at the point £ — 0. Therefore, we consider the problem It is easy to show that there exists 70 > 0 such that this problem has a unique solution v ( t , £ ) G Wp^Rf), 7 ^ 70 for any f ( t , £ ) 6 L p , 7 (M+). The
76
2. Equations not Solved Relative to Higher-Order Derivative
solution can be written in the form (5.2)
t)dT, 0
•"'•<)=s /!(&)<"•
(5 3)
-
r(0
where F(£) is a contour in the complex plane surrounding the roots of the equation L("MO=0,
£eMn\{0},
(5.4)
relative to 77. This assertion is verified with the help of the following assertion which is proved in the same way as Lemma 3.1. Lemma 5.1. For £ 6 M n \{0} the contour integral J ( t , £ ) is a solution to the Cauchy problem L(Dt,i£)v = 0, t > 0,
In the case of the Cauchy problem for a simple Sobolev-type equation, we can take a contour F(£) independent of £. This is true because the symbol of the operator LQ(DX] satisfies the estimate 2
n
c i < O ^ M « O l ^ c 2 ( 0 , <0 = £**2/afc' c 2 ^ C l > o . fc=l
(5.5)
Since the homogeneity vector of the symbol L(in,i£} takes the form a = (0,a), the symbols of the operators Li-k(Dx) satisfy the inequality
Consequently, the roots of the equation
are bounded. We also note that for t > 0 the following estimate holds: 1^c(0-V<,
(5.7)
§ 5. Approximate Solutions
77
where d, c > 0 are constants. In the case of the Cauchy problem (4.1) for a pseudoparabolic equation, for F(0 in (5.3) we can take the boundary of the domain G(£) = {Im A > 5{£}a°}n{|A| < 2r(£) a °}, where r > 8 > 0 are some constants. Indeed, from the definition of the paseudoparabolic operator L(Dt, Dx) it follows that the roots 771 (0> . . . , r//(0 of equation (5.4) lie in the upper half-plane {Im 77 > 0}. Since they are homogeneous relative to the vector o; = (Q\, . . . ,an) with exponent aQ, i.e., rjk(ca£] = ca°r]k(^), c > 0, there exist r and 6 such that 0 < 2£<0 ao ^ Im ^(0 ^ MO I ^ r(0 ao ,
*=!,...,/.
(5.8)
Therefore, r)k(t) £ <7(0By the above arguments, for the pseudoparabolic operator L(Dt,Dx) the contour integral (5.2) for t > 0 satisfies the estimate
WU)l^c(O a °- 1 e-"«> 00 , £eM n \{o},
(5.9)
where the constant c > 0 is independent of £. We proceed by constructing a solution to the Cauchy problem (4.1). Applying the inverse Fourier operator with respect to £ to the function v(t,£) in formula (5.2), we can obtain a formal solution to the Cauchy problem (4.1) for simple Sobolev-type equations or pseudoparabolic equations without lower-order terms. However, the estimates (5.7) and (5.9) provide us with a precise rate of growth of the contour integral J ( t , £ ) as |£| —> 0. Therefore, the function v(t,0 has, in general, nonintegrable singularity at £ = 0. This fact demonstrates the essential difference between the classes of equations without lower-order terms and classes of equations considered in Section 3. Therefore, to obtain a formula for the solution to the problem (4.1), it is necessary to regularize the inverse Fourier operator. For this purpose, we use the integral representation of
v(x) = iim(27r)- n j v-w-1 t / »/ h
*/ »/ KnKn
(5.10) where (7(0 = 2N(£)2N exp( — (02JV) and the limit is understood in the sense of convergence in L p (M n ). Recall that the natural number JV can be taken so large as desired. We set N > n + 1.
78
2. Equations not Solved Relative to Higher-Order Derivative
Consider a sequence of functions {um(t, x)} of the form
l/m
0 Kn.
(5.11) By the estimates (5.7), (5.9) and the definition of the kernel G(£), the functions um(t,x) are well defined. Since formula (5.2) expresses the solution to the problem (5.1), um(t,x) is a solution to the Cauchy problem
L(Dt,Dx)um = fm(t,x), Dktu t = 0
< > 0 , x^mn,
where
l/m
Inffin
By the representation (5.10), we have lim f m ( t , x ) = f ( t , x ) . Consem—>oo quently, the function um(t,x) can be regarded as an approximate solution to the problem (4.1). In the following two sections, we establish estimates for the functions um(t} x] which imply the existence of a solution to the Cauchy problem (4.1) in the corresponding spaces. In the proof of these estimates, we essentially use the assertion about the symbols of the differential operators L(Dt,Dx) and the contour integrals J(2,£). Lemma 5.2. Let L(Dt,Dx) be a simple Sobolev-type operator. Then there exists a constant 70 > 0 such that the symbol of this operator satisfies the two-sided estimate + l),
(5.12)
where £ £ M.n, Re r > 70, and c2 > c\ > 0 are constants. PROOF. For f e M n \{0}, Re r > 0 we have
Taking into account the inequalities (5.5) and (5.6), for sufficiently large Re r we obtain (5.12). For £ = 0 this estimate is obvious. D
§ 5. Approximate Solutions
79
Lemma 5.3. Let L(Dt,Dx) be a pseudoparabolic operator. symbol of this operator satisfies the two-sided estimate
Then the
where £ G M n; Re r > 0, and C2 > GI > 0 are constants. PROOF. For £ G M n \{0) we have /-I
k=Q
/-AN
r
uv
^'
By the conditions on the symbols L(in,i£), LQ(I^) in Section 2, we have fli^)1"'"0
^ IM'OI
£ e ^n,
(5.14)
where 0,1 ^ a\ > 0 are constants. On the other hand, by definition, for the pseudoparabolic operator L(Dt,Dx] we have l-l
^
,.£,
l
~k^)rk + 0, Re r ^ 0, ^ € ^n, |r| + |^| / 0.
Since the function M(r, £) is homogeneous with respect to the vector a with exponent /ao, the following two-sided estimate holds:
where 62 ^ 61 > 0 are constants. Taking into account the inequality (5.14), we obtain (5.13). For £ = 0 this estimate is obvious. D Lemma 5.4. Let L(Dt,Dx) be a simple Sobolev-type operator, and let 70 > 0 be the constant in Lemma 5.2. Then the functions £o£^(^( z £o + 7) z '£))~ 1 > 7 ^ 7o, ft a — 1, fc 0, /?ao + ^a ^ 1, /: ^ /, 1 — lao ^ /?a ^ 1; are multipliers on L p (M n + i). We note that Lemmas 5.4 and 5.5 follow from Lemmas 5.2 and 5.3 since the corresponding functions satisfy the assumptions of the Lizorkin theorem about multipliers (cf. Chapter 1, Section 4). Lemma 5.6. Let L(Dt,Dx) be a simple Sobolev-type operator, and let 70 > 0 be the constant in Lemma 5.2. Then for 7 ^> 70, £ G M n \{0) we
80
2. Equations not Solved Relative to Higher-Order Derivative
have
= (in + i)k(L(ir, + 7, i*))- 1 , k = 0, . . . , / - 1, (5.15) = (iq + 7 )'(Lto + 7, tf))-1 - (M*))'1 • (5-16)
Lemina 5.7. Le£ L(Dt,Dx) be a pseudoparabohc operator. 7 > 0, £ G K n \{0} Me identities (5.15) anrf (5.16) /ioW.
Then for
Lemmas 5.6 and 5.7 are proved in the same way as Lemmas 3.4 and 3.5. Lemma 5.8. Let L(Dt,Dx) be a simple Sobolev-type operator, and let 7o > 0 be the constant in Lemma 5.2. Then for t ^ 0 the contour integral J ( t , £ ] satisfies the estimates
where 7 ^ 7o; ^ € M n \{0} ; k = 0, 1, . . . , / ? = (/?i, . . . , /?„) and c k i p , 8>0 are constants. Lemma 5.9. Le^ L(Dt,Dx) be a pseudoparabohc operator. Then for t ^> 0 Me contour integral J ( t , £ ) satisfies the following estimates:
7 > 0, ^ € M n \{0} ; ^ - 0 , 1 , . . . , J3 = (0lt... , / 3 n ) , A = 7 1 Q ° + (0, and Cktp, & > 0 are constants. Lemmas 5.8 and 5.9 are proved in the same way as Lemma 5.9. PROOF OF LEMMA 5.9. By Lemma 5.1, the contour integral J ( t , £ ) can be regarded as a solution to the Cauchy problem
t=o=
0
'
J = 0,...I
§ 6. Estimates for Approximate Solutions
81
where 771 ( £ ) , - • • ,ty(0 are roots of (5.4). Then u 7 ,/(i,0 = is a solution to the problem
t
I'1 t=0
'
•*
)••• i
>
Consider the solutions u7 m(t,£}, m = 1 , . . . , / — 1, to the Cauchy problems
= 0, j = 0 , . . . , m - 2 , It is obvious that
t 7.
// e\ _ / ^.mV 1 ) s; — /
m — —
Since the roots ^(0 satisfy the estimates (5.8), from these recurrent formulas we obtain the inequalities 771— 1
-y,m',
, I771
.
,
.
- 1 /-
Taking into account the equality e~ 7t
)! ^ '(7 + {O ) -'^ ' ' ^^ ^ " ^. * 2 o.
which implies the required estimate for k = |/?| = 0. The general case is considered in a similar way. D
§ 6. Estimates for Approximate Solutions In Section 5, we constructed the sequence of functions {um(t,x)} of the form (5.11) regarded as approximate solutions to the Cauchy problem (4.1) for simple Sobolev-type equations and pseudoparabolic equations without lower-order terms. In this section, we estimate the sequence
82
2. Equations not Solved Relative to Higher-Order Derivative
{um(t,x)}. We also prove that this sequence is a Cauchy sequence in the We divide the proof of estimates for um(t,x) in several lemmas. All estimates for approximate solutions to the equations under consideration are established in the same method. Therefore, the corresponding lemmas are formulated for both classes. We will indicate differences in the proof. In the sequel, we extend f ( t , x ) by zero for t < 0 and preserve the same notation for the extended function f ( t , x ) . We denote by f-y(rj^) the Fourier transform of e ~ ' y t f ( t , x ) with respect to ( t , x ] and by / 7 (£,£) the Fourier transform of e~~itf(t,x] with respect to x. Hereinafter, 70 is the parameter defined in Lemma 5.2 for simple Sobolev-type equations and 70 = 0 for pseudoparabolic equations. Lemma 6.1. L e t f ( t , x ) E L p , 7 (M+ +1 ) ; 7 > To- Then for (3 = (ft , . . . ,f3n) /3a = I , the following estimate holds: ^c||/(*,z),Z, p , 7 (IR+ + 1 )||,
(6.1)
where the constant c > 0 is independent of m and f ( t , x ) . Furthermore, ^ n + l ) l l ~~* 0>
m
!'
m
2 ~~^ °°-
(6-2)
PROOF. It suffices to consider f ( t , x ) (E Co°(M^ +1 ). By the definition (5.11), um(t,x] is infinitely differentiable with respect to x. Moreover,
I/TO r
e
0 lnKn
7T
r
x (i^Ye~^^~ ^ J(t — r, £) ~ /( > y) d^dydrdv.
(6-3)
Using the above notation, the properties of the Fourier transform, and the Heaviside function 0(t), we can write this expression in the form
l/m
x
Ereln
eiys(isfe(t-T}e-^t-^J(t-T,s}J^(T,s}dsdT\d^dydv.
§ 6. Estimates for Approximate Solutions
83
The function in the brackets will be denoted by $ 7ii g(i, y). Then
= (27r)- n / v~l j I ei(x-yKG(£va)^!f3(t,y)d£dydv.
(6.4)
ffinKn
l/m
We establish the inequality \\*i,tt(t, x), Lp(Rn+1)\\
(6.5)
where the constant cp > 0 is independent of f ( t , x ) . Using the Fourier transform, we write the function $^$(t,y) in the form - (27r)-( n+1 )/ 2 /
fe^tso+y^(isf
Ml ffin OO f
x f
t
\
e-(iso+^TJ(T,s)dT]f^(sQ,s)dsds0.
\ J
/
Taking into account Lemmas 5.6 and 5.7, we can write the last formula in the form I f e^tso ffii Kn
x (L(is0 +i,is))~1 f ^ ( s 0 , By Lemmas 5.4 and 5.5, the functions (j,p(sQ,s} = (is)P (L(iso + j,is))~l , 7 > 70, fto. = 1, are multipliers, which implies the inequality (6.5). Using (6.5), it is easy to establish the estimate (6.1) and the convergence (6.2). Indeed, using the identity m
J v-iG(sv«)dv = e-m-2N(^N-e-m2N(^N,
(6.6)
l/m
we can write formula (6.4) in the form = C2rr}-n f(
J \J
En
En
-(27r)-» [( J
f e^-^e-™-™^
\ J
fe^-^se-m2N^N
ds^i/3(t, y] dy
84
2. Equations not Solved Relative to Higher-Order Derivative
Using the Minkowski inequality, the Young inequality, and (6.5), we find
\\Jlim(t, x),
£ let
+U
It is obvious that the constant c > 0 is independent of m and f ( t , x ) . The convergence (6.2) holds because, by the integral representations (5.10) and (6.4), the sequence {e~~ftD^um(t,x)} converges to the function $ g £ x i n t h eI M - n o r m . D Lemma 6.2. Let f ( t , x ) e L P?7 (IR+ +1 ), 7 > 70- Then for k ^ / and (3 — ( / ? i , . . . ,/3n) such that kaQ + /3a = I , I - Ia0 ^ /3a; ^e following estimates hold: ||D*^« m (* ) x) > L p , 7 (M+ + 1 )|| ^ c||/(/,x),L p , 7 (M+ + 1 )||,
(6.7)
where the constant c > 0 is independent of m and f ( t , x ) . Furthermore, )|l->0
(6.8)
as mi , m2 —> oo.
. Consider the most
PROOF. It suffices to show that /(/, x) G interesting case k = I. By Lemma 5.1,
l/m
m
t
l/m
= ^(t.x) +
(6.9)
85
§ 6. Estimates for Approximate Solutions
We estimate each term on the right-hand of (6.9) separately. Using the properties of the Fourier transform, we can write the first term v ^ f t ^ x ) in the form vlm(t,x) = C2Krn I v-1 J J e^ l/m
)-"/ 2
Enln
J
feiys-^LQ(IS)
We denote by Fp(t, y] the function in the brackets. Then m
-1 j j ei(x l/m
(6.10)
EnEn
By Assumptions 1 and 2 in Section 2, the functions LQ(IS)
(6.11)
,
are multipliers. Therefore,
where the constant cp > 0 is independent of f ( t , x ) . Using this estimate, it is easy to obtain the inequality t
,x,
P ) 7 + 1
c
,x,
pi7+1
)||,
(6.12)
where the constant c > 0 is independent of m and f ( t , x ) . Indeed, using (6.6), we can write formula (6.10) in the form
Arguing in the same way as in the proof of Lemma 6.1, we obtain the inequality fei
86
2. Equations not Solved Relative to Higher-Order Derivative
which implies (6.12). Consider the second term f ^ ( t , x) on the right-hand of (6.9). Using the properties of the Fourier transform and the Heaviside function, we can write e~'ytv^n(t, x) in the form
l/m
lnffin
eiys(is)f3d(t-r}e-^t-T^DltJ(t-T,s}f^(T,s}dsdT^d£dydv.
x / f Kl Kn
We denote by $7 /j ;(t, y) the functions in the brackets. Then ee'^v"(2; m(t ^ t , x} xj m
n
= (27r)- y v~l f f e i ( x - y K G ( t v a } 3 > ^ p i l ( t , y } d t d y d v . l/m
(6.13)
lnffin
We prove the inequality )!!,
(6.14)
where the constant cpj > 0 is independent of f ( t , x ) . Using the Fourier transform and Lemmas 5.6 and 5.7, we can write $ 7 i / g i /(^, y] in the form
OO
x ( f \
J
e-(-is°+^TDlTJ(T,S)dr]f^(s0,S)dsds /
0 (zs))
1
)J F 7 (so,
By Lemmas 5.4 and 5.5, the functions t4pti(so,s) = (is}13(iso + *y}1 (L(isQ + 7, is))" 1 , 7 > 70, k <^. I, kctQ + (3a = I , I — IctQ ^ /?«, are multipliers on Lp(M n -)_i). Since the functions (6.11) are also multipliers, we obtain the inequality (6.14). Using (6.6) and (6.14), we obtain the following estimate for the function (6.13):
87
§ 6. Estimates for Approximate Solutions
where the constant c > 0 is independent of m and f ( t , x}. From (6.9), (6.12), and the above inequality we obtain the estimate (6.7) for k = I. The convergence (6.8) takes place because, in view of the integral representations (5.10), (6.10), and (6.13), we have
as m —>• oo. The case k < / is considered in a similar way.
n
Lemma 6.3. Let f ( t , x) 6 L P)7 (M+ +1 ) n L p / > (M+; £i l f f (ia 0 -i)OM), 7 > 70- // |a| > 1 - Ia0) \a\/p > (7(1 - /a 0 ) > I - Ia0 - \a\/p', then for f3 = (/?!,. . . , / 3 n ] , f3a < 1 — locQ, the following estimate holds:
where the constant c > 0 is independent of m and / ( £ , x ) ; moreover,
(6.16)
-Df «„,,(/, x ) ) , L p i 7 ( M + + 1 ) | | - > 0
as mi, m2 —>• oo. PROOF. By the Minkowski inequality, we have
t
l/m
v-i
+
j-(7(l-/ao-/9a)
J
i , y)
+ 1>
(it;
(6.17)
2. Equations not Solved Relative to Higher-Order Derivative
We estimate each term on the right-hand of (6.17) separately. Using the Heaviside function 0(t), we write the first term 7° m in the form
1/m
, y) d^dydr,
x 0(t -
dv.
Taking into account that a ^ 0, ft a < l—lao and using the Young inequality, we find dv 1/m
(6.18) Introduce the notation
AM = Making the change of variables Sk = write AQ(V) in the form
— x/^v
ak
, k = 1, . . . , n, we
I elzsG(s}(isYe-^J(t, sv~a) ds, LI (M++
sn
(i + M2*)-1 «//((i + (-i)*AV")G( S )(» fi y 3 Consider pseudoparabolic equations. Using the estimates for the contour integrals in Lemma 5.9, the definition of the function G(s) and integrating by parts, we find
§ 6. Estimates for Approximate Solutions
x ZA
'
\s)
&
89
ds, jL/i^iK-j I
where A = 7 1/a ° + v~l(s). Choosing k ^ [n/2] + 1, we find
AO(V) ^ y^
acvy~'
a
/?a
oo
°~ y y i ^
By the definition of the kernel G(s), we have ^o(^) ^ cvl~la°~P°t, where the constant c > 0 is independent of v. Using this inequality, from (6.18) we find
i
f v 1/m
Since /?a < 1 — lao, we find (6.19)
where the constant c\ > 0 is independent of m and f ( t , y). We estimate the second term /° m on the right-hand of (6.17). Using the inequality (x - y)(l + (x))- 1 ^ a (1 + (y)},
a = const,
the Minkowski inequality, and the Young inequality, we find
j
/"
J(x-
<(*-
dydr,
(6.20)
90
2. Equations not Solved Relative to Higher-Order Derivative
dv (6.21) Introduce the notation
Making the change of variables s/. = £,kVak, z/^ = x^v can write B o f in the form .—a(l-la0—/3a)
ak
, k = 1,. . . , n, we
izs fI eizs G(t
, sv~a) ds, Lp(Rn]
Using Lemma 5.9, in the same way as in the proof of the estimate for AQ(V), we find
A (l-/)a0/syao-^a-l,)-<5AQfot
Z-i
ft
L
where A = ^l/a° -\- v l ( s ) . Taking into account the condition \ a \ / p > a(l — /ao) and setting k ^ [n/(1p]} + 1, in accordance with the definition of the kernel G(s) we find B0(v) ^ c v -|«|/p'+(i-a)(i-/«o-/3«) j w here the
§ 6. Estimates for Approximate Solutions
91
constant c > 0 is independent o f f . Using this estimate, from (6.21) we find m
/2,m ^ C f
v-l-\«\/P'
+ (l-*)(l-l«o-P«)dv
1
x I! ||(1 + (y)}a(l-la°-Pa}f(T,
y), Li (M n )||, Lp
By the condition |a|/p' > (1 — cr)(l — /OJQ), we have I",m ^ c2|| ||(1 + (y}r(l-la°-^f(r,
y), Lx 0MI
where the constant c-2 > 0 is independent of m and f ( r , y ) . By the last inequality and (6.19), from (6.17) we obtain (6.15). We can establish (6.16) in a similar way. The case of simple Sobolev-type equations is treated in a similar way, but Lemmas 5.8 should be used instead of Lemma 5.9. D Lemma 6.4. Let f ( t , x) e £ P , 7 (M+ +1 ) n I P|7 (M+; Li i f f ( t a o -i)(M n )), 7 > 70- // |a| > 1 - Ia0, \Q\/p > cr(l - /a 0 ) > 1 - I&Q - \a\/p', then
+ 1111(1 + (x)rl-aof(t,
x), L^E,)!!, L P ,,(M+)||),
(6.22)
where the constant c > 0 is independent of m and f ( t , x ) ; moreover, ||(1 + (z})-^1-^^^, x) - &tum,(tt x)), L P17 (R+ +1 )|| -> 0 (6.23) as mi, m-2 —>• co.
we find
l/m
InKn
+ (27T)-" l/m
0
92
2. Equations not Solved Relative to Higher-Order Derivative —
7,1 (/ Tf)
\
)
T
} _L /
"^
2 7;
(i
7T7 V
r\
)
/ *
(((
94^
lWt^"l
We estimate each term on the right-hand of (6.24) separately. We first consider the function f ^ ( t , x ) . By the Minkowski inequality, the Young inequality, and (6.20), as in the proof of Lemma 6.3, we have \\(l + (x})-a^-la^v1m(t,x),Lp
I
dv\\f(t,y),L
l/m
I'v~l (I + (z))-^ 1 -' 00 ) /'e ia *G(£w a J J
dv 2,m.
Ki
By the definition of the kernel G(^) and the fact that the symbol LQ(I£) is quasihornogeneous, the term I\ m can be written in the form
l/m
x ||/(*, I/), L P ,
Since /a 0 < 1, we have /i ) m ^ c \ \ \ f ( t , y ) , L P)7 (M^ +1 )||, where the constant GI > 0 is independent of m and f ( t , y ) . Similarly, the term /2 >m can be written in the form =C
x HIK1 Since \ct\/p' > (1 — a] (I — /OQ), &\/P > °"(1 — ^o), we have /2,m ^ C2
where the constant C2 > 0 is independent of m and f ( t , y ) . From these two estimates we find x),
c2
(6-25)
93
§ 6. Estimates for Approximate Solutions
We estimate the function v^(t,:e) arguing in the same way as in the proof of Lemma 6.3. By the Minkowski inequality, we have n+U ,
,,,-<,(!-<„„)
. + <*»
f
f
f
JJJe
i(
0 !„!„
l/m
x G(£>va}DltJ(t — r, £)/(T, y) d£dydr, Lp>
1
m
1
dv
t
+yv (i+(x}}-°^- ^ j j jj(*x
G(^va}D\J(t-T,^f(T,y} (6.26)
Using the Young inequality, for the first term /[ m on the right-hand of (6.26) we derive the estimate
(6-27) l/m
where
For definiteness, we consider pseudoparabolic equations. We write the norm AI(V) as follows:
Ai(v) =
f
,-7«
94
2. Equations not Solved Relative to Higher-Order Derivative
Using the estimates of the contour integrals in Lemma 5.9 and the definition of the function G(s) and integrating by parts, we find
I2*)-1 (1
where A = ^l/a° + s). Choosing k ^ [n/2] + 1 and taking into account the kernel G(s), we can write this estimate in the form
Therefore, ^4;(^j) ^ cu 1 "'" 0 , where the constant c > 0 is independent of v. From (6.27) it follows that
1/m
Since /QIQ < 1, we have l{,m^c(l-la0)-i\\f(r,y),Lpn(R++1}\\.
(6.28)
Arguing in the same way as in the proof of the inequality (6.21), it is easy to obtain the following estimate for the second term I12 m on the right-hand side of (6.26):
(6.29)
where
§ 6. Estimates for Approximate Solutions
95
We write this norm in the form Bl(v) = v-WP'-^-1^
i
} f " J
f
x e"1"£>[,/(<, siT") ds,Lp(M n ) Using Lemma 5.9 and arguing in the same way as in the proof of the estimate for A I ( V ) , we find Bi(v)
Taking into account the condition \ot\/p > a (I — IQQ) and choosing a sufficiently large fc > 0, from the definition of the kernel G(s) we conclude that BI(V) ^ c 2 t;~ | a | / p ' + ( 1 ~ C T : ) ( 1 ~ / a o : ) , where the constant c2 > 0 is independent of v. From (6.29) we find
x || 11(1 + < y > r 1 - ' a o / ( r , y), Since jal/p' > (1 — <J)(1 — /ao), we have
1
-' ao) /(r, y), Ii(M n )||, LP
By the last inequality and the relations (6.25), (6.26), (6.28), from (6.24) we obtain (6.22). We establish (6.23) in a similar way. The case of simple Sobolev-type equations is treated in the same way but Lemma 5.8 should be used instead of Lemma 5.9. D
96
2. Equations not Solved Relative to Higher-Order Derivative
Introduce the function x(ri) e C°°(Kf), 0 ^ X(n] ^ l>
1
{; rir ' 10
for r) ^ 2.
^
Lemma 6.5. Let the assumptions of Theorem 4.6 or Theorem 4.7 6e satisfied. Then for any 0 = (0i,... ,/?„), 0 ^ /?a ^ 1 - /a 0 , 0 <: a ^ 1 t/ie following limit relation holds:
(6.31) 05 p —> CO.
PROOF. If J3 is the zero multi-index, then from the definition of the function x('n) it follows that <: \\(l + (x)ro(1-lao}um(t,x),
L P , 7 (R+ x {(x) > p } ) \ \ .
Taking into account Lemma 6.3, we obtain (6.31) for /3a = 0. Consider the case 1 — /QQ ^ fta > 0. We have (*,x) - u m ( t , x } X ( ( x } 2 / p 2 ) ) = D?um(t,x}(l - X ((
moreover, the sum in $3 p (2, x) is taken over 0 and 9 such that Oa + qa = f3a,
\0\>o, kl>o.
By the definition of the functions <$i ) p (tf,a:) and xl 7 ?)) we nave < 11(1 + (x)ra(1-lao-fta)D^um(t, x), L p , 7 (E+ x {(x) > p})\\.
By Lemmas 6.2 and 6.3, we have ||(1 + <*»- ff < 1 -' fl ">-*'> 0
(6.32)
as p —> co. We prove the following limit relations as p —>• co: ||(l + <x»-^ 1 -' a °-^)$ 2 i ^ > x) > I P l 7 (M+ + 1 )||->0, 1
0
(1(1 + (x))-^ -' '-**)*^,*), I P ,,(M+ + 1 )||^0.
(6.33) (6.34)
§ 6. Estimates for Approximate Solutions
97
We first establish the estimate |£>*X((*> 2 /P 2 )K cp-'a,
« = («!,...,«»),
H>0,
(6.35)
where the constant c > 0 is independent of x £ M n and p > 0. n
Since (z) 2 = ^ x{
.
, for 1/a/c ^ s^ ^ 1 we have
J=l Therefore, by the definition of the functions xl 7 ?). ^ suffices to prove the estimate (6.35) only on the compact set Kp = {x £ R n : /? ^ (x) ^ \/2/>}. We note that for the functions gj(xk] — (xk p~^Yx^Sk, 1 ^ j ^ s^, k — 1, . . . , n, on the compact set A'p the following estimate holds: -Skakp-Skak.
(6.36)
Indeed, \9j(xk}\ = |x fc |W-*«*)/«*p- 2j ' ^ (x) 2j '- Sfca >- 2j ' ^ (>/2p) 2j '"' fcafc p~ 2j ^ (V2) 2 j - S f c a f c p~* f c a f c Since the derivative D£x({x) 2 / p*} can be represented as the sum
from the inequality (6.36) we obtain the estimate (6.35). We prove that the convergence (6.33) holds as p —>• oo. By the definition of $2,p(^, x) and x(^)) we nave
= 11(1 + < * ) ) - ' 7 1 - ' a 0 - / * * « m ( * ,
Since /?a ^ 1 — lao, from (6.35) and the definition of Kp we conclude that ||(1 + (x))-^-la^^2ip(t,x), -
---
L P|7 (RJ +1 )II x ' - x
2. Equations not Solved Relative to Higher-Order Derivative
98
Taking into account the inequalities 0 0 we have
x K
Cl
Since 0
D
Arguing as above, it is possible to prove the following lemma. Lemma 6.6. Let the assumptions of Theorem 4.6 or Theorem 4.7 be satisfied. Then for 0 <^ co :
|(1 + (x))-a(1-lao)Dkt (um(t, x} - um(t, x \D%D*(um(t, x) - um(t,
0 0-
We note that Lemmas 6.5 and 6.6 imply the convergence \(um(t,x)-um(t,x)X(t/p)x((x)*/P2)),
^'.^(Kj+iJII-^
§ 7. Existence and Uniqueness of a Solution to the Cauchy Problem for Equations without Lower-Order Terms In this section, we prove the solvability theorems in Section 4 for the Cauchy problem (4.1) for simple Sobolev-type equations and pseudoparabolic ones without lower-order terms. PROOF OF THEOREMS 4.6 and 4.7. We assume that the right-hand side f ( t , x) satisfies the assumptions of Theorem 4.6 in the case of a simple
§ 7. Existence and Uniqueness
99
Sobolev-type equation and the assumptions of Theorem 4.7 in the case of a pseudoparabolic equation. By Lemmas 6.1-6.6, the sequence of approximate solutions {um(t, x)} is a Cauchy sequence in the space Wl>r a (M^ +1 ), 7 > 70, 1 ^ a ^ 0, and um(t, x] satisfies the estimate um(t,x)t
, 7iff (K+ +1 )|| <: c ( \ \ f ( t , x ) ,
where the constant c > 0 is independent of m and f ( t , x ) . Since the space Wp')7 i(T (R+ +1 ) is complete, there exists a function u ( t , x ) £ W^'^|(7(]R^+1) such that \\um(t,x) — u ( t , x ) , Wp^aO^n+i)!! —>• 0 as m —>• oo, and the following inequality holds: , x), L pi7 (R+ +1 )||
By Lemma 6.2 and the properties of the weak differentiation operator, we obtain the existence of the weak derivatives D^.D^u(t,x)} k&Q + (3a = 1, 0 ^ k ^ /; moreover, \\D^xDktum(t, x) - D^xDktu(t} x ) , I P17 (M+ +1 )|| -> 0,
m -> oo,
where the constant c > 0 is independent of f ( t , x ) . Taking into account the conditions in Section 2 on the differential operator L(Dt, Dx), we have \\L(Dt,Dx)um(t, x} - L(Dt,Dx)u(t, x), L P , 7 (M+ +1 )|| -> 0 as m —> co. By the construction of the functions um(t, x} (cf. Section 5), we have ||£(A, Dx)um(t} x} — f ( t , x), Lp i7 (M^ +1 )|| —>• 0 as m -> oo. Therefore, the function u(t,x) is a solution to the equation L(Dt, Dx}u(t, x} = f ( t , x ) . Recall that D^um(Q,x) = 0, k = 0, . . . , / — 1. By the properties of the trace operator, we have Dku\t_Q— 0, k = 0, . . . , / — 1. Consequently, the limitr function u(t, x} is a solution to the Cauchy problem (4.1) in the space TA/'i ("n3>+ ^ ^p, al M n + l ) 7)
The uniqueness of a solution to the problem (4.1) in Wp' 7CT (R^ +1 ) is obtained from the assertion below. D Lemma 7.1. If u ( t , x ] is a solution to the Cauchy problem (4.1) in the space Wj; 7ity (lRj +1 ) with f ( t , x ) - 0, then u(t,x) = 0. PROOF. The arguments are the same as in the case of simple Sobolevtype equations and pseudoparabolic equations. We first assume that the
100
2. Equations not Solved Relative to Higher-Order Derivative
solution u ( t , x ) has compact support relative to x. Then the Fourier transform of u with respect to x is a solution to the Cauchy problem /-i LQ(i£)Dltv + ]T Li-k(it,)Dktv = 0,
t > 0,
k=0
By assumption, LQ(i£) ^ 0 for f £ M n \{0}. Hence v(<,0 = 0 for £ € M n \{0). By continuity, v(t,£) = 0, £ £ ffin- But, in this case, w ( t f , x ) is the zero solution. The above arguments mean the uniqueness of compactly supported in x solutions to the problem (4.1) in the space Wp'^a(M.^+l). It is obvious that such solutions can be regarded as the limit of the sequence of approximate solutions {um(t,x}}. In view of (7.1), this means the following remarkable fact: any function w ( t , x ) G W p ' ^ f f ^ n + i ) w ^h compact support in x, D3tw t_Q= 0, j = 0, . . . , / — 1, having the weak derivatives G £pp (7 (]R*+11 ), kaQ + /3a — 1, 0 ^ k ^ /, satisfies the inequal( 7(]R* ity )|| ^ c\\L(Dt,Dx)w(t,x), I Pl7 (Mj +1 )||, (7.2) where the constant c > 0 is independent of w ( t , x ) . Consider the general case. Since u ( t , x ) G Wp'^ tion to the Cauchy problem (4.1), we repeat the arguments of the proof of Lemma 6.5 and for any e > 0 construct a compactly supported in x function u £ ( t , x ) E W^;^(1R+ +1 ), D{ue t=Q= 0, j - 0, ... ,1- 1, such that G L P ) 7 ( M + 1 ) , ka0 + pa = l, k = I , . . . ,/, and
(*, x) - D£Due(t, x), Lp By (7.2), for any /? = (A , . . . ,/?„), /?a = 1 - / a 0 , we have 3,7 I
where the constant c > 0 is independent of £ and u £ ( t , x } . Since L(Dt, Dx)u(t, x) = 0, we find
§ 7. Existence and Uniqueness
101
l-l Recall that the symbol L(«M'0 = M*0("7)' + E Li-k(i£)(ir)}k
of the
fc=0
operator L(Dt , Dx] is homogeneous relative to the vector a — (a 0 , a), i.e., L(c a °z77, c a z£) = c L ( i r ] , i ^ ) , c > 0. By the above inequality, we have
E
E Since £ is arbitrary, we have \\D%Dltu(t,x), L P|7 (IRJ +1 )|| - 0, /a 0 + 0a = 1, and I)Ju(t, x) = 0 for /fa = 1 - /o 0 because D{U t=Q= 0, j = 0, . . . , / - 1, almost everywhere. On the other hand, taking into account the definition of oo. However, by the assumptions of Theorems 4.6 and 4.7, \a\/p > cr(l — IQQ). Consequently, the function u(t,x] vanishes almost everywhere. D PROOF OF THEOREMS 4.1 and 4.3. Setting a = 0, from Theorems 4.6 and 4.7 we obtain the assertions of Theorems 4.1 and 4.3 respectively. D We pass to the proof of Theorems 4.2 and 4.4. To prove the existence of a solution to the problem (4.1), we derive estimates for approximate solutions um(t,x) which imply that the sequence {um(t,x}} converges in the space Wp'^I&n+i) and the limit function u(t, x] is a solution to the problem (4.1). First of all, we note that from the proof of Lemma 6.1 it follows that Lemmas 6.1 and 6.2 are valid for any values of |a|. However, in the proof of Lemmas 6.3 and 6.4, the condition \a\/p' > 1 — /ao is essentially used in the case a = 0. If this condition is not satisfied, the inequalities (6.15) and (6.22) and the convergence (6.16) and (6.23) do not hold, in general, even for infinitely differentiate compactly supported functions f ( t , x). This assertion can be easily justified by considering the special case p — 2, a = 0. In this case, using the Paley-Wiener theorem and the Plancherel theorem and arguing in the same way as in the proof of Theorem 4.5, we conclude that some additional conditions on f ( t , x) are necessary to obtain estimates similar to (6.15) or (6.22) for um(t,x) with constants independent of m. These conditions are similar to the orthogonality conditions (4.2). We prove assertions similar to Lemmas 6.3 and 6.4 under the assumption
102
2. Equations not Solved Relative to Higher-Order Derivative
a\/p' ^ 1 - Ia0.
Lemma 7.2. Let \a fp' + lct$ ^ 1, and let N be a number such that /p' + Namm > l — la0 ^ \0i\lp'-}-(N — l)a m m . We assume that f ( t , x) £ L pi7 (IK+ +1 ) n L p i 7 (IR^; Li,-N|«|0^n)), 7 > 7o, satisfies the orthogonality conditions (4.2). Then for (3 = (/?i,. . . , /3n), (3a < 1 — Ia0, the following estimate holds: c(\\f(ttx), (7.3)
where the constant c > 0 is independent of m and f ( t , x ) . Moreover,
PROOF. We write the estimate (6.17) for a = 0:
-i l/m
-i
From the proof of Lemma 6.3 (cf. the estimate (6.19)) it follows that I^ satisfies the inequality If m <J c i \ \ f ( t , x ) , for any We estimate the second term 7° m on the right-hand of the above relation. By the orthogonality conditions (4.2), for the partial Fourier transform with respect to x we have
. . .A
(7.5)
103
§ 7. Existence and Uniqueness
Therefore, 1
1
/•••/ 0
0
t 0
-
dv.
Using the Minkowski inequality and the Young inequality, we find
/ -> \P\=N
l
Introduce the notation , 0 de,
Arguing as in the proof of Lemma 6.3, it is easy to derive the estimate BO,P(V) ^ c v -|«l/p'+(i-'«o-/3a)-P« > where the constant c > 0 is independent of v. Since v ^ 1, we have
However, by assumption, we have 0 > 1 — /c*o — M/V ~~ ^ a min- Consequently, I^m ^ c 2 ||||(l + (y}}N\a\ f ( t , y ) , Li(K n )||, Lp, 7 (Rf)||, where the constant 02 > 0 is independent of m and f ( t , x ) . Taking into account the estimate for the first term /° m , we obtain (7.3). We prove (7.4) in the same way. D Lemma 7.3. Let the assumptions of Lemma 7.2 hold. Then c(\\f(t,x), (7.6)
104
2. Equations not Solved Relative to Higher-Order Derivative
where the constant c > 0 is independent of m and f ( t , x ) . \\Dltumi(t,x)-Dltum,(t,x),Lpn(l&++1)\\->Q,
Moreover,
mi, m 2 -> oo.
(7.7)
PROOF. We estimate the terms on the right-hand of (6.24). Regarding the first term f^(i, x), it is obvious that
.-i l/m
+
v-i
, y)
dv
As was mentioned in the proof of Lemma 6.4, from the condition /QQ < 1 we can obtain the estimate I\>m 0 is independent of m and f ( t , x ) . To estimate /2, m , we write this term taking into account the orthogonality conditions (4.2). By (7.5), we have i
i
h,m — 0
0
J x A7V-1 ; Using the Minkowski inequality and the Young inequality, we find
v^
Co / V
-1
dv
By the quasihomogeneity of the symbol LQ(I£) and the definition of the
§ 7. Existence and Uniqueness
105
kernel G(£), we find
f eizsG(s}(LQ(is}}-l(isy
ds,
m
(N) f v-i-Wp'
1 Since 0 > 1 — Ia0 — \a\/p' — Nam(n, we have
where the constant 0-2 > 0 is independent of m and f ( t , x ) . estimate for /i, m , we find
Using the
We estimate the second term v^(t,x) on the right-hand of (6.24). We write the inequality (6.26) for this term:
• /•-///•"
2 l I?) \vm\It > xx\ /'
l/m
t
m
1
0 KnKn
0
From the proof of Lemma 6.4 it follows that for any value of |a| the estimate (6.28) holds.
106
2. Equations not Solved Relative to Higher-Order Derivative
Consider I\,>m. Using (7.5), we write J2 m in the form m
/
1
r r "" /-7 l
0
i
0
t
0
i . . . d\N,
dv.
As in the proof of Lemma 7.2, we have
where
Repeating the proof of Lemma 6.4, we obtain the estimate
where the constant c > 0 is independent of v. As above, we obtain the estimate
where the constant c2 > 0 is independent of m and f ( t , x ) . estimate for /[ m , we find
Using the
The last estimate, together with (7.8), implies (7.6). We can prove (7.7) in the same way. D PROOF OF THEOREMS 4.2 and 4.4. We assume that the right-hand side f ( t , x ) of the equation satisfies the assumptions of Theorem 4.2 in the case of a simple Sobolev-type equation and the assumptions of Theorem 4.4 in the case of a paseudoparabolic equation. By Lemmas 6.1, 6.2, 7.2, and
§ 7. Existence and Uniqueness
107
7.3, the sequence of approximate solutions {um(t,x)} converges to some function u(t,x) e Wp' 7 (M+ +1 ) having the weak derivatives D^D^u(t,x) G ka0 + /fo = l , 0 ^ f c < C / . Moreover, ( t , x } , L P17 (M+ +1 )|| -> 0 as m —> co. By the assumptions on the operator L(Dt,Dx), we have \\L(Dt,Dx)um(t,x)-L(DttDx)u(t,x), L P|7 (M+ +1 )|| -+ 0 as m -» co. Taking into account that D^w m (0,x) = 0, A; = 0 , . . . ,/ — 1, and using the properties of the trace operator, we find D^u^t_Q= 0, k — 0 , . . . , / — 1. This means that the limit function u(t,x) is a solution to the Cauchy problem (4.1) under the assumptions of Theorem 4.2 or Theorem 4.4. The uniqueness of a solution to the problem (4.1) in Wp' i7 (lR+ +1 ) follows from Lemma 7.1 since this lemma is valid under the assumptions of Theorems 4.2 or 4.4. Indeed, analyzing the proof of Lemma 7.1, we see that only the condition on the operator Lo(Ar), the estimate (7.1), and the restriction \a\/p > cr(l — I a 0 ) were essentially used. D PROOF OF THEOREMS 4.8 and 4.9. Under the assumptions of these theorems, Lemmas 6.1, 6.2, and 7.1 hold. Therefore, it suffices to prove analogs of Lemmas 6.3 and 6.4. D Lemma 7.4. The following estimate holds: \\um(t,x), L P17 (K+ +1 )|| < c
£
\ \ F p ( t , x ) , I P17 (M+ +1 )||,
(7-9)
/9a = l-/a 0
where the constant c > 0 is independent of m and F p ( t , x } . Furthermore, ||wmi(t,z)-Mma(*,x)>Lp)7(M++1)||-»0
asmi,m2-»oo.
(7.10)
PROOF. It suffices to consider the case Fp(t, x] 6 C*o°(M^+1). By (5.11), the function um(t, x) can be written in the form um(t,x)= /3a=l-la0
1/m
108
2. Equations not Solved Relative to Higher-Order Derivative
However, for each function u^(tf, x) the assumptions of Lemma 6.2 are satisfied. Consequently, (7.9) and (7.10) hold. D Lemma 7.5. The following estimate holds: )ll
V / j
\ \ F p ( t , x ) , L P , 7 (M+ +1 )||,
where the constant c > 0 is independent of m and F p ( t , x } . l mi(t,x)-D tum2(t,x)),Lp^(mt+1)\\-^0
(7.11)
Moreover, (7.12)
as m\, m-2 —>• oo.
PROOF. It suffices to consider the case F p ( t , x } £ C^°(M^+1). As in the proof of Lemmas 6.2 and 6.4, we have Dltum(t,x}=
]T
l/m
Since F p ( t , x ) 6 L pi7 (R^ +1 ), each term u\^(t, x} -f w^(2, x) takes the form (6.9). Therefore, (7.11) and (7.12) are established by repeating the computations in the proof of Lemma 6.2. D
§ 8. Equations with Variable Coefficients In this section, we consider the Cauchy problem for simple Sobolev-type equations and pseudoeparabolic equations with variable coefficients and special lower-order terms. We suppose that the coefficients of all equations are continuous and are constant outside some ball (|x| < p } .
§ 8. Equations with Variable Coefficients
109
We first focus on the Cauchy problem for simple Sobolev-type equations 1 C(x- Dt D" Dx}u = F(t x} t ' ' ' ' k
t > 0 '
x £ Mn "'
(8.1)
where the operator C(x; Dt, D^1, Dx) takes the form /-i £(x; Dt,D^l,Dx] = L0(x; Dx)Dlt + J^ L,_ f c (x; Dx}Dkt k=0
q=0
Lj(x]Dx) = ^a/3J(x)D^,
j = 0 , 1 , . . . ,/ + m+ 1.
Recall that D^1 is the ^-integration operator: t D^1u(t,x}= I u(s,x)ds. o
We assume that for any fixed x° £ Wn /-i L(x°; A, £>«) = MZ°; ^)^! + 5] Li-k (x°\ Dx}Dkt
is a simple Sobolev-type operator without lower-order terms, i.e., the symbol L ( x ° ; i r ] , i ^ } satisfies Assumptions 1 and 2 in Section 2 for a 0 = 0. Let a = (0, a) = (0, a i , . . . , a n ), I/a,- £ N, be the corresponding homogeneity vector. We assume that the symbols L/+ g +i (x; z'£) are homogeneous relative to a, i.e., /,,+,+!(x°;c a iO - cI /+ , + 1 (x°;ie),
c > 0.
We look for a solution to the Cauchy problem (8.1) in Wlp'r^ 1< p < oo, r = ( 1 / a i , . . . , l/a n ), 0 ^ (r ^ 1.
(8.2) <7 (K^ +1 ),
Theorem 8.1. Lei |a|/p' > 1. Then there exist 71 > 0 and e > 0 such that, if the coefficients of the principal part satisfy the condition \a/3,j(x)-al3j(x0)\^e,
\x°\>p,
j = Q,l,...,l,
(8.3)
110
2. Equations not Solved Relative to Higher-Order Derivative
then the Cauchy problem (8.1) is uniquely solvable in Wp'!^(IR* +1 ), 7 ^ 7i ; for any right-hand side F ( t , x ] E L pi7 (M^ +1 ) fl Lp i 7 (IR^; Li(IR n )) and the following estimate holds:
),Ll(Rn)\\,Lp<,(R+)\\},
(8.4)
where the constant c > 0 is independent of F ( t , x ) . Theorem 8.2. Let equation (8.1) contain no integrodifferential terms. Let \Q\ > I , and let \a\/p > a > 1 — \a\/p'. Then there exist 71 > 0 and e > 0 such that, if the coefficients satisfy the condition (8.3), then the Cauchy problem (8.1) is uniquely solvable in Wp'^a(W^+1), 7 ^ 71, for any right-hand side F(t,x) E L P|7 (M+ +1 ) fl Lp i 7 (E^; L i t - f f ( ^ n ) ) , and the following estimate holds: t
u , x,
>
< c ( \ \ F ( t , x), L p i 7 (R+ + 1 ) | + || ||F(/, x), ^.^(Mn)!!, Lp.^Rj-)!!), where the constant c > 0 zs independent of F(t,x). PROOF OF THEOREMS 8.1 and 8.2. We apply the perturbation method. For the sake of definiteness, let the assumptions of Theorem 8.1 hold. In the previous sections, we established the unique solvability of the Cauchy problem for simple Sobolev-type equations with constant coefficients without lower-order terms. The solution u(t, x} G Wl
(8.5)
where the operator P : L pi7 (M+ +1 )nL p , 7 (IR}-; Li,_ C T (E n )) -» Wfci(T(R++1)) 7 > 7 o , 0 ^ c r ^ l , is linear and continuous; moreover, for any multiindex /3 = (/3i, . . . , /? n ), (3 a — 1 and k = 0, 1, . . . , / the linear operators D* o D% oP : L P)7 (M^ +1 ) -> L pi - 7 (IR+ +1 ) are also continuous. Therefore, we look for a solution to the problem (8.1) in the form (8.5). For the unknown function /(/, x) we obtain the operator equation (/ - T ) f ( t , x ) = F ( t , x ) ,
(8.6)
where
Tf(t, x) = ( L ( x ° ; D t , D x ) - L(x; Dt, Dx))Pf(t, x) m
(x; D x ) P f ( t , x) = TJ(t, x) + T 2 f ( t , x),
|x°| > p.
§ 8. Equations with Variable Coefficients
111
Since the coefficients of the operator C(x] Dt, -Of 1 , Ac) are constant for \x\ > p, from (8.6) we conclude that f ( t , x) = F(t, x), \x\ > p, i.e., it suffices to consider equation (8.6) only on a compact set relative to x. By the properties of the operator P and the homogeneity conditions L(x°; ir], cai£) = cL(x°] irj, z'£), c > 0, for 7 > 70 we have
I
Y, \ \ ( a ^ - k ( x Q } - a p , i - k ( x } } D k t D P p f ( t , x } ,
Lp,
where the constant c > 0 is independent of f ( t , x ] and 7. Consequently, there is e > 0 such that, under the condition (8.3), the following estimate holds:
liri/^xj.^^^!)^^^.^.^^!)!!,
7 >7o.
Taking into account the homogeneity conditions (8.2), for 7 > 70 we find
/
(t-T
J/ Q
By the continuity of apj(x) and apj(x] = a/3,j, for |x| ^ p we have m
E E ™J x l a /M+
n
Therefore, by the Young inequality and the properties of the operator P, we obtain the estimate T 2 f ( t , x ] , L pi7 (M+ +1 )|| ^ c
^9"1 H/^^)' ^P,7(Rj+i)ll.
7 > 70,
where the constant c > 0 is independent o f / ( / , x ) , 7. Consequently, there are 71 > 70 such that for 7 ^ 71 we have
\\T2f(t,x), I Pl7 (M+ +1 )|| ^ \ \ \ f ( t , x ) , LP
112
2. Equations not Solved Relative to Higher-Order Derivative
The above arguments mean that for e > 0 and 71 > 70 we have \ \ T f ( t , x ) , I Pl7 (K+ +1 )|| ^ \ \ \ f ( t , x ) , L P l 7 (R+ + i)ll,
7 £ 71-
Consequently, equation (8.6) is uniquely solvable: f ( t , x) = (I- T)-1^*, x) € L Pl7 (Kj +1 ),
7 £ 7i-
Since f ( t , x ) = F ( t , x ) as |x| ^ p, by the assumptions of the theorem, we have /(*,z) € L P)7 (1R+ +1 ) n Lp ( 7 (Mj";Li(M n )). By the definition and properties of the operator P, the function u ( t , x ) = P(I — T } ~ l F ( t , x ] £ Wp' 7 (H&n+i)> 7 ^ 7i) ig a solution to the Cauchy problem (8.1) and the estimate (8.4) holds. We prove the uniqueness of a solution. Let u ( t , x ) £ Wp' 7 (IR* +1 ) be a solution to the Cauchy problem (8.1) for F ( t , x ) — 0. Then L(x°;Dt,Dx)u(t, x] = (L(x°; Dt, Dx) - L(x; Dt, Dx))u(t,x) °; Dx) - Ll+q+1 (x; Dx}}u(t, x) ( x ° ; D x ) u ( t , x),
\x°\ > p.
Since the coefficients of the operators are constant for |x| > p, the members of the first two groups satisfy the assumptions of Theorem 4.1. The members of the third group satisfy the assumptions of Theorem 4.8. Therefore, in view of the unique solvability in 14/?^(]R^+1) of the Cauchy problem for simple Sobolev-type equations, we obtain the representation u(t, x) = P(L(x°' Dt>Dx) - L(x\ Dt,Dx))u(t, x) (x; Dx)}u(t, x) J l
~ Li+q+i (x°; Dx)u(t, x} = Tiu(t, x) + T2u(t, x) + T3u(t, x).
q=Q
Taking into account the properties of the operator P, we deduce the in-
§ 8. Equations with Variable Coefficients
113
equality
\\u(t,x), W
where the constants c\(p), C2(/>), 03 are independent of u ( t , x ) and 7. Hence we can indicate 71 > 70 and e > 0 such that for 7 ^ 71 from the condition (8.3) we have
\\u(t,x), W
^ \(\\u(t,x), which implies the uniqueness of a solution Theorem 8.1. The proof of Theorem 8.2 is similar. D We proceed by considering the Cauchy problem for pseudoparabolic equations. We assume that the operator £(x; Dt, D^1 , Dx] takes the form m
C(x- Dt, D~\ Dx) = £(x; Dt, Dx) +
D^'1 Li+q+1 (x; Dx],
and the leading part of L(x;Dt,Dx} — L0(x;Dx}Dlt + £] Li-k(^\ k=0
for any x 0 G M n is a pseudoparabolic-type operator without lower-order terms (cf. Section 2). Recall that a = (ao,a), «o > 0, I/a,- 6 N, is the homogeneity vector of the symbol L(x; irj, i£). We assume that the symbols L/ + 9 + i(z;if) satisfy the equality L; +g+1 (x; cai£,) = cl~lc*0 Li+q+i(x\ i£), c > 0. We write the Cauchy problem for a pseudoparabolic equation: LQ(X; Dx}D[u + ]T Lt.k (x; Dx}Dktu k=0
114
2. Equations not Solved Relative to Higher-Order Derivative m ft T / (t-s)qLi+q+i(x]Dx)u(syx)ds
= F(t,x),
=o •'o
(8.7) A*
t==o=°.
* = <>,.. .,/-!.
We look for a solution to the Cauchy problem (8.7) in < p < oo, r = Theorem 8.3. Let £/ie equation in (8.7) /mt>e no terms, and let all the coefficient be constant. Let |a| + / a 0 > l ,
integrodifferential
|a lp> ff(l-la0) > I - Ia0 - \a\/p'.
(8.8)
Then there exists 70 > 0 such that the Cauchy problem (8.7) is uniquely solvable in the space Wp'^ a(^^+l), 7 ^ 70, for any right-hand side F(t,x) 6 L p i 7 (M+ + 1 ) n L p i 7 (Mj;L 1 | C r (; a o _ 1 ) (IR n )) ; and the following estimate holds:
\u(t, x), ^;; fc = l /ca a +/3a = l
,^,^,,^,)!! (8.9) where the constant c > 0 is independent of F ( t , x ) . Theorem 8.4. Let the equation in (8.7) have no integrodifferential terms, and let all the coefficients be constant. Then there exists 70 > 0 such that the Cauchy problem (8.7) is uniquely solvable in the space Wp'^(l&n + l ) , 7 ^ 70, for any right-hand side of the form F(t,x) = ) 6 L P , 7 (M+ + 1 ), ^(t,x) e L P , 7 (M+ + 1 ) ; anrf Me following estimate holds: \\u(t, x } ,
where the constant c > 0 z's independent of F(t,x). Theorem 8.5. Z/e^ |a|/p' + /ao > 1- Then there exist 71 > 0 ana? £ > 0 if the coefficients of the principal part of the operator sat-
§ 8. Equations with Variable Coefficients
115
isfy the condition (8.3), then the Cauchy problem (8.7) is uniquely solvable in the space W^(IR++1), 7 ^ 71, for any right-hand side F(t,x) £ L P)7 (IR* +1 ) n L p) -y(]Ri~; Li(IR n )), and the following estimate holds: l r
<
n+1
--
k=i
(8.10)
where the constant c > 0 /s independent of F(t,x). Theorem 8.6. Let the equation in (8.7) have no integrodifferential terms, and let the condition (8.8) hold. Then there exist 71 > 0 and e > 0 such that if the coefficients of the principal part of the operator satisfy the condition (8.3), then the Cauchy problem (8.7) is uniquely solvable in the space Wp'^iCr(R^+l}, 7 ^ 71, for any right-hand side F(t,x) £ L p i 7 ( M + + J n L p i 7 ( E f ; L i i f f ( | a o _ i ) ( R n ) ) , and the estimate (8.9) holds. PROOF OF THEOREM 8.3. The idea of the proof is the same as in Theorem 4.7 (cf. Sections 5-7). So, we explain only differences. Recall that the key of the proof of the existence of a solution to the Cauchy problem for pseudoparabolic equations without lower-order terms is the construction of special approximate solutions and estimates. In the case pseudoparabolic equations with constant coefficients and special lower-order terms jC(Dt}Dx}u = L(Dt,Dx)u +
a
p(x}DxU = F(t, x ) ,
we act in the same way. We first consider the Cauchy problem for the ordinary differential equation with parameter £ £ M n \{0} which is obtained by formal application of the Fourier operator with respect to x to the problem (8.7) for the equation with constant coefficients £.(Dt,i^)v = F(t,^}, t > 0, D^v t=Q= 0, k = 0, . . . , / — 1. The solution to this problem can be written in the form
where F(£) is a contour in the complex plane surrounding all the roots of the equation £(in, z'£) = 0, £ £ M n \{0}, relative to 77. Taking into account the form of the lower-order terms of C(Dt,Dx}} it is easy to derive the corresponding estimates. The contour integral
116
2. Equations not Solved Relative to Higher-Order Derivative
J ( t , £ ) for t > 0 satisfies an estimate of the form (5.9), i.e., c{f) o r o - 1 e-* t tf>° r o ) f € Kn\{0}, where the constants c, 6 > 0 are independent of £. Since the principal part of the symbol satisfies (5.13), there exists 70 > 0 such that for Re T ^> 70, £ G K n we have
The first estimate allows us to write the sequence of approximate solutions to the Cauchy problem (8.7) of the form (5.11). The second estimate implies analogs of Lemmas 5.5, 5.7, 5.9 with 7 ^ 70 instead of 7 > 0. Repeating word- by- word the computations for um(t, x) in Section 6, we can establish assertions similar to Lemmas 6.1-6.6 for 7 ^ 70- As in Section 7, we obtain the existence of a solution to the Cauchy problem (8.7) in the space W^CT(IR*+1), 7 ^ 70, and the estimate (8.9). We emphasize that a solution is constructed in the following integral form: u ( t , x ) = PF(t,x) = lim um(t,x}. m —K5O
(8.11)
The uniqueness of a solution can be proved in the same way as it was done in the proof of Theorem 4.7. D PROOF OF THEOREM 8.4. It is easy to see that analogs of Lemmas 7.4 and 7.5 hold for 7 ^ 70- As in the case 4.9, we obtain Theorem 8.4 using analogs of Lemmas 6.1, 6.2, and 7.1, D PROOF OF THEOREM 8.5. As in the case of Theorem 8.1 and 8.2, we use the perturbation method. We look for a solution to the problem (8.7) in the form u ( t , x ) = P f ( t , x ] , where the operator P is defined in (8.11). Then the unknown function f ( t , x) is found from the equation (I-T)f(ttx)
= F(ttx)t
(8.12)
where T f ( t , x) = (£(x°; Dt , Dx) - £(z; A, D x ) ) P f ( t , x)
Repeating the proof of Theorem 8.1, we can show that there exist e > 0 and 71 ^ 70 (70 in Theorem 8.3) such that if the coefficients of the principal part of the operator jC(x;Dt,Dx) satisfies the condition (8.3), then the norm of the linear operator T : L P|7 (IR^ +1 ) —>• Lp
§ 8. Equations with Variable Coefficients
117
7 ^ 71, is strictly less than 1. Then equation (8.12) is uniquely solvable: f ( t , x ) = (I - T)~lF(t,x) € L pj7 (E+ +1 ), 7 ^ 71. Since the coefficients of the equation are constant outside the ball {|x| < p}, we conclude that f ( t , x ] belongs to the space L P|7 (M^"; Li(M n )). Consequently, the function u ( t , x ] — P(I — T)~lF(t,x) is a solution to the Cauchy problem (8.7) in the space W^(IR+ +1 ), 7 ^ 71, and the estimate (8.10) is valid. As in the proof of Theorem 8.1, to prove the uniqueness of a solution, we consider the solution u(t,x) £ W.^(]R*+1) to the problem (8.7) for F(t,x) = Q. Then for x°, |x°| > p we have C(x°- A, Dx}u(t, x) = (jC(x°- A, A) - £(*; A, Ar))«(*, x}
q=0
q=0
Since the coefficients of the operators are constant for |x| > p, the members of the first two groups satisfy the assumptions of Theorems 8.3, and the members of the third group satisfy the assumptions of Theorem 8.4. Using the operator P, we find u(*, x) = P(C(x°; A, Ac) - £(*; A, Dx))u(t, x) °; Dt) - Ll+q+1 (x; Dx))u(t, x) 9=0
q=0
Therefore, for sufficiently large 71 > 70 and sufficiently small £ > 0, by the properties of the operator P, the following estimate holds:
OIK
fc=i
1 / I \\iid T\>> wPl
9 I II ^ '
VV
^
7 ^ 7i> which implies the uniqueness of the solution. Theorem 8.6 is proved in a similar way.
D
118
2. Equations not Solved Relative to Higher-Order Derivative
We emphasize one more interesting property of equations that are not solved relative to the higher-order derivative. Namely, the presence of lowerorder terms in the equation can essentially affect the solvability of the problem. For example, the number of the solvability conditions can considerably grow. To illustrate this assertion, we consider the Cauchy problem for the Sobolev equation with perturbed lower-order terms .9u + d « _ / ( t , x ) ,
t > 0 , zeIR3,
(gi3)
As we know, in the case d = 0, for the solvability of the problem in the space W 2 ' 7 (M4~) the orthogonality conditions
f I f(t,x}dx = 0 J
is necessary and sufficient. We show that for d > 0 the Cauchy problem is not necessarily solvable in the space W2 1 (K^) for the right-hand side f ( t , x] satisfying only the finite number of the orthogonality conditions f x P f ( t , x ) d x = Q,
\p\ = 0 , . . . . 2 A T - 1 .
(8.14)
5&3
Indeed, consider, for example, the function f ( t , x ) = ( — A ) N e ~ ' x ' / 2 . This function satisfies (8.14). We assume that for this right-hand side the problem (8.13) has a solution u ( t , x ) 6 W^Rj), 7 > 0. Then
Therefore, c,
G£ = (0,T) x {£ < K| < ^-}.
(8.15)
We write an explicit expression for the solution u ( t , £ ) to the Cauchy problem
For e < |£j < d/(2o;) we have
§ 9. Pseudohyperbolic Equations
119
where AI(£) = -\/d 2 - w 2 f|/|£|, A 2 (£) = -Ai(£). It is clear that sup 11^(^,0, L 2 , 7 (Cr e )||,
^ ci < oo.
£>0
By (8.15), c2 < oo
or
On the other hand, \\(d2 - a,2
oo
as e —> 0, and we arrive at a contradiction. Therefore, the Cauchy problem (8.13) for d > 0 can have no solutions in the space W 2 2 ^OR+) if the righthand side of the equation satisfies the finite number of the orthogonality conditions. Similar examples can be given for any simple Sobolev-type equations and pseudoparabolic equations defined by the differential operators
/-i L0(Dx)Dlt k=Q
where Lo(Dx) is aquasielliptic operator and Lo(c a z'£) = c1 ' a °Lo(z'£)i c > 0The reason is that the operator LQ(DX) : Wf(Rn) ->• L p (M n ), s = ((1 lao}/ai,.. . , (1 — /ao)/a n ), 1 < p < oo, is not continuously invertible.
§ 9. Pseudohyperbolic Equations In this section, we consider the Cauchy problem
i-i , V^ L0(Dx}Dl->/,. tu + k=0
(9.1)
120
2. Equations not Solved Relative to Higher-Order Derivative
for strictly pseudohyperbolic equations without lower-order terms. According to the definition in Section 2, this means that the symbol of the differential operator L(Dt , Dx) = L0(Dx)Dlt k-Q
defining the equation is homogeneous relative to a — (QQ, oe\,... , a n ), where QQ > 0 and I/ on are natural numbers. Recall that the differential equation L(Dt,Dx)u = f ( t , x ) is said to be pseudohyperbolic if the operator at the higher-order time-derivative Lo(Dx) is quasielliptic and the equation (9.2)
has only real roots r?i(£), . . . , ty(0- ^ tne ro°ts are distinct, the equation is said to be strictly pseudohyperbolic. We note that the class of equations under consideration contains, in particular, quasihyperbolic equations (c*o — a\ =• . . . — an — introduced by S. A. Galpern [2]. We give two examples of strictly pseudohyperbolic equations: n
n a
4
A A« + i Yl kD Xk u = f ( t , x ) ,
AD?u - ^ akD*k u = f ( t , x
where ^ a/c£^ > 0, ^ G M n \{0). For the first equation the symbol of k-i the differential operator L(Dt,Dx] takes the form L(irjji^) = — |^| 2 z/y + n
i XT ak£k'> moreover, OQ — 1/2, OL\ — . . . = an — 1/4, and the second fc=i n
equation we have L(iry,z'£) = |<^| 2 ry 2 — ^ ak£k
an<
^ a o = a i — • • • — an —
k =l
1/4. We look for a solution to the problem (9.1) in the weight Sobolev spaces Wft^n+i), r = (l/ a i ! . . . , l/a n ), 7 > 0, if Dtu(t, x) € ^ 2 % (1 ~" Q ° )r (M+ +1 ), Ar = 0 , . . . ,/. As we will see below, the formulations of the solvability theorems differ from those for simple Sobolev-type equations and pseudoparabolic equations (cf. Section 4). Namely, the right-hand side of the equation should satisfy conditions that are similar to the conditions appearing in the study of the Cauchy problem for hyperbolic equations (cf., for example, J. Leray [3]
§ 9. Pseudohyperbolic Equations
121
and I. G. Petrovsky [1]). However, unlike the hyperbolic theory, for the existence of a solution to the problem (9.1) in the spaces WV-v(^n+i) ^ ^s necessary that the right-hand side of the equation satisfies the orthogonality conditions of type (4.2). n
We introduce the notation |a| = ]T) a,-, a m i n = m i n j a i , . . . ,an}- We denote by u7 (??,£) the Fourier transform of the function u^(t, x) = e~"*tu(t, x) and by u ^ ( t , £ ) the partial Fourier transform of this function with respect to x. Theorem 9.1. Let |a|/2 + /a 0 > 1, and let f ( t , x ) € W L 2 , 7 (Mf;Li(lR r i )), s = (a0/ai,... ,a0/an), 7 > 0. Then the Cauchy problem (9.1) is uniquely solvable in WV-vO^n+i) ana the estimate holds \\u(t, *), W | 7 (R+ + 1 )|| +
\\Du(t, x), k=o
<: c(\\f(t, x), W2 where the constant c> 0 is independent of f ( t , x ) . Theorem 9.2. Let |a|/2 + /ao ^ 1; and let N be a natural number such that \a\/2 + Ia0 + Nam[n > 1 ^ |a|/2 + Ia0 + (N - l)a m i n . Then the problem (9.1) has a unique solution u ( t , x ) 6 ^'^(^n+i); 7 > 0, /or any function f ( t , x ) E W$£(R++1) such that f
(t,x)dx = Q,
(9.3)
Kn
where \0\ — 0, . . . , N — 1; moreover, \\u(t, *)X , x),
i;(M+ +1 )|| + || 11(1 + ( x ) ) f ( t , x), L!(]R n )||, L 2
where the constant c > 0 is independent of f ( t , x ) . By Theorem 9.1, the Cauchy problem (9.1) is unconditionally solvable in the space W2;7(Mj+1) if |a|/2 + /a 0 > 1- In the case |a|/2 + /a 0 < 1, by Theorem 9.2, the problem is solvable in the space 'W^'.-yC^n+i) under the orthogonality conditions (9.3) on the right-hand side of the equation. A similar situation appears in the case of the Cauchy problem for simple
122
2. Equations not Solved Relative to Higher-Order Derivative
Sobolev-type equations and pseudoparabolic equations. Such additional conditions are essential for these equations (cf. Section 4). At the end of the section, we show that, in the case of pseudohyperbolic equations, the orthogonality conditions (9.3) are also necessary for the solvability of the Cauchy problem in these classes. We divide the proof of Theorems 9.1 and 9.2 into several lemmas. We first establish some properties of symbols of strictly pseudohyperbolic operators which are similar to the corresponding properties of symbols of strictly hyperbolic operators. Lemma 9.1. There exist constants c i , C 2 > 0 such that for cr ^> Q, (?7,f) 6E M n +i the following estimates hold: M(ir/ -f cr, if) = - Im (L(irj + a, i£)DrjL(irj + a, if)) c7| + <0 Q o ) 2 '- 2 )
(9.4)
(f
PROOF. By definition, we have M(iri +
2-
n
i
'
- n' n Since the roots of (9.2) are homogeneous with respect to (a\, . . . , an) with
§ 9. Pseudohyperbolic Equations
123
homogeneity exponent ao, the function i
\irj + a - irjj (f ) |2
m(ir) + cr, if) = T
is homogeneous with respect to the vector (c*o, ai, • • • , Qn) with homogeneity exponent 2ao(/ — 1). Consequently, for any (77, a, £) G ffin+2 we have m(i77 + cr, if) £ a(|i»j +
(9.6)
where a > 0 is some constant. Indeed, by homogeneity, to verify (9.6), it suffices to establish that m(ir] + cr, i£) ^ 0, (77,cr,£) 6 M n +2\{0}. However, this assertion is valid for cr ^ 0 since the roots TJJ(£) are real and for a = 0 since ty(0 ^ ^(0, j ^ *, ^ 6 M n \{0}, ^-(0) = 0, j = 1, . . . , /. Taking into account the identity M(irj + cr, i£) = (T(Lo(i^))2 the inequality (5.14), and the estimate (9.6), we obtain (9.4). To prove the estimate (9.5), we note that \M(i-n + o-, iOI ^ \L(ir) + a,
/ ^ |L(ii7 + (r, «)| 2 |L 0 (iOI *=1
^ c|L(zr7 + ^ «OI(0 ( 1 " / a o ) (l»'7 + ^1 + (O 00 )'- 1 • By the inequality (9.4), we obtain (9.5).
D
Lemma 9.2. For any function u ( t , x ) 6 W2'^(Rn+i), 7 > 0, such that D^u(t,x) £ W2'^ ~ a (M n+ i); A; = 0, . . . ,/ ; the following estimate holds:
7 + {O ao )"%(»7, 0, ^2(Mn + i)||, (9-7) where c2 is the constant in Lemma 9.1. PROOF. By the Parseval identity, we have \\L(Dt,Dx)u(t,x),L2^(Rn+1)\\ = \\L(Dt + 7, Dx)u^(t, x),L2(mn+1}\\ Recalling the estimate (9.5), we obtain (9.7).
D
Lemma 9.2 implies the uniqueness of a solution to the problem (9.1) in the above-indicated class of functions. Indeed, if u ( t , x ) is a solution to the
124
2. Equations not Solved Relative to Higher-Order Derivative
Cauchy problem with f ( t , x ) — 0, then, extending it by zero to t < 0, we obtain a function u(t, x) satisfying the assumptions of Lemma 9.2. Consequently, from the estimate (9.7) we obtain the equality IKf/ 1 ""' 00 ^! 7 /! +7 + {£) a °) / ~%(7?,0> L 2( M n+i)ll = 0 which implies u(t,x]) = 0. Remark 9.1. The estimate (9.7) is similar to the energy inequality for strictly hyperbolic operators (cf. J. Leray [3] and I. G. Petrovsky [1]). We proceed by constructing a solution to the problem (9.1). We consider the Cauchy problem for the ordinary differential equations with real parameter £ which is obtained by applying the Fourier operator with respect to x to the problem (9.1):
Since the coefficient Lo(i£) in the equation /-i L(Dt,%)v = L0(it)D>tv + '£iLi-k k=0
is degenerate at £ = 0, the problem (9.8) is considered for £ e M n \{0). The solution to this problem can be written in the form
£)dr,
(9.9)
o
-
l
[ &itX 2?r J L(i\, z£) r(0
where F(£) is a contour in the complex plane surrounding the roots of equation (9.2). It is easy to verify formula (9.9) since for the contour integral J ( t , £ ) Lemma 5.1 holds, i.e., L ( D t , i £ ) J ( t , £ ) = 0, Df J(0,0 = 0, j = 0, . . . , / - 2, Dlt~1J(Q,£) = Lo(i£)~l. We write an explicit expression for the contour integral (9.10). Lemma 9.3. For £ e M n \{0} the following representation holds:
i
§ 9. Pseudohyperbolic Equations
125
where the coefficients afc(£) for I > 1 have the form a,k(£) = H (^(0 ~~
PROOF. By the definition of the contour integral (9.10), for / > 1 we have 1 r p^\ ^ d\.
Since the roots 77i(£), . . . , *7/(£) are distinct, we have the equality '
which implies (9.11). For / = 1 formula (9.11) is obvious.
D
By Lemma 9.3, since the roots r)k(£} are real, we conclude that for any 7 > 0, £ € M n \{0} the following integrals are defined: oo
f
Therefore, repeating the proof of Lemmas 3.4 and 3.5, we obtain the following analogs of these lemmas. Lemma 9.4. For 7 > 0, £ € M n \{0} the following identities hold:
o
= (),. . . , / - ! ,
J(t, t}dt = (irj + ^(LCii, + 7,
Introduce the functions x m (£) such that x m (0 — 1 f°r (0 > l/ m X (£) = 0 for {£) < 1/TTi. Consider a sequence of functions {vm(t,£)}, where vm(t,£) = x m (£H*>0- By (9.9)-(9.11), the function v m (*,0 has no singularities. Consequently, if the right-hand side of the equation in (9.1) m
126
2. Equations not Solved Relative to Higher-Order Derivative
f ( t , x ) satisfies the assumptions of Theorem 9.1, we can apply the inverse Fourier operator to vm(t, £) with respect to £. Thereby we define a sequence of functions um(t, x) = F ~ l [ v m ] ( t , x). We estimate the sequence {um(t, x } } in the W 2 '^(M^ +1 )-norm and prove that this sequence is converging; moreover, the limit function u ( t , x ) is a solution to the Cauchy problem (9.1). , ao/a n ). Then
Lemma 9.5. Let s = (0,
1 )||
(9.14)
^ c\\f(t,x),
where the constant c > 0 is independent of m and f ( t , x ) ; moreover, for any q ^ 1
E
x -
(9.15)
as m —>• oo. PROOF. By the Parseval identity, we have
Using the Heaviside function 6(t] and the properties of the Fourier transform, we can write this expression in the form
r)J(t -
dr
By Lemma 9.4,
We use the estimate (9.5) in Lemma 9.1
(9.16)
§ 9. Pseudohyperbolic Equations
127
For j3a = 1 we find
\\D?um(t,x}, L 2
If ft a = I — lao, then \\D?um(t,x),
These two estimates imply the inequality (9.14). Arguing in the same way, for any q ^ 1 we find
7
x (|r/| + 7 + (O ao ) 1 "'A(»/.0, ^2(Mn+i)|| -> 0 as m —>• oo, where 1 — /ao ^ 0a ^ 1.
D
Lemma 9.6. Le£ |a|/2 + /ao > 1- T/ien the following estimate holds:
<: c ( \ \ f ( t , x ) , L 2>7 (K+ +1 )|| + || \\f(tt x ) , Li(M n )||, L 2 , 7 (E+)||),
(9.17)
where the constant c > 0 is independent of m and f ( t , x ) ] moreover, for any q^ I \\DPUm+*(t, x) - D?um(t, x ] , L 2|7 (R+ +1 )|| -> 0 -/a 0
as m —>• oo.
PROOF. Setting /3 = 0 in (9.16), we have
(9.18)
128
2. Equations not Solved Relative to Higher-Order Derivative
Taking into account the condition |«|/2 > 1 — I&Q > 0 and using the Parseval identity and the Riemann—Lebesgue theorem, we find \\um(t,x), L 2
where the constant c\ > 0 depends only on n, /, a. Similarly,
Since |a|/2 > 1 - Ia0, we find \\um+q(t, x) - um(t} x), ^2 l7 (M+ +1 )|| -> 0 as m —> oo. Similarly, for any /?, 0 < /?a < 1 — /ao, we have J+i)ll
+^(^ x) - DUm(t, x } , L 2|7 (Rj + i)|| -> 0,
m -> oo.
By the above arguments, (9.17) and (9.18) hold.
D
Lemma 9.7. Le^ |a|/2 + /a 0 > 1. Then \\Dltum(t,x), L 2 l 7 (M+ + 1 )|| ^ c d l / ^ . x ) , L 2l7 (Mj + i)|| + ||||/(<>a:)>Ii(]Rn)||,L2)7(M+)|)l
(9.19)
where the constant c > 0 is independent of m and f ( t , x ) . Moreover, for any q^ I \\Dium+'(t, x) - Dltum(t, x } , I 2l7 (M+ +1 )|| -> oo,
m -> oo.
(9.20)
§ 9. Pseudohyperbolic Equations
129
PROOF. By the Parseval identity and Lemma 5.1, we have
M«0' _
rm ii rm +
Since |Z, 0 (»OI ^ ^O1"'"0) ° > 0 and H/2 + /ao > 1, arguing in the same way as in the proof of the inequality (9.17), we find +)||) >
|| \ \ f ( t , x ) t
(9.21)
where the constant c^ > 0 is independent of m and /(<, a?). To estimate the second term 7™, we use formula (9.11). We have 1
r
^(0 / e'^-
Since the roots ^(0 are distinct and are homogeneous relative to the vector a with exponent a 0 , we find
c = const.
MO I =
Therefore,
ca — 1
X m (0(0 ( ' +1)ar °~ 1
,--yr
130
2. Equations not Solved Relative to Higher-Order Derivative
[the Young inequality is used] i+i!\\-
Since (/ + l)ao ^ 1 an d |a|/2 + ^o > 1, we, as above, obtain a similar inequality
where the constant c 2 > 0 is independent of m and f ( t , x ) . This estimate, together with (9.21), implies the inequality (9.19). We establish (9.20) in a similar way. D Lemma 9.8. Let s = (0, OQ/QI, . . . , ao/an), \a\f1 + IQQ > 1. Then \\D*um(t,x), ^ 1 -* ao) (K+ +1 )|| ^ (9.22)
|| \ \ f ( t , x ) ,
where the constant c > 0 is independent of m and f ( t , x ) . Moreover, for any q ^ 1 we have oo as m —> CXD.
PROOF. Consider the norms
If k = /, then, arguing as in the proof of Lemma 9.7, we have
(9.23)
§ 9. Pseudohyperbolic Equations
131
By the condition |a|/2 > 1 — lot® ^ /?a, we have )\\ + || \ \ f ( t , *), Li(M n )||, L 2 where the constant c\ > 0 is independent of m and f ( t , x ) . In the case k < /, using properties of the Fourier operator and Lemmas 5.1, 9.4 and arguing as in the proof of Lemma 9.5, we find Am — ~~
Using Lemma 9.1, we find
Taking into account the conditions |a|/2 > 1 — /c*o, 1 — fcao ^ /^«) it is easy to derive the estimate O ^ C 2 ( \ \ f ( t , x ) , ^2%'(M++1)|| + || ||/(/, x), ^(M^H, L 2 , 7 (M+)||), where the constant 02 > 0 is independent of m and /(<, a?). The above arguments imply the inequality (9.22). We establish (9.23) in a similar way. D PROOF OF THEOREM 9.1. Since |a|/2 + /a 0 > 1, from Lemmas 9.5-9.7 it follows that the sequence of functions {um(t,x}} is a Cauchy sequence in the space W l E & n + i ) ; moreover,
where the constant c > 0 is independent of m and f ( t , x ) . Since the space W 2 '^(^n+i) ^s complete, there exists the limit function u(t,x] G iy 2 '^(^n+i)- ^y Lemma 9.8 and the properties of the weak differentiation operators, we conclude that the weak derivatives D^D^u(t} x), kaQ-\-(3a ^ 1, Q^k ^ I, exist; moreover, \\D^Dfum(t,x)-D^D^u(t,x), L 2 , 7 (Kj +1 )|| -> 0 as m —> oo and
)|| ^ c(\\f(t,x),
132
2. Equations not Solved Relative to Higher-Order Derivative
By the construction of the functions um(t,x), we have \\L(Dt,Dx)um(t,x)-f(t,x),L2^(R++l)\\
as ?7i —> co. Therefore, u(t, x] is a solution to the Cauchy problem (9.1) and the corresponding estimates holds. D Consider the case |a|/2 + /ao ^ 1- Let the assumptions of Theorem 9.2 hold. Arguing as above, we can prove the existence of a solution to the Cauchy problem (9.1) and derive the corresponding estimates for the sequence {um(t,x)}. We first note that the arguments of the proof of Lemma 9.5 imply that the lemma remains valid for any value of |a|/2 + laQ. However, in the proof of Lemmas 9.6-9.8, we essentially used the condition | a |/2 -f la® > 1. We establish assertions similar to those lemmas under the assumption |a|/2 + /o;o ^ 1Lemma 9.9. Under the assumptions of Theorem 9.2, the following estimate holds:
+ II 11(1 + ( x ) } N l a l f ( t , x ) , Li(Mn)||, L 2 i 7 (M+)||),
(9.24)
where the consta.nt c > 0 is independent of m and f ( t , x ) ; moreover, for any q ^ 1
0<:(3a
}\\->U
(9.25)
as m —> oo. PROOF. Let us show how the orthogonality conditions (9.3) are used in the proof of (9.24) and (9.25). Consider, for example, the norm of the
§ 9. Pseudohyperbolic Equations
133
function um(t, x). In the proof of Lemma 9.6, we obtained the estimate
(9.26) which implies an estimate of the form (9.17) if |a|/2 + /ao > 1. If this inequality fails, then the second term in (9.26) cannot be, in general, estimated from above by a constant independent of m. However, if the orthogonality conditions (9.3) in Theorem 9.2 are satisfied, the function /(t,£) can be represented in the form
f ( t , y) 0 1
0
ffin
2
x Af - . . . X N_2XN-id\i ...d\N.
(9.27)
From this representation we see that, in view of the factor (iy£)N , the function ( £ } l a ° ~ l f ( t , £ ) is summable in a neighborhood of £ = 0. Consequently, \\um(t,x), L 2
where the constants c, c\ > 0 are independent of m and f ( t , x } . In the same way, we establish the convergence \\um+q(t,x) — um(t,x), L2 )7 (K^" +1 )|| —>• 0 as m —> oo. To estimate the remaining terms on the left-hand side of (9.24), we argue as in the proof of Lemmas 9.6-9.8 and use the representation (9.27). D PROOF OF THEOREM 9.2. As in the proof of Theorem 9.1, from Lemmas 9.5 and 9.9 we obtain the convergence of the sequence um(t, x) to a solution to the problem (9.1), and the estimate in Theorem 9.2. D We illustrate Theorems 9.1 and 9.2 by an example of the Cauchy problem for two equations mentioned at the beginning of this section
2^akD*ku = f ( t , x ) , /e=i
ut=0=0,
Dtu\t=Q=Q.
(9.29)
134
2. Equations not Solved Relative to Higher-Order Derivative
To simplify the exposition, we assume that the function f ( t , x) has compact support. In both cases, we have \a\/2 + lao — n/S -f 1/2. Therefore, for n ^ 5, by Theorem 9.1, the problem (9.28) has a unique solution u ( t , x ) £ ^2 7 (^n+i) f°r anv function f ( t , x ) £ ^2 7(^^+1) and ^he problem (9.29) is uniquely solvable in the space W2 '7 (M+ +1 ) if f ( t , x ) £ H^' (R* +1 ). If n = 3 , 4 , then, by Theorem 9.2, both problems are uniquely solvable in the corresponding classes if the orthogonality condition is valid:
f I f ( t , x ) d x - 0, J
and, in the case n — 2, the following additional conditions hold:
[ I f ( t , x)xj dx\ dx<2 = 0,
j = 1,2.
K2
We show that for a\/2 + lao ^ 1 the Cauchy problem (9.1) is not necessarily solvable in the classes under consideration. Theorem 9.3. Let 1 - a mm < a|/2 + Ia0 ^ 1, and let f ( t , x ) £ C r ^°(IR^ +1 ). Then for the solvability of the problem (9.1) in the space WV-y^n+i) the following orthogonality condition is necessary: r
I f ( t , x ) d x = 0.
(9.30)
J
ffin PROOF. Assume the contrary: if the condition (9.30) fails, then there exists a solution to the problem (9.1) u ( t , x ) £ VV 2 '^(^n+i)i moreover, D^u(t, x) £ W2 '^
a
° (M^, j ) , k — 0 , . . . , /, and
c(f) k =0
By the properties of the Laplace operator C and the Fourier operator F, the function V(T, a] = jCF[u](r,£} satisfies the equation L(T, i ^ } v ( r , £} = g ( r , ^), where g ( r , £ ) = CF[f}(r,£), r = ir/ + a, a > 7, ^ 6 M n . Furthermore, by the Paley-Wiener theorem, we have sup \\v(ir) + cr,0, L2(Rn+1)\\ ^ c(f).
(9.31)
§ 10. Applications of Sobolev-Type Equations
135
Taking into account the assumptions on the operator L(Dt,Dx), we can write the function i>(r, £) for £ 6 M n \{0}, cr > 7 in the form i>(r, £) = '0- Consequently,
where a > 0 is a constant. By this estimate, from (9.31) it follows that sup IKO'"0"1^ + To,0, L 2 ( ( a , b)x{e< (0 < 1})||
Cl c(/)
< oo, 7o > 7,
where GI > 0 is a constant depending on a, 6, 70. Therefore, 6
7o,0, L 2 ( ( a , b)x{£< (0 < 1})|| ^ <; c?c 2 (/). We represent the function g(irj + 70, 0 in ^ ne f°rm n
I
#("7 + 70,0 =9(iri + 70,0) + ^^- / DSjg(ir) + 70, J=i o Then 6
°-1, ^2({e < (0 1+ami
", L 2 ({ £ < (0
Since |a|/2+/o;o-f a m i n > 1 , the right-hand side is estimated from above by a constant ca(/) independent of e > 0. On the other hand, for |a|/2 + /a:o ^ 1 we have IKO'" 0 " 1 , L2({£ < (0 < 1})|| -> oo as e -> 0. If the condition (9.30) fails, there exists an interval (a, b] such that
Hence we arrive at a contradiction. Consequently, the condition (9.30) is necessary for solvability. D
136
2. Equations not Solved Relative to Higher-Order Derivative
§ 10. Applications of Sobolev-Type Equations to the Solution of a Hyperbolic System In this section, we consider a hyperbolic system described small-amplitude oscillations of a compressible stratificated rotating fluid. Solving the Cauchy problem for this system, we establish an interesting relationship with simple Sobolev-type equations. Namely, the proof of L p -estimates for the solution to the Cauchy problem for this system is based on the use of the Lp-estimated proved in the previous sections. We write a system of equations describing the dynamics of an exponentially stratificated compressible fluid with po — Ae~2f3xz , (3 > 0, in an unperturbed stationary state (cf. L. M. Brekhovskikh and V. V. Goncharov [1], S. A. Gabov and A. G. Sveshnikov [1]). If the fluid is rotated around the £3-axis with constant angular velocity a/2, this system is reduced to the form DXlu4 = 0, Dtu-2 - Q.UI + DX2U4 = 0, Dtu3 + DX3u4 + gu5 = 0, Dtu5 -I- DXlu\ + Ac 2 u 2 -f DX3u3 = 0, 1 c
DtU5 -- 7 2 DtU4
--
N2 1/3 g
MOD
= 0,
where c is the velocity of sound, g is the acceleration of gravity, and TV2 = 2(3g— g2 / ( c 2 ) ^ 0 is the squared Vaisala-Brunt frequency. It is easy to verify that this system is hyperbolic with multiple characteristics. We consider the Cauchy problem with the initial conditions « f c t = 0 = «£(*),
* = ! , . . . , 5.
(10.2)
Since the system (10.1) is hyperbolic, it is easy to obtain the /-^-estimates for a solution to the Cauchy problem by standard methods using the FourierLaplace transform. However, the proof of the Lp-estimates for a solution for p ^ 2 is more complicated. We will show how to solve this problem using the above results. Theorem 10.1. Let e^ 3 u°(x) £ W™+l(R3), i = 1, . . . ,5, 1 < p < oo, p ^ 1, where m is natural. Then for a sufficiently large 7 > 0 there exists a unique solution to the Cauchy problem (10.1), (10.2) such that e / 3 x 3 U i ( t , x ) G W£\f(1&~i)> i — I , . . . ,5, and the following estimate holds: 5
5
/3x3 U L m /3x3a L YV c e V \\e u-(t x] vv Wp,i\^4 nR~hll < cc /Y^ u°(x} Wm+1l ^(Ri}\\ / ^ \\ I \ J *)•> )\\
(103) \iv.0)
§ 10. Applications of Sobolev-Type Equations
137
where the constant c > 0 is independent of u®(x), i — 1,.. . ,5. Remark 10.1. In the case p = 2, for the solvability of the problem (10.1), (10.2) in the corresponding class of functions it suffices to require the conditions e^X3u^(x) E W^db)) z = 1 , . . . ,5, i.e., the order of the smoothness of the initial functions is less by 1 than that indicated in the theorem for p ^ 2. We note that this situation is typical for hyperbolic equations and systems in obtaining the //^-estimates for solutions (cf., for example, E. M. Stein [1]). We divide the proof of the theorem into several lemmas. It is easy to prove the following assertion. Lemma 10.1 The components Uj(t,x), j = 1,2,3,5, of a sufficiently smooth solution to the problem (10.1), (10.2) can be expressed in terms of the function u^(t, x} as follows: u\(t, x) = cos(at)u®(x) — sin(at)u2(x) t — I (cos(a(/ — s))DXl 1*4(5, x) — sin(a(t — s))DX2U4(s, x)) ds, 0
t
— I (sin(a(i — s)}Dx. u^is, x} + cos(a(t — S}}DX^UA(S, x}} ds, J 0
,.
(4
v\
_
rr-f
Us(l, X) — O
/
JV7-JN- - 0 / _ \
9 _;.
t / \ f cos(JV(* - s))DX3u4(s, x) + -~ sin(N(t - s))Dsu4(s, x) J ds, o
N u5(t, x} - — s 9
t
0
— sin(N(t — s)}DX3u4(s, x) -- c - cos(N(t — s))D 5 w 4 (s, x) } ds. 9 * J
Lemma 10.2. If Uj(t,x), j = 1 , . . . ,5, is a sufficiently smooth solution to the problem (10.1), (10.2), then the function v ( t , x ) — e / 3 x 3 U 4 ( t , x ) is a
138
2. Equations not Solved Relative to Higher-Order Derivative
solution to the integrodifferential
equation
-jDtv - I cos(a(t - s))(D^ + D*3)v(s, x) ds o t - / cos(N(t - s}) I D32 - /?2 + — )v(s,x)ds =F(t,x), J V cj )
(10.4)
0
wAere e~PX3 F(t, x) = — L>XI (cos(off)u°(x) — sin(at)u2(x}) ( D ^ N2\ ] ^(z) — DX2 (sin(at)u^(x) + cos(a^)«2(x)) — cos(Nt] | X3
+ sm(^) f A/?,, + N] (u°5(x) - \u°4(x)} .
(10.5)
PROOF. We substitute the expressions for the components U k ( t , x ] , j — 1,2,3, 5, in Lemma 10.1 into the fourth equation of the system (10.1). Making the necessary transformations and taking into account the relation a N2 4^2 + — = 2/3, c
g
(10.6)
it is easy to obtain the following integrodifferential equation for the function u4(t,x): t
~Dtu4- t cos(a(t - s))(D2Xi + D22)u4(s, x) ds o t - J cos(N(t - s}} (D23 + WDX, + —} u4(s, x) ds = e~^F(t, x } . o Multiplying both sides by e^ 3 , we obtain (10.4) for v ( t , x ) .
D
Lemma 10.3. If Uj(t, x), j = 1,. . . , 5, 25 a sufficiently smooth solution to the problem (10.1), (10.2), then the function v ( t , x ) = e ^ X z u ^ ( t } x ] is a solution to the Cauchy problem
3
i-
\
9
v
+°2
§ 10. Applications of Sobolev-Type Equations t r
2
/
/
+ c N I sm(N(t ~ s}}\Dl 3 -
J
139
V
V
- c2
/? - ),
(10.7)
where F(t,x) is defined in (10.5). PROOF. Equation (10.7) follows from (10.4) by differentiating with respect to t and taking into account the relation (10.6). D Lemmas 10.1 and 10.3 show how to construct a solution to the problem (10.1), (10.2) in the classes of sufficiently smooth functions. The formulas obtained in such a way will be used in the proof of the solvability of the Cauchy problem in the weight Sobolev space indicated in the theorem. We will need the L p -estimates, p / 2, for a solution to the Cauchy problem for the Klein-Gordon-Fock equation Dlu - c2Au + du = f ( t , x ) , u
t=o=
t>0, x G Ma,
DtU
\t=o= ^( x )'
where d ^ 0 is a constant. Such estimates can be obtained by using multipliers. However, in the case n = 3, the proof is significantly simplified if we write explicit expressions for the solution to the problem (10.8) using the Bessel function «/o(0- Setting c = 1 for the sake of simplicity, we can write formula (10.8) as follows:
\x-y\
\x-y\
\x-y\
2
= u (t,x} + u (t,x) + u3(t,x).
(10.9)
In the case d = 0, it is the Kirchhoff formula. In the case p ^ 2, the Lp-estimates for the function u(t,x) are given in the following lemma. Lemma 10.4. If
140
2. Equations not Solved Relative to Higher-Order Derivative
s = ( Q , k , k , k ) , p ^ . ' 2 , 7 > 0 ; k ^ Q is integer, then H U ^ . Z ) , < 7 (Mf)|| ^ c|M*), W* +1 (R 3 )||,
(10-10)
2
(10.11)
3
(10.12)
\\u (t, x), W^(R+}\\ <: c||V(*), W*(K 3 )||, ||u (t,z), ^(Mj)|| ^ c\\f(t)X), WpyKj)!!,
where the constant c = 0(7) > 0 is independent of (p(x), i/;(x), f ( t , x ) . PROOF. We write the function u 1 ^ , ^ ) in the form
0 \x-y\
=r
d where J 0 (£) = -T~
f
1
t 4?r Ul = i
o
x
/f i<
—
\47T
/" ,
x ,
I (p(x -\- rz) as H
J |0| = 1
IT
[ . „ .
v, , \ ,
/ (z, VCCMX + rz)) ds dr.
47T 7 |z| = l
/
Using the properties of the function Jo(0 and the generalized Minkowski inequality, for 7 > 0 we obtain the estimate Similarly, we can prove the estimate (10.10) for k > 0. To derive the estimate (10.11), we represent the function u 2 ( t , x ] in the form
x
f / ijj(x + rz] dsdr. J
§ 10. Applications of Sobolev-Type Equations
141
Thus, the inequality (10.11) is obtained in the same way as (10.10). We prove the estimate (10.12). As above, we represent the function u3(t, x] in the form
t u (t,x) = I —^ 3
r
n[[ vd I I
r J Til
J0 0J
j
/ IJl(4
o ( v d ,( ( t ~
V(t-T
47T
f(r,x+l
Using the Minkowski inequality, the Young inequality, and the properties of JQ(£,), we
J J 0
0
2
vV-r) -
-^2^
Since
we have
The above estimates imply (10.12) for k = 0. The case k > 0 is considered in a similar way. D
142
2. Equations not Solved Relative to Higher-Order Derivative
We note that to estimate the functions u , J ( t , x ) , j = 1 , 2 , 3 , in the V^2 )7 (B&4")~ norms > k ^> 1) ^ suffices that the order of the smoothness of the data of the problem (10.8) is less by 1 than that indicated in Lemma 10.4. This reflects the essential difference between the L2-theory L p -theory, p ^ 2, of the well-posedness of the Cauchy problem for hyperbolic equations. We return to the Cauchy problem (10.7). For the sake of simplicity, we set c — 1. We look for a solution to the problem (10.7) in the form (10.9), where d = ((3 - g)2 , (p(x) = e^u^x), ^ ( x ] = F(0, x), and f ( t , x) is the unknown function. Using such a method of solving the problem (10.7), we should choose a function f ( t , x ] in such a way that the function v(t, x) defined by the right-hand side of (10.9) satisfies the integrodifferential equation (10.7). This requirement leads to the following integrodifferential equation f o r f ( t , x ) : f ( t , x ) + Tf(t,x)=$(t,x),
(10.13)
where
t $(<,ar) = DtF(t,x)-a j sm(a(t - s))(D2Xi + D2J(ul(s, x) + u2(s, x ) ) d s o t - N I sin(N(t - s})(D2E3 - d } ( u 1 ( s , x) + u2(s, x ) ) d s ,
t
T f ( t , x) =
a*m(a(t
- rj))(D2Xi + D^}
o
_
(10.14)
o , y) dy x-y\
/ x A, f \
r / «v
\ \ Jv(\/d((r)- r) 2 - \x - y| 2 ))/(r, y } d y \ d r \ d r ] / /
and the functions w 1 ^ , ^ ) , u2(t,x] are defined in (10.9). Our goal is to show that for any $(t,x) G Wp 7 (M|), s = ( Q , k , k , k ) , equation (10.13) is uniquely solvable in the class W p s |7 (Mj) for sufficiently
§ 10. Applications of Sobolev-Type Equations
143
large 7 > 0. It suffices to establish that the linear operators T; : W*^ (M*) —. W' (l^.^), i — 1,2, are continuous; moreover, \\Ti\\ <:c(i),
c(7)->0,
7~>°o.
(10.15)
By the definition of the operator T, equation (10.13) is uniquely solvable in the space W*i7 (M*) for sufficiently large 7 > 0. To prove (10.15), we need the following assertion. Lemma 10.5. Let u3(t,x) be the function in (10.9). Then
t
t 3
(A — c?) / sin(a(£ — rj))u (rj, x) dr\ — — I sin(a(^ — ^))/(?y, x] drj
o
o t
+ au3(t, z) - a 2 / sin(a(t - r,))u3(n, x) dn.
(10.16)
PROOF. By definition, the function u3(t,x] is a solution to the Cauchy problem for the Klein-Gordon-Fock equation (cf. (10.8)) for
t (A - cf) / sin(a(^ - r)})u3(n, x] drj o t
=-
sin(a(t - r7))(/(/7, x) - D*u3(ri, x))drj.
Integrating by parts and using the equality w 3 | _ = Dnu3\ tain (10.16).
= 0, we obD
From Lemma 10.5 it follows that the function t v(t,x}—
sin(a(t —
rj))u3(rj,x)drj
can be regarded as a solution to the elliptic equation (A — d}v = g ( t , x ] , where the function g(t,x) is the right-hand of (10.16). Lemma 10.6. 7 / / ( t , x ) ' e W*^(R%), s= ( Q , k , k , k ) , p ^ 2; 7 > 0, where k ^> 2 is integer, then (10-17)
144
2. Equations not Solved Relative to Higher-Order Derivative
where i = 1,2 ; r — (0, / , / , / ) , 0 ^ / ^ k and the constant 0(7) is independent of f ( t , x), and c(j) —>• 0 as 7 —> oo. PROOF. Using the definition of u3(t,x] in (10.9), we can write the function T\f(t,x] in the form J(t,x) = a(D2Xi
m(a(t - rj)}u3(n, x) dn.
Since d ^> 0, from Lemma 10.5 we conclude that this function can be represented as follows: T i f ( t , x) = (A - d}-1 (a(D2^ + D2X2)(au3(t, x) u3(n,x) + f ( n , x ) ) d n Using estimates for solutions to the second-order elliptic equations, the Minkowski inequality, and the Young inequality, we find \\Tif ( t , x ) , sin(a(t -
3
Cl\\u
(t,x),
r)))f(r),x)dr),
sm(a(t-rj))u3(i1,x)dn,
where the constants GJ > 0 are independent of f ( t , x ) and 7. Taking into account the estimates obtained in the proof of Lemma 10.4, we find
where c( estimate
0 as 7 —> cxo. Arguing in the same way, we can establish the
where r = (0, \/3\, \(3\, \/3\). This implies (10.17) for i = 1. The case i = 2 is treated in a similar way. D
§ 10. Applications of Sobolev-Type Equations
145
From Lemma 10.6 we can obtain the estimates (10.15). As was mentioned above, this means that the operator equation (10.13) is uniquely solvable in the space W£^(R%) for any function $(t,x) e WJ i7 (Mj), 7 ^ 70, where 70 > 0 is sufficiently large. From the proof of Lemma 10.6 it follows that the solution f ( t , x ) = (I + T)-l*(ttx)
(10.18)
II/C*,*), w;,7(Mf)n ^ c||*(*,*), w;i7(R+)||,
(10.19)
satisfies the estimate
where r = (0, / , / , / ) , 0 ^ / ^ k. Lemma 10.7. Let e^ 3 u?(x) G Wj+HRa), * = 1 , . . . ,5, 1 < p < oo, p ^ 2, k ^ 2 is integer. Then the Cauchy problem (10.7) is uniquely solvable in the class W^ (j£+^ 7 ^ 7o; one? the following estimate holds: , (10.20)
1=1 where the constant c is independent of u®(x). PROOF. By Lemma 10.4, the solution to the problem (10.7) in the space W£ 7 (Rj) is represented in the form (10.9) if (p(x) = e^ 3 u5(x) e VV^+^Ms), (0, A;, AT, A;), where F(t, x) is denned in (10.5) and <£(/, x) is defined in (10.14). The estimates (10.10)-(10.12), (10.19) imply the inequality
where r = (0, m, TTZ, m), 0 ^ m ^ A;. Since 5
by the definition of the function F ( t , x ] , it suffices to verify the inequality H,
(10-21)
146
2. Equations not Solved Relative to Higher-Order Derivative
where r — (0, m, m, m). Therefore, it is necessary to estimate the functions
t Gi(t,x) = f sm(a(t - s})(D2Xi + Dljui(s,x)ds, J
o t
F i ( t , x ) = I sm(N(t-s))(D23
- d ) u i ( s , x) ds,
•J
o
where i = 1,2 and u^^x), u2(t,x) are defined in (10.9); moreover, (x) = e ^ X 3 u ^ ( x ) , 4>(x] — F(0,x). These functions are estimated by the same method. For the sake of defmiteness, we consider Gi(t,x}. We show that t sin(a(* - s ) ) u 1 ( s , x ) d s
o t sm(a(t — s))w 1 (s, x) ds.
(10.22)
o 1
Indeed, u ^ , ^ ) is a solution to the Cauchy problem (10.8) for -0(x) = /(t,x) = 0. Therefore, (A - d^^z) = D^ul(t,x). Consequently,
t
t r sin(a(^ — s))u (s, x) ds = I sin(a(/ — s ) ) D 2 u 1 ( s J x} ds. 1
/
0
0
Integrating with respect to tf, we obtain (10.22). Since d ^> 0, arguing in the same way as in the proof of Lemma 10.6 and taking into account (10.22), we find
t
-f a
2
r
\
/ sin(a;(£ — s))w 1 (s, x) ds } J J
Consequently,
sm(a(t - s ) ) u l ( s , x ] ds,
§ 10. Applications of Sobolev-Type Equations
147
Using the Young inequality and (10.10), we find ||Gi(*>x)>Lp Similarly,
Estimating functions G2(t,x), Fi(t,x), F2(t,x) by the above method, we obtain (10.21). It is easy to show that the solution to the problem (10.7) is uniquely determined. D From the above arguments and Lemma 10.4 we obtain the following theorem about the solvability of the Cauchy problem
t - At; + dv + a I sm(a(t - s}}(D2Xi + Dl2)v(s, x) ds
t
(10.23) sin(N(t - s)}(D*3 - d)v(s, x) ds = g ( t , x } ,
t > 0,
0
Theorem 10.2. Suppose that y?(x) 6 Wf + 1(R3), ^(x) G Wf(R3), g ( t , x ) G W p s ]7 (Mj), 7 ^ 70, s = ( Q , k , k , k ) , 1 < p < oo, p / 2, fc ^ 2. T/ien ^/iere exists a unique solution to the problem (10.23) v(t,x) G W' moreover,
Remark 10.2. For the well-posedness of the problem (10.23) in the class W2n(B&4} the following equations are sufficient: 3), we can cnoose a sequence of functions {^(x)}, i = 1 , . . . ,5, y>?(z) e C^°(M3) such that | c^'^f (x) - ti?(x)), jy p m+1 (M 3 )|| -^0,
Jb -> oo.
(10.24)
148
2. Equations not Solved Relative to Higher-Order Derivative
We construct a sequence { F k ( t , x } } such that each function Fk(t,x) is defined in the same way as F ( t , x ] by formula (10.5) with u^(x] replaced with i(x), i — 1 , . . . ,5. We consider the Cauchy problem (10.7) for F ( t , x ) = F k ( t , x ) , w°(x) =
v ( t , x ) , W^(R+)\\ <: c
||C^V?(*), Wj +1 (R 3 )||,
/ £ 0,
(10.25)
where the constant c is independent of k and • oo. Consequently, from (10.24) and the estimates (10.26) for Uj(t,x], j = I , . . . ,5, we obtain the inequality (10.3). It is obvious that the constructed vector-valued function u(t, x} is a solution to the Cauchy problem (10.1), (10.2). The proof of the uniqueness of a solution is not difficult. D Lemma 10.8. For any k the following estimates hold:
R 3 )||,
i
/ ^ 0,
i=i
(10.26) where j = 1,. . . , 5 and the constant c > 0 is independent of k,
§ 10. Applications of Sobolev-Type Equations
149
By the definition of uk (t, x), we have 11*1(1, x) = cos(at)(pk(x) — sin(at)(f>2(x) t — I (cos(a(t — s))DXl1/4(5, x) — sin(o;(t — s))/)^ 1*4(5, x)} ds. Therefore, the main difficulty in the proof of (10.26) is the proof of the estimates
(10.27) t
/ Sm(a(t - s ) } D 2 X y X . v k ( s , x} ds, Lp>
? P X 3 p k ( x ) , W2(R3)\\,
(10.28)
where j = I , 2, 3 and the constant c > 0 is independent of k and
t j
cos(a(t-s}}D2XiX]Vk(s,x)ds
o t
t 2
=I D
s
iXjv
k
(s,x}ds-a I'sin(a(*-s)) j D 2 X i X j v k ( r , x ] dr ds.
Consequently, to obtain the estimate (10.27,) it suffices to establish the inequality
r
D2XlX]vk(s,
|.
(10.29)
The proof of the estimate (10.29) is reduced to the proof of the weight Lpestimates (with weight e 1t) for solutions to the Cauchy problem for the equation that is not solved with respect to the second-order t-derivative.
150
2. Equations not Solved Relative to Higher-Order Derivative
Indeed, using the definition of vk(t,x) and setting c = 1 for simplicity, we have the identity
t 2 k
k
D v (t, x) - (A - d)v (t, x} + a I sin(a(t - s}}(D2Xi -f D2X2)vk (s, x) ds o t
+ N fsm(N(t-s))(Dl3 o
-d}vk(s,x)ds = DtFk(t,x},
d = (/? - g}2 .
Therefore, for the function w(t,x) — D2 x.vk(t,x) we have
t sin(a(t - s } ) ( D X i + D2Jw(s, x) ds
o t sin(N(t - s))(D2C3 - d)w(s, x} ds = g ( t , x), where g ( t , x ) = D? D2iX]vk ( t , x) - DtD2XiJ:jFk(t,x). Applying the differential operator (D2 -fa 2 ) (D2 + N2} to this identity and integrating twice with respect to 2, after some transformations we can conclude that the function w(t,x) is a solution to the simple Sobolev-type equation (A - d) D2w -f N2(D2Xi + D22}w + a2(D2X3 - d}w = G(t,x), where t
D * X l X . F k ( r , x) dr
0
By definition, the function w(t,x) satisfies the initial conditions w t_Q= 2 D2xiXj\ _ ( e ^ X 3 u ^ x )/ . From the results of the prei vious sections we obtain the estimate (10.29). As was already mentioned, this implies (10.27). We can establish (10.28) in the same way. Using these estimates, it is easy to obtain the inequality (10.26). D
Chapter 3 The Cauchy Problem for non-Cauchy—Kovalevskaya Type Systems
In this chapter, we begin to study systems of differential equations Ai(Dx)u = f ( t t x ) ,
x = ( x i , . . . ,xn),
(0.1)
where AQ is a singular matrix. Such systems, called non-Cauchy-Kovalevskaya-type systems, were first discovered in hydrodynamics. The systems (0.1) attracted a special interest after works of J. Leray [1, 2], J. Leray and J. Schauder [1], F. K. G. Odqvist [1, 2], S. W. Oseen [1], E. Hopf [1] devoted to the study of the Navier-Stokes system and the works of S. L. Sobolev who studied small-amplitude oscillations of a rotating fluid [4-7]. As was mentioned in Introduction, after the publication of results by S. L. Sobolev, I. G. Petrovsky emphasized the importance of the study of general differential equations and systems not solved relative to the higherorder time-derivative. The first deep investigation in this direction was due to S. A. Galpern [1-3] who constructed the L2-theory for the Cauchy problem for systems of the form DtM(t, Dx)u + £(t, Dx}u = 0, Later, K. K. Golovkin [1, 2], 0. A. Ladyzhenskaya [1, 2], V. N. Maslennikova [1, 2], V. A. Solonnikov [1, 3], and others established the Lp -estimates and estimates in the Holder norms for solutions to the Cauchy problem for some systems in hydrodynamics of the form (0.1). 151
152
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
The main difficulties arise when we try to choose suitable function spaces, find the solvability conditions, and derive estimates for solutions. As can be seen from examples, this problem turns out to be more complicated for systems than for scalar equations discussed in Chapter 2. We emphasize that, at present, the theory of Sobolev-type equations is developed better than the theory of systems (0.1), although the sources of such equations are systems; moreover, almost all Sobolev-type equations appearing in practice are derived from systems of differential equations of the form (0.1). To construct the complete theory of the Cauchy problem and the boundary-valued problems for systems (0.1), it is necessary to classify them. In this chapter, we consider two classes: Sobolev-type systems and pseudoparabolic systems (cf. Definitions 2.1 and 2.2). Examples of Sobolev-type systems are, in particular, the Soboiev system, the internal wave equation and the gravity-gyroscopic wave equation in the Boussinesq approximation (cf. Section 1). The linearized Navier-Stokes system can serve as an example of pseudoparabolic system. We describe both classes of systems in Section 2. We study the Cauchy problem for Sobolev-type system in Sections 3-6, and for pseudoparabolic systems in Sections 7-13. In this chapter, we develop a method of obtaining the L p -estimates for solutions to equations not solved relative to the higher-order derivative.
§ 1. Examples of non-Cauchy-Kovalevskaya Type Systems In this section, we give some examples of system of the form (0.1) appearing in hydrodynamics. We consider only equations describing the dynamics of small internal motions of a stratificated fluid in the equilibrium state. Such equations are linear. We start with equations describing small oscillations of an ideal imcompressible stratificated fluid that uniformly rotates about the vertical OxaOaxis. We write the equation in the Cartesian coordinates ( x i , x - 2 , x ^ ) which rotate together with the fluid. Relative to these coordinates, the unperturbed fluid is in the rest state. We assume that the fluid is exponentially stratificated along the Oxa-axis, i.e., the density of the fluid in unperturbed state is expressed as po(x3) = Ae~2/3x3, A, /3 > 0. Then small-amplitude oscillations of the fluid in the field of gravity force without other exterior forces is described by the system of differential equations dv
Q,
(1-1)
§ 1. Examples
153 -—- - 2/3pQv3 = 0, at
divv = 0, v = (vi,V2,v3y is the velocity of particles of the fluid, p\ is a change of the density caused by the motion of the fluid, p is the dynamic pressure, g is the acceleration of gravity, o7 = ue3 = (0, 0,0;)', o; is the double angular velocity (cf., for example, L. M. Brekhovskikh and V. V. Goncharov [1, Chapter 10]). Definition 1.1. The system (1.1) is called the internal gravity wave system of equations if u> = 0 and the gravity-gyroscopic wave system if o; > 0. In the study of small-amplitude oscillations of a fluid, the system (1.1) are often considered with the help of the Boussinesq approximation. In this case, this means that the density po(x3) in (1.1) is constant (po(x3) — p0). Making the change u = PQV, we can formally reduce the system (1.1) to one of the following systems:
dt du2
dp ox i dp
-UJU-2 + -- = 0,
= 0,
d2p = 0, x3
divu = 0, (1.2) dp
^— = 0, ox i
dt
du-2 —l dt
dp
0
divw = 0, where ./V2 = 1(3g is the squared Vaisala-Brunt frequency. Unlike the problem about small-amplitude oscillations of a fluid where we can assume that the density is constant in an unperturbed state po(x) = o
A, i.e., 0 — 0, in this case, we have - — = 0 in (1.1).
154
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
Then the system (1.1) is reduced to the form r\
c\
nii.i
nn
dt
"JU2
du-2
|
dt '
UJUi
dp dx-2
= o, = 0,
(1.3)
dp dt ' dx3 = 0, div u -= 0.
duz
i
This system was intensively investigated by many mathematicians and mechanicians. This problem was considered by H. Poincare [1] in 1885. The first deep investigation of the system (1.3) was made by S. L. Sobolev in the 1940's. Results of his study were published in S. L. Sobolev [4-7]. In particular, S. L. Sobolev studied the Cauchy problem, the first and second boundary-value problems in a cylindrical domain for the system (1.3), and stated a number of new problems in mathematical physics. That is why the system of equations (1.3) is referred to as the Sobolev system in the literature. In the authors opinion, it is convenient to write the internal gravity wave system (uj = 0) and the gravity-gyroscopic wave system (ui > 0) in the integrodifferential form (1.2) since the system in this notation differs from the Sobolev system (1.3) only by the integral additive term in the third equation. It will be clear from the further consideration that this notation allows us to study different boundary-value problems for all three systems using the same scheme. Similar systems of differential equations are written for small-amplitude oscillations of a viscous imcompressible fluid rotating about the vertical axis. In particular, if the density po and the kinematic viscosity v are constant, the equations describing the motion of a fluid relative to the rotating coordinates are the well-known linearized Navier-Stokes equations du
—
i^Au - [u, uJ] + Vp = 0,
\J 6
div u — 0.
The physical basis of the above systems, as well as statements and solutions of some problems for such systems can be found in S. M. Belonosov and K. A. Chernous [1], L. M. Brekhovskikh and V. V. Goncharov [1], S. A. Gabov and A. G. Sveshnikov [1, 2], V. M. Kamenkovich [1], N. D. Kopachevsky, S. G. Krein, and Ngo Zuy Can [1], O. A. Ladyzhenskaya [1], J. Lighthill [1], J. Serrin [1], R. Temam [1], C. Truesdell [1].
§ 2. Classes of non-Cauchy-Kovalevskaya Type Systems
155
§ 2. Classes of non-Cauchy-Kovalevskaya Type Systems In this section, we describe classes of systems of differential equation that are not solved relative to the time-derivative. We consider the Cauchy problem and mixed boundary-value problems for such systems. We consider the matrix differential operator +K - \K° ° Dt ) uDx)}— M(D } ^
0
where K\(DX}} L ( D X ) , and M(DX] are mxm-, mx (v— m)-, and (i/— m) xmmatrix differential operators with respect to x with constant coefficients and A'o is a nonsingular number m x m-matrix. This operator can be written in the form
where K
°
Oj'
A in\-
Al(Dx
*
> - [M(DX)
0
We consider two classes of matrix differential operators £(Dt, Dx}. For the first class, Ki(Dx] is a number matrix denoted by K\. For the second class, there are differential operators with respect to x among elements I. Sobolev-Type Operators We describe the class of Sobolev-type operators. We indicate assumptions on entries //-^(z'ry, z'£), £ = ( £ 1 , . . . , £ n )i of the matrix £(ir), i£) playing the role of the symbol of the operator £(Dt, Dx). Assumption I.I. There exist numbers s i , . . . , s^ and t i , . . . , t,, such that max Sfc = 0 , tj ^> 0, j = 1 , . . . , i/, and t z ^ ti, s,- ^ si, i = m + 1,. . . , i/, and a vector a = ( a i , . . . , a n ), a,- > 0, such that for k,j— 1 , . . . , v we have JfcjO' 7 ?) z'0 — 0 f°r s fc + tj < 0 and lkij(if],cai^) = cSk +t^lkj(ir], z^), c > 0, for s/c + tj J> 0. We assume that all the numbers tj /a z - are natural. We write the equalities in the matrix notation
where 5(c) = (<Jfc S i ) and T(c) = (^'c4''). By Assumption I.I, ti = . . . =
156
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
Assumption 1.2. The equality det(M(i£)KQl L(i£)) = 0, f 6 M n , holds if and only if £ — 0. By Assumptions I.I and 1.2, for c*i = . . . = an the operator [ KQ [M(DX]
L(DX 0
is elliptic in the sense of Douglas-Nirenberg. For sufficiently large Rer > 0 we have det£(r,z£) = det(rK0 + Ki)det(-M(i^)(rK0 + I
where 0
operator £(Dt,Dx) satisfies Assumptions I.I and 1.2, I\2 is a number m x m-matrix, and Df 1 is the integration operator: t
,x)= I u(s,x}ds. The Sobolev system (1-3), the internal wave system, and the gravitygyroscopic wave system in the Boussinesq approximation (1.2) can serve as examples of Sobolev-type systems. We give explicit expressions for the corresponding operators of these systems. EXAMPLE 2.1 (the Sobolev system). For the Sobolev system we have
'1 0 0' 0 1 0 0 0 1
L(DX) =
(DDX:, x
\D^
X)
= (DX1DX2DX3),
'0 -u u 0 0 0
det
§ 2. Classes of non-Cauchy-Kovalevskaya Type Systems
157
EXAMPLE 2.2 (internal waves and gravity-gyroscopic waves equation). For the internal wave system and the gravity-gyroscopic wave system
K0 =
"1 0 0" 0 1 0 , 0 0 1
(Dx
A
L ( D X ] = \DX 2
"0 A'i = w _0
,
— UJ
0 0
0" 0 0_
j
^2
"0 0 0 " 0 0 0 2 0 0 JV
M(DX] = (^«i DX2DX3),
For all three systems (cf. (1.2), (1.3)) Si = S 2 = S3 = -1,
S4 = 0,
tl = t2 = ts = 1, t 4 = 2,
We present one more class of Sobolev-type systems: n
A0Dtu + ^ AjDXju + An+lu = f ( t , x), where
K0 ol
o
O '
KI ol 0
O'
A'o and A'i are number m x m-matrices, LJ is an m x (y — m)-matrix, and Mj is an (y — m) x m-matrix. This class of Sobolev-type systems contains, in particular, a special class of symmetric hyperbolic systems with a positive semidefinite matrix AQ such that Ak = A*k, k — 0 , 1 , . . . , n. We note that, in this case, such systems are not hyperbolic in the sense of Friedrichs since detAo = 0. II. Pseudoparabolic Operators We consider the class of operators £(Dt,Dx), called pseudoparabolic, with some conditions (cf. below) on the elements /fc,j(iJ7, i£) of the symbol of the operator £(Dt, Dx). Assumption II. 1. The following equalities hold: si = . . . = sm = 0, tl — . .. = t m = 1. There exist numbers s m + i , . . . ,8^, t m + i , . . . , t^ such
158
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
that 0 ^ Sj ^ — 1, tj ;> 0, j = m + 1 , . . . , z/, and a vector a = ( c * i , . . . , a n ), a; > 0, i — 1, . . . , n, such that for k,j= 1, . . . , v we have l k j ( i r ) , i£) = 0 if s / c + t j < 0 and l k j ( c i r j , c a i £ ) = cSk+^lktj(irj,i^}, c > 0, for s/j +1j ^ 0. We assume that all the numbers tj /<*,• are natural. We write the equalities in Assumption II. 1 in the matrix notation
where S(c) = (^c Si ) and T(c) = (^c t j ). Assumption II.2. det(rA' 0 + K\ (if)) ^ 0, Rer ^ 0, f 6 M n , |r| + |f | 7^ 0, moreover, det(M(is)(rA'o + A' 1 (z^))~ 1 L(zs)) = 0, s £ M n , if and only if s = 0. Definition 2.2. A system of differential equations £(Dt, Dx)u = f ( t , x) is said to be pseudoparabolic if the operator C(Dt,Dx) satisfies Assumptions II.1 and II.2. In what follows, we will also consider pseudoparabolic systems with lower-order terms £(Dt, Dx)u + A2(Dx)u = f ( t , x), where K
A,(D)-\ A u *( ')-[
^D 0^
°1J ' O
and the following condition is satisfied. Assumption II.3. There exist numbers qij, 0 ^
< 1 such that
EXAMPLE 2.4 (Linearized Navier-Stokes system). For an example of pseudoparabolic systems we can consider the linearized Navier-Stokes system vt — i/At; — [v,U] + Vp — f+ (t, x),
divu = f ~ ( t , x), where v > 0, o7 — (0, 0, w)*. In this case,
A'n =
'1 0 0
0 0" 1 0 0 1
-i/A 0 0
0 -i/A 0
0 0 -i/A
159
§ 3. The Cauchy Problem for Sobolev-Type Systems
!<•>. =
'0 a;
-w 0
.°
°
0' 0 ,
L(DX] =
°.
M(DX) = (DX1DX,DX3), s4 = -1/2, t 4 = 1/2,
= a2 = a3= 1/2,
§ 3. The Cauchy Problem for Sobolev-Type Systems In this section, we formulate the main results concerning the solvability of the Cauchy problem for Sobolev-type systems K0Dtu+ + K
K-2 I u+ds + L(Dx)u-
=f+(t,x),
(3.1) M(Dx)u+ =
where the operator )~l
A'o
M(D2
L(DX] 0
satisfies Assumptions I.I and 1.2 in Section 2. We assume that the compatibility condition holds: /~ (0, x) = 0, x £ E n . We note that the study of the Cauchy problem for Sobolev-type systems with nonzero initial conditions u + | t =o = ^(x) is reduced to the study of a problem of the form (3.1) by replacing v+(t,x) = u+(t,x) — WQ"(X), v ~ ( t , x ) = u~(t,x). As above, we denote by W^(R+ +1 ), / ^ 0, r = (n, . . . ,r n ), 1 < p < oo, 7 > 0, the Sobolev space of functions with exponential weight e 1t } i.e., u ( t , x ) G Wj;^(M+ +1 ) if e-^ W (i,x) € V^' r (E+ +1 ). We introduce the norm
We denote by Y\ j=i tions u(t,x) = (MI^.
the Sobolev space of vector- valued func-
160
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
In the case of the same smoothness exponents, the space Y[ Wp'^ will be written as Wp'^(M^ +1 ) for brevity, and the norm of a vector-valued function u(t,x) = ( u i ( t , x } , . . . ,um(t,x)Y £ Wp'^O^n+i) is written in the form m
\\u(t,x), W Let p' = p/(p- 1), \a\ = £ a,-, t max = max{t m + i,.. . tiv}. Suppose that »=i H > tmax-
Theorem 3.1. Let f ~ ( t , x ) = 0. // jaj/p' > t max — ti, then there exists 7i > 0 such that the Cauchy problem (3.1) has a unique solution «+(*, x) € Wp1;;+ (»++,),
r+ = (t!/,.!, . . . , t,/o n ),
t
'=
j=m+l
(3 2)
'
for any vector-valued function f + ( t , x ] e Wp 0 ^ + (M+ +1 )nL pi7 (IRf ; The solution satisfies the estimate
j=m+l +
^c(||/+(/,x) ) ^ (M+ + J|| + | | | | / + ( < ) x ) , L 1 ( E O I I , ^ J 7 ( ^ ) l l ) . (3-3) where the constant c > 0 is independent of f+(t,x). Theorem 3.2. Let f ~ ( t , x ) = 0. Then there exists 71 > 0 such that the Cauchy problem (3.1) has a solution u+(t,x) e W^+(^+1),
D£'uJ(t,x) e L Pl7 (M+ +1 ),
(3-4)
where tj ^ foot. ^ tj -ti, j = m + 1, . . . ,i/ ; 1 < p < oo; 7 > 71, for any compactly supported vector-valued function f + ( t , x ] G Wp£ O&n+i)T/ie solution satisfies the estimate
X] j=m+l
)|| >
(3.5)
§ 3. The Cauchy Problem for Sobolev-Type Systems
161
where the constant c > 0 25 independent of f+(t,x). Theorem 3.3. Let f ~ ( t , x ) ^ 0. // \a\/p' > t max; then there exists 71 > 0 such that the Cauchy problem (3.1) has a unique solution (3.2) for any right-hand side f + ( t , x ) 6 W p °; 7 r+ (M+ +1 )nL pi7 (E+; Li(M n )), f ~ ( t , x ) e f[
^ m9 ~(lRj
that Dtfq(t,x) 6 w£'™'(R;Ui) H L P , 7 (IR+; Li(M n )), q = m+l,...,v, 7 > 71 . TAe solution satisfies the estimate
j=m+l
z),^^
(3.6) where the constant c> Q is independent of f ( t , x ) . Theorem 3.4. £e£ /"(*,i) ^ 0. // |a|/p' > max{tmax - ti, ti}, then there exists 71 > 0 such that the Cauchy problem (3.1) has a solution (3.4) for any right-hand side /+(*,*) € Wp°;7r+(K£+1); f + ( t , x ) = 0, |t| + g=m+l
( K ?; L i( M «)); 9 = m + l , . . . ,i/, 7 > 7 i - The solution satisfies the estimate
j=m+l
\\\\Dtf-(t,x), Li(M n )||, I Pl7 (Rf)||),
(3.7)
162
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
where the constant c > 0 is independent of f ( t , x } . These theorems are proved in Sections 4-6. We illustrate Theorems 3.1-3.4 by an example of the Cauchy problem 0
-w
0'
Dtu+ + divw + = f ~ ( t , x ) ,
0 0 0' u+ + 0 0 0 0 0 N2
r = f+(t,x),
t > 0,
(3.8)
We recall that the system (3.8) is the Sobolev system (1.3) for N = 0, the internal gravity wave system (1.2) for u = 0, and the gravity-gyroscopic wave system (1.2) in the Boussinesq approximation for u =£ 0, N =£ 0. Assuming that the compatibility condition holds, / ~ ( 0 , x ) = 0, x £ MS, we formulate the assertions of Theorems 3.1-3.4 for the system (3.8) as the following corollaries. Corollary 1. Let f ~ ( t , x ) = 0. Then there exists 71 > 0 such that the Cauchy problem (3.8) has a unique solution (t,x)
u~(t,x) 6
(3.9)
with p > 3/2, 7 > 71 for any vector-valued function f+(t, x) 6 ; Li(M3)) ; and the following estimate holds:
where the constant c > 0 is independent of f+(t,x). Corollary 2. Let f ~ ( t , x ) = 0. Then there exists 71 > 0 such that for any compactly supported function f + ( t , x ] G W^'^(M^") ; p > 1, 7 > 71, the Cauchy problem (3.8) has a solution u+(t,x) G W^(R+),
D^U-(t
and the following estimate holds:
where the constant c > 0 is independent of f+(t,x).
\P\ ^ 2,
(3.10)
§ 3. The Cauchy Problem for Sobolev-Type Systems
163
Corollary 3. Let f ~ ( t , x ) ^ 0. There exists 71 > 0 such that the Cauchy problem (3.8) has a unique solution (3.9) with p > 3, 7 > 71 for any right-hand side f + ( t , x ) € ^^(Mj) n L pi7 (R?; Li(R 3 )), / ~ ( t , x ) € W p 1 ; 7 0 (Mj)nL p>7 (M+;Li(]R3)), DtfJ(t,x) 6 Lp,7'(]R+; Li(M 3 )) ; and f/ie /o/lowmg estimate holds:
where the constant c > 0 is independent of f ( t , x ) . Corollary 4. Lei f ~ ( t , x ) ^ 0. There ezzsts 71 > 0 sucA Cauchy problem (3.8) Aas a solution (3.10) with p > 3/2, 7 > 71 for any right-hand side f + ( t , x ) £ Wrp°;71(lR+), / + ( / , x ) = 0, |t| + |x| ^ p, /-(*,x) 6 % 1 ;7°( M J) nL P.7( IR f;^i(i3)), Dtf-(t,x) € L p , 7 (Rf;Li(R 3 )), ana7 i/ie following estimate holds:
where the constant c > 0 is independent of f ( t , x } . We note that the existence of a solution to the Cauchy problem (3.8) is guaranteed in not all Sobolev weight spaces iy p ' 7 (M^). In Corollaries 1, 3, 4, some restrictions on the power degree p > po > 1 are required. The necessity of these restriction is shown by the following assertions. Theorem 3.5. Let f + ( t , x ] € C£°(Mj), f ~ ( t , x } = 0. For the existence of a solution to the problem (3.8) in the class (3.9) for p ^ 3/2 the following orthogonality conditions are necessary: fff(ttx)dx
= 0t
j = 1,2,3.
(3.11)
Theorem 3.6. Let f + ( t , x ] , f ~ ( t , x ) 6 Cg°(Mj). For the existence of a solution to the problem (3.8) in the class (3.10) for p ^ 3/2 the following
164
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
condition is necessary:
r
I f ~ ( t , x ) d x = 0.
(3.12)
PROOF OF THEOREM 3.5. Assume that for some vector-valued function f+ (t, x) that does not satisfy the condition (3.11) the Cauchy problem (3.8) has a solution in the class (3.9) for p ^ 3/2. Then |K(*,z), W^(R+}\\ + \\u-(t,x), Wtf(BL+)\\
7 > 7l .
Since p
where rt
v-2
l x
^ u (t,x)dxdt,
r — irj + 7,
7 > 71-
o K3
Consequently, for any a, b
b sup / e>o 7
/ J
(3.13)
|«
We write an explicit formula for the component w~(r, £). For this purpose, it is necessary to solve the system of linear equations ru\ -umj-f z£iu~ = / + ( r , £ ) , /V 2 — r -f
It is easy to verify that for 7 > 71, £ € M n \{0}
'^2
x J 3 ( r ) 66
66
66
66'
£2 66
66 £3
165
§ 3. The Cauchy Problem for Sobolev-Type Systems
where T
V a; 0
-w r 0
0 ' 0 r+^
T
~\~ LiJ
7-2.1-^2
0
(jj T
0
-4" UJ
T2+U2
0
0 r r 2 + 7V 2 J
Introduce the notation 70, £ G M3\{0} the following estimate holds: 1
Taking into account (3.13), we have L
3
(3.14) Using the Hadamard lemma, we write
where 1
00
2
F*(r,0 = (27r)- / / j e-Tt-iX**(-ixk)f+(t, x] dx dt d\. 0
0
It is obvious that J_
3
l^l 2 3
E
166
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
where Ge = {(rj, £) £ E 4 : a < T] < 6, e < [£| < I } . Taking into account the inequality (3.14) and the definition of the vector-valued functions we obtain the estimate 1
sup
c < oo.
(3.15)
e>0
We consider the integral 6
and make the change of variables £1 = p sin 0i sin 02, £2 = psin 0i cos #2, £3 = p cos 91. Then 6 2rr TT
r r r I / s l u l / i am 1/2 \ \ / / s i n ^ i ( ^ ( r . O ) , sin 0! cos 02 }
2 p
/
P
- 'dp
J J J a 0 0
e
\
\ \
cosfli COS P l
/ ' /
By (3.15) sup 7£ ^ c < oo. Since for p
p2
p
dp -> oo,
e -> 0,
for the validity of (3.15) it is necessary that 6
t
/" a
0
sin i sn
sm
2
n
rns COS
0
Since the integrand is nonnegative, this condition is equivalent to the following condition:
(
sin 0i sin 02\ sin 0i cos 02 I COS 01 //
\ } = 0, / '
where a < r] < b, 0 < 0i < TT, 0 < 02 < 2^ Therefore, g+(r, 0) = 0. Since the matrix rB(r} is nonsingular, we have f+(r, 0) E 0 or e-Ttf+(t,x)dxdt
= Q.
§ 3. The Cauchy Problem for Sobolev-Type Systems
167
By the arbitrariness of the interval (a, 6) and properties of the Laplace transform, the vector-valued function f+(t,x) satisfies the condition (3.11). We obtain a contradiction, which means the necessity of the orthogonality conditions (3.11) for the solvability of the Cauchy problem (3.8) in the class (3.9) for p ^ 3/2. D PROOF OF THEOREM 3.6. We follow the above scheme. We assume that for f+(t,x) = 0 and some function f ~ ( t , x ) £ (^(M^) that does not satisfy the condition (3.12), the Cauchy problem (3.8) has a solution in the class (3.10) with p ^ 3/2. Then
where 7 > 71 . By the Hausdorff- Young inequality, we have \\Z+(T,t),Lpl(R4)\\^c(f-,p}«x>, where oo
w+(r,0 = (27r)-
2
I I
e-Tt-^xu+(t,x)dxdt,
0 13
r — ir) + 7, 7 > 71, and, consequently, CO
sup / 0 J
I
J
|«t(r,Ol p < ded'Kc(/-,p), J
j = 1,2,3.
(3.16)
We write an explicit formula for the components u+(r, £). For this purpose, we need to solve the system of equations TU* — (jju^ + i£iu~ = 0, uju^ + TU\ + i^u~ = 0,
N2 , (r+ — )u+ + i^3u- = 0 , T
if ! wf + if 2 «2" + Z'653 = 7" ( r ' 0-
168
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
For 7 > 71, £ £ M n \{0} we find
(l + tfVr2)(tf + g) + (l+wV*- 2 Kr There exists 70 ^ 71 such that for 7 = Rer > 70, £ 6 R3\{0) the following estimates hold: \ui ( r ) s ) l ^ TTTTT!/ ( r > s ) L
J — 1)2,3.
Therefore, from (3.16) it follows that (3.17)
We write the function /~(r, ^) in the form
where 1
00
2
= (2;r)- f I I e-Tt-{^x(-ixk)f-(t,x) 0
dx dtd\.
0
By the Minkowski inequality, we have
1 ~
Ui/ KO, where Ge - {(r/,0 € M 4 : -oo < ry < oo, £ < |£[ < 1}. Taking into account the inequality (3.17) and the definition of the vector-valued functions < ^/c( r >Oi we obtain the estimate sup
< C < 00.
§ 4. Approximate Solution to Sobolev-Type Systems
169
Since for p ^ 3/2, n = 3 the function <£>(£) = |£|~ p is not summable in a neighborhood of the origin, for /~(T, 0) ^ 0 we have i* iim
1 ~
J^~~ / O\ T I f~^ \ 7TTJ v^")"/i -^p'v^fj
~~~ CO,
which leads to a contradiction. Consequently, the orthogonalit- condition (3.12) is necessary for the solvability of the Cauchy problem (.-.8) in the class (3.10) for p^ 3/2. D Remark 3.1. Arguing as in Chapter 2, Section 7, one can e.-tablish the sufficiency of the orthogonality conditions for the solvability of the Cauchy problem (3.8) in the classes (3.9) and (3.10) for p
§ 4. Approximate Solutions to Sobolev-Type Systems In this section, we sketch the proof of the theorems formulated in Section 3. We begin by considering the Cauchy problem (3.1) for the Sobolevtype system
Taking into account the compatibility condition /~(0, x) = 0, we write this problem in the form
K0Dtu+ + Kiu+ + K2Dt1u+ + L(Dx}u~ = t M(Dx}u+ = f DJ-(s,x)ds, * > 0 , x 6 «+| t=0 = 0. For this problem, we construct a sequence of approximate solutions. Let f + ( t , x ) , f - ( t , x ] £ C£°(Mj+1). We consider the Cauchy problem for the system of integrodifferential equations with parameter £ £ M n which is obtained by application of the Fourier transform with respect to x to the problem (3.1): K0Dtv + i£)v+ = J DJ-(s,t)ds, |t=0
(4.1)
170
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
We apply the Laplace operator with respect to the variable t to this problem:
rK
where Rer > 0, £ £ M n . Since the matrix A'o is nonsingular, there exists a number 70 > 0 such that for Rer > 70 the inverse matrix (rAo + K\ + ^A^)" 1 exists. We can express u> + (r, £) from the first m equations of the system (4.2) as follows: w + ( r , 0 = (rA 0 + A! + -A 2 )- 1 (7 + (r,0 - L(^-(r^)).
(4.3)
Substituting the expression (4.3) in the last (y — m] equations i (4.2), we obtain the system of equations (M(i£)(TKQ + A! + i/ra)- 1 ^))"-
By Assumption 1.1, there exists a number 71 ^ 70 such that for Rer ^ 7i, £ G Mn\{0} this system is uniquely solvable: u;~(r, £) = (M(i^)(rK0 + A'! 4- ^A 2 )- 1 L(^))- 1 (^K)(^ 0 + A! + ^rV+Cr.O - ^ A T ^ O ) Using this formula, from (4.3) we also obtain an explicit expression for
w + ( r , 0 = (rA 0 + A t + -A^)-^/ r
x (rA 0 + A! + i/TaJ-^tiO)- 1 ° M(i£)(rKQ + K1 + ^K,}-l + -(TK0 + A! + -A 2 )- 1 L(f£) o (M(i<e)(rA 0 + A! + -K2}~1 T
T
T
For the sake of simplicity, we introduce the notation
K(T) = rK0 + KI + -A' 2) r
Then
(4-5)
§ 4. Approximate Solution to Sobolev-Type Systems
171
(4.6)
where Rer > 71, £ £ R n \{0}. By definition, the entries of the matrices A' -1 (r) and N § I ( T , £), £ £ M n \{0} are analytic bounded functions for Rer > 71. Since / + ( t , x ) , D t f ~ ( t , x ) £ C^°(Rj+1), we can apply Theorem 5.2 in Chapter 1 to the vector- valued functions w + (r, £), c<;~(r, £), £ £ K n \{0}. Consequently, the vector- valued functions v+(t,£) = (27T)- 1 / 2 f
i1
e(
' +^tu+(ir1 + -f,t)dr),
(4.7)
— CX) 00
1 2
f e^+^tu-(irj-^f^}dri
v - ( < > 0 = (27r)- /
(4.8)
— oo
are independent of 7 > 71 . Furthermore, they solve the Cauchy problem (4.1)for££Mn\{0}. We proceed by constructing an approximate solution to the Cauchy problem (3.1). Applying the inverse Fourier operator to the vector- valued functions v+(t,£) and u ~ ( t , £ ) , we can obtain a formal solution to the Cauchy problem (3.1). However, by Assumptions I.I and 1.2 and the definition of the matrix NQ(T,£), the components of these vector- valued functions have, generally speaking, nonintegrable singularities at £ = 0. As in Chapter 2, Section 5, in order to construct a solution to the Cauchy problem for equations that are not solved relative to the higher-order time-derivative, it is necessary to introduce some regularization of the Fourier operator. For this purpose, we use the integral representation of functions (p(x) £ Lp(Rn] in Chapter 1, Section 6:
where G(£) = 27V(£) 2JV exp(-(£) 2N ) and <£}2 = E£ t ? / a '. »=i Chapter 2, we construct the vector- valued functions
j l/k
In
As in
Section 5,
(4.10)
172
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
n 2
u ~ ( t , x ) = (2rr)- /
k / v~l I eix*G(tva)v- (<,£) d£dv, J
J
l/k
!„
(4.11)
where f + ( ^ , £ ) and v ~ ( t , £ ) are defined in (4.7) and (4.8) respectively. These vector-valued functions have singularities only at the point £ = 0. Therefore, there is NQ such that for N ^ NO the vector- valued functions u £ ( t , x ) , u^ (t, x] are infinitely differentiable and summable with any power. We will assume that N ^> NQ. By the construction of u£(t,x} and u^(t,x), we have
t u+(s,x 0
M(Dx)u+(tyx) = f k ( t , x ) ,
u+(0,x) = 0,
where
l/k
k
l/k
Inffin
By the integral representation (4.9), we have
as k —» oo. Consequently, the vector-valued functions (4.10) and (4.11) can be regarded as an approximate solution to the Cauchy problem (3.1). In Section 5, we establish estimates for the components u£(t, x}, u^ (t, x) which imply the existence of a solution to the Cauchy problem (3.1) if the assumptions of the corresponding theorems in Section 3 are satisfied. § 5.
Estimates for Approximate Solutions to Sobolev-Type Systems
In Section 4, we constructed the sequences {u^(t,x)} and { u ^ ( t , x ) } of vector-valued functions of the form (4.10), (4.11) which can be regarded as
§ 5. Estimates for Approximate Solutions to Sobolev-Type Systems
173
approximate solutions to the Cauchy problem (3.1) for Sobolev-type systems. In this section, we derive estimates for {u^(t,x}} and {u^(t,x}} and prove the convergence of in the weight Sobolev spaces. In the proof of estimates, it is convenient to write the vector-valued functions u£(t,x) and u^(t, x} in the form K V *
/
K
\
'
/
K
\
'
/
k
(5.2) I/A;
Kn
where, by formulas (4.5)-(4.11), oo + +
1 2
v ' (t,t) = (27T)- / j
e^+^a; + - + (z-r 7 + 7 ,0^ !
(5.3)
— oo
+ +
1
u; - (r ! 0-A'- (r)(/-L(z-OAr o - 1 (r,OM(zOA'- 1 (^))7 + (r,0, (5-4) oo
v+'-(t,0 - (2rr)-
1/2
j
e^+^'w+'-^ + ^.Orfi?,
(5.5)
— oo
(5.6) (5.7) (5.8)
hO-
(5-10)
This notation is convenient because the vector- valued functions u~£'+ (t, x) and u^'+(t, x) depend only on f+(t, x), whereas the vector-valued functions u~£'~ ( t , x) and u^'~ ( t , x ) depend only on /~ (t,x).
174
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems To estimate the approximate solutions, we need two algebraic lemmas. Lemma 5.1. For any For any c > 0 for Rer > 71, £ £ M n \{0} it}.
(5.11)
PROOF. We denote by Sm(c] and T m (c) diagonal matrices with diagonal entries cSl , . . . , cSm and ctl , . . . , c tm and by Su-m(c) and T^_ m (c) diagonal matrices with diagonal entries cSm+i , . . . , cs" and c tm+1 , . . . , c1" . By Assumption I.I, the following identities hold:
Recall that ti = . . . — t m = — si = . . . = — s m . Therefore, these identities can be written as follows: ,
(5-12)
).
(5.13)
Taking into account the definition (4.4), for Rer > 71, £ G M n \{0} we find ^-'(r.c-O^r-^^JVo-Hr.O^mlc),
c> 0.
From (5.12)-(5.14) we obtain (5.11).
(5.14) D
Lemma 5.2. For any c > 0 for Rer > 71; £ 6 IR n \{0} we
PROOF. The identities follow form (5.12)-(5.14). Indeed,
In what follows, we extend the vector- valued function f ( t , x] by zero for t < 0, and use the same notation for the extended function. We denote by
§ 5. Estimates for Approximate Solutions to Sobolev-Type Systems
175
/( r ) 0> r — ^+7) the Fourier transform with respect to (t, x) of the function e ~ i t f ( t , x ) and by / 7 (t,£) the Fourier transform of the same function with respect to x. Introduce the notation f ( t , £ ) = fo(t,£}. The proof of the estimates for the vector-valued functions (5.1), (5.2) is divided into several lemmas. LeminaS.3. Let r+ = ( t i / a i , . . . , ti/a n ). Then the vector-valued function u£'+(t,x) satisfies the estimates \\Dtu+'+(t,x), LP
where 7 > 71, j3a 71, ana1 independent of k and f+(t,x). Furthermore, .. , ,.
.
,L .
. ..., _ ._ ,
constant c > Q is
^ 00.
PROOF. From (5.1), (5.3), (5.4) it follows that f v~l I l/k
f eir>tu>+'+(i
t v~l f l/fe
By Lemma 5.1 and the definition (4.4) of the matrix K ( r ] , the entries of the matrices rK~l(r) and A'~ 1 (r)L(z^)7V 0 ~ 1 (r,£)M(z£) are multiplier. Repeating the proof of Lemma 6.1 in Chapter 2, we find ,
(3a ^ ti,
where the constant c is independent of k and f + ( t , x ) . Furthermore,
176
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems We consider the /-derivative k l/k
By Lemma 5.1 and the definition of A'(r), the entries of the matrix
are multipliers. As above, we arrive at the inequality \\Dtu+'+(t,x), I P17 (R+ +1 )|| ^ c \ \ f + ( t , x ] , L pi7 (R+ +1 )||, where the constant c is independent of k and f+(t,x). (*, x) - Dtu+(t, z), L p i 7 ( M + 1 ) | | -> 0,
Furthermore, klt k2 -> oo.
D
Lemma 5.4. The vector-valued function u^'~(t,x) satisfies the estimate
q=m+l aicf=\ s, |
(5.15) where f3a = ti, 1 < p < oo, 7 > 71; one? the constant c > 0 is independent of k a n d f ~ ( t , x } . Furthermore, n*,*) - £>f u + ' - C / . x ) , L Pl7 (R+ +1 )|| -» 0 as k\ , Ar2 —> oo.
PROOF. From (5.1), (5.5), (5.6) it follows that k
l/k
(5.16)
§ 5. Estimates for Approximate Solutions to Sobolev-Type Systems
177
Taking into account Lemma 5.2, this expression can be written in the form k
v~l f eix \/k
(5.17) The entries of the matrix Rer > 71,
(5.18)
are multipliers. Therefore, for (3a = ti, as in the proof of Lemma 6.1 in Chapter 2, we find (5.15) and (5.16). D Lemma 5.5. // a\/p' > ti, then \\u+>-(t,x), L P , 7 (R+ +1 )||
(5.20)
where 7 > 71 and £/ie constant c > Q is independent of k and f ~ ( t , x ) . Furthermore, 0, II^O
(5.21) (5.22)
as ki, k2 —>• oo.
PROOF. We consider formula (5.17) for /? = 0. Recall that the entries of the matrix (5.18) are multipliers. By the definition of the matrix 5 l /_ m (c) and the condition si ^ Si ^ max s,- = 0, the integrand in the brackets j^m+l
has singularity at £ = 0 of the form l/^)*1 . As in the proof of Lemma 6.3 in Chapter 2, the condition \a\/p' > ti implies the estimate (5.19) and the convergence (5.21).
178
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
To prove (5.20) and (5.22), we write an analog of formula (5.17):
l/k
•^DJ^frfidr^dtdv. In this formula, the expression in brackets also has singularity at £ = 0 of the form l/^)* 1 . As above, repeating the arguments of the proof of Lemma 6.3 in Chapter 2 and assuming \a\/p' > t i , we obtain (5.20) and (5.22). D Lernina5.6. The jth component u^'^+-(t,x) of the vector-valued function u^'+(t,x) satisfies the estimate a L f+(tT x\ \ln
\\ 'xJ
fTU + Ml ( > )> LTp,-y("^n+}.)\\'
- t m+J (5.23) where t m+J - — ti ^ J3a ^ t m + j, 1 < p < oo; 7 > 71 and the constant c > 0 25 independent of k and f+(t,x}. Furthermore,
(5.24) as ki, k-2 —>• oo.
PROOF. From (5.2), (5.7), (5.8) we have
l/k
Taking into account Lemma 5.2, this expression can be written in the form k
i/k
§ 5. Estimates for Approximate Solutions to Sobolev-Type Systems
179
>r°
J-V
^.
(5.25)
The entries of the matrix
are multipliers. Therefore, if tm+j — ti ^ /?a ^ t m +j, then, repeating the arguments of the proof of Lemma 6.1 in Chapter 2 for the jth component of the vector-valued function u^' ( t , x ) , we obtain the estimate (5.23) and the convergence (5.24). D Lemma 5.7. // |a|/V > t m +j — ti, j = 1, . . . , v — m, then (5.27) where 7 > 71 and the constant c > 0 is independent of k and Furthermore, )|| -^ 0
f+(t,x). (5.28)
as k\, k-2 -> oo.
PROOF. We consider formula (5.25) for (3 = 0. Since the entries of the matrix (5.26) are multipliers and ti = — si ^ tm+j, taking into account that the expression in brackets has singularity at £ = 0 of the form l/{£) tm+J ~ ll > as in the proof of Lemma 6.3 in Chapter 2, for l^l/p' > t m +j — ti we obtain (5.27) and (5.28). D Lemma 5. 8. The jth component u^'^+,(t,x] of the vector-valued function u^'~(t,x) satisfies the estimate
)|| >
(5.29)
q=m+l aia = \ s, |
where /3a = tm+j, 1 < p < oo, 7 > 71 and the constant c > 0 is independent of k and f ~ ( t , x } . Furthermore, | ^0
(5.30)
180
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
as ki, k-2 —> oo.
PROOF. From (5.2), (5.9), (5.10) we have k
/V1 f ei J
J
l/fc
Taking into account (5.14), we write this expression in the following form: k
/ v~l f e l/k
•. £) dn } d £ d v .
(5.31)
By the definition (4.4) of the matrix N Q I ( T , £ ) , the entries of the matrix (5.32) are multipliers. Therefore, if @a — tm+j, then, repeating the arguments of the proof of Lemma 6.1 in Chapter 2, for the jth component of the vectorvalued function u^'~ (t, x] we obtain the estimate (5.29) and the convergence (5.30). D Lemma 5.9. // \a\/p' > t i , then the jth component u^'^+At,x) of the vector-valued function u^'"(t,x) satisfies the estimate
=m+l
U K D t f q ( t , x), ^ ( M n J H , L P , 7 (K+)|
,
(5.33)
§ 6. Solvability of the Cauchy Problem for Sobolev-Type Systems
181
where m~ = (\sq\/a\, . . . \sq\/an), 7 > 71 and the constant c > 0 is independent of k and f ~ ( t , x ) . Furthermore, ,*), L P , 7 (M+ +1 )|| -* 0 (5.34) as
PROOF. We consider formula (5.31) for tm+j — ti ^ /?<* ^ tm+j- The expression in the brackets has singularity of the form l/(£) t m + J ~l s m+, \~Pa at £ = 0. Taking into account that the entries of the matrix (5.32) are multipliers and max s,- = 0, we obtain (5.33) and (5.34) for \a\/p' > ti in j ^ m +l
the same way as in the proof of Lemma 6.3 in Chapter 2.
D
Lemma 5.10. // |a|/p' > km+j, j — 1, • • • , v — m, then \\u-'-(t,x), L P|7 (M+ +1 )|| ^ c ( \ \ D t f - ( t , x ) , L P|7 (M+ +1 )|| (5-35) where 7 > 71 and the constant c > 0 is independent of k and f ~ ( t , x ) . Furthermore, }\\ -* 0
(5.36)
as ki, k-2 —>• oo.
PROOF. We consider (5.31) for (3 — 0. The expression in the bracket has singularity of the form l/(£) tm+J ~' S m + t ' at £ — 0. Since the entries of the matrix (5.32) are multipliers, we obtain (5.35) and (5.36) for |cv|/p' > t m +j in the same way as in the proof of Lemma 6.3 in Chapter 2. D
§ 6. Solvability of the Cauchy Problem for Sobolev-Type Systems In this section, we prove the theorem in Section 3 about the solvability of the Cauchy problem (3.1) for Sobolev-type systems. By formulas (5.1)(5.10), the approximate solutions to the Cauchy problem (3.1) have the form u£(t, x} = u~£'+(t, x)+u%'~(t, x) and u^(t, x) = u^'+(t, x)+u^'~(t, x] or, in the operator notation, u~£'+(t,x) = P f f + ( t , x } , u~£'~(t,x) = Q % f ~ ( t , x ] , u^'+(t,x) = P^f+(t,x], u^'~(t,x] = Q k f ~ ( t , x ) . The linear operators P£ , P^ , Q~£ and Q^ are defined on vector- valued functions f + ( t , x ) , f~ (t , x) 6 Co°(M+ +1 ). By Lemmas 5. 3-5. 10, the sequences {u^'+(t,x)}, {uj'~(i,x)},
182
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
{u^1 (t} x)}, {u^'~ ( t , x}} are Cauchy sequences in the corresponding weight Sobolev spaces. Therefore, we can define the linear operators P+f+(t,x)= P~f+(t,x}=
lim P+f+(t,x),
Q+f~(t,x)=
Hm P fc -/ + (M),
Q-f-(t,x)=
k —loo
lim
Q+f-(t,x),
k—Kx>
hmQ-/-(t,x),
k— >-oo
where the limit is understood in the corresponding norm. We consider the definition of operators in detail. By Lemma 5.3, the operator P+ is defined by the formula
and the following estimates hold:
I ft c T S . T* Uj I C4 O . J-/T)
i TUP '
ill
~
ft
/~\
r^
(r\ 1 i/ \
)
o I I n P+ f+ /'y f\ j T ^TO+ ^ 1 1 + 111 ^ •/ \ ' ) ' * ' / ! ^ p " y l-^n-t-l J ^; \\J \^ i "^ / 1 •'-'p 7 \ n + l / l '
('fi 9"\ V /
where 1 < p < oo, 7 > 71, and the constant c > 0 is independent of f+(t,x}. Since (7^°(]R++1) is dense in W°£+(R++1), the operator P+ can be uniquely extended to the entire space Wp'I. (M^"+1) by continuity. The extended operator is denoted by P + . The estimates (6.1) and (6.2) hold for this operator. We consider the operator P~. By Lemma 5.6, for tj — ti x ) - D f J ' U - ' + ( < > x ) > L p > 7 ( M + + 1 ) | ->0,
Ar^co,
(6.3)
where 1 < p < oo, 7 > 71. If |a| > t max — t i , then there exists a number 1 < q < oo such that \a\/q' > t m a x — t i , l/q + l / q f = 1. By Lemma 5.7, the sequences {u^'^(t,x)} converge in L 9 ) 7 (M^ + 1 ), i.e., \\uj'+(t,x) — u ^ " ^ ( i , x ) , Lg i7 (E^ +1 )|| —>• 0 as k -^ oo, where 7 > 71. Taking into account (6.3) and properties of the weak derivatives (cf. Chapter 1, Section 7), we find v- ( t , x ) = D%3 u~'+ ( t , x ) , j = m+l,...,i'. Consequently, the linear operator P~ is defined by the limit
§ 6. Solvability of the Cauchy Problem for Sobolev-Type Systems
183
By Lemma 5.6, it satisfies the estimate
where tj — ti ^ j3a ^ tj, j = m + 1, . . . , is, 1 < p < oo, 7 > 71, and, by Lemma 5.7, we have \\p-f+(t,x), Lg
(6.5) where |a|/V > t ma x ~ ti and the constants Cj > 0 are independent of f+(t,x). Since C0°°(M++1) is dense in iy p 0 ;; + (M^ 1 )n J L,, 7 (IR+ +1 )n^ i7 (M+; L, (!„)) the operator P~ can be uniquely extended to the entire space by continuity. The extended operator is denoted by the same symbol P~ and satisfies the estimates (6.4) and (6.5). We consider the operator Q+ . By Lemma 5.4, {D%u~£'~ (t, x ) } , /3a — ti, converges in LP|7(1R^+1), i.e., there exists a vector-valued function v&(t, x] 6 ) such that ||^(t, x) - D£U+'- (t, x), I P17 (M+ +1 )|| ^ 0 ,
k -> oo,
(6.6)
where l < p < o o , 7 > 7 i . I f | a | > t 1 ; then there exists a number 1 < q < oo such that \a\/q' > t 1; 1/q + l/q' = 1. By Lemma 5.5, the sequence {u~£'~(t,x}} converges in the space W^OR^+J, i-e., there exists a vector-valued function u+>~(t,x) 6 W^'°(R^+1) such that \\u+'~(t,x] — u j ' ~ ( t , x ) , VF^0(Mj+1)|| -> 0, 7 > 71, as A; -> oo. Taking into account (6.6) and properties of the weak derivatives, we find v ^ ( t , x ) = D%u+'~(t,x). Consequently, the linear operator Q+ is denned by the formula \\Q+f-(i,*) ~ Qtr(t,x], £, By Lemma 5.4,
E j=m +
E
IPf/^^^'^^n+i)!!,
(6-7)
184
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
where {3a = ti, 1 < p < oo, 7 > 71. By Lemma 5.5, ( t , x ) , L 9i7 (M+ +1 )||
f6 9)
where |a|/ ti and the constants Cj > 0 are independent of f ~ ( t , x ) . It is clear that the operator Q+ can be extended by continuity to the entire space of vector-valued functions iF~(t x} ' F-(t x} £• W
>m]
(M"^" } D l4^1'°flR"'" } fl L
DtFj(t,X) eL^^MfjL^E^),
Fj| t= o = 0,
fM"^"' Lt fM ))
j = m + 1, . . . ,i/}.
The extended operator, denoted by the same symbol, satisfies the estimates We consider the operator Q~ . By Lemma 5.8, {D% u k ' - ( t , x ) } , foot = tj, j = m + 1, . . . , v , converges in L P|7 (M^ +1 ), i.e., there exists ujj (t, x) £ L P?7 (IR^ +1 ) such that ||Wf (t,x)-Dl]u-j-(t)x),Lp^(R++l}\\-^Q,
Ar-^oo,
(6.10)
where l < p < o o , 7 > 7 1 . If a| > t m a x , then there exists a number 1 < q < oo such that \a\/q' > t m a x , 1/q -f- 1/g' = 1. By Lemma 5.10, the sequence { u ^ ' ~ ( t , x ) } converges in the space Lq^(R*+l), i.e., there exists a vector-valued function u ~ ' ~ ( t , x ) G L q)7 (M^ +1 ) such that | | w ~ ' ~ ( t , x ) — u^'~ (t, x], Lq^(]&n+1)\\ -> 0, 7 > 71, as k —>• oo. Taking into account (6.10) and properties of the weak derivatives, we find u/ ( t , x ) = D^ u ~ ' ~ ( t } x), j = m + 1, . . . , i / . Consequently, the linear operator Q~ is defined by the formula \\Q~f- ( t , x ) - Qif-(t,x), I,, 7 (R+ +1 )|| -> 0,
A: -> co.
By Lemma 5.8,
E
E
H^A/,-(*,ar) > L p i 7 (M+ + 1 )|| l
(6.11)
§ 6. Solvability of the Cauchy Problem for Sobolev-Type Systems
185
where (3a = t j , I < p < oo, 7 > 71. By Lemma 5.10, \\Q-f~(i
x]
£, flj£+ )|| \6.12)
where |a|/V > t max and the constants Cj > 0 are independent of f ~ ( t , x ) . The operator Q~ is extended to the entire space of vector-valued functions {F(t x) : DtF-(t x} G W°'^' fM"^" ) 0 L "• :,(»„)).
i = m+l
v}
by continuity. The extended operator, denoted by the same symbol, satisfies the estimates (6.11) and (6.12). PROOF OF THEOREM 3.3. By condition, we have \a\fp1 > t m a x . By the definition of the operators P+ , P~ , Q+ , Q~ , for any function f ( t , x ) = ( f + ( t , x ) , f ~ ( t , x ) Y satisfying the assumptions of the theorem, the vectorvalued function u(t, x) = (u+ (t , x) , u~ (t , x}Y such that u+(t,x) = P+f+(t,x) + Q+f~ ( t , x ) €
j=m+l
7 > 71 , is a solution to the Cauchy problem (3.1). Furthermore, the inequalities (6.1), (6.2), (6.4), (6.5), (6.7)-(6.9), (6.11), (6.12) imply the estimate (3.6). To prove the uniqueness, we need the following assertion. Lemma 6.1. The Cauchy problem K0Dtu+ + L(Dx)u~ = 0 , M(Dx)u+=0,
t>Q,
xeRn,
(6.13)
has only the zero solution in the class (3.2) : u+(t,x) 6
); 7 > o. PROOF. We assume that a solution to the Cauchy problem (6.13) is compactly supported relative to x. Then the partial Fourier transform
186
3. The Cauchy Problem for nori-Cauchy-Kovalevskaya Type Systems
with respect to x satisfies the conditions KoDtu+(t^) + L(i£}u (t,£) = 0, M ( z O u + ( < , 0 = 0, t > 0, u+\t=0 = 0 or = 0.
o
t=0
^) = 0, f € M n , if and only if £ = 0. Con-
By Assumption 1.2, sequently, the matrix
, ^ G ffi.n\{0}, is nonsingular. There-
fore, for t > 0, £ 7^ 0 we have
which implies u (t,£) = 0. By continuity, Consequently, u ( i , x ) = 0.
= 0 for all £ 6 D
The above arguments imply the uniqueness of a solution to the Cauchy problem (3.1) in the class of vector- valued functions with compact support with respect to x. From the estimates (6.1) and (6.7) it follows that such solutions satisfy the inequality
/3a=t, t
f D?f+(S,x)dS,Lp
++1}
•J
0 f
y E ^
/
^1
-J
(6.14)
j=m+l aia~\ s_,
The estimates (6.4) and (6.11) imply
E
'a=t1
+
E
E
(6.15)
§ 6. Solvability of the Cauchy Problem for Sobolev-Type Systems
187
where the constants ci,C2 > 0 are independent of f ( t , x ) . Let u(t, x ) = (u+(t, x), u~(t, x))1 be an arbitrary solution to the Cauchy problem (6.13) in the class under consideration. Let us show that u(t,x) = 0. It is obvious that I\oDtu+(t,x} = —L(Dx)u~(t,x) and u+\t=o = 0. Therefore, t I
(6.16)
o
Since M(Dx)u+(t, x} — 0, we have t t M(Dx)I
(6.17)
Let e > 0, and let u~(t, x) = (u~m+1(t,x), . . . , u~>t/(t, x))* be a vectorvalued function with components u~ - ( i , x} E C^^n+i)' J ~ rn + ^ > • • • > vLet
t/
_
' ) | | ^ e.
(6.18)
j=m+l
We consider the vector-valued function t = - I Ko1L(Dx)u-(s,x)ds.
(6.19)
It is obvious that the vector-valued function u+(t, x) E C roo (M^ +1 ) has compact support with respect to x. Furthermore, KoDtu+(t,x) + L(Dx)u-(t,x) = 0, t IM(Dx}K-lL(Dx}u-(s}x}ds,
M(Dx)u+(t,x) = -
o w+(0,a?) = 0. Since solutions to the Cauchy problem (3.1) with compact support with respect to x satisfy the inequalities (6.14) and (6.15), we have
f=m+l
i+i;n-
188
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
Taking into account (6.17), we find V
E
t'=m +
\\Dax3(M(Dx}K^L(Dx)(u-(t,x
E
^E
j=m+l aJa—\ s7
-«-(<,*)),•, L p i 7 (R+ + 1 )||.
(6.20)
As was already mentioned, by Assumption I.I, we have
where c > 0, Sj,_ m (c), Tl,_ m (c) are diagonal matrices with diagonal entries c Srn+1 , . . . , cs" and c tm+1 , . . . , c1" respectively. Consequently,
which means that any vector-valued function v(x) = (u m + i(ar), . . . ^^ Vi(x) 6 Wp l (Mn), 1 < P < oo, satisfies the estimate
E /?'a=t (
where cr-^a = | Sj , j — m+ 1, . . . , i/, and the constant CQ > 0 is independent of t>(x). Using these estimates, from (6.20) we find
E E
i=m+l /9'ant
E E
/=m + l /3'a=t
where the constant c > 0 is independent of e, u £ ( t , x ) , u ( ^ , x ) . Then from (6.18) we obtain the estimate V
Y^
V
iin/ 3 \/-.r* -^ r._
§ 6. Solvability of the Cauchy Problem for Sobolev-Type Systems
189
Taking into account the approximation (6.18), for the components of the vector-valued function u~(t,x) we have
/3'a=t,
and, since £ > 0 is arbitrary, we obtain the equalities
/?'a=t,
Consequently, the Cauchy problem (6.13) has a unique solution.
D
Thus, Theorem 3.3 is proved in a special case K\_ = 0 and A'2 = 0. Let us prove the uniqueness of a solution the Cauchy problem (3.1) for systems with lower-order terms. Let u ( t , x ) be a solution to the problem KQDtu+ + Kiu+ + K2D~lu+ + L(Dx)u~ = 0, M(Dx)u+ = 0, t>0,
xeRn,
t=o = 0
in the class (3.2). Since Kiu+(t,x) + K2D^u+(t,x) 6 the proved assertion we find u+(t,x) = -P+KlU+(t,x)u~(t,x) = -P~K1u+(t,x)By the estimate (6.1), we have
t J
Klu+(s,x)ds,
i l
(t-s)K2u+(s,x)ds,
I
o
\\u
n+l>
from
190
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
Consequently, there exists 71 > 0 such that for 7 > 71 i +
+
! + 2
+
Therefore, u+(t,x) = 0, but, in this case, we have u ~ ( t , x ) = 0. Thus, we have proved the uniqueness of a solution the Cauchy problem (3.1). D PROOF OF THEOREM 3.1. By condition, we have \a\/p' > t m a x - t i . By the definition of the operators P+ and P~ for any right-hand side of the system f ( t , x ) = (/+(*, a:), 0) f such that f + ( t , x ) <E Wp0;7r+(M++1) 0 Lp)7(Mf; LI(M n )), the function u+(t, x) = P+f+(t, x), u~(t, x) = P ~ f + ( t , x } is a solution to the Cauchy problem (3.1). From the estimates (6.1), (6.2), (6.4), (6.5) for this solution we obtain the estimate (3.3). By Lemma 6.2 below which generalizes Lemma 6.1, we can conclude that a solution to the Cauchy problem (3.1) is unique in the class (3.2). D Lemma 6.2. There exists 71 > 0 such that the Cauchy problem KoDtu^ 4- A'iw 4- K^D^ u"^ + L(Dx]u~ = 0, M(Dx)u+ = 0,
t > 0,
zEMrc,
(6.21)
has only the zero solution in the class (3.2) : u+(t,x] G
PROOF. We follow the proof of Lemma 6.1. We first consider a solution u ( t , x ) to the problem (6.21) with compact support with respect to x. For the partial Fourier transform with respect to x of this solution we have i
0] „ „ , \K2
0
u+(t where v(t,£) r - _ / >\ » I • Since the matrix ,/,.^ ,£ v 1S/ = I J u (s,£)dsl [M(i£) 0n \o , is nonsingular for ^ € M n \{0}, we have w(t,^) = 0. Consequently, u+(t,£) = 0, w - ( f , 0 = 0, < > 0, ^ 6 M n \{0}, which implies u ( t , x ) = 0. From the above arguments we establish the uniqueness of a solution to the Cauchy problem (3.1) in the class of vector- valued functions (3.2) with
§ 6. Solvability of the Cauchy Problem for Sobolev-Type Systems
191
compact support with respect to x. Such solutions satisfy the estimates (6.14) nd (6.15). We consider an arbitrary solution u ( t , x ) to the problem (6.21) in the class (3.2). We have K0Dtu+(t,x} + Kiu+(t,x) + K2D^lu+(t,x} = -L(Dx)u-(t,x), u+
t=o=
0-
Since the matrix A'o is nonsingular, there exists a smooth (m x m)-matrix k(t) such that
t u (t,x) = I k(t-s}L(Dx}u~(s,x}ds. o +
(6.22)
Taking into account the condition M(Dx}u+(t, x) = 0, we find t [ M(Dx}k(t - s)L(Dx)u- (s, x) ds = 0. o
(6.23)
As in the proof of Lemma 6.1, we take a vector-valued function u^(t, x) £ C^°(Ej+1), e > 0, such that the condition (6.18) holds. We consider the smooth vector-valued function t u+(t,x)= k(t-s)L(Dx)u~(s,x)ds. (6.24) o By the definition of the matrix k ( t ) , we have
t K0Dtu+(t, x} + Kiu+(t, x} + A'2 / w+(s, x) ds + L(Dx)u~(t, x) = 0, o t
M(Dx)u+(t,x)=
f o
M(Dx)k(t-s)L(Dx)u-(s,x)ds,
u+(0,i) = 0. Since the solution to the Cauchy problem (3.1) with compact support with respect to x satisfies (6.14) and (6.15), taking into account (6.23), we obtain for the vector-valued function u ~ ( t , x ) the following inequality which is similar to (6.20):
192
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
E
?1 u V nn/ m + MI / ^ \\ x U7/e,i\al^XT\ h LTP,-i\^n-\-l)\\ t I
(M(Dx}k(t-
s}L(Dx)(u-(s,x)
=m + l c r a = | s j |
-u-(5li)))jd5>Lpi7(M++1)||l
(6.25)
where the constant c > 0 is independent of E, u£(t,x), u(t,x). Taking into account the definition of the matrix k(t) and the identities L(i^) = c^L(cai^Tj2m(c)t M(it)k(t)L(it)
00,
= SJ^COMCc^'OACO^C^iOr^Cc),
we can show that there exists 7 > 71 such that for any vector- valued function W
P] ( M n + l ) .
V~ (t, X) e
K P < 00,
7 > 7l
j=m + l
the following estimates hold:
/ = m + l /3>a=t,
E
E
/=m + l 3'a=t
where crja — |sj|, j = m + 1, . . . , v, and c0 > 0 is an absolute constant. Using these estimates and taking into account the approximation (6.18), from (6.25) for 7 > 71 we obtain the estimates «',-(*, x), L P , 7 (M+ +1 )|| ^ cet which imply
E
i = m + 1, . . . , i/,
§ 7. Solvability of the Cauchy Problem for Pseudoparabolic Systems
193
Since 6 > 0 is arbitrary and the solution u(t,x) belongs to the class (3.2), we find u~(t,x) = 0. By (6.22), we have u+(t,x) = Q. D PROOF OF THEOREM 3.2. By assumption, the vector-valued function f+(t,x] of class Wp'^ (M*+1) is compactly supported. Therefore, we can apply the operator P~ : u~(ttx) = P~f+(t,x) £ L< ?i7 (M+ +1 ), \a\/q' > tmax — ti, 1/9+ 1/V = 1. 7 > 71, By the estimate (6.4), we have
where l < p < o o , 7 > 7 i . The operator P+ can be applied to any vectorvalued functions f+(t,x) € % 0 1 if + (M+ +1 ). By the estimates (6.1) and (6.2) for u+(t,x) — P+f+(t,x), we have \\ii+(-t T\ W 1 ' r " f CTU+ \\\ + ^11 ||W ( I , X J , ^p|7 U ^ n + l J H ^ C||/ ( > )> p,'i lMn+lJII>
where 1 < p < oo, 7 > 71- By the definitions of the operators P+ and P~ , the vector- valued function u(t, x) = (u+(t, x ] , u~(t} x)}* is a solution to the Cauchy problem (3.1) in the class (3.4) and satisfies (3.5). D PROOF OF THEOREM 3. 4. The solution to the problem (3.1) in (3.4) takes the form u+(t,x) - P+f+(t, z) + Q + /~(t, x), u~(t, x) - P~f+(t,x) + Q~ f~ ( t , x ) . This assertion can be proved by the above scheme with Lemma 5.9. D
§ 7. Solvability of the Cauchy Problem for Pseudoparabolic Systems In this section, we present some results about the solvability of the Cauchy problem for pseudoparabolic systems: K0Dtu+ + Ki(Dx)u+ + K2(Dx)u+ + L(Dx}u~ = f+(t, x), M(Dx)u+= f(t,x},
t > 0 , x6Mn,
(7.1)
where the operator '.QoDt+K^Ds)
M(DX)
L(DX)
0
194
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
satisfies Assumptions II. 1 and II. 2, whereas the operator K-2.(DX} satisfies Assumption II. 3 in Section 2. Let the compatibility condition hold: / - ( 0 , z ) = 0, z G M n . We note that the study of the Cauchy problem for pseudoparabolic systems with nonzero initial conditions u+\t=o = u£ (x) is reduced to the study of a problem of the form (7.1) by replacing v+(t,x) = u+(t,x) — u^(x), v~ (t, x) — u~ (t, x). n
Let p' - p/(p- 1), \a\ = JD "i, tmax = max{t m + i, . . . , t,,},
min{|s m+ i |, . . . , &v \}. Suppose that |a| > 1 + t max - 0. // \a\/p' > tm&x, then the Cauchy problem (7.1) has a unique solution u+(t, x} £ W^(R++l), v
t|-(t,x)G
r+ = ( I / a , , . . . , l/a n ),
_
In c\\
H Wp^1 (M+ +1 ),
r~ =
(tj/al,...,tj/an),
j-m + i
and the following estimate holds:
)!!), (7.3) where the constant c> 0 zs independent of f+(t,x). Theorem 7.2. Let f~ (t, x) = 0, /+(*, x) G L P , 7 (M+ +1 ) ; / + ( t , x ) = 0/or |i| + |a?| ^ p, 7 ^> 70 > 0. T/zen ^Ae Cauchy problem (7.1) /ias a solution ),
«7(/>x)GLpf7(M++1)l
(7.4)
where fia = tj, j — m + 1, . . . , i/, anc? t/ie following estimate holds:
j=m+l >
where the constant c > 0 is independent of f+(t,x).
(7.5)
§ 7. Solvability of the Cauchy Problem for Pseudoparabolic Systems
195
Theorem 7.3. Let f - ( t , x ) £ 0, f + ( t , x ) £ L p , 7 (M+ +1 )nL p , 7 (]R+; Li(M n )), 1 (IR n )) ;
m~ = (\ sq \/alt . . . ,
\sq\/an), D t f q ( t , x ) G L p ) 7 (Mf;Li(R n )), g = m + 1, . . . , v, 7 ^ 7o > 0. // \a\/P' > 1 + t max — crmin; f/ien the Cauchy problem (7.1) /ms a unique solution (7.2), and the following estimate holds:
qr=m+l
4-||||A/-(^^) ) Li(Mn)IU P , 7 (M+)||) )
(7.6)
where the constant c > 0 is independent of f ( t , x ) . Theorem 7.4. Let f ~ ( t , x ) £ 0, /+(*,*) € L p , 7 (M+ +1 ) ; / + ( t , x ) = 0 for \t\ + \x\ ^ p, f ~ ( t , x ) 6
fl
Wp1,'
D t f g ( t , x ) £ L p i 7 (M+;Ii(K n )), «/ = m + 1, . . . , i/, 7 ^ 7o > 0. If \a\/p' > 1 — o"min; then the Cauchy problem (7.1) has a solution (7.4) and the following estimate holds:
j=m+l
(7.7)
where the constant c > 0 25 independent of f ( t , x ) . Now, we formulate results concerning the solvability of the Cauchy problem (7.1) in the case K2(DX) ^ 0. Theorem 7.5. Let f ~ ( t , x ) = 0. Then there exists 71 > 0 such that the Cauchy problem (7.1) has a unique solution in the class (7.2) with \a\/Pf > t m a x , 7 > 7i for any f + ( t , x ) 6 LP solution satisfies the estimate (7.3).
196
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
Theorem 7.6. Let f~ ( t , x ) = 0. Then there exists 71 > 0 such that for any compactly supported vector-valued function f + ( t , x ) G Lp^(M.^+l) the Cauchy problem (7.1) has a solution (7.4) for 1 < p < oo; 7 > 71. The solution satisfies the estimate (7.5). Theorem 7.7. There exists 71 > 0 such that the Cauchy problem (7.1) has a unique solution in the class (7.2) for \a\/p' > l + t max — 7i for any right-hand side f ( t , x ) , satisfying the assumptions of Theorem 7.3. The solution satisfies the estimate (7.6). Theorem 7.8. There exists 71 > 0 such that the Cauchy problem (7.1) has a solution in the class (7.4) for \a\/p' > 1 — o-m-m, 7 > 7i for any right-hand side f ( t , x ) satisfying the assumptions of Theorem 7.4. The solution satisfies the estimate (7.7). In Section 12, we sketch the proof of Theorems 7.5-7.8. We illustrate these results by the Cauchy problem for the linearized Navier-Stokes equations uf - i/Aw+ - [u+,u} + Vu~ = f + ( t , x ) , divu+ = f - ( t , x ) ,
00,
zeM3,
(7.8)
+
w |t=o = 0, where i/ > 0, u7 = (0,0, w)*, / ~ ( 0 , z ) = 0, x e R3. We formulate the assertions of Theorems 7.1-7.8 for the problem (7.8) as corollaries. Corollary 1. Let f ~ ( t , x ) = 0. Then there exists 71 > 0 such that the Cauchy problem (7.8) has a unique solution u+(t,x) 6 Wftjpjj),
u-(t,x) 6 W
for p> 3/2, 7 > 7 i for any f + ( t , x ) £ L P | 7 (M+) n L P , 7 (E+; Li(E 3 ))- The solution satisfies the estimate
where the constant c > 0 is independent of f + ( t , x ) . Corollary 2. Let f ~ ( t , x ) = 0. Then there exists 71 > 0 sucA that for any compactly supported vector-valued function f + ( t , x ) E LP 1 < p < oo; 7 > 7i; fAe Cauchy problem (7.8) /ms a solution x) € W J O R t ) ,
Vw
"(^. ^) e ^ P l 7 (Kf ),
(7-10)
§ 7. Solvability of the Cauchy Problem for Pseudoparabolic Systems
197
and the following estimate holds: \\u+(t,x), ^(MfJIMIVtr (*,*), WKf)ll ^ c \ \ f + ( t , x ) , LP where the constant c > 0 is independent of f+(t,x). Corollary 3. There exists 71 > 0 such that the Cauchy problem (7.8) has a unique solution (7.9) for p > 3, 7 > 71 for any f + ( t , x ] 6 Lp 7 (M^)D Lp.^R+^CRg)), f - ( t , x ) 6 W£^R?)nLp, 7 (Rf;Ii(R 3 )), Dtf-(t,x) e L p) .y(]Ri~; Z/i(R 3 )), anrf the following estimate holds:
where the constant c> 0 z's independent of f ( t , x ) . Corollary 4. There exists 71 > 0 such that the Cauchy problem (7.8) has a solution (7.10) for p > 3/2, 7 > 71 /or an^/ right-hand side f + ( t , x ) E L p , 7 (R+) ; /+(*,*) = 0 /or |/|+ k| ^ P, /-(t.x) G
\\f-(t,x),
where the constant c> 0 z's independent of f ( t , x ) . As in the case of the Cauchy problem for Sobolev-type system (3.8), the corollaries assert the solvability of the Cauchy problem (7.8) in the weight Sobolev spaces Wp'7(IR;j~) for p > p0 > 1. We formulate theorems which show that such restrictions on the exponent p are necessary. Theorem 7.9. Let f + ( t , x ) £ C^ORj), f ~ ( t t x ) = 0. Then for the existence of a solution to the problem (7.8) in the class (7.9) for p
.7 = 1,2,3,
(7.11)
198
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
Theorem 7.10. Let f + ( t , x ) , f ~ ( t , x ) € Cg°(Mj). Then for the existence of a solution to the problem (7.8) in the class (7.10) for p
(7.12)
I f - ( t , x ) d x = 0.
PROOF OF THEOREM 7.9. We argue as in the proof of Theorem 3.5. We assume that for some vector-valued function f + ( t , x ] G CQ°(M^) that does not satisfy the condition (7.11) there exists a solution to the Cauchy problem (7.8) in the class (7.9) for p <J 3/2. For this solution we have \u-(t,x)t
oo,
7
By the Hausdorff-Young theorem, ||u~(r, £), Lp/(M 4 )|| ^ c ( f + , p ) < oo, where 'u
(t,x)dxdt,
T = IT] + 7, 7 > 7!. Consequently, for any a and 6 we have (7.13)
sup 00
We write an explicit formula for u (r,£). For this purpose, we need to solve the system of equations (r + v\t\2}u+ - u>u+ + i^u~ = /+(r,0, uu+ + (r + H<e| 2 )<4 + i&u- = f?(r, 0, (r + ^ie| 2 )«3 + » 6 « " = / 3 " ( r l O . ^1^1" + ^2^3" + ^3^3" = 0. It is obvious that for 7 > 71, £ 6 M 3 \{0) r+
+e)+
+
+
§ 7. Solvability of the Cauchy Problem for Pseudoparabolic Systems
199
where -\ -1
Ul
(r 0
Introduce the notation ^ + (r,0 - (r + j / | ^ | 2 ) 5 ( r , ^ ) + ( r , ^ ) . From the explicitly expression for w~(r, £) we see that there exists 70 ^ 71 such that for 7 = Rer > 70, £ E M3\{0} the following estimate holds:
Taking into account the inequality (7.13), we find 6
/
/
(7.14)
j^p
Using the Hadamard lemma, we can represent the vector-valued function <7+(r,0 a s 0 + ( r , 0 = 0 + ( r , 0 ) +
(r, 0 , where Gfc (r, £) is a smooth
vector-valued function. Then 1
, Lpi(Ge)
iei:
200
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
where G£ ~ {(77, f) E M 4 : a < T] < b, e < \£\ < 1}. By (7.14), we find 6
o
/ / J
/
"
I 777^7 y^6<7?M)
p
d£d77^c
(7.15)
Repeating the corresponding arguments of the proof of Theorem 3.5, we conclude that for the validity of the inequality (7.15) for p
r2
r 2
UJT 2
r -f a;
\ (jj 9 T 1 -\-
r r +u;2 0 2
0
0
2
0
/+(r,0)EEO.
1.
This expression is equivalent to the identity f+(r, 0) EE 0, a < Rer < 6. By the arbitrariness of the interval (a, b) and properties of the Laplace transform, the vector- valued function f+(t,x) satisfies the orthogonality condition (7.11); we arrive at a contradiction. Consequently, the condition (7.11) is necessary for the solvability of the problem (7.8) in the class (7.9) for p ^ 3/2. D PROOF OF THEOREM 7.10. We follow the above scheme. We assume that for f + ( t , x ) = 0 for some function f ~ ( t , x ) G Cfo°(IR|') that does not satisfy the orthogonality condition (7.12), the problem (7.8) has a solution in the class (7.10) for p ^ 3/2. Then co,
c(
where 7 > 71- In particular, by the Hausdorff- Young inequality, we have ||K|u-(r,0,MK4)|| ^ c(/-,p) < oo, where Tt x ~-/ t\/ _— I'OTT'I"^ t* I rJ ^ \ / I/ I/ e~ ~^ u~(t l i / j «3-} * j | idx * u / i dl *(/j
,
7>7i,
o 13
and, consequently, oo
sup / £>0
J
/ J
KlP'ltl-hOr'^^^cCr.p),
Rer > 71.
(7.16)
§ 7. Solvability of the Cauchy Problem for Pseudoparabolic Systems
201
We write a formula for the components u~(r, £). For this purpose, we need to solve the system
uu+ + (r + H£| 2 )u+ + ifctT = 0, (r + v\£\2)u+ + i&tT = 0 , *fl«l" + *6«2" + »6«3 = 7 ~ ( r i O -
For 7 > 71, £ E M3\{0} we have
r
^ ' There exists 70 ^ 7i such that for 7 = Rer > 70, £ G M3\{0) the following estimate holds:
By the inequality (7.16), we have c1(/-,p)
e>o
We write the function Dtf
(f,£) in the form
___ ^_ ___ ^_ A/~(r,0 = A/~(r,0) where 1 CO
0
0 K3
3
(7.17)
202
3. The Cauchy Problem for rion-Cauchy-Kovalevskaya Type Systems
Since j£JDtf
-Dtf
(r,0),
+ E 11^,0, MGC)\\, where Gt — {(^,0 G IU : —oo < rj < oo, e < |£| < 1}, from the inequality (7.17) we obtain the estimate sup e>0
A/ ( r , 0 ) , V ( G e ) < c < oo.
But since p
/ e - T t D t f ~ ( t , x ) d t = 0. By the compatibility condition, we see that this condition is equivalent to the orthogonality condition (7.12). We obtain a contradiction which shows that this condition is necessary for the solvability of the Cauchy problem (7.8) in the class (7.10) for p^ 3/2. D Remark 7.1. Arguing as in Chapter 2, Section 7, we can establish the sufficiency of the orthogonality conditions (7.11) and (7.12) for the solvability of the problem (7.8) in the classes (7.9) and (7.10) for p
00,
x
(8.1)
where A'o is a nonsingular mxm-matrix and the operator K(DX] — K]_(DX) + K-2(DX] satisfies Assumptions II.1-II.3 in Section 2, i.e., det(rA' 0
c>0, , Rer^O,
(8.2) 0, (8.3) 1, 0 0. (8.4)
§ 8. Parabolic Systems
203
Theorem 8.1. Let K-2(DX) — 0. Then for any right-hand side f ( t , x) G L pi7 (IR* +1 ) ; 7 > 0; the Cauchy problem (8.1) has a unique solution u ( t , x ] G W££+(R++1), r+ = (I/a!,... , l/a n ), and the following estimate holds: )!! ^ c \ \ f ( t , x ) , L P , 7 (MJ +1 )||,
(8.5)
where the constant c> 0 zs independent of f ( t , x ) . Corollary. ///(*,*) G ^(Mj+i), * = (l/alt.. . ,l/an), 7 > 0, Men Me solution to the Cauchy problem (8.1) /ias Me weaA; derivatives D%u(t,x} G L P|7 (R* +1 ) ; /3a ^ 1 +/. Furthermore, )||.
(8.6)
Theorem 8.2. Le£ A'2(-Dr) ^ 0. TAen Mere 62:25^5 a number 70 > 0 swc/i Ma^ Me Cauchy problem (8.1) z's uniquely solvable in Wp£ (M^"+1); 7 > 70, /or any right-hand side f ( t , x ) G Lp i7 (E^ +1 ), anrf the solution satisfies the estimate (8.5). Corollary. If f ( t , x ) 6 PVp°;7*(Mj+1); 7 > 70, Men Me solution to the Cauchy problem (8.1) has the weak derivatives D%u(t,x] G Z/P|7(M^"+1); /?a ^ 1 4- SQ, and the inequality (8.6) holds. PROOF OF THEOREM 8.1. We first establish the existence of a solution to the Cauchy problem. Let /(£, x) G Co^M^.J. We consider the system of equations obtained by formal application of the Fourier-Laplace operator to the problem (8.1) (rAr 0 + tfi(*'OMr,£) = 7(r,0, Her > 0 , £ G Mn- By the condition (8.3), the inverse matrix (rA'o + A'i(z^))" 1 exists and its entries are analytic functions of (r, £), Re r > 0, £ G M n . Taking into account the condition (8.2), we can represent this matrix in the form ,
where A = |r| + {£), (£) 2 = ]T) £j
(8.7)
• Consequently, for any 70 > 0 the
entries of the matrix (rA'o + Ki(i£))~l are bounded functions for Rer ^ 70, £ € M n . Since f ( t , x ) G Co0(M^'+1), we can apply Theorem 5.2 in Chapter 1 to the vector-valued function u>(r, £) = (rA'o -4- A'i(z'£))~ 1 /(r, £).
204
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
Consequently, the vector-valued function (8.8)
is independent of 7 ^ 70. Furthermore, it is a solution to the Cauchy problem
v = (t,t),
t>0,
t=o = 0.
Since f ( t , x) £ C^°(M+ +1 ), from the properties of (rA'0-f A'i(^))" 1 it follows that we can apply the inverse Fourier operator with respect to £ to the vector-valued function (8.8). We obtain the function
(8.9) which is an infinitely differentiable solution to the Cauchy problem (8.1). We establish the L p -estimates for the solution (8.9). Since for any 70 > 0 the vector-valued function (8.8) is independent of 7 ^ 70, we have
x ((iri + i)K0 +
• 1 /(«f/ + 7,0^^,^P,7(Kj+i)
By the Lizorkin theorem about multipliers (cf. Chapter 1, Section 4), the entries of the matrix (i£)P ((irj + 7)A'o + A'i(z'£))~ 1 , Pa ^ 1) are multipliers and, by (8.7), we find
where (3a ^ 1, 1 < p < oo, 7 ^ 70 > 0 and the constant GI > 0 is independent of f ( t , x ) , 7. Similarly,
where 1 < p < o o , 7 ^ 7 o and the constant 02 > 0 is independent of f ( t } x}} 7From formulas (8.8) and (8.9) it follows that for f ( t , x) G C^°(Mj +1 ) we can construct a solution in the operator form u ( t , x ] = S f ( t , x } . By the
§ 8. Parabolic Systems
205
estimates and the fact that C*o°(IR*+1) is dense in LP|7(IR*+1), the linear operator S can be extended to the entire space LP)7(]R*+1) in such a way that the norm of the extended operator is preserved. The extended operator is denoted in the same way S: Lpi^(R++1)->W(R++1),
Kp
7 > 0.
(8.10)
The extended operator satisfies the estimates \\DtS f ( t , x ) , L Pl7 (R+ +1 )|| ^ c 2 ||/(*,x), L P l 7 (M+ + 1 )H,
(8.H)
\\D£Sf (t,x), L Pl7 (R+ +1 )|| ^
(8.12)
zdl/(*,*), W^n + i)U>
where 0 ^ / ? a ^ l , l < p < o o , 7 > 0 . Hence u(t, x) = S f ( t , x) is a solution to the Cauchy problem (8.1) in the class W*£+(SL++1) and the estimate (8.5) holds. We prove the uniqueness of a solution. Let a vector- valued function u(t,x) of class Wp£ (M*+1) with compact support with respect to x be a solution to the problem (8.1) with the zero right-hand side. It is obvious that K0Dtu(t,£) + Ki(i£)u(t,£) - 0, t > 0, and u\t_Q= 0. Therefore, u ( t , £ ) = 0, £ 6 M n , Hence u(t,x) — 0. Consequently, a solution to the Cauchy problem (8.1) with compact support with respect to x is uniquely defined. Hence any vector-valued function u(t,x} £ Wp£ O&n+i) w ith compact support with respect to x such that u t _ 0 = 0 satisfies the estimate
^ c\\(KQDt + IC 1 (D,)M*,x) > L P>7 (KJ +1 )||,
(8.13)
where l < p < o o , 7 > 0 and c > 0 is an absolute constant. Let u(t, x) G Wp£ (M^"+1) be an arbitrary solution to the Cauchy problem with the zero right-hand side. For an arbitrary e > 0 we find a vectorvalued function ue(t,x) 6 CQ°(M^ +I ) such that )!! ^ e By (8.13), for ue(t,x) we have
J3a=l
(8-14)
206
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
^cl\\Dtue(t,x)-Dtu(t,x),Lp^(1BL++1
where the constants c; > 0 are independent of e, u£(t,x), u ( t , x ) . By (8.14), we have P U e ( t , x ) , L pi7 (K+ +1 )|| <: ( Cl + c 2 )e, Pa-l
and, consequently,
Since e: > 0 is arbitrary, we have D^u(t,x) — 0, /9a = 1. Consequently, u ( t , x ) = 0, i.e., a solution to the Cauchy problem (8.1) in W*'^ (^n+i) *s determined in a unique way. D PROOF OF COROLLARY TO THEOREM 8.1. The assertion directly follows from the theorem because the above arguments imply the equality D£ o S f ( t , x) = So D%f(t, x) which, in turn, implies (8.6). D PROOF OF THEOREM 8.2. We use the perturbation method to construct a solution to the Cauchy problem (8.1) for a system with lower-order terms. We look for a solution in the form UU
x\
_ g/nU
x\
(g 15}
For the unknown vector-valued function p(t,x) we obtain the equation
(8.16)
By (8.4) and (8.12), we have
where the constant c(j) > 0 is independent of 0 if 7 —> +00. Consequently, there exists 70 > 0 such that the norm of the operator I\2(Dx}oS : Lp^(R^+l) —> L pi7 (E^ +1 ), 7 ^> 70, is strictly less than 1.
§ 9. Approximate Solutions to Pseudoparabolic Systems
207
Then equation (8.16) with any right-hand side /(/, x] £ Z/ P)7 (M* +1 ), 7 > 70, is uniquely solvable :
Substituting this vector- valued function into (8.15), we obtain the solution to the Cauchy problem (8.1) in the form u ( t , x ) = S o (/ + K^(Dx) o S ) ~ l f ( t , x} e W p 1 i ^ + (R+ +1 ) > 1 < p < oo, 7 > 70- This solution satisfies the inequality (8.5). The uniqueness of a solution to the problem (8.1) can be proved by repeating the proof of Theorem 8.1. D PROOF OF COROLLARY TO THEOREM 8.2. It suffice to note that D^ o S o (/ + K2(DX) o S ) ~ l f ( t , x) = So(I + K2(DX) o 5)-1 £>£/(*, x). D Remark 8.1. A review of results concerning parabolic equations and systems in contained in S. D. Eidel'man [1].
§ 9. Approximate Solutions to Pseudoparabolic Systems In this section, we construct a sequence of approximate solutions to the Cauchy problem (7.1) for pseudoparabolic systems without lower-order terms (K-2(DX] = 0). Recall that the compatibility condition holds: f~\t—o = 0. We construct a sequence of approximate solutions for this problem in accordance to the scheme presented in Section 4. We assume that f + ( t , x ) , f~ ( t , x ) € C^°(Mj+1). We consider the Cauchy problem for the system of ordinary differential equations with parameter £ £ K n which is obtained by application of the Fourier operator with respect to x to the problem (7.1): K0Dtv+ + K^iQv* + L(i$v- = / + (*,0, M(it)v+=f-(t,t),
* > °' (9.1)
Applying the integral Laplace operator with respect to t, we find (rK0 + tf!(iO)w+ + L(i$u- = 7 + M),
M(i^+ = /'M), (9.2)
where Rer > 0, £ £ R n . By Assumption II. 2, the matrix (rA'o + K\(i£)) is nonsingular. Therefore, the first m equations of the system (9.2) imply ^i(^))- 1 (7 + (r > 0-^(«'0w-(r,0).
(9-3)
208
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
Substituting this expression in the last (i> — m) equations (9.2), we obtain the system (M (i
By Assumption II. 2, for Her > 0, £ € M n \{0} this system is uniquely solvable. In the notation A'(r,£) = rK0 + A^), tf (r,0 - M^OA'^MWO,
(9.4)
the solution can be written in the form w-(r,0 = N-l(r^)M(i^K-1(r^)f+(r,^
- N-l(Ttt)f-(r,t).
(9.5)
Substituting (9.5) into (9.3), we find
(Ttt)f-(T,t).
(9.6) 1
By definition, the entries of the matrices K ~ ^ ( T , £), N~ (r,^}, £ £ M n \{0), are analytic bounded functions for Re r ^ 70 > 0. Since f+(t, x ] , f ~ ( t , x ) e (7o°(IR^ +1 ), we can apply Theorem 5.2 in Chapter 1 to the vector- valued functions u> + (r, £), w ~ ( r , ^), ^ 6 M n \{0). Consequently, the vector- valued functions
(9-8)
are independent of 7 ^ 70. Furthermore, they form a solution to the Cauchy problem (9.1) for f e M n \{0}. As in the case the Cauchy problem for Sobolev-type systems, it is possible to construct a sequence of approximate solutions to the problem (7.1). Using the integral representation (4.9), we construct the vector- valued functions u+(t,x] = (2rr)- n / 2 f v~l f ei^G(^va}v+ ( t , ^ ) d^dv, l/fc
(9.9)
Kn
k
u^(t,x] = (27r)- n / 2 / v-1 t eix^G(^va}v-(t^}d^dv}
(9.10)
§ 10. Estimates for Approximate Solutions to Pseudoparabolic Systems
209
where v+(t,£) and v ~ ( t , £ ) are defined in (9.7) and (9.8). We note that the vector-valued functions v+ (t, f ) and v~ (t,£) have singularities only at £ = 0. Therefore, we can indicate NQ such that for N ^ NQ the vector- valued functions u£(t,x) and u^(t,x) are infinitely differentiable and summable with any power. In what follows, we assume that N ^ NQ. It is easy to show that KoDtu£(t, x) + Ki(Dx)u£(t, x) + L(Dx)u^ (t, x) = f f ( t , x ) , M(Dt)u+(ttx) = /n*,*), u+(Q,x) = 0, where
l/k
A:
B
(*,*) = (2ir)- I v-1 j I Jt'-MGftv")/l/k
(t, y] d£ dydv.
KnKn
By the integral representation (4.9), we have
as k -4- oo. Consequently, the vector- valued functions (9.9) and (9.10) can be taken for an approximate solution to the Cauchy problem (7.1). In Section 10, we establish estimates for the vector- valued functions u^(t, x], u^ (t, x) which imply the existence theorems for the Cauchy problem for pseudoparabolic systems.
§ 10. Estimates for Approximate Solutions to Pseudoparabolic Systems In Section 9, we constructed the sequences {u~£(t,x}} and { u ^ ( t , x ) } of vector- valued functions of the form (9.9) and (9.10) respectively that are approximate solutions to the Cauchy problem (7.1) for pseudoparabolic systems without lower-order terms. In this section, we obtain estimates for this sequence and show that it converges in the weight Sobolev spaces. As in Section 5, to derive estimates, it is convenient to write the vectorvalued function u % ( t , x ) , u^(t,x) in the form
l/k
210
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems u
k (*> x} = uk'(^ x} + uk '"(*. x} k
= (27r)- n / 2 j v-1 f eixtG(tva)(v->+(t,t) + v - > - ( t , t ) } d £ d v , l/fc
(10.2)
Kn
where, in accordance with formulas (9.5)-(9.10), +'+(in + 1,t)dr!,
(10.3)
(^0.
(10.4) (10.5)
(10.6) (10.7) (10-8) (10.9)
(10.10) In this notation, the vector-valued functions u% ( t , x ) and u^' ( t , x ) depend only on / + ( / , x ) , whereas the vector-valued functions uk'~(t,x) and w^'""(t,x) depend only on f ~ ( t , x ) . To obtain estimates for approximate solutions, we need the following lemmas. Lemma 10.1. For any numbers c\ and c2 > 0 for Rer ^> 70 > 0,
(10.11)
§ 10. Estimates for Approximate Solutions to Pseudoparabolic Systems
211
PROOF. Let Sv-m(c) and T l/ _ m (c) be diagonal matrices with diagonal entries c Sm+1 , . . . , cs" and c tm + 1 , . . . ,ctv respectively. By Assumption II. 1, for any c > 0 we have K(cT,cat) = cK(T,t),
(10.12) ,
(10.13) (10.14)
Taking into account (9.4), for Rer ^ 70, £ 6 M n \{0} we find
m(c2}.
Using (10.12)-(10.14), we obtain (10.11).
(10.15) D
Lemma 10.2. For any numbers c\ and c2 > 0 for Her ^ 70 > 0, £ G M n \{0} ^/ze following identities hold:
PROOF. By (10.12)-(10.15), we have
The lemma is proved.
D
In what follows, to obtain estimates for the vector- valued functions (10.1) and (10.2), we extend the vector- valued function f ( t , x) to the domain t < 0 by zero and use the same notation for the extended function. We denote by /( r )0> r ~ zr ?+7) the Fourier transform with respect to (£, x] of the function e~^tf(t,x} and by / 7 (tf,£) the Fourier transform of the same function with respect to x. We introduce the notation /(^,£) = f o ( t , £ ) .
212
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
Lemma 10.3. The vector-valued function u^'+(t,x) satisfies the estimate /1ir+/'P+ f\\f+!i T /TO+ Ml ( l , Xr\ ), M ""p-y v^n + l /Ml l l << x; M l / ( l , Xv\ ) , -t>p,-y V^n+l/ II'
\\ii~^'~^(-t
\\Ufc
1]^ ]P.\
^lU.lDJ
where 1 < p < oo, 7 ^ 7o; and the constant c > 0 is independent of k andf+(t,x). Furthermore,
05 & i , k-2 —> oo.
PROOF. From (10.1), (10.3), (10.4) we find
/ t;-1 /'e"*G(tva)( ( I/A;
By Assumptions II. 1 and II. 2, from Lemmas 10.1 and 10.2 it follows that the entries of the matrices
are multipliers. Repeating the proof of Lemma 6.1 in Chapter 2, we obtain (10.16) and (10.17). D Lemma 10.4. The vector-valued function u*'~(t,x] satisfies the estimate x k
Y.
E
q=m+\ aia = \ s, |
\Wfq(t,x),Lpi^++l)\\,
(10.18)
§ 10. Estimates for Approximate Solutions to Pseudoparabolic Systems
213
where 1 < p < oo, 7 ^ 70, and the constant c > 0 is independent of k a n d f ~ ( t , x ) . Furthermore, )||-*0
(10.19)
/3a=l
as ki, k-2 —>• oo.
PROOF. From (10.1), (10.5), (10.6) it follows that K
/'-/
l/k
Introduce the notation A = |r| + (£). By Lemma 10.2, we can write
l/k
o ,_
ln
j /- (r, 0 drj} d£ dv.
(10.20)
The entries of the matrix
(10.21) are multipliers. Therefore, for fta — 1, repeating the proof of Lemma 6.1 in Chapter 2, we obtain (10.18) and (10.19). D Lemma 10. 5. I f \ a \ / p ' > 1 — \sm+j |, j = 1, . . . , i/— m; iAen the following estimates hold:
(10.22) \\Dtu'-(t,x), L P|7 (M+ +1 )|| ^ c(\\Dtf~ ( t , x ) , L pi7 (M+ +1 )|| (10.23)
214
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
where 7 ^ 70 and the constant c > 0 is independent of k and f ~ ( t , x ) . Furthermore )il - > 0 ,
(10.24)
\\Dtu%-(ttx)-Dtu%-(t>x))Lpii(®++l)\\-+0
(10.25)
as ki, k-2 —>• oo. PROOF. We consider (10.20) for (3 = 0. By Assumption II. 1, we have — 1 I. Since the entries of the matrix (10.21) are multipliers, arguing as in the proof of Lemma 6.3 in Chapter 2, we conclude that, under the condition \ot\/p' > 1 — | sm+J- |, the estimate (10.22) and the convergence (10.24) hold. We prove (10.23), (10.25). It is obvious that
i/k
Taking into account Lemma 10.2 and arguing in the same way as in the proof of (10.20), we can write this expression in the form
l/k
As before, under the condition |a|/p' > 1 — | $m+j |, we obtain (10.23) and (10.25). D Lemma 10.6. The jth component u^'*i+.(t,x] function u^'+(t,x) satisfies the estimate E
of the vector-valued
II^Vm+^^^.^K+i)! ^ c | | / + ( i ) x ) > L p ) 7 ( M + + 1 ) | | ,
(10.26)
§ 10. Estimates for Approximate Solutions to Pseudoparabolic Systems
215
where 1 < p < oo; 7 ^ 70; and the constant c > 0 is independent of k a n d f + ( t , x ) . Furthermore,
as i, 2 —> oo.
PROOF. From (10.2), (10.7), (10.8) we have k
/v l/fe
Taking into account Lemma 10.2, we can write this expression in the form k
f v~l f ei l/k
(10.28) The entries of the matrix
(10.29) Rer ^ 70, are multipliers. Therefore, for ft a = t m+J -, repeating the proof of Lemma 6.1 in Chapter 2, for the jth component of the vector- valued function u^'+(t,x) we obtain (10.26) and (10.27). D Lemma 10.7. If \a\/p' > ^m+j , J — 1, . . . , ^ — m; then the following estimate holds:
(10.30)
216
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
where 7 ^ 70 and the constant c > 0 is independent of k and Furthermore, )||->0,
*!, £ 2 ->oo.
f+(t,x). (10.31)
PROOF. We consider formula (10.28) for j3 = 0. Since the entries of the matrix (10.29) are multipliers and the expression in brackets has singularity of the form l/(£) tm+ -' at £ = 0, arguing as in the proof of Lemma 6.3 in Chapter 2, for \a\/p' > tm+j we obtain (10.30) and (10.31). D Lemma 10.8. // \a\/p' > 1 — \sm+i \, i = 1,.. . ,v — m, then for the jth component u^'^+.(t,x} of the vector-valued function u^'~(t,x) the following estimate holds:
/3a:=t m -|_j
<j=m+l
+ \ \ \ \ D t f q ( t , x ) , Li(M n )||, L p i 7 (Kf)||),
(10.32)
where 7 ^ 70 and the constant c > 0 is independent Furthermore,
\/ _,^
\\ni3ii~'~ T\ ~ w ll-^i An, m + j ldZ ' ^J
of k and f ~ ( t , x ) .
nPn~'~ T 1K(i®+ —v Un UU.OC5J nn^^'i x k2,m+j(t>d Xr\ )> LP,-y( -n + l )\\\ \\ ^
JJ u
as ki, k-2 —>• co. PROOF. From (10.2), (10.9), (10.10) we have k
v'1 l/k
Taking into account (10.15), we write this expression in the form
k I/A:
ln -1
§ 10. Estimates for Approximate Solutions to Pseudoparabolic Systems A = M + <0-
217 (10-34)
If 0a = t m+ ,- , then the expression in the brackets has singularity of the form l/(£) 1 "~' Sm +' I. at £ = 0 because —1 $C Sj ^ 0. Since the entries of the matrix 1
M
[-£- }K- -r,-f- L TTfV(ov VA'A«; v
(10.35)
are multipliers, as in the proof of Lemma 6. 3 in Chapter 2, under the condition l^l/p' > 1 — | sm+t- | for the jth component of the vector-valued function u^'~ (t, x) we obtain the estimate (10.32) and the convergence (10.33). The lemma is proved. D Lemma 10.9. // \a\/p' > 1 + | tm+j \ - \ s m+i |, i, j = 1, . . . , v - m, then the following estimate holds:
q-m+1
,(tla:)>Li(RB)||>Lpl7(Rf)||)l
(10.36)
where 7 ^ 70 and the constant c > 0 is independent of k and f ~ ( t , x ) . Furthermore, )ll ^ 0
(10.37)
05 ATI, k-2 —> oo.
PROOF. We consider formula (10.34) for (3 = 0. The expression in brackets has singularity of the form l/(£) 1+tm+J' H S m + « I at £ = 0, and, since the entries of the matrix (10.35) are multipliers, for \a\/p' > 1 + tm+j — |sm+,- we can obtain (10.36) and (10.37) arguing in the same way as in the proof of Lemma 6.3 in Chapter 2. D To conclude the section, we prove two lemmas which will be used in the proof of the uniqueness of a solution to the Cauchy problem (7.1) (cf. Section 11). Lemma 10.10. For the jth component u^'^.At, x) of the vector-valued function u^'~(t,x) the following estimate holds:
/3or=l+tm+>
218
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
E
E a:=| s
(10.38) where 1 < p < oo; 7 ^ 70 and the constant c > 0 is independent of k and f ~ ( t , x ) . Furthermore, \ ^
I r\8 . . —i—
(A -\
j-\3
—.—
I,
\
T
/Tn-i-f
\ll
. rv
(10 ^CA
as ki, k-2 —> oo. PROOF. Since the entries of the matrix (10.35) are multipliers, from (10.34) for /3a = 1 + t m + J , as in the proof of Lemma 6.1 in Chapter 2, we obtain (10.38) and (10.39). D Lemma 10.11. Let S : L P , 7 (E+ +1 ) -» Wj£(R++1), 1 < p < oo; 7 ^ 70, 6e a linear operator of the form (8.10). Then every vector-valued function Ulim+j(t,x) = (D? V~ + ^, x), . . . , D%* V'+^^x))*, ^a = . . . — /?ma = 1 + t m+J - ; j = 1, . . . , v — m, satisfies the estimate
^c E E ii^v,^,*), q=m + l pioi-\ sq |
where I < p < oo; 7 ^ 70 and the constant c > 0 is independent of k and f ~ ( t , x ) . Furthermore, l l < , m + j ( < > * ) - U£3im+j(t,x), L P , 7 (K+ +1 )|| ^ 0
(10.41)
as &i , A?2 —> oo.
PROOF. From (10.2), (10.9), (10.10) for the /th component of the vectorvalued function U^m+At,x) we have
§ 11. Cauchy Problem for Pseudoparabolic Systems
219
where / = 1 , . . . , m and K'1 (i»7 + 7,0 = (("7 + 7)^0 + A'i(O)" 1 - By the definition of the kernel G(£], we can explicitly write the result of application of the operator S to the vector- valued function U^ m+j(t, #) using formulas of the form (8.8) and (8.9). Then
v-1 J J e l/k
A'"1 (277+7,0
(*)"' :
drjd£dv.
Taking into account the identities (10.12) and (10.15), we write this expression in the form
1/fc M
i
if-
\
, I in -4- 'v
Since the entries of the matrices
are multipliers and 0qa = l + tm_|.j, we obtain (10.40) and (10.41).
D
§ 1 1 . Solvability of the Cauchy Problem for Pseudoparabolic Systems In this section, we prove Theorem 7.1-7A about the solvability of the Cauchy problem (7.1) for pseudoparabolic systems without lower-order terms:
K0Dtu++ Kl(Dx)u+ + L(Dx}u~ = f + ( t , x ) , M(Dx)u+ = f ~ ( t , x), t>Q, x e M n ,
220
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems t=o = 0.
By formulas (10.1)-(10.10), the approximate solutions to the Cauchy problem (7.1) have the form u£(t,x) — u^'+(t,x) + u^'~(t,x), u^(t,x] = u^'+(t, x) + u^'~(t, x) or, in the operator form, u^'+(t,x] = P*f+(t,x), u+'-(t,x} = Q+f-(t,x),u-'+(tlX) = p-f+(t,x),u^-(t,x) = Q-f-(t,x). The linear operators P^~ , Pk , Q^ and Qk are defined on vector- valued functions f + ( t , x ] , f ~ ( t , x ) e C^°(M++1). By Lemmas 10.1-10.10, the sequences {u£'+(t,x}}, {u£'~(t,x)}, {u^'+(t,x)}, {u^'~(t,x)} are Cauchy sequences in the corresponding weight Sobolev spaces. As in Section 6, we can define the linear operators t,x)= lim P£f+(t,x),
Q+f~(t,x)=
K—>OO
P~f+(t,x}=
lim P~f+(t,x),
k—too
lim Qr(t)X),
K—KX>
Q-f-(t,x)=
lim
k—too
(fif-fax),
where the limit is understood in the corresponding norm. We give the definitions of these operators. By Lemma 10.3, the operator P+ is defined by the limit \\P+f+(t,x] — P + / + ( t , z ) , W p 1 ^ + (M+ +1 )|| -4 0 as k ->• oo. This operator satisfies the estimate \\P+f+(t,x), W}:(R++1)\\ <: c \ \ f + ( t , x ) , L P17 (M+ +1 )||,
(H.l)
where 1 < p < oo, 7 ^ 70 and the constant c > 0 is independent of / + (t, x ) . In what follows, we need the L p -estimates for the lower-order derivatives D%P+f+(t,x], j3a < 1. From the proof of Lemma 10.3 and the Lizorkin theorem about multipliers (cf. Chapter 1, Section 4) we find \\nPp+f+(-t T\ T /"p+ 'HI <• ll^x* J (l,x), Lp,-t(Kn+l)\\ ^
c
\\f +
i-pa\\J
where 0 ^ /3a < l , l < p < o o , 7 ^ 7 o and c > 0 is an absolute constant. Since C^°(M^"+1) is dense in L pi7 (M^ +1 ), the operator P+ can be uniquely extended to the entire space L pi7 (IR^" +1 ) in such a way that the norm is preserved. The extended operator is denoted by P+ : L P>7 (M^ +1 ) —> w i£+K+i)- lt satisfies the estimates (11.1) and (11.2). Introduce the operator P~ . By Lemma 10.6, {D^u^'^(t, x ) } , (33 a = tj, j = m+lt... ,v, converge in Lp^R^j), i.e., there exist functions v^ ( t , x ) e L pi7 (R+ +1 ) such that ||«
(t>x)-D?]u+(t,x),Lpn(R++1)\\->Q,
k->oo,
(11.3)
§11. Cauchy Problem for Pseudoparabolic Systems
221
where l < p < o o , 7 ^ > 7 o - If |c*| > t max , then there exists a number q, I < q < oo, such that \a\/q' > t ma x, 1/9 + l/q' — 1. By Lemma 10.7, the sequence {u^'+(t,x}} converges in the space L gi7 (IR+ +1 ), i.e., there exists a vector-valued function u~t+(t,x) € I/9)7(]R^+1) such that
j=m+l
By (11.3), we have v? (t, x} = D% uj '+ (t , x) , j - m + 1, . . . ,i/. Consequently, the linear operator P~ is defined by the limit ||P-/+ (t, x) - P~f+(t, x ) , L,l7(R++1)|| -X),
k -> oo.
From Lemmas 10.6 and 10.7 we obtain the estimates
E
E
j=m+l /3Ja=tj )||,
(11.4)
where 1 < p < oo, 7 ^ 70, \\P~f+(t,x), L,|7(M++1)|| ^ c ( \ \ f + ( t , x ) , L,,7(M++1)|| (11.5) where laj/g' > t max and the constant c > 0 is independent of f+(t, x } . Since C£°(M++1) is dense in L pj7 (Mj +1 )nL q , 7 (Ej +1 ) nL 9 , 7 (Mf ; Ii(R n )), we can extend by continuity the operator P~ to the entire space, the extended operator is denoted by the same symbol. This operator also satisfies the estimates (11.4) and (11.5). Introduce the operator Q + . By Lemma 10.4, {D^u^'~(t, x ) } , J3a — 1, converge in LP|7(]R^"+1), i.e., there are v^(t,x] 6 Lpi-y(IR^"+1) such that \\vP ( t t x ) - ££«+•- (t, x), L P , 7 (M+ +1 )|| -> 0,
k -»• oo,
(11.6)
where 1 < p < oo, 7 ^ 70- If |a| > 1 — crm|n, then there exists a number 1 < q < oo such that \a\/q' > 1 — ~ (t, x) £ W^C^n+i) sucn ^at ll u + '~(^' x ) ~ u+'~(t,x}, W*$(R++i)\\ -> 0 as A; -> oo, where 7 ^ 70- By properties of the weak derivatives in (11.6), we have v^(t,x) = DP.u+'~(t, x } . Consequently, the linear operator Q+ is defined by means of the limit
222
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
\\Q+f'(t,x) - Q+f-(t,x), Lg, 7 (M+ + i)|| -> 0 as k -> oo. From Lemmas 10.4 and 10.5 we obtain the estimates ^
/3a = l
E
E II^-M). w^++1)ii,
en.?)
where 1 < p < oo, 7 ^> 70,
(11.8)
where \ct\/q' > 1 — crmm and the constants Cj > 0 are independent of/ (t, x } . The operator Q+ can be extended by continuity to the entire space of vectorvalued functions:
D t F j ( t , x ) £ Lq^(R+;Li(Rn)),
Fj\t=o = Q,
j = m + 1, . . . , i/}.
For the extended operator, denoted by the same symbol, the estimates (11.7)-(11.9) hold. We define the operator Q~. By Lemma 10.10, {D^ u^'~(t, x } } , $ a. — 1 4- tj, j — m + 1, . . . , i/, converge in the space L p i 7 (M^ + 1 ), i.e., there exist functions u? ( t , x ) £ L P)7 (M* +1 ) such that ||wf ( * , z ) - D f V~ (*,o:), LP,7(1R++ 0 |-> 0,
& -» oo,
(11.10)
where 1 < p < oo, 7 ^ 70- If |a| > 1 -f t m a x — 1 + t max - - O a s A ; - ^ o o , 7 ^ 7 o . Taking into account the properties of the weak derivatives and (11.10), we have u>j ( t , x ) = D^ u~'~(t, x), j — m -(- 1, . . . , v. Consequently, the linear operator Q~ is
§ 11. Cauchy Problem for Pseudoparabolic Systems
223
defined by means of the limit \\Q~f~(t,x) - Q ^ f ~ ( t , x ) , Lq^(R++l)\\ -> 0 as k —} oo; moreover, HDj^Q-rCt.x^-I^^Q^rCt.x^.Lp.^K+^II^O
(11.11)
J
as k —>• oo, where /3 'a = 1 + tj, j = m+1, . . . , z/, 1 < p < oo. From Lemmas 10.8-10.10 the following estimates hold:
»=m+l
I, W]R+)||),
(11.12)
where \a\fq' > 1-f t max -
(11.13) where \a\fr' > 1 -
t = m + l >'a=| s, |
iwV.-^*), ^,7(^)11), ,|
(n.14)
'
where l < p < o o , j = m+ I , . . . ,v and the constants Ck > 0 are independent of f ~ ( t , x ) . The operator Q~ can be extended by continuity to the entire space:
K = { F ( t , x) : Fi(t, x) e WJ;?' ( DtFi(t,x) G L 9 i 7 (E+;L 1 (IR n ))nL r F.-|t=o = 0, i = m+ 1 , . . . ,i/}. The extended operator, denoted by the same symbol, satisfies the estimates (11.12) and (11.13). In a narrower space, we have F2 = {F(t,x} € Ti :
Df^F^x)
k* + p*a ^ 1 + | s i | , i = m + 1, . . . ,
224
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
and the estimate (11.14) holds. We indicate one more useful inequality for the operator Q~ , which is essentially used in the proof of the uniqueness of a solution to the Cauchy problem (7.1). Lemma 11.1. Let f ~ ( t , x ] be a vector-valued function in TI, and let U f ( t , x ) = (Df>1(Q-f-(t,x))jt...,D$m(Q-f-(t,x))jy, /?'« = ... = 0ma= 1 + i j , j = m-f 1 , . . . ,*/. Then \\SUf
(t,x},Lp<,(R++l)\\ V
^c E
E
ll^/.-(<,x),I p i 7 (R+ + 1 )||,
(11-15)
i=m+l p'a = \ s, |
where 1 < p < oo, 7 ^ 70, S is a linear operator of the form (8.10) £Ae constant c > 0 zs independent of f ~ ( t , x ) . PROOF. Let a sequence {^~(2,£)} e ^^(^n+i) approximate f ~ ( t , x ) E . By the definition of the operator Q~ , we have
1=1
where l < p < o o , 7 ^ 7 o . From the arguments in Section 8 it follows that the operator (8.10) is continuous. Consequently, for a sequence of vector-valued functions {[/• ' ( t , x ) } such that
we have \\SUf <>l(t,x}-SUJ>(t,x),Lp^(R++l}\\-*Q,
/->oo.
(11.16)
We construct the same sequence of vector- valued functions {U^j(t, x}} as in Lemma 10. l l ) i . e . ) C / ( t ) x ) By (11.11) \\u£t(ttx) - U^'l(t,x), L pi7 (R+ +1 )|| -> 0 as A; -». oo. By the properties of the operator S, we have \\SU^(t, x] - S U f ' l ( t , x), L P , 7 (K+ +1 )|| -> 0,
k -» oo.
(11.17)
Using the Minkowski inequality, we find ), L pi7 (R+ +1 )|| ^ \\SU?(t,x)-SU?'l(t)x), ^, x) - SUH'fax), L P|7 (K+ +1 )|| + 115^'!, (11.18)
§ 11. Cauchy Problem for Pseudoparabolic Systems
225
By Lemma 10.11, the third term satisfies the estimate
i=m+l p'a=\Si\
where the constant c > 0 is independent of k, I,
\\SU (t,x), WRJ+1)|| ^ \\SU (t,x) -
E
i=m+l p«a=|s;|
E i=m+l p'a=\st\
Since
i=m+l p«a=|*i|
taking into account (11.16), (11.17) and passing to the limit on the righthand of this inequality, we obtain the estimate (11.15). D PROOF OF THEOREM 7.3. By condition, we have \a\/p' > l+t max -cr min . By the definition of the operators P+ , P~ , Q+ , Q~ , for any right-hand side /(i, x) = ( f + ( t , x), f~ (t, x)Y satisfying the assumptions of the theorem, the vector-valued function u(t,x) = (u+(t, x ] , u~(t, x))* such that < , x ) + g+/- ( t , x ) 6 W p 1 ;;(Rj +1 ),
(11.19)
where r+ = (1/ai, . . . , l/a n ), (M+ +1 ),
(11.20)
j=m+l
where r~ = (tj/a\,... , tj/an}} 7 ^ 70, is a solution to the Cauchy problem (7.1) for the system without lower-order terms, i.e., Ki(Dx} = 0. The inequalities (11.1), (11.4), (11.5), (11.7)-(11.9), (11.12), (11.13) imply the estimate (7.6). By Lemma 11.2 (below), we can establish the uniqueness of a solution to the Cauchy problem (7.1) for systems without lower-order terms in each
226
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
space of the scale W^^ O^n+i)
x
'
70.
Fl^m+i ^P.V (^n+i)> 1 < P < oo, 7 ^ J
n
Lemma 11.2. The Cauchy problem K0Dtu+ + Ki(Dx}u+ + L(Dx}u~ = 0, = 0, u+
t > 0,
x G Mn,
(11.21)
«=o-) = °
has only the zero solution in the class (7.2) : u+(t,x) G Wp1'^ (^n + i);
PROOF. We assume that u(t,x) = (u+(t, x), u~ ( t , x)Y is a solution to the problem (11.21). Let u~(t,x) G Qf^n+i)- Since (KQDt + K^D^u+^x) = -L(Dx}u-(t,x],
u+
t=Q=
0,
+
from Theorem 8.1 for any / G N, q G (1, oo), 7 > 0 we have u (t, x} = —S o L(Dx}u~(t,x] G Wq1^ (^n + i ) > r/ — ( V ^ i i ' - - J/Qn), and, in particular, this is true for q = 2. Applying the Fourier-Laplace operator, we find
Re T > 0, £ G R n . Therefore,
But, by Assumption II.2, for Rer > 0, f G M n \{0) the matrix A'i(z'^))~ 1 L(z'^) is nonsingular. Therefore, w ~ ( r , £ ) = 0 and, consequently, u + ( r , £ ) = 0. Thus, u ( t , x ) = 0. The above arguments mean the uniqueness of the solution to the Cauchy problem (7.1) in the class of vector-valued functions (7.2) with components u~(t,x) G Qj>0(]R^+1). Consequently, such solutions have the form (11.19), (11.20). In particular, for f+(t,x) = 0 we have u+(t,x) = Q+f~(t,x), u~ ( t , x ) = Q~ f~ (t, x) and, in the case /~ ( t , x) G J-~2, from Lemma 11.1 it follows that for the vector-valued function
where /^a = . . . = /3ma = 1 + tj, j = m + 1 , . . . ,i/, the following inequalities hold: ||5t/f(t,x),L p ^(]R+ + 1 )|l ^
E
—.—^
Z .- •
t
§ 11. Cauchy Problem for Pseudoparabolic Systems
227
where l < p < o o , 7 ^ 7 o and the constant CQ > 0 is independent of /-(*,*)• We consider an arbitrary solution u(t,x) = (u+(t,x},u (t,x)Y to the problem (11.21) in the class (7.2). We show that u(t,x) vanishes. We have (A'oA + Ki(Dx))u+(t,x) = —L(Dx)u~(t,x) and u+\t=o = 0. Using operator 5* in Section 8, the vector-valued function u+(t,x] can be represented in the form u+(t,x) = -SoL(Dx)u-(t,x).
(11.23)
M(Dx)oSoL(Dx)u-(t,x)
(11.24)
Consequently, = Q.
Let £ > 0. We find a vector-valued function u~ ( t , x ] € Co^O^n+i) such that uJ(t,*)-u7,j(t>*),WP(R++l)\\^e.
(11-25)
We consider vector-valued function u-(t,x).
(11.26)
Taking into account the definition of the operator 5, we find (K0Dt + Ki(Dx}}u+(t,x] + L(Dx)u-(t,x) = 0, M(Dx)u+ ( t , x ) = -M(DX) o S o L(Dx}u~ (t, x) for t > 0, x 6 M n , and u+ t_0= 0. Consequently, by the estimate (11.22) and the equality (11.24), the vector- valued functions
where {3la=... = /3ma=l + t j , j = m + l,... ,v, satisfy the inequalities
i—m + l p'a = \ s,
i=m+l p»a=| s,
228
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
where 1 < p < oo, 7 ^ 70- As was already mentioned, from Assumption II. 1 we obtain the identities M(i£) = c-lS~lm(c)M(cai^) and L(if) = L(cQ!i£)T~^m(c) . Taking into account the properties of the operator 5 and the approximation (11.25), we find \\SUf l'e(t,x},Lp^(R++1)\\ E
IPrlK/(<,a:)-wr(<>a?))>Lpi7(R++1)||^c£>
(11.27)
cr'a=t|
where the constant c > 0 is independent of e, ue(t,x), u(t,x}. Consider the vector-valued function V p > e ( t , x ] = (Dx u ~ j ( t , x ) , . . . , Dx™ u~ j ( t , x ) Y , pla=.... = pma = t j , j = m + l,...,i'. By the properties of the operator 5, for multi-indices (3l = 0 + pl , . . . , fim = 0 + pm , 9a - 1, we have D9xoSVf'e(t, x) = S U f ' € ( t , x } . By (11.27) and the properties of the operator S, for V f ( t , x ) = (Dpxu-(t,x),... ,D?muJ(t,x}y we have
)|| ^ \\SUf'' ( t , x ) , L \D xoSVf(t)x)-D xoSVf>e(t,x),Lp>,(R++l)\ e
e
where the constant c\ > 0 depends on the norm \\S\\. Taking into account (11.25), we find \\DBX o S V f ( t , x), £ P , 7 (IR+ +1 )|| ^ ( c + c i ) e . Since e > 0 is arbitrary, for any multi-indices Q, da = 1, we have ||^o5^(* > x) l L p i 7 (R+ + 1 )|| = 0.
(11.28)
By condition, we have u~(t,x) e Wp'^1 (M^ +1 ), r~ = ( t j / a i , . . . ,tj/a>n). By definition, we have V f ( t , x ) G L P)7 (M^ +1 ). Therefore, by the properties of the operator 5, we have S V f ( t , x ) € H^;7r+(Mj+1). From (11.28) it follows that S V f ( t , x ) = 0 and, consequently, V f ( t t x ) - 0. Therefore, for any multi-indices p such that pa = tj we have Dxii~(t,x) = 0. Then u^(t,x) = 0. Since j = m + 1, . . . , v is arbitrary, we have u~(t,x) = 0. Hence u+(t,x) = 0 in view of (11.23). Consequently, the Cauchy problem (11.21) has only the zero solution in the class (7.2). D PROOF OF THEOREM 7.1. By assumption, we have \a\fp' > t max . By the definition of the operators P+ and P~ , for any right-hand side
§ 12. Pseudoparabolic Systems with Lower-Order Terms
229
,0), / * , * ) G L p ^ M ^ n L p M j L i R U ) , 7 ^ 70, the vector-valued function u(t,x) = (u+(t,x),u (t,x)Y, u+(t,x) — P+f+(t>x), u~(t,x) = P~f+(t,x), is a solution to the Cauchy problem (7.1). The estimates (11.1), (11.4), and (11.5) imply the estimate (7.3). The uniqueness of a solution to the problem (7.1) in the class (7.2) follows from Lemma 11.2. D PROOF OF THEOREM 7.2. By assumption, the function f+(t,x) G Z/P|7(IR*+1) has compact support. Therefore, we can apply the operator P-. Consequently, u~(t,x) = P-f+(t,x) G L<,,7(K++1), \a\/q' > t max , l/q + l/q' = 1, 7 ^> 70, and, in the view of the inequality (11.4), E
\\DfuJ(t,x), L P|7 (K+ +1 )|| ^ c \ \ f + ( t , x ) ,
j=m + l /3Ja=t
l < p < o o , 7 ^ > 7 o . The operator P+ can be applied to any vectorvalued function f + ( t , x ] £ Z/ P)7 (]R^ +1 ), 1 < p < oo, 7 ^ 70; moreover, for the vector-valued function u+(t,x) = P+f+(t,x) G W p 1 > iJ + (Mj +1 ), by the estimate (11.1), we have
By the definitions of the operators P+ and P~ , we find that the constructed vector-valued function u ( t , x ] = (u+(t, x ) , u~(t, x)Y is a solution to the Cauchy problem (7.1) in the class (7.4) and (7.5) holds. D PROOF OF THEOREM 7.4. From the proof of Theorems 7.1-7.3 we immediately conclude that the solution to the problem (7.1) from the class (7.4) takes the form (11.19), (11.20):
and (7.7) holds.
D
§ 12. The Cauchy Problem for Pseudoparabolic Systems with Lower-Order Terms In this section, we sketch the proof of Theorems 7.5-7.8 asserting the solvability of the Cauchy problem (7.1) for pseudoparabolic systems with lower-order terms
K0Dtu+ + Kl(Dx)u+ + K2(Dx)u+ + L(Dx)u~ = f+(t, x), M(Dx)u+ = f - ( t , x ) , 00, x G M n ,
230
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
The proof of Theorems 7.5-7.8 follows the scheme presented in Sections 9-11. We emphasize the necessary modifications caused by the presence of lower-order terms. We construct a sequence of approximate solutions to the Cauchy problem for f + ( t , x ) , f ~ ( t , x ) £ Q°(M+ +1 ). Consider the system of equations with parameters Rer > 0, £ G M n which is obtained by application of the Fourier-Laplace operator to the problem (7.1): (rA'o + A-^zO + tf2(»0)"+ + £(''0"~ = 7 + (r,0,
Consider the matrix /C(r, £) = rA'o + A'i(z£) + A'2(z£)- Bu Assumption II. 2 in Section 2, the matrix A'(r, £) = rA'o + A'i(z£) possesses the inverse matrix for Rer ^ 0, £ G K n , |r + |f| / 0. Consequently, /C(r,0 = A'(r,£)(/ + A'~ 1 (r, ^)A'2(i^)). Taking into account Assumptions II. 1 and II. 3, we find K(cT,cat) = cK(T,t), /C 2 (*0 - (fcj,j(*0). Mctt«'0 - c^'^fcu^), 0 ^ 9;j < 1, c > 0. Therefore, \\K-l(rtt)K2(it)\
^ A - ^ l / C - ^ r / A ^ / A ^ I I I I A T a C i O I I ^ c( 7 ),
(12.2)
where A = |r| + (^}, 0(7) —> 0, 7 = Rer —> -foo. Hence there exists 70 > 0 such that for Rer ^ 70, <^ G M n the matrix /C(r,£) possesses the inverse matrix; moreover, /C~1(^0-(/+A'-1(r,OA2(Z-0)-1A-1(r;0.
(12.3)
By the above arguments, we can conclude that for Rer ^ 70, £ G M n the system (12.1) is reduced to the following;
Consider the matrix Af(r,£) — M(z£)/C (T,£)L(i£). By Assumption II.1 and (12.3) for Rer ^ 70, £ G M n \{0}, as in Section 10, we have
_ (0 A
By Assumption II. 2, the matrix M(is)(rKo + Ki(i^))~1L(is) for Rer ^ 0, ^ G M n , |r| + |^| ^ 0, s G M n \{0} is nonsingular. By (12.2), there exists 71 ^ 70 such that for Rer ^ 71, £ G M n \{0} the inverse matrix
§ 12. Pseudoparabolic Systems with Lower-Order Terms
231
, £)L(z'£))~ 1 exists. Therefore, for such values of parameters (r, £) the system (12.4) (consequently, the system (12.1)) is uniquely solvable. From (12.1) and (12.4) we find r ) 0, (12-5)
where Rer ^ 71, £ € M n \{0}. The entries of the matrices /C~ 1 (r, £), M~l(r, £), £ G M n \{0}, are analytic bounded functions for Rer ^ 71. Since f + ( t , x ) , f ~ ( t , x ] € C^°(Rj +1 ), Theorem 5.2 in Chapter 1 is applicable to each of the terms in (12.5), (12.6):
Consequently, the vector-valued functions
oo
/" 00 00
/
are independent of 7 ^ 71. By construction, it is clear that the vector- valued function v(/,£) = >(w+(/,0, """('.O)*. where u+ (*.0 = v + 1 + (<,0+ v + 1 ~ (*.0 + and v ~ ( < , 0 = v ~ ( < , 0 + v~'~(t,£], is a solution to the Cauchy problem K0Dtv+ + tfi(iO«+ + ^ 2 (te)v +
232
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
As in Section 9, we can construct a sequence of approximate solutions to the Cauchy problem (7.1) for K^(DX] ^ 0. Using the integral representation in Chapter 1, Section 6
where G(£) = 2N(£)2N exp(-(£) 2N ), <£) 2 = ££? / Q ', »=i vector-valued functions
we
construct the
= (27r)- n / 2 /V1 f eixtG(tva)v+< l/k /c
,x =
7r
v-
e
>
- *,
v,
l/k
k
l/k A;
l/k
In
Further, we assume that N is such that these vector-valued functions are infinitely differentiate and summable with any power. Such an N exists since v+l+(t,£), f + ' ~ (*,£), v~'+ (t,£), v ~ ' ~ ( t , ^ } have singularities only at the point £ = 0. Consider a sequence {iik(t, x ) } such that Uk(t, x) = (u~£(t, x } , u^(t, z)Y, easy to verify that
M(Dx)u+(t,x) = f-(t,x),
§ 13. Pseudoparabolic Systems with Variable Coefficients
233
where f f ( t , x ) = (27T)-" I v-1 f
fe**
l/k
k
v-1
/fe- ( t , x ) = (27r)-» l/k
e*-G(tva)r (tt y] d£ dy dv. KnIRn
Consequently, {uk(t, x)} is a sequence of approximate solutions to the Cauchy problem (7.1) in the case K^(DX} ^ 0. For the vector-valued functions u^(t,x) and u^(t,x) all the estimates in Section 10 hold with a single modification that 7 > 71 should be taken in the formulations of Lemmas 10.1-10.11. Furthermore, the proof differs by a slight additional arguments concerning the fact that in approximate solutions, instead of the matrices A'(r, £) and N ( T , £ ) , we have matrices /C(r, £) and A/^r, £) that are close for Rer ^> 0. Using these lemmas and arguing as in Section 11, we can define the operators P+ , P~ , Q+ , Q~ acting in the spaces indicated in Section 11 with a single modification 7 ^ 71 instead of 7 ^ 70. Thus, Theorems 7.5-7.8 can be established by repeating the proof of Theorems 7.1-7 .4 respectively; moreover, the solution to the Cauchy problem (7.1) for K^(DX] ^ 0 is also represented in the operator form (12.7)
JJ
(12.8)
j=m+l
where 7 > 71 .
§ 13. Pseudoparabolic Systems with Variable Coefficients In this section, on the basis of the results presented in Section 12, we study the Cauchy problem for pseudoparabolic systems with variable coefficients £(x;Dt,Dx)u=:f(t,x),
t> 0, x E R n ,
where the operator jC(x°; Dt, Ac), #° E M n satisfies Assumptions II.1-II.3 in Section 2. For this purpose, we use the classical approach of the theory of elliptic and parabolic equations.
234
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
To simplify the exposition, we assume that only differential operators A'i and A' 2 have variable coefficients, whereas L and M are operators with constant coefficients, i.e., we consider the Cauchy problem K0Dtu+ + Ki (x; Dx)u+ + K2(x\ Dx)u+ + L(Dx)u~ = f+ (t, x), M(Dx)u+ = f - ( t , x ) ,
t > 0,
xeMn,
(13.1)
We further assume that the coefficients of the operators KI(X; Dx) and I\2(x; Dx) are continuous and constant outside some ball {\x\ < p } , and for the homogeneity vector of the symbol of the principal part of the matrix operator £(x; Dt, Dx) the following condition holds: |a| > 1 + t max — t m a x . // the coefficients at the derivatives in the operator K\(x\Dx} are close to constants, then there exists 72 > 0 such that the Cauchy problem (13.1) has a unique solution in the class (7.2) : u+(t,x) G Wp£ (R+ +1 ), r+ = (I/a,, . . . , l/an), u~(t, x) G
ft
Wp% (R+ +1 ), rj = (tj/alt ..., ljl<*n),
j=m + \
7 ^ 72, for any f + ( t , x ) e L P , 7 (R+ + I ) H L P]7 (1RJ; Li(M n )). The solution satisfies the estimate (7.3). Theorem 13.2. Let f ~ ( t , x ) = 0. // the coefficients at the derivatives in the operator Ki(x;Dx) are close to constants, then there exists 72 > 0 such that the Cauchy problem (13.1) has a solution in the class (7.4) : u+(t,x) e Wplf(R++l), D?uJ(t,x) G L Pl7 (M+ +1 ), /3ja = ij, j = m + 1, . . . , z/; 7 ^> 72, /or any compactly supported vector-valued function f + ( t , x ) G Lp j 7 (K^ + 1 ) ; ana1 ^Ae estimate (7.5) /ioWs. Theorems 13.1 and 13.2 can be proved by the perturbation method. We consider the proof of Theorem 13.1 in detail. The arguments are similar in the case of Theorem 13.2. PROOF OF THEOREM 13.1. We consider the Cauchy problem (13.1) with frozen coefficients at the point. x° , \x°\ > p. By the above theorems, the solution is uniquely found by formulas (12.7) and (12.8). Taking into account these formulas, we look for a solution to the problem (13.1) with variable coefficients in the form u + ( t , x ) = P+p+(t,x),
u ~ ( t , x ) = p-p+(t,x).
(13.2)
§ 13. Pseudoparabolic Systems with Variable Coefficients
235
For the unknown vector-valued function we obtain the operator equation p+(t, x) + ( K i ( x ; Dx) - Kl(x°]Dx))P+^+(t, x) + (K2(x-Dx)-K2(x°]Dx))P+v+(t,x)
= f+(t,x).
(13.3)
By the conditions on the coefficients of the operators K\(x\ Dx), K2(x; Dx), we have p. Therefore, it suffices to consider equation (13.3) in the ball {\x\ < p}. Recall that the operator P+ satisfies the estimates (11.1) and (11.2) for 7 > 71- Taking into account assumptions II. 1 and II. 3 on the symbols of the operators K\(x\Dx], K2(x;Dx), we find \\(Kl(x-Dx)-Kl(x°;Dx))P+^+(t,x),Lpi^^n+1 % •>
. j, .
^ C ^
j . O\HI
rXlcLX J G E — . ( 3 ? J — Q,(j.\X
+ /
) j [^
\
\ j *^/ >
||(A' 2 (^iD x )-A' 2 (^°;I
where a},(x} are the coefficients at the derivatives in the differential operator KI(X; Dx) and 0(7) —>• 0 as 7 —> +00. From these estimates it follows that there exists a number 72 ^ 71 such that, in the case c^ max |a^(a;) — X a \ \^P
a^(x°)| + c(7) < 1, 7 ^ 72, the operator equation (13.3) is uniquely solvable: +(t,x) = (I + T)-lf+(ttx) e Ip, 7 (M+ +1 ), 7 ^ 72, T = (#!(*;£>,)Ki(x°\Dx))P+ + (K2(x;Dx) - K2(x°\ DX}}P+. Substituting p+(t,x) into (13.2), we obtain a solution to the Cauchy problem (13.1) of the class (7.2) for 7 ^ 72 in the operator form u+(t, x} = P + o(J-f T)~l f+(t, x), u~(t, x) = p-o(I + T}-lf+(t,x}. Let us prove that a solution is unique. Let u(t,x) be a solution to the Cauchy problem (13.1) with f ( t , x ) = 0. Then K0Dtu+(t,x) +
Kl(x0]Dx)u+(t}x)-^K2(x0-Dx}u+(t,x)
= -(Ki(x- Dx] - K1(x°-Dx)}u+(t, x) - (K2(x- Dx) - K2(x°; Dx})u+(t, x), M(Dx)u+(t, x) = 0,
t > 0, x e R n , |x°| > p,
+
u (t,x)\t=0 = Q. Since the coefficients at the derivatives in the operators Ki(x\Dx) and K2(x; Dx) are constant for |x| > p, the vector- valued function ,x) =
-(K1(x;Dx)-Ki(x°;Dx))u+(t,x)-(K2(x;Dx)
236
3. The Cauchy Problem for non-Cauchy-Kovalevskaya Type Systems
satisfies the assumptions of Theorems 7.1 and 7.5. By the unique solvability of the Cauchy problem (7.1) in the class (7.2) for 7 > 71, we obtain the following representation: u+(t,x) = P+F+(t,x), u~(t,x) = P~F+(t,x). As above, by the estimates (11.1) and (11.2) for the operator P+, we have
8a<\
<1 E /3a = l
E where the constant c > 0 is independent of 7, u(t,x) and a^(ar) are the coefficients at the derivatives in the operator Kj(x; Dx). The above estimates imply that if, for example, c ]P max \a^ (x) — ala (x°)\ ^ 1/2, then there exists 72 > 71 s^h that \\u+(t,x), Wrp1;7r+(Rj+1)|| ^ ilh+Ct.^.^ where 7 ^ 72, i.e., w + (t, x) — 0. Taking into account the estimate (11.4) for the operator P~ , we deduce the inequality
E /^ E / -j
j-m + l p}a=t
— A'i(x°; Dx)}u+(t, x} -f (A 2 (x; Dx] — A' 2 (x°; Dx)}u+(t, x), Lp which implies u~ ( t , x) = 0. Thus, if the coefficients of the operator K\(x\ Dx) are close to constants, then there exists 72 > 0 such that the Cauchy problem (13.1) is uniquely solvable in (7.2) for 7 ^ 72D Remark 13.1. Theorems 13.1 and 13.2 are valid without the assumption that the coefficients at the derivatives in the operator A'i(x; Dx] are small. We can prove this assertion in the same way as an analogous assertion is proved in the theory of elliptic and parabolic equations, using the theorem about the unit partition and the above results.
Chapter 4 Mixed Problems in the Quarter of the Space In this chapter, we consider the mixed boundary-value problems in the quarter of the space ^^+i — {(^> x ) : t > 0, x = (x',xn) E M+} for equations not solved relative to the higher-order time-derivative /-i k=0
and systems with a singular matrix at the time-derivative
A0Dtu + Ai(D^l,Dx)u = f ( t , x ] ,
detA0 = 0.
We study the mixed problems for two classes of equations mentioned in Chapter 2 (simple Sobolev-type equations and pseudoparabolic equations) and for two classes of systems mentioned in Chapter 3. The theory of boundary-value problems for pseudohyperbolic equations is more complicated and is not developed yet. To study mixed problems, we modify the method of integral representations of solutions represented in Chapters 2 and 3. In this case, the technique of obtaining approximate solutions is slightly more complicated.
§ 1. Statement of the Mixed Boundary- Value Problems for Simple Sobolev-Type Equations In this section, we consider the mixed boundary- value problem in the quarter of the space
(1.1)
237
238
4. Mixed Problems in a Quarter of Space
where
i-i L ( D t , D x ) = L0(Dx)Dlt + Y^Li-k(D^)Dt k =0
is a simple Sobolev-type operator with constant coefficients. By definition (cf. Chapter 2, Section 2), the symbol of such an operator L(irj,i^) should satisfy the following conditions:
• ASSUMPTIONS ON THE SYMBOL 1. The symbol L(ir), i£) is homogeneous with respect to the vector a = (0,a) = ( 0 , a i , . . . ,an), I/a,- G N, i.e., L(irj,cai^ =cL(ir],i^, c> 0. 2. The operator Lo(Dx) is quasielliptic. Further, we impose conditions on the boundary operators Bj(Dt,Dx}. But we first note that, in view of the conditions on the symbol L(ir},i^), there exists 70 > 0 such that the equation L ( r , z s , z A ) = 0,
Rer^7o,
s 6 M n -i\{0},
(1.2)
has no real roots A. We assume that the number of the boundary conditions in (1.1) at xn — 0 is equal to the number of the roots with positive imaginary part. • ASSUMPTIONS ON THE BOUNDARY OPERATORS The boundary operators have the form
moreover, deg^ B j ( i r i , Dx) — degr)bj(ir]) ^ /, mj < m = l/a n , and the symbols Bj(ir],i£) are homogeneous relative to the vector a = (0,a) from condition 1 with exponents /3j , 0 ^ /3j < 1, i.e., Bj (irj , ca'i£) — c^^Bj^rj, z£), 00. •
LOPATINSKII CONDITION
The boundary- value problem with parameters (r, s), Re r ^ 70, s £ M n _i\{0}, L(r, is, DXn}v — 0, xn > 0, Bj(T,is,DXn)v\Xn=0 = Vj, j = l,...,n, sup \v(r, s, xn)\ < oo
(! 3)
xn>0
is uniquely solvable for any data (f>j, j = 1, . . . ,[i, or the Lopatinskii condition (cf. Chapter 1, Section 9) holds.
§ 1. Sobolev-Type Equations. Statement of Problems
239
Definition 1.1. We say that the mixed boundary- value problem (1.1) satisfies the Lopatinskii condition if there exists 70 > 0 such that for Re T ^ 70, s € M n -i\{0} the problem (1.3) is uniquely solvable. For an example we consider the classical statements of the boundaryvalue problems for the Sobolev equation and the internal wave equations in the Boussinesq approximation. For a distinguished variable we can take any Xk. For the sake of defmiteness, we distinguish #3. Example 1.1 (the Sobolev equation). For the Sobolev equation
we have a single condition at £3 = 0, i.e., // = 1. The first boundary-value problem takes the form
u\X3=Q = 0,
t> 0, x1 G M 2 ,
u| t =o = Dtu\t-Q -0,
xEMj.
For this problem, we have Bi(Dt, Dx] = I and ft — 0. The second boundary-value problem takes the form AD2u + uj2D2X3u = f ( t , x}, 2
(Dj. + w )A;3«U=o = 0, u\t=o = Dtu\t=o = 0,
t > 0, x3 > 0, t > 0, x' e M 2 >
xeMj.
For this problem, we have Bi(Dt,Dx) = (Df +u2)DX3 and ft = 1/2. Example 1.2 (the internal wave equation in the Boussinesq approximation) . For the internal wave equations
at £3 = 0 the following additional condition is imposed: // = 1. The first boundary-value problem takes the form 2
u + N2(D2Xi + D2Ju = f ( t , x } ,
u\X3=0 = 0,
t> 0, x3 > 0,
t > 0, x' e M 2 ,
u\t=0 = Dtu\t-Q = Q,
xGMj.
For this problem we have B\(Dt, Dx] = 7 and ft = 0.
240
4. Mixed Problems in a Quarter of Space
The second boundary-value problem takes the form 2
2
2
2
V I'D Ai-DL-'t?"/ 4-i^ J/V D }n — — 'f (v i ' r\/ > *V jL/ 27 1 4-' ^£2' D? Ap 3 wU3=o = 0 , < > 0, i' G M 2 ,
i "> D ' TI3 ~> 0'
For this problem, Bi(Dt}Dx] - D2DX3 and ft = 1/2. It is easy to verify that the above boundary-valued problems satisfy the Lopatinskii condition.
§ 2. Solvability of Mixed Problems for Simple Sobolev-Type Equations In this section, we formulate the main assertion concerning the solvability of the mixed problem for simple Sobolev-type equations in the quarter of the space (cf. Section 1). We start by formulating the solvability theorems for the problem (1.1) in the weight Sobolev spaces Wp'^ with exponential weight with respect to t. Formulations of these theorems are similar to those of the solvability theorems for the Cauchy problem for simple Sobolev-type equations given in Chapter 2, Section 4. The theorems formulated below imply that the mixed problem (1.1) does not necessarily unconditionally solvable; moreover, for the solvability of this problem some additional assumptions on the righthand side of the equation are, in general, required. These assumptions mean the orthogonality of the right-hand sides to polynomials. Then we present the solvability theorems in weight Sobolev spaces Wp'^a with exponential weight in time variable and power weight in spatial variables. By these theorems, we can significantly weaken the requirements on the right-hand side of the equation by a suitable choice of weight functions. We denote by W^^^^), I ^ 0, r — ( r ^ , . .. ,rn], I < p < oo, 7 > 0, the Sobolev space with exponential weight e~ 7 t , i.e., u ( t , x ) G Wp'^O^n+i) if e ~ ^ t u ( t ^ x ] G Wp' r (ffiLn+i)- We introduce the norm
n
and use the notation p' — pf(p— 1), |a| = X^ on, a m i n = min {QI, . . . , an}, f=i
Definition 2.1. By a solution to the problem (1.1) in ^ ' ^ ( M ) we mean a function u(t,x) & W^OR^-fi), r = U / a i > - • • > l/ a n), that has the
§ 2. Simple Sobolev-Type Equations. Solvability
241
weak derivatives D f u ( t , x ) E V7p°;^(IR++1), k = 1 , . . . ,/, and satisfies (1.1) almost everywhere. Below, we formulate assertions about the solvability of the mixed problem (1.1). Theorem 2.1. Let \a\/p' > 1. Then for any f ( t , x ) G Lp L P]7 (IR^; Li(M+)) ; 7 > 70 the problem (1.1) is uniquely solvable in the space Wp'^(lSin+i) and the solution u(t,x) satisfies the estimate \\u(t, x ) , WftKRj+JII + \\Dltu(t, x), <; c (||/(t, *), WlR+i)!! + || ||/(*, x), Li(M+)||, Lp, 7 (E+)||) ,
(2.1)
where c > 0 is a constant independent of f ( t , x } . Theorem 2.2. Let \a\/p' ^ 1, and let N be a natural number such that \a\/p' + Namin > 1 ^ \&\/p' + (N — l)a m j n - Then the mixed problem (1.1) has a unique solution u(t,x] E Wp'^^n+i); 7 > 7o for any function f(t,x) €^
I
(t, x)dx = Q,
\p\ = Q,...,N-l.
(2-2)
T/ie solution u(t,x) satisfies the estimate
+ II 11(1 + where the constant c > 0 is independent of f ( t , x } . In the case |a|/p' ^ 1, Theorem 2.2 yields sufficient conditions on the right-hand side f ( t , x ) under which the mixed problem (1.1) is solvable in the space ^^(MjJ"^). This result is similar to Theorem 4.2 in Chapter 2 about the solvability of the Cauchy problem in the space Wp'^(M.*+1), where, as we know, the additional orthogonality conditions on the right-hand side of the equation are necessary in the isotropic case. In the case of the mixed problem, the same question arises: If the orthogonality conditions (2.2) are necessary for solvability? This question is rather complicated in the general case. However, as is shown by the following theorem, these conditions are necessary for certain mixed problems, i.e., the orthogonality conditions (2.2)
242
4. Mixed Problems in a Quarter of Space
are close to the necessary solvability conditions for the mixed problems (1.1) in the class of functions. In Section 8, we prove the following assertion.
Theorem 2.3. Let F(s) be a contour in the complex plane surrounding all the roots of the equation L(r,is,i\) = 0, Re r ^ 7! > 70, s E M n _i\{0} ; relative to A. // \a\/p' + a min > 1 ^ \a\/p', l
r( s )
L(r \s U)
dX
(2 4)
* °'
'
then for the solvability of the mixed problem (1.1) in the space W^'^IR*^) 7 > 71 it is necessary that
I
f(t,x)dx-Q,
t>Q.
We illustrate Theorems 2.1-2.3 by an example of the second boundaryvalue problem for the internal wave equations (cf. Example 1.2)
+ N2(D^ +D2X2)u = f ( t , x ) ,
t >0, z3>0,
1
D*DX3u\X3=0 = 0, t > 0, x G M 2 , u\t=o - Dtu\t-Q = 0, x E l R j .
(2-5)
Since a = (0, 1/2, 1/2, 1/2) is the homogeneity vector of the symbol of the differential operator, for p > 3 Theorem 2.1 shows that the problem (2.5) is uniquely solvable in the space Wp'^(^+), r = ( 2 , 2 , 2 ) , 7 > 70 > 0, for any f ( t , x ) E L p , 7 (IRj + ) n L p / y ( R + ; Li(Mj)). If 3/2 < p ^ 3, then Theorem 2.2 guarantees the solvability of the problem (2.5) in the space W p 2 ^(]Rj + ) under the additional condition f f ( t , x ) d x = Q,
t>Q.
(2.6)
ffi+
On the other hand, for the problem (2.5) the condition (2.4) holds since for a sufficiently large 70 > 0 for Re r > 70, s G M2\{0) we can choose the contour independently of r\ moreover,
I(r,s)=
I
J
\ '. '
L(T,lS,lX)
d\=- I — J
T<
r 2X
§ 2. Simple Sobolev-Type Equations. Solvability
=
=
_
243
f
/ A2 +
!
l(
x
-
*
A + ( r , S ) - A - ( r , s ) J \X-X+(r,s)
}
X-X-(r,S)J
_
~~^'
Consequently, by Theorem 2.3, for p ^ 2 the orthogonality condition (2.6) is necessary for the solvability of the problem (2.5) in the space Wp'^(R^+). We introduce a wider scale of weight Sobolev spaces Wp'1^ C T ( M * + i ) , l < p < o o ,
7>0, l^cr^O,
as the completion of the set of functions in C00^^^) vanishing for large (It| + |ar|) by the norm (x))-°Dltu(t, x } ,
+ £ By the definition of the spaces W^^R*^) and W^£(R++i), we have P,7,a
n+l •
Further, LI 4 (Mn) denotes the space of summable functions y>(x) with the norm ||^(xj, L l i f (M+)|| - ||(1 + (x))-¥>(x)iLl(R+)\\. Definition 2.2. By a solution to the mixed problem (1.1) in the space ^^^(M+Ji) we mean a function u(t, x) 6 ^^^(Mjjj) that has the weak derivatives D^D^u(t,x) € Lp^^^), / ? a = l , fc = 1, . . . , / , and satisfies (1.1) almost everywhere. Theorem 2.4. Le^ |a| > 1 and \a\/p > cr > 1 - |a|/p'. Then the mixed problem (1.1) 25 uniquely solvable in the space W1'1^ CT(]R^1); 7 > 7o; /or any function f ( t , x) E Lp>-,(lSi++l) H LP,7(1R+; Li,_ f f (M+)), anrf ^e following estimate holds: \\ + \\D>u(t,x)^ ^ c (\\f(t, x), Lp^Rj+JH + || ||(1 + (^» a /(^ x), where c > 0 zs a constant independent of f ( t , x ) . We illustrate Theorem 2.4 by the second boundary- value problem for the internal wave equations (2.5). As was already mentioned, the problem (2.5) is unconditionally solvable in each space of the scale of weight Sobolev spaces for p > 3, whereas, in the case p
244
4. Mixed Problems in a Quarter of Space
problem the following additional orthogonality conditions (2.6) on the righthand side of the equation are necessary. On the other hand, since |a| = 3/2, the problem (2.5) is unconditionally solvable in the space Wp'^ ff(l&^+} for any p e (l,oo), a 6 (1 - 3/(2p / ), 3/(2p)). Theorems 2.1, 2.2, 2.4 can be proved on the basis of the results due to G. V. Demidenko [9, 10] about the solvability of the boundary- value problems in the half-space for quasielliptic equations. However, we used here other arguments, which allows us to generalize the developed technique to more complicated problems for pseudoparabolic equations (cf. Section 9). § 3. Approximate Solutions to Simple Sobolev-Type Equations In this section, we construct a sequence of approximate solutions to the mixed boundary- value problems (1.1) satisfying the conditions formulated in Section 1. In Sections 4-6, we derive estimates for this sequence in the weight norms which imply the existence theorems for the problem (1.1). Let the right-hand side f ( t , x ) of equation in (1.1) belong to Co>0(]R*+1). Consider the mixed boundary-value problem for the differential equations with parameter s 6 M n _ i which is obtained by formal application of the Fourier operator with respect to "tangent" variables x':
i-i L 0 (zs, DXn}D[v + ]T L,_ f c (zs, DXn}Dktv = f ( t , s, xn],
t > 0, xn > 0,
k =0
Bj(Dt,is,DXn)v\Xn=o 0*4=0 = 0,
= 0,
j = 1 , . . . ,/i, t > 0 ,
*= (),...,/-I,
(3.1)
zn>0.
We apply the integral Laplace operator with respect to / to the problem (3.1): L0(is,DXn}rlw + ^Li-k(is,DXn}rkw = /(r, s,z n ), k=o Bj(r,is,DXn}w\Xn=Q -Q, j - 1,. . . ,/i.
xn > 0, ^-2>
We consider the problem (3.2) for Re r ^> 70, s G M n -i\{0) and additional condition sup \W(T, s, xn)\ < co.
(3-3)
§ 3. Simple Sobolev-Type Equations. Approximate Solutions
245
For these values of the parameters (r,s] the problem (3.2), (3.3) satisfies the Lopatinskii condition (cf. Section 1) and, consequently, has a unique solution. We indicate explicit formulas for solutions using the corresponding formulas in Chapter 1, Section 9. Recall that the equation L ( r , i s , i X ) = 0, Re r ^ 70, s G R n _i\{0}, has no real roots relative to A. We denote by X£(T,S), k — 1 , . . . ,// the roots with positive imaginary part and by A~(r, s), i = 1 , . . . , m — //, m — l/an, the roots with negative imaginary part. We set M+(T,S,X)=f[(X-X+(T,S}).
k=i +
We denote by F (r, s) a contour in the complex plane surrounding all the roots X ^ ( r , s) and by r~(r, s) a contour surrounding all the roots X~(r, s). As in Chapter 1, Section 9, we define the contour integrals J+(T)S)Xn) =
J_
/
ex
2iTT
J
i>( Tj IS) ZAJ
l
7 (r s,x s r )\ -- -— J.(T, n LT\
J-(T S,Xn) = —
fi
J
f
P(**"A) dX, ex
(3.4)
P( z ' x " A ) dx Lt\T) t a , l/\)
6XP
,-\^
( 5) n '
'
^ n A ) " '- - ^ ^ -^
^ ^
where j = 1, . . . , / / , s 6 M n _i\{0} and A/j (r, s, A) are polynomials in A such that the following equalities hold: _L
f
2;ri
7
flfcfo'g.'AMfcg.A)
->
M+(r,S,A)
efc j
'
r+(r,s)
* is the Kronecker symbol. To satisfy these equalities, we set
1=1
where 6 l J (r, s) are the entries of the inverse matrix 6 M (r, s) + 6 fei2 (r, s)A + . . . + & f c ) M (r, , is, iA) mod (M+(r, s, A)),
246
4. Mixed Problems in a Quarter of Space
and the polynomials M^"_,.(r, s, A) are determined by the formulas
n
p
M+(T \\I — \^ n-(r c\\^~'i . 1V1 I\ T. ' o A / — /7 ^ CIj I\ T.J o /I A '
M+(-r c \\ — \ ^ n • ( T *\\P~i, JW», p I\ T,> S,' A >I — / 7 j Cl? IvT,' o /I A '
i=0
i=0
where p = 0 , . . . ,/j — 1. By the definition (3.4) and (3.5), the contour integrals J+(r, s, x n ) and J_(r, s, x n ) are analytic functions of r for Re r > 70- The integrals J j ( r , s , x n } are analytic functions of r for Re r > 70 since, in view of the preliminary Weierstrass theorem (cf., for example, B. V. Shabat [1]) the coefficients a;(r, s) are analytic. Consequently, the Lopatinskii condition implies that 6 tJ '(r, s), are analytic functions, i.e., the integrand in (3.6) is an analytic function with respect to T for Re r > 70Since the contour F + (r, s) can be locally chosen independently of (r, s ) , we conclude that JJ(T, s, xn) is an analytic function. We introduce the functions x
_
n
/
J+(r,s,xn-yn)f(T,s,yn)dyn,
(3.7)
~ J_(r, s,xn - yn)f(r, s,yn)dyn,
(3.8)
o CO
/
CO
U! (T, s,xn) = JJ(T, s,xn} If JJ(T, s, y n )/(r, s, yn) dyn, J
(3.9)
J
o
where j =1,... ,fjt and I j ( T , s , y n ) = -Bj(r, is, DZn) J_(r, s, zn - y n )| 2 n = 0 Since the contour integrals (3.4)-(3.6) are analytic bounded functions for Re T ;> 70, we apply Theorem 5.2 in Chapter 1 to the functions U+(T, s, xn), u}~(r,s,xn), u > i ( T , s , x n } . Consequently, the integrals (t, s, xn] = ( 2 7 T ) - 1 2
e + w + t o + 7, s, x n ) drj,
(3.10)
-(^ + 7 , s , x n ) ^ ,
(3.11)
(iri + - f , s , x n ) d r i ,
(3.12)
§ 3. Simple Sobolev-Type Equations. Approximate Solutions
247
where j = 1 , . . . , / / , are independent of 7 ^ 7 0 - Furthermore, the function v(t,s,xn) = v+(t,s,xn] + v~(t,s,xn) + V~ is a solution to the mixed problem (3.1) for s 6 M n -i\{0). We proceed by constructing approximate solutions to the mixed problem (1.1). We note that the solution to the problem (1.1) can be formally obtained by applying the inverse Fourier operator with respect to s to the function u ( t , s , £ n ) . However, the contour integrals (3.4)-(3.6) (consequently, the functions (3.10)-(3.12)) have, in general, nonintegrable singularities at s = 0. Therefore, as in the case of constructing approximate solutions to the Cauchy problem for equations and systems of non-Cauchy-Kovalevskaya type, in order to regularize the inverse Fourier operator, we use the integral representation of functions p ( x ' ) 6 L p (M n _i) in Chapter 1, Section 6:
p ( x ' ) — lim(27r) h-+o
f _i 'i-
/ v~\a ' J h
f/
J
x>
f
/ J
exp(z
~ ;y'—s)G(sWy') dsdy1
va
dv,
ln_iln_i
(3.13) where G(s) = 2N(s)2N exp(-(s)2N),
(s)2 = ^s^0', \a'\ = *£>«•• i=l
As in
i=l
the case of constructing approximate solutions to the Cauchy problem, we introduce the functions k 1 n
u+(t,x) = (2 7 r)( ~ )/
2
/ v~l k f
II
(3.14)
Kn_i
l/k l n 2 ii~ I(ti, A -r\I — — IOir\( llr Z7TI ~ H
f eix>sG(sva')v+(t,s,xn)dsdv,
t,~l V
J
f
jI J
ix p G
is
i
a rLi(sm « r \fidf1ii J l o t / \i>~ ) I) (i \^L^OjJi,jij(JLc>LL(J^
H ^"lJ ^ O .1I O
ln-l
l/k
k ,.3(4
I/ J l/k
V -1
II J
ix s 01 f, sn \r1 C ' C( vjl o V '\ij i u (i l t , o Q, X |T^ I C lt>Uc/,
C\ ^ G . - L}fi\ U^
ln-1
where j = I , . .. ,n and v+(t,s, xn), v~(t,s,xn], u j (i,s, xn) are determined in (3.10)-(3.12). Taking into account that the contour integrals (3.4)-(3.6) have only power-type singularities at s = 0 and using (3.7)-(3.12), we can indicate a number NQ such that for N ^. N0 the functions u£(t, x), u^ ( t , x ) ,
248
4. Mixed Problems in a Quarter of Space
u3k ( t , x) are infinitely differentiable and summable with respect to x with any power. We set N ^> NQ. We consider the functions
n uk(t,x) = u+(t,x) + u-(t,x) +
ui(t,x).
(3.17)
By the above constructions, we have L(Dt,Dx)uk(t,x} = f k ( t , x ] for t > 0, x 6 M+; 5,(A,ArM*X,0) = 0, j = 1 , . . . ,/;, for < > 0, x' e M n _ i ; Dju f c (0,ar) = 0, i = 0, . . . , / - 1, for x <E M+, where
l/k
By the integral representation (3.13), we have
fk(t, x) - f(t, x}, L^ORJ+JH -> o, k -> oo. Consequently, the function (3.17) can be regarded as an approximate solution to the mixed problem (1.1). In Sections 4-6, we establish estimates for the functions (3.10)-(3.12) which imply the existence of a solution to the mixed problem (1.1) under the assumptions of the corresponding theorems in Section 2.
§ 4. Properties of the Contour Integrals In this section, we establish some properties of the contour integrals (3.4)-(3.6) which are useful in the proof of estimates for approximate solutions to the mixed problem (1.1). Lemma 4.1. For any c > 0 the following relations hold: J+(r,s,xn) = c1~0lnJ+(r, ca s,c~anxn), J^(r,s,xn} = c 1 ~ a "J_(r, ca s,c~anxn), Jj(r,s,xn) = cp>' J j ( r , ca s,c~anxn),
j = 1, . . . ,/*.
PROOF. The assertion is true in view of the homogeneity conditions on the symbols L(r,ca is,cani\) = cL(r,is,i\) and Bj(T,ca is,cani\) = c^BTisi\. D
§ 4. Properties of the Contour Integrals
249
Lemma 4.2. For xn > 0, Re r ^ 70, s € M n _i\{0} for any k, q, ft = (/21; . . . ,/? n _i) the following estimates hold:
n-l
where 77 = Im r, /j is the power of the polynomial Bj(ir/, is, iX) in r/ and c, 8 > 0 are constants. PROOF. By the homogeneity condition of the symbol L(r, is, iX) relative to the vector a = (0, a', an), we can write m-l
L(T,is,i\) = lQ(T,is)(i\)m +
\k lm-k(r, is)(i\), m = l/an,
k=0
where / m _fc(r, is) are homogeneous polynomials relative to the vector (0, a'} with homogeneity exponent 1 — kan, i.e., / m _fc(r, ca is) = cl~kanlm-k(r,is), c > 0. Therefore, the roots \k(r, s) of the equation L(r,is,i\) = 0, Re r ^ 70, are homogeneous relative to the vector (0, a') with homogeneity exponent an, i.e., Xk(r, ca s) — c a n Afc(r, s), c > 0. Consequently, there exist constants A > S > 0 such that the following estimates hold: 26(s)«« ^ |Im X k ( r , s ) \ ^ \Xk(r, s)\ <: A< S ) a «,
k=l,...,m,
which imply that the groups of roots A^(r, s), k = 1 , . . . , //, and A~(r, s), i — 1 , . . . ,m — /j,, are at distance not less than 4J(s) a n . Therefore, in a small neighborhood of any point SQ £ M n _i\{0} the contour integrals J+(T, s, xn), J-(r,s,xn), Jj(r,s,xn) can be taken along contours independent of (r, s). Consequently, one can differentiate only integrands in order to compute the derivatives. Respectively all estimates are established by the same method. For example, we estimate Dkn J+(T, s, xn). By Lemma 4.1,
27r
4. Mixed Problems in a Quarter of Space
250
where F + is the boundary of the domain G+ = {X £ C : |A| < 2A, Im A > 8}. Therefore, \DkXnJ+(r,s,Xn)\
max
Thus, the first estimate for q = \(3\ = 0 is proved. The case q ^ 0, |/?| 7^ 0, is considered in the same way. D Lemma 4. 3. For Re r ^ 7o; s G M n _i\{0) the following identities hold: ,s,xn] - J-(r,s,xn))\Xn=Q = 8^n_l/am(r), where S^n_1 is the Kronecker symbol and am(r) •= QQTI + QITI~I + . . . + ai is the coefficient at the higher-order term (i\)m of the polynomial L(r, is, z'A). PROOF. Introduce the notation T(s) = {A G C : |A| = 2A(s) a "}. Then 1
(i\}k
f
ZTT J
L(r, is, i\)
d\.
Since the contour F(s) surrounds the roots of the equation L(T, is, iX) — 0, the value of the integrals /A;(T, s) remains unchanged under the replacement the contour with a circle surrounding this contour. Consequently, if k ^ m - 2, for any s E M n -i\{0}
L(r, is, iX)
dX
pk+1 max|L(r, is,i
as /? —> oo, i.e., //c(r, s) = 0. In the case k = m — I , we have ~l
r( s )
am(r}(iX)m+
0
§ 4. Properties of the Contour Integrals
1 f dX a m (r)27rz J X
251
*^-v aq(r, s) f (iX)q 1 4^ a m (r)2;r J L(r, is,L\)
am(r)'
Lemma 4.4. For Re r ^ JQ, s G M n _i\{0} the following identity holds: oo f
-«'A*n
*
£(T, 2s! *A)
7 — oo
(4.1) where 0(xn) is the Heaviside function. PROOF. Let (xn),
sup \V(T, s, xn}\ < oo.
The equation L(r, is, iX) = 0 has no real roots relative to A. As was mentioned in Chapter 1, Section 9, this problem is uniquely solvable, and the solution can be represented in the form 00
v(r, s, xn] -
I (0(xn - yn)J+(r, s, xn ~ yn] — oo
By Lemma 4.2, it is possible to apply the Fourier operator to this function. Using the formula of the Fourier transform of convolution, we find 00
I
On the other hand, L(r, is, iX}v(r, s, A) = £>(A). Since <£>(A) is arbitrary, we obtain (4.1). D Lemma 4. 5. For Re r ^ 70, s G K n _i\{0} the following identity holds: , ..v Lr( r , i s , i X )
252
4. Mixed Problems in a Quarter of Space
PROOF. The assertion is directly obtained from (4.1) by applying the inverse Fourier operator. D Lemma 4.6. Let < p ( x ' , x n ) € Li(M+), and let fi(s,xn] be the partial Fourier transform of the function ( p ( x ' , x n ) with respect to x' . Then for Re r ^ JQ, s 6 M n _!\{0} the following identities hold: 00
-Jj(T,s,xn}
r
I
Ij(T,s,yn}fi(s,yn)dyn
J
0 oo
r
= I D2n(Jj(r,s,xn
oo
r
+ zn) I Ij(T,s,yn + z n ) f i ( s , y n )
dyn)dzn,
PROOF. The assertion directly follows from the Newton-Leibniz formula and Lemma 4. 2. D Lemma 4.7. Let v = (i/,i/ n ) ; vet — v'a' -f vn&n ^ 1- For 7 ^ 70 the functions oo
/i+, g («7 + 7,0 = to + 7)'(^ o 0
oo
x / elt»x"Bj(iri + ~ f , i s , D y n ) J - ( i r ) + j,is,yn - xn)\yn=0 dxn, 0
9 = 0 , . . . , / , j = I , . . . ,n, are multipliers in the space Lp(Rn+i). PROOF. By the Lizorkin theorem on multipliers (Chapter 1, Section 4), it suffices to show that for any vector x — (*to,xi,... ,-^n) — (>*0)^)> where Xi = 0 or Xi — 1, for rj ^ 0, £/ ^ 0, / = 1 , . . . , n, 7 ^ 70 the following
§ 5. Estimates for Approximate Solutions. Nonhomogeneous Equations
253
estimates hold: WFDDu+
+ ^W <: c,
\Trt"DD^(il + 1,t)\ ^ c,
where c > 0 is a constant independent of (77,^). The proof of these estimates uses the estimates from Lemma 4.2 in the same way as in the proof of Theorem 4.4 in Chapter 1. D
§ 5. Estimates for Approximate Solutions to Nonhomogeneous Simple Sobolev-Type Equations In Section 3, we constructed the sequence {uk(t,x}} of approximate solutions to the mixed boundary-value problem (1.1). The functions U k ( t , x ) n have the form (3.17) U k ( t , x ) = u^(t, x) + u^ (t, x] + ^ ^(2, x) or, in opj=i + P j ^ f ( t t x ) + E P3J(t,x), f ( t , x ) £ j=i ^(Mjji), where the operators P+, P~ , P3k are denned by formulas (3.7)(3.12), (3.14)-(3.16) as follows:
erator form, uk(t,x) = P^f(t,x)
k l n
2
,x) = (27r)( - V
f v-1 j
eix'sG(sva')v+(t,s,xn)dsdv,
(5.1)
f eix'sG(Sva')v-(t,S,xn)dSdv,
(5.2)
where oo
f
x \
o
,x) = (27r}^-n^2
f v-1 l/fc
where 00
1 2 /e^' U-(i,S,xn)-(27r)- /
254
4. Mixed Problems in a Quarter of Space
~ + l,s,yn)dyn \\ dn, _(ir} + ~ f , s , x n - yn)f(ir]
K
J
1 n
P k f ( t , x ) = (2 7 r)( - )/
2
t v~l l/k
t eix/sG(sva')vJ(t,s,xn)dsdv,
(5.3)
ln-l
where oo 1 2 ij(t s ^ r*L n \t — (9'7r}~ '' U {L ) d \&i\)
f
J
i t II p( s j J^Yl T }) C '?+'T) JT-dn-^-v n\lll ^ J j O
X
o
Recall that
To-
It is convenient to regard this function as the partial Fourier transform with respect to ( t , x ' ) of the function e ~ ' y t f ( t , x' , xn) extended by zero to t < 0, xn < 0. In this section, we study properties of the operators P^~ and P^ . Operators P3k will be treated in the following section. Lemma 5.1. For (3a = 1 the following estimates hold:
ip^ + ^rmM), w^^
(5.4)
7 ;> 70, where c > 0 zs independent of k and f ( t , x ) . Furthermore, ll -+ 0 (5.5) 05 ki^k-2 —>• CO.
PROOF. We first consider the case /3 = (/3',0). By the definition (5.1),
§ 5. Estimates for Approximate Solutions. Nonhomogeneous Equations
255
for 7 ^ 70 we have 0-lt
n/3 p+ f ( 4 ~.\ = — O-\-n/2 (27T)-"/ 2 jI ,,-1 v-1
mn_i
i/k 00
ix s e^x'irtf^a'\f-\p' ' G(sva>}(is
I/
X
I elT1t I J + (z77 + 7,s,z n - yn}J(ir) 0
-oo
Using the Heaviside function 0(xn), we can write the last expression in the form !,z) = (27r)- n / 2 f v~l
eix's G(sva'}(isf
f »„-!
l/k
OO
00
/
/ eir}t9(xn — yn] J+(ir) + 7, s, xn — yn)f(i'n + 7> *> yn)dyn dr) \dsdv.
—oo —oo
Using the formula 00
00
/ K(xn-yn)p(yn)dyn=
I
elXn^K(£,n}$(£n]
we find v~l
[
eix'sG(sva>)
l/fc OO
OO
x — 00 —OO
where oo
'
4
^- J+(:i7 + 7, «, y«) dyn
and /7(^,s,Cn) is the Fourier transform of the function e~~*tf(t, x' , xn) extended by zero for t < 0, xn < 0.
256
4. Mixed Problems in a Quarter of Space
By Lemma 4.7, the functions (irj + 7)™fj,~t (irj -f 7, s,£ n ), m = 0 , . . . , /, are multipliers. Consequently,
I
l/k
Using the identity
i/k we obtain the estimate \\nPp+f(t U l x^ LT \\ x^k J ( > ) >
(]$>++ \\\ <•
P^(M-n+l)\\
( /•"
c
^ ~l
+
where the constant GI is independent of k and f ( t , x ) . Making similar transformations, for 7 ^ 70 we find k
= (27r)-"/
2
l"^1 J
i/k
x
oo
oo
' Jf 1jei -oo —oo
where
By Lemma 4.7, the functions (Z'T; + 7)m/^g (ZT? + 7, s,^ n ), m = 0, multipliers. Arguing as above, we obtain the estimate ++
. , /, are
§ 5. Estimates for Approximate Solutions. Nonhomogeneous Equations
257
where the constant 02 is independent of k and f ( t , x ) . The estimates obtained for /? = (/?', 0) imply the inequality (5.4). Consider the case j3 = (/?',/?„), [3n > 0. For 7 ^ 70 we have
I/A: oo X
J
f
Kn-i
xn
f
ir)t
J
-oo
i
i
n
n
~ •
A J
0
k
+ (27r)-
n/2
/V1 /" l/k
OO
eixl'G(8val)(isf
ffiU-!
00
~ • + 7, s, yn)dyn drj \ } ds dv1\. x fI f/ e"•7* f/ «/_ (irj + 7, s, o;n — yn)f(ir) \ J
J
J
Xn
— OO
Adding and subtracting the expression A:
v_n/2
f
-i
y
l/fc
oo
f
y
ix's
N/3'
a'
In-!
a;n
vy y
X ^
f
il*
f
0
"-oo
we find
l/k oo
xn
/ ^ I (J+(ir) + l>s>xn -Vn) ~ J-(irj + ~f,s,xn -yn}) 0
-oo
l//c oo
X
IR n -i
oo
( J// *e . /J/• DP J-(ir} + i,s,x ~y }f(irj~ t7?t
n
n
-oo
v-1 f eix'sG J
0
n
+ 'y,s,yn)dyn
J
258
4. Mixed Problems in a Quarter of Space
Using Lemma 4. 3, for /3n < m = l/an we find k
l/k oo
(
xn
f • f ~ \ / eir]t I D^J+(ir] + - f , s , x n - yn)f(irj + - / , s , y n ) d y n d r j \dsdv \ J J /
~l
t elx'sG(sva'}(is
,
c
v^oX^-^'
J
Kn-l
l/k 00
X
OO
/ /• . . f I I
I I
P C-
'
\ J
J
— 00
~.
g
I /^ n 7 -*-^ 3* t/
I f
( 7 T\ ™\^ '"V C T* V / "^ i ) ) n
n
N
\
?/ I * I 1 T\ -' i • *V Q ?/ \ ft 11 HT) \ H <» /^r?) L/M / J V / "^ i ) J i/Tl / tyTl *-*''/ I L t o U i l / .
J
In
(5.6)
For /?„ = m, by Lemma 4.3, we have A:
l/k OO
(
f
Xn
•
f
~
\
~
\
f / e Z7?f / D^n J + (zr7 + 7,s,x n - yn)f(iri + i,s,yn)dyndr)\dsdv \ j j / o k
( v~l
i
eix'sG(sva>}
l/k oo
oo
7 (777 -4- 'y s x
1
( v~l
I
J
J
l/k
En_!
w ) / (27? "4~ *y s ij tdij dfi \ ds dv
J
e**'sG(sva')
y — oo
where a m (zr? + 7) = OOT' + air'" 1 + . . . + «/, Q:Q 7^ 0. From the above formulas we see that for each term on the right-hand side we can obtain
§ 5. Estimates for Approximate Solutions. Nonhomogeneous Equations
259
estimates similar to the estimates in the case (3n = 0. By Lemma 4.7, the functions (ir, + 7, s,£ n ) - (l^'^isf
j e-^D^J^ir, + 7,*, yn)dyn, o 0
and the functions
A; = 0 , 1 , . . . , / , are multipliers. Arguing as above, we obtain the inequality (5.4). Similarly, we obtain (5.5). D Arguing as above, we can prove the following assertion. Lemma 5.2. For /3a — I , I ^ m ^ 1 the following estimate holds:
where 7 ^ 70 anrf c > 0 zs a constant independent Furthermore,
of k and f ( t , x ) .
Lemma 5.3. Lef |a| > 1, \a\/p > 1 — |a|/p'. T/ien /or /?a < 1 following estimate holds:
r 1 - ^ / ( t l x ) > Li(M+)||, L P , 7 (E+)||), where 7 ^ 70 a^c? c > 0 is a constant independent Furthermore,
(5.7)
of k and f ( t , x ) .
}-a(l-fta}(D^(P^Pk~}f(t^}
(5.8)
260
4. Mixed Problems in a Quarter of Space
PROOF. Since 0a < 1, we have 0 ^ /3n < m = l/an. Consequently, for 7 ^ 7 0 , by (5.6), we have e^ D% (P+ + P k ) f ( t , x) = F+(t, x) + F^(t, x), where eix'sG(sva'}(isf l/k X
00
• irjt — oo
0
v~l
s a a ei ixx's'G(sv ''}(is
l/k 00
— 00
00
X,
Using the Minkowski inequality, we find
(5.9) where oo
ik
= (2;r)- n / 2 I v~l
^
f
ix s
a
e ' G(sv '}(isf'
J
J
( f e1'"' \ J
l/fc
7, s, y n )dy n dt; Jcls, /
i /2,fc - (27T)-"/ 2 j
V~l
1//C
ds, LP(E+|1) dv,
D1 J- (irj + 7, s, x n - yn)f(ir) + 7, «, yn /
§ 5. Estimates for Approximate Solutions. Nonhomogeneous Equations
261
I3,k = (27r)-n/2 f v-1 1i
I
00 f
ir>t
/
Xn f
/ e ( I D^ J \J 0
-oo oo r
+ / D^J-(ir] + ~f,s,
dv.
J
Recall that we consider a function f ( t , x ) in the space Q^O^+i) and preserve the notation f ( t , x] for the function extended by zero to the entire space M n +i. We consider the first term I\tk on the right-hand of (5.9). Using the Heaviside function and properties of the Fourier transform, we write I\^ in the form
i / v~l
n 2
= (27r)- / oo
— oo
f
Ilk oo
—oo
ds,
dv
I
i/fc oo
oo
/
/
/
itrj + ixnsn)G(sva )(z'
, s,yn)dyn
262
4. Mixed Problems in a Quarter of Space
where
exp(-i(y0ri + y'
By Lemma 4.7, the function + 7, «, yn
is a multiplier in the space L p (M n following estimate holds:
Since a(l — (3a) ^> 0, for I\^ the
where the constant c > 0 is independent of A; and f ( t , x ) . By the Young inequality, we have i
h,k ^ c j v-1
j
e^sG(sva'}(sfa-1
l/k
++ or, making the change of variables,
i/fc Since @a < 1, the following estimate holds: c
1
ds, L i ( M n _ i ) dv
§ 5. Estimates for Approximate Solutions. Nonhomogeneous Equations
263
By the definition of the kernel G(£), we have /ilfc^c1||/(/>ar)>Lp,7(M++1)||>
(5.10)
where c\ > 0 is a constant independent of k and f ( t , x ) . We consider the second term htk °n the right-hand of (5.9). Using the Heaviside function and properties of the Fourier transform, we find i n
fc))-"7*1-**) I
J2|fc = (27r)- / OO
eix'sG(sva')(iSf'
00
iTlt — CO
—OO
, L p (M n + i) dv
= (27T)-<»
1/k
•// 00
00
ix's + itr) + ixnsn}G(sva
) —C
0
/ X
By Lemma 4.7 the function o l-/J.«.
/" C-"-"«Dj : J_(
is a multiplier in the space L p (lR n +i). Therefore, as in the case of the first term I\tk, we obtain the estimate (5.11) where 02 > 0 is a constant independent of k and f ( t , x ) .
264
4. Mixed Problems in a Quarter of Space
We consider the third term I^tk on the right-hand of (5.9). To simplify the exposition, we set (3n = 0. Using properties of the Fourier transform and Lemma 4. 4, we write 1$^ in the form oo
f f J
J
-oo -oo K n _ i
yn) + 6(yn - xn}j^\
x (0(xn - yn)J+(ir)
7, s, xn - yn)}
dv
= ( 27r
-< n + 1 >/ 2
x G(sva
1
s«))
fi(r),s,sn)dsdsn
By the definition of the kernel G(s) and the condition f ( t , y ) € Co°(E n+ i), we can exchange the integral under the sign of norm. Using Lemma 5.6 in Chapter 2, Section 5 and properties of the Fourier transform, we can write /3/e as follows: s+itrj + ixns — oo —oo E n _ i
x /y(77,s, sn)dsdsn dr), k n 2
= (27r)- /
OO
CT 1
a
( -^ ) f J
OO
f J
f
eix's+iXnSn
J
— OO —CO l K n _ j
x 0(< - r)e
dv,
r, s , s n ) # ( r , s,s n )dsc/s n rfr, .
where g(r,s,sn) is the Fourier transform of the function f ( r , x ' , x n ) with respect to (x1 , x n ): , s, zsn
d\
§ 5. Estimates for Approximate Solutions. Nonhomogeneous Equations
265
and F is a contour in the complex plane surrounding the roots of the equation L(iX,is,isn) = 0. Recall that we can take this contour independent of (s,s n ), and the contour integral satisfies the following estimate for £ = (s,sn) 6 R n \{0} (cf. Chapter 2, Section 5): -^e-6t,
6>0,
7 ^ 7o, i >0
(5.12)
a = const,
(5.13)
Using the inequality (x - y}(l + (x))- 1
and the Young inequality, for Iztk we obtain the estimate
dv c
f v~l
We consider the norm
B(v} =
Making the change of variables ("fc = write B(v) in the form
= Xf,v
ak
, k = 1, . . . , n, we
266
4. Mixed Problems in a Quarter of Space
Since a\/p > cr(l — IOCQ), from the definition of the kernel G(s) and the inequality (5.12), arguing in the same way as in the proof of 6.3 in Chapter 2, we derive the estimate B ( v ) ^ CQV-\^\/p'-o(i-fta)-f5a+i ^ where the constant CQ > 0 is independent of v. Using this estimate, we find
p i 7 (M+)||.
(5.14)
Taking into account the condition a > 1 — \ot /p' , we find
where the constant 03 > 0 is independent of k and /(i, y). The case f3n / 0 is treated by the same method. From the estimates (5.10), (5.11), (5.14) we obtain the inequality (5.7). We establish (5.8) in a similar way. D Arguing as above, we can prove the following assertions. Lemma 5.4. Let a| > I , \a\/p > cr > 1 — \a\/p'. estimate holds:
Then the following
\f(t,x), / where 7 ^ 70 and c > 0 is a constant independent of k and f ( t , x ) . Furthermore,
as ki , k-2 —> cxo.
Consider the function x(n] € C°°(lRj) such that 0 ^ x(n) ^ 1, x(n) - 1 for 0 ^ 77 ^ 1 and x(r?) = 0 for 77 ^ 2.
Lemma 5.5. Let the assumptions of Theorem 2.4 hold. Then the following convergence takes place:
+ P k ~ ) f ( i , x ) , W;;it7(R
011 -> 0,
p ^ co
§ 6. Estimates for Approximate Solutions. Homogeneous Equations
267
Furthermore, for any q = 1,. . . , / and 0a = I x)*/p2) H -> 0,
p -> oo.
The proof is similar to that of Lemma 6.5 in Chapter 2, but Lemmas 5.1-5.4 should be used here. D
§ 6. Estimates for Approximate Solutions to Homogeneous Simple Sobolev-Type Equations In this section, we study some properties of integral operators PJk defined by formulas (5.3). Lemma 6.1. For vo. — 1 the following estimate holds: \\D»PJJ(t,x}, Lp.^R+i)!! ^ ^\\f(t,x), WM+^II,
7^70,
(6.1)
where j — 1, . . . , // and the constant c > 0 is independent of k and f ( t , x ) . Furthermore, \\->Q}
kltk2^oo.
(6.2)
PROOF. By the definition (5.3), for 7 ^ 70 we have k
= ('27r)-»/i /V1 / J
eix'sG(sva'}(isy'
J
l/k
CO
x /f Ij(ir) + i , s , y n ) f ( i r ) +
\ ~f,s,yn)dyndr)\dsdv.
•J
/
o
Taking into account Lemma 4.6, we can write this expression in the form k f
e~~" ~" "
9
oo
l
v~ I
! e^x'
268
4. Mixed Problems in a Quarter of Space
j r ] + i,s,xn + z \
0
0 s
x f(if] + 7, , yn}dyn k n 2
-(27r)- /
oo l
fv~ l/k
dzn ds drj dv
I
I
e^x>s+tll)G(sva>)(isf
-00ln_!
00
r(
x / ( D^Jj(irj + i,s,xn + z 0
x
0
(ir] + j}s,yn)dyn
dzndsdr)dv
= 3>lk(t,x) + 3 > t ( t , x ) .
(6.3)
We consider the first term lk(t,x) on the right-hand of (6.3). Since xn > 0, we can write
l/k
,s,yn + zn)f(irj + - f , s , y n ) d y n
dzn \dsdr] dv.
Using the formula of the Fourier transform of convolution, we can write <&l(t, x) in the form
— oo fe
l/kJ
§ 6. Estimates for Approximate Solutions. Homogeneous Equations
269
where ^irj + 7, 8, xn)dxn.
7, 0 - (i
By Lemma 4.7, the functions /^>("7 + 7 > £ ) Consequently, if we prove the estimate
j j
are
multipliers in Lp(Rn+i).
s+t
e^'
^\e(xn) j (isf (Sy
°O K n - l
~ °°
x (ir) + 7)~ /J Ij (irj + 7, s, yn + z n )
(
fe /• / t;~1G(st;Q')^ J
(6.4)
where the constant c > 0 is independent of k > 0 and f ( t , x ) , then for the function $^(t,x) the following inequality is valid:
where the constant c\ > 0 is independent of k and f ( t , x). We denote by Kj^ the norm on the left-hand side of the inequality (6.4). It is obvious that
k X
Jj(«7 + 7 i * > y n +
a:
x f(ir) + 7, s, yn)dyn
-OoKn-1
(
f -1
n)( V \ J l/k
ds drj, Lp(Rn+i)
270
4. Mixed Problems in a Quarter of Space
x (
k f
1
/ v~1G(sv i/k
,s,yn)dyn \dsdrj, Lp(Rn+1)
By the formula of the Fourier transform of convolution, this expression can be written in the form
x I I v
1
G(sva )dv j [if] 4 7)
fi(n, s, sn}dsdsn dry, L p (lR n + ^
i/*
where is
7, 0 -
dZn,
£ = ( s , s n ) . By Lemma 4.7, the functions L p (IR n + i). Therefore,
4-7,0
Oi(x'-y')i
i/k x
are
multipliers in
''G(sva
ffi,
/(^, 2/, xn)ds dy' dv,
where the constant c > 0 is independent of k and f ( t , x ) . Consequently, by the definition of the kernel G ( s ) , as in the proof of Lemma 5.1, we obtain the inequality (6.4). From the inequality (6.4) we obtain the estimate (6.5). Similarly, for the second term $%.(t} x) on the right-hand side of (6.3) we can establish the estimate \ y
where the constant c-^ > 0 is independent of k and /(t, x). From the estimates (6.5) and (6.6) we immediately obtain the inequality (6.1). Similarly, the convergence (6.2) takes place. D Arguing as above, we can prove the following assertion.
271
§ 6. Estimates for Approximate Solutions. Homogeneous Equations Lemma 6.2. For i/a = I , I ^> m ~>> I , the following estimate holds: n +l
\ y_m
'
>
P,1
n +1
^
where j = 1 , . . . , // ana7 Me constant c > Q is independent of k and /(/, x} Furthermore, 0.
oo.
'. T/ien /or ^a < 1
Lemma 6.3. Let \a\ > I, \a\/p > 1 following estimates hold:
(6.7)
where 7 ^ 7o; j = 1, . . . , / / antf Me constant c > 0 25 independent of k and f ( t , x ) . Furthermore, ||(1 + (x))-^ 1 — )(D^;/(t, x) - D^ a /(t, x)),
0 (6.8)
OO.
PROOF. Taking into account the definition (5.3), for 7 ^ 70 we have
f
eix'sG(sva'}(isY'
l/k
M
e*ltL%l
x f(irJ + 7, «, yn)c?yn dnjds, / ei:r'sG(s7;
-^) TCP
lKn_i 00
f
I Ij(irj + 7, s, yn] J
272
4. Mixed Problems in a Quarter of Space
7:s, Vn}dyn d r j d s , Lp(R
(6.9)
We estimate each of the terms A\tk and A^tk on the right-hand of (6.9), We first consider A\tk- Applying Lemma 4.6, we find 1 n 2
Ai>k ^ (27r)- /
/ v~l 1/fc
00
,yn)dyn \dzndsdr],
/
dv
l/k
oo
/f (lD^ */
\
dv
(6.10) Both terms ^4} fc and A\ k on the right-hand of (6.10) are estimated by the same scheme. For definiteness, we consider ^4} k. Since cr(l — voi) ^ 0, we have (27r - n / 2 l/k
-ooln_!
dv.
Using the formula of the Fourier transform of convolution, we write this
§ 6. Estimates for Approximate Solutions. Homogeneous Equations
273
inequality in the form
!lr) -
1
/2
v -1
OC
J ~ J /
[
00
/ Ij(ir) + i,s,yn + zn) x f(irj + 7, s, yn)dyn ) dzn ds dr), L p (Rj +1 ) dv /'c l ' (rl
, L p (IR n +i) dv, — (s, s n ), where, as in the proof of Lemma 6.1,
7,
7, «, xn
=
Recall that, in view of Lemma 4.7, this function is a multiplier in the space Lp(R n +i). Consequently, f v~l 1/fc
x (irj + 7)
lj
Ij (n
,s,yn)dyn }dsdr),
dv.
Using the Heaviside function and the formula for the Fourier transform of
274
4. Mixed Problems in a Quarter of Space
convolution, we find
A\>k ^c i v i/k
~
\
dv
x f(irj + 1,s,yn}dyn j d s d r j ,
l/k
-oo -oo ffin_]
sn dr/, Lp(Rn+1) dv where, as in the proof of Lemma 6.1,
, is,yn - xn)\yn-Qdxn,
7, s,
. Hence
— (s, sn). By Lemma 4.7, this function is a multiplier in
i>/c
i/•
^ 7' y dv.
By th Young inequality, we have
i i * ^ ^7 A'1
i J
i/k
f
J
e^aG(sva'}(sYa-ldS,
dv
Making the change of variables (/c = s/cVak, y/t = zi
§ 6. Estimates for Approximate Solutions. Homogeneous Equations
275
we can write this inequality in the form i A\
By the condition 0 <^ VOL < 1 and the definition of the kernel G(s), we have
where the constant C2 > 0 is independent of k and f ( t , x ) . The following estimate is established in the same way: "V1
•"•>"•
These estimates and (6.10) imply
2c
(6.11)
Consider the second term A^^ on the right-hand side of (6.9). Using the Heaviside function and taking into account that f ( t , y] = 0 for yn < 0, we can write ,, = (27r)-"/ 2 /V1 oo
oo f
f
- lTjt
e e(xn)Dxnn
( / \ •/ — 00
/" el'«'
—00
dyn ds,
dv.
Using the formula of the Fourier transform of convolution, we write this expression in the form k
,k = (27r)-( 00
00
*/
00
t( \
— OO —CO
x e
1+n
—CXD
2
)/ /V1
CT 1 t a
( -' ) /
eix''G(sva')(is)"'
276
4. Mixed Problems in a Quarter of Space
Because of a(l — va] ^ 0 and the inequality (5.13), we have
/''"'""'' ds
dv.
Using the notation F ( y 0 , y ] = (1 + (y}) a(1 ~ t/a) e- 72/0 /(y 0 , y), we write the last inequality in the form
1
-00
dv.
Applying the Holder inequality, we obtain the estimate
oo
/ f f / \ */
i e
(
1/p
l/p' dv.
§ 6. Estimates for Approximate Solutions. Homogeneous Equations
277
By the generalized Minkowski inequality, we have k
co 1
c
t vJ
/ [ / / < * - y)-pa(1-va} «/
\,J
I e^'-
J
J
-oo
oo
7, s} xn
X
(I* x \F(y0,y)\dydx
dy0,
\\F(y0,z),
ds
7, a, yn
dv.
By the Tonelli theorem, this expression can be written as follows: k
f
J/> 7/ (1
e^x'-^sG(sva'}(isY'
oo
(/"
7) s > xn,
dx\\F(y0,y}\dy
dv
(x)-
f
eix'aG(sva')(is)vl
I/P dx\\F(y0ly)\dy
We denote by K j ( v , ^ / , t — yo, y n ) the expression in the square brackets, i.e.,
278
4. Mixed Problems in a Quarter of Space
(6.12)
dz.
Then k /
I v
F(y0,z))
oo -1
i/P
1f (( 1f Kj(v,i,t-yQ,yn)\F(yQ,y)
d y \j
(6.13)
dv.
We establish some properties of the integral (6.12). Making the change of variables ^ = S k V a k , k = 1, ... ,n— 1, xz- = z ; f ~ a ' , z — 1, ... , n , and taking into account Lemma 4.1, we write (6.12) in the form
.,7, i - yo,ynv (6.14) By Lemma 4.2 for Re r ^> 70, <^ G M n _i\{0} the following estimates hold:
Since both contour integrals are analytic functions, we can apply Theorem 5.2 in Chapter 1 to the function 6(xn)D^Jj(ir] + j,^, 7 , £ , y n ) . Hence for 7 > 70, f G M n _i\{0} the integral
vanishes for i < 0 and B(j,£,t,xn,yn] = B(^0^,t, xn, y n ) e ~ 7 ~ 7 o t , t > 0. Taking into account the definition of the kernel G(£) and the equality (6.14), we can write the estimate (6.13) as follows:
§ 6. Estimates for Approximate Solutions. Homogeneous Equations
x (I /f Kj(l^0,t-y0,ynv \J
-a
279
")|F(yo, y)\dy \ 1 / dv.
(6.15)
We show that for /, yn > 0 the following estimate holds: (6.16) Indeed,
c/x.
Integrating by parts, we find
dx. Since |o;|/p > cr, we obtain (6.16) in view of the estimates for the contour integrals and the definition of the kernel G(£). Using the above estimate, from (6.15) we find
280
4. Mixed Problems in a Quarter of Space
where the constant c\ is independent of k and f ( t , x } . By condition, we have a > 1 — \a\/p'. Therefore, CO
^Xl-fc*)^ <
fv-l-\a\/p' / - 1 + Or
J
Using the Heaviside function and the Young inequality, we find
H/y-^Pl^l,
Cl
By the definition of the function F(t, y), we find
where the constant c2 > 0 is independent of k and f ( t , y}. Taking into account the inequality (6.11), from (6.9) we obtain the estimate (6.7). The convergence (6.8) is proved in the same way. D Repeating the above arguments, we can establish the following assertion. Lemma 6.4. Let \a\ > 1, \a\/p > a > 1 — \ot\/p' . Then
where 7 ^ 7o; j — 1, . . . ,/u and the constant c > 0 zs independent of k a n d f ( t , x ) . Furthermore, }
as
x) - Dtpj(t,x)}, LP,,(R}\\ -> 0
i ^ - 2 —> oo.
°° Consider a function x(n) G C°°(lRj) such that 0 ^ x(n) ^ 1> x(n) = l r for 0 ^ ry ^ 1 and x(^) — 0 f° f] ^ 2. The following assertion is proved in the same way as Lemma 6.5 in Chapter 2, but Lemmas 6.1-6.4 should be used here.
§ 7. solvability of Mixed Problems
281
Lemma 6.5. Let the assumptions of Theorem 2.4 hold. Then
\\Pif (t,*) - x(t/p)x((x)*/p3)Pif(t,x),
w^CR+i)!! -> o
as p —)• oo. Furthermore, for any q = 1, . . . , / and /?a = 1
as p -^ oo.
§ 7. Solvability of Mixed Problems In this section, we prove the theorems formulated in Section 2 about the solvability of the mixed boundary- value problem (1.1) for simple Sobolevtype equations in the quarter of the space
L(Dt, Dx)u = f ( t , x), t > 0, x e R+, Bj(Dt}Dx)u\Xn=,o = 0 , j = I , . . . ,v, t > 0, z ' e R n - i ,
By (3.14)-(3.17), approximate solutions to the mixed problem (1.1) have the form uk(t,x) = P f f ( t , x ) + Pfc~ /(*,*) + X] ^fe/(*,*)- The linear operators Pfc+, P fc ~, Pk were defined on functions f ( t , x ) 6 C7^0(1R+J1). Using the estimates derived in Sections 5 and 6, we see that { u k ( t , x ) } is a Cauchy sequence in the Sobolev spaces Wp'^R^J. Therefore, we can introduce a linear continuous operator P such that P f ( t , x ) = lim A*
+ 5D Pk}f(t'x}>
an<
^ extend it by continuity to the subspaces of the
space L P)7 (M^ 1 ) indicated in Theorems 2.1, 2.2, and 2.4 for the right-hand sides of the equation. We consider the definition of operators in detail. If |a| > 1, then from Lemmas 5.1, 5.3, and 5.5 it follows that the sequence {(Pf -f -P^~}/(^,2;)} converges in the space ^'^^(M^), l < p < o o , 7 > 7 0 , \a\/p > cr > I — \a\/p' . Consequently, it is possible to define a linear continuous operator (P+ + P-) : Lp.^M+i) H L p , 7 (M+;L l i _ a (M+)) -> W^.JK^) with the domain ^(R+tj by the formula (P+ + P~)/(/,i) = lim fc— >00
P k - ) f ( t , x ) . Since Q00 (M+^) is dense in ^ ( R + f ^ n ^ ^ f ; !!,_, the operator (P+ + P~) is uniquely extended by continuity to the entire space I p ? 7 (Ej^ 1 )nLp i 7 (]Rj";Li i _ C T (E+)). We denote the extended operator
282
4. Mixed Problems in a Quarter of Space
in the same way. By Lemmas 5.1 and 5.3, the extended operator satisfies the estimates
P - ) f ( t , z), )
f ( t , x ) , Ll(m+)\ , L P , 7 (M+)|| ),
0 ^ pa < 1,
where the constant c > 0 is independent of f ( t , x } . By Lemma 5. 2 and properties of the derivatives, u ( t , x ) = (P+ + P ~ ) f ( t , x ) , f ( t , x ) 6 ^^(M+Jj n L p i 7 ( I R j ; L i > _ f J ( I R + ) ) , have the weak derivatives DfD% u(t, x), j3a = 1, / ^ m ^ 1, in the quarter of the space M^+\. Moreover, P~\j Jf (\l, i T} J -y (1U+ r X J , i/p I icK + . I "Ii l l T ^\/ > 1L'p ~
7
where the constant c > 0 is independent of f ( t , x ) . Similarly, using Lemmas 6.1, 6.3, 6.5, for |a| > 1 we can define linear continuous operators P3 : L p i 7 (E++ 1 )nL p i 7 (Ef; L i ? _ a ( M + ) ) -> W^^R*^), j — 1, . . . , / / , 1 < p < oo, 7 > 70, a\/p > a > 1 - (aj/p', by the formula p i f ( t , x ) = lirri P ] k f ( t , x } . These operators satisfy the estimates k—>oo
++ \\nPpifd T\ T^p,-y(t& (]\$j.-] \\ x J( , )> 1J
r
l x
n
where the constant c > 0 is independent of f ( t , x } . By Lemma 6.2 and properties of weak derivatives, the function Uj(t, x} = P j f ( t , x), f ( t , x) 6 L p i 7 (M++ 1 )nL p i 7 (R^; L i , _ C T ( M + ) ) ! have the weak derivatives D™DP.Uj(t, x), (3a = 1, / ^ m ^ 1, in the quarter of the space IE Moreover,
where the constant c > 0 is independent of f ( t , x ) . We pass to the proof of the theorem. PROOF OF THEOREM 2.1. By definition, W^;!;(JR++1) = W^J i 0 (Kj+i). Therefore, Theorem 2.1 directly follows from Theorem 2.4. D
§ 7. solvability of Mixed Problems
283
PROOF OF THEOREM 2.4. Since \a\ > 1, for any right-hand side of the equation f ( t , x ) G Lp^M^) H L p , 7 (M+; L1^a(R+)) from the definition of operators (P+ -f -P~), PJ , j — 1, • • • , A 4 , we conclude that the function u(t, x) = (P+ + P~ + X^=1 P i ) f ( t , x) is a solution to the mixed boundaryvalue problem (1.1) in the space Wp:')^(7(]R++1), 1 < p < oo, 7 > 70, \a\/p > a > 1 — |a|/p'. For a function u ( t , x ) the following estimates hold: ll + \ \ D l t u ( t , x ] , W
where j3o. = 1, / ^> m ^ 0 and the constant c > 0 is independent of /(/, x). The uniqueness of a solution is established by the following assertion. D Lemma 7.1. The mixed boundary-value problem L(Dt,Dx}u = Q,
t>Q,
x€R+,
(7.1)
on/y f/ze ^rero solution in the space Wp'^ PROOF. We first consider a solution to the problem (7.1) such that u ( t , x ] — 0, t > r, |x'| > r. Acting on this solution by the integral Laplace operator with respect to t and the Fourier operator with respect to x' , we find /-i L0(is,DXn}Tlu(T,s,xn} + ^Li-k(is,DXn)Tku(r,s,xn)
= 0,
xn > 0,
fc=0
B j ( T , i s , D X n ) u ( T , s , Q ) = 0 , j - I , . . . ,/./, Re r > 70, s G E n _ i Furthermore, by the embedding theorem, we have sup |w(r, s,x n )| < oo. xn>Q
Consequently, for Re r > 70, s £ M n _i\{0} the function u(r, s,xn) is a solution to the homogeneous boundary- value problem on the half-axis xn > 0 which satisfies the Lopatinskii condition. By Theorem 9.1 in Chapter 1, we have u(r, s, xn) = 0. Hence u(t, x} = 0. Arguing as in the proof of the uniqueness of a solution to the Cauchy problem for equations (cf. Chapter 2, Section 7) or for systems (cf. Chapter 3, Section 6), it is easy to show that the problem (7.1) has only the zero solution in W'^^M^.^). D
284
4. Mixed Problems in a Quarter of Space
PROOF OF THEOREM 2.2. To establish the existence of a solution to the mixed problem (1.1), we estimate the sequence of approximate solutions { u k ( t , x } } . These estimates imply the convergence of this sequence in the space Wp^(M++ 1 ) to the solution u(t,x). First of all, we note that the arguments of Sections 5 and 6 imply Lemmas 5.1, 5.2, 6.1, 6.2 for any values of |a|. However, in the proof of Lemmas 5.3, 5.4, 6.3, 6.4 we essentially used the conditions on |a|. We establish analogs of these lemmas under the conditions |a| ^ p' . Lemma 7.2. Let \a\ 1 •£ \a\/p' + (N - l)a m i n . // the function f ( t , x ) e L pi7 (R^ 1 )nL pi7 (IR] 1 "; Li^TviaKB&n)) satisfies the orthogonality conditions (2.2), then for 1 > /3a ^> 0, 7 > 70 the following estimate holds:
+ 1 11(1 + ( x ) ) N W f ( t , x ) ,
Li(M+)||, ^(M+JH),
(7.2)
where the constant c > 0 is independent of k and f ( t , x ) . Furthermore, ) , LP!,(R^)\\ -> 0 (7.3) as k i , k % —>• cxo. PROOF. As it is seen from Lemma 5.3, the main difficulties in the proof of (7.2) and (7.3) appear in the case f3a = 0. We consider this case in detail. Let ,/V = 1 in the assumptions of the lemma, i.e., lo^/p' + cvmin > 1 ^ \a\/p' . From the orthogonality conditions we have oo
f f(t,s,xn)\s=0dxn = Q. J
0
We write the function e~~1t(P^ + P ^ ) f ( t , x ) in the form i l/k
x -oo
0
yr)- n / 2 f v-1 j i/k J
J
elx'sG(sva'
(7.4)
§ 7. solvability of Mixed Problems
285
oo oo
/
/ e*tr}J-(ir
—-oo oo xn k
oo
7r)- n / 2 fv~l j
oo
I
1
-oo -oo K n _ i
)8,xn
-yn) +d(yn - xn)J,(ir) + 7, s} xn - yn)) = uitk(ttx) + uztk(t,x} + u3>k(t,x). (7.5)
Repeating the proof of Lemma 5.3, It is easy to establish estimates for the first two terms on the right-hand of (7.5): \\ui,k(t,x), Lp(R£MH ^ c||/(*,x), WMj+OII,
(7.6)
||ti 2ifc (*,*), LpCRjJJH ^ c \ \ f ( t t x ) , LP.^R+J!)!!,
(7.7)
where the constant c > 0 is independent of k and f ( t , x}. We consider the third term 113^(2, z) on the right-hand of (7.5). Using properties of the Fourier transform and Lemma 4.4, we find k
oo
oo
/V1 -oo -oo ffin_i
1 1
r
t, is, z'fin))" f-y( ), s, s n ) ds dsn dr)dv,
where f-y(rj, s, s n ) is the Fourier transform of the function e~~it f ( t , x', xn] extended by zero for t < 0, xn < 0. By the condition (7.4) and the Hadamard lemma, we have k
oo
oo
r r,ix's
1
G(sv
+ itT)+iznSnr!(c*,a'\
J J -00 - 0 0 f f i n _ !
^(77, s, s n ) ~ /y( 7 ? ) 0) 0)) ds dsn drj dv 1
0
oo
-oo
x (-ifj)(L(ir} + i,is,isn)) Iyj9-1(r], y) d£ dy dr) \d\dv,
286
4. Mixed Problems in a Quarter of Space
Repeating the proof of Lemmas 5.3, 6.3 and taking into account the condition \a\/p' + am'm > 1, we obtain the estimate
f \ II
T
/TCP+MI
^ )\\, -^p,7v
1 /ll>
(7 %} V
/
where the constant c > 0 is independent of k and f ( t , x } . By (7.5), from (7.6)-(7.8) we obtain (7.2) for /3a = 0, N = 1. The general case is treated in a similar way. The convergence (7.3) is proved in the same way. D Repeating the above arguments, we can prove the following assertion. Lemma 7.3. Let the assumptions of Lemma 7.2 hold. Then for 7 > 70 the following estimate holds:
< f(\\ fd r\ T
f i H > + + i n - i - n u n 4-/T\^l a l / f/vv i >r\/ > T 1 \f p +n ^/ nII > j Pi7 Vfiu+iin I / II;'
5v C ^ | | y ^ t , .C J , L/p ^ V n + l / l t ' l l l l \ . " T " \ / /
where the constant c > 0 ts independent of k and f ( t , x ) . Furthermore, :), I n - v ( M t ^ i ) l l -> 0 05
'
T2 —> 00.
Lemma 7.4. Let the assumptions of Lemma 7.2 hold. Then for I > (3 a ^ 0, 7 > 70 the following estimate holds:
+ 1111(1 + (x})Nf(t,x),Ll(m+)\lLpn(Rl)\\),
(7.9)
where j = 1, ... , fj, and the constant c > 0 is independent of k and f ( t , x ) . Furthermore, \\^xPJklf(t,x}-DPp^f(t,x),
Lp.^K+i)! ^ 0
(7.10)
as ki , k-2 —>• co.
PROOF. From the proof of Lemma 6.3 it can be seen that the main difficulties in the proof of (7.9), (7.10) appear in (3a = 0. We consider this case in detail.
§ 7. solvability of Mixed Problems
287
At the beginning of the proof of Lemma 6.3, we wrote the obvious inequality (6.9) which for cr — 0, (3a — 0 takes the form
?x'aG(sva')( I ^ \J
(27T) -"/ 2
^ Pi f ( t , x ] ,
\/k oo
Jj (irl + 7, s, xn) Ij (iri + J,s, yn)f(ir) + ~f,s,yn) dyn drjjds,
dv
exsG(sva
(27T -"/ 2
dv
From the proof of Lemma 6.3 we find that the first term A\^ on the righthand of the last equality satisfies the estimate (6.11) for any |a| without any additional orthogonality conditions on the function f ( t , x ) . Therefore, to prove the estimate (7.9), it suffices to estimate the second term A?^Let N = 1 in the assumptions of the lemma, i.e., |a|/p' + a m in > 1 ^ \a\/p'. As was mentioned above, the orthogonality conditions imply the relation (7.4) or the equality
which allows us to estimate A<2,k as follows:
/ J ' v~l
n 2
Ij(irj + 7, s, yn}(f(irj + 7, 0 , 2 / n ) )
exsG(sva
7, 5, xn
+ 7, s, yn
dyndrj } d s , Lf
dv
288
4. Mixed Problems in a Quarter of Space oo
/ «/
7, s,0)) eft; +
The first term A\ k on the right-hand of the last equality can be written in the form
t eix>sG(sva')(
n 2
= (27r) - /
J
f \
J
where
and the second term A\ k can be written in the form
00
0
1
0 j>n u/// i t.4o . x>r)i ir^ri -4- I I
U-V .
J Repeating the proof of Lemma 6.3 and taking into account the inequality
§ 8. Necessary Solvability Conditions
289
\Q\/P' + a min > 1, we find n-1 1=1
where the constant 02 > 0 is independent of A: and f ( t , x ) . These inequalities and the estimate AI^ imply (7.9) for (3a = 0, TV = 1. The general case is considered in a similar way. The convergence (7.10) is establish by the same arguments. D Repeating the above arguments, we can prove the following lemma. Lemma 7.5. Let the assumptions of 7.2 be satisfied. Then for 7 > 70 the following estimate holds:
+ 1111(1 + < * » a / ( ' , *), £ i ) l l , WK)!!).
3 = 1, - - - ,0,
where the constant c > 0 is independent of k and f ( t , x } . Furthermore, \\DltPij(t,x} - DltPij(t,x), Lp^CR+i)!! -^ 0,
*!,*;, -> oo.
From Lemmas 5.1, 5.2, 6.1, 6.2, 7. 2-7. 5 it follows that, under the assumptions of Theorem 2.2, the approximating sequence { u k ( t , x } } , where
converges in the space Wjij^Rj^), 7 > 70, H/P'+ ^"min > 1 ^ |"l/p' + (TV — l)of m i n . By construction and the properties of the operators P£ , P^ , P^, the limit function u(t,x) is a solution to the mixed problem (1.1) and satisfies the estimate (2.3). The uniqueness of a solution in the space W R follows from Lemma 7.1. D
§ 8. Necessary Solvability Conditions for Mixed Problems In Section 7, we proved the unique solvability of the mixed boundaryvalue problems (1.1) in the scale of weight Sobolev spaces Wlp'^
290
4. Mixed Problems in a Quarter of Space
1 < p < oo, 7 > 7o> \a\/P > 1 — \ a \ / p ' , for any right-hand side f ( t , x] £ ^ P l - 7 (M++ 1 )nLp i 7 (]R+;L 1 | _ ( 7 (]R+)). In particular, if cr = 0, then for \a\/p'> I the problem is unconditionally solvable in the Sobolev spaces Wl>r ((0, T) x 1R+). For \a\/p' ^ 1 in Theorem 2.2, we obtain the solvability in the space Wp'r ((0, T) x IR+) only under the additional orthogonality conditions on the right-hand side of (2.2): / x P f ( t , x ) d x = Q,
\(3 = 0 , . . . ,N-l,
where a|/p' + Nam}n > 1 ^ M/V + (N — l)a m in) i - e - > the unconditional solvability of the problem (1.1) is not guaranteed in the whole scale of Sobolev spaces. By Theorem 2.3, the unconditional solvability in the whole scale Wp'r((Q,T) x I R + ) , 1 < p < oo, does not necessarily hold. Furthermore, the orthogonality conditions (2.2) are close to necessary ones. We pass to the proof of this theorem. PROOF OF THEOREM 2.3. We show that if
(8.1)
then, even in the case of compactly supported and infinitely differentiate functions f ( t , x ) , for the solvability of the mixed boundary-value problem (1.1) in Wp'^(1&n+i)i under the condition
it is necessary that
r
I f ( t , x ) d x = 0.
(8.3)
Assume the contrary. We assume that f ( t , x ) G Co°(M^ 1 ), does not satisfy the assumption (8.3) and the problem (1.1) has a solution u ( t , x ) £ Wp^ORj+i), 1 < p ^ 2. By the Hausdorff- Young inequality, the following estimate holds: u(T,s,xn), Lpl(Rn)\\, L p (Mf)|| ^ clK/.x), ^^(K+i) |,
(8.4)
Re T > 70, where U(T, s, xn) is the partial Fourier transform of the function e~'1tu(t, x', xn] with respect to ( t , x ' } .
§ 8. Necessary Solvability Conditions
291
It is obvious that for Re r > 70, s £ M n _i\{0} the function u(r, s,xn) is a solution to the boundary-value problem
Bj(r,is, DXn)u\Xn=o = 0, sup \u(r, s,xn)\ < oo.
j = I , . . . ,/j,,
xn>0
Since this problem satisfies the Lopatinskii condition, its solution can be represented in the form
/
^ J+(r, s, xn - y n ) f ( r , s, yn) dyn
o
oo
+ / J-(r, s,xn - yn}J(r, s , y n ) d y n xn ,,
°°
E
,Jj(r,s,xn) I
Ij(T,s,yn}J(r,s,yn}dyn.
o
We define the function U V^"> ^' "^n ' y™ ) ~ " -\-\TI $ •> En
yn ) i /
Jk\Tj ^ j •^n)^k\^'i & •> 2/r
From Lemma 4.1 we obtain the following assertion. Lemma 8.1. U(r, ca s, c~anxn, 0) = c~i+anU(r, s, xn, 0), c > 0. Further we introduce the notation M+(T,s,DXn)=f[(riDXn-X+(T,S)), A = max |A,-(r, s')|, R e r ^ 7 o , (s') = 1, Bj(r,is,i\}
Lemma 8.2. The function U ( r y s , x n , Q ) is a solution to the boundary-
292
4. Mixed Problems in a Quarter of Space
value problem M+(r,s,DXn)U = Q, xn > 0, Bj(r,is,DXn)U\Xn=o - ^j(r,s), U(r, s,xn) -> 0, xn->+00.
j - 1 , . . . ,//,
PROOF. From the definition of the contour integrals J+(s,xn), and Lemma 4.2 it follows that
IU [ T S X
0 1 '"'• J ^ 0
X
^ ~4~OO
Let us verify the boundary conditions. Taking into account the definition of the integrals Jk(r, s, xn] and /fc(r, s, xn), we have BJ(T, is, DXn)U(r, s, xn, 0)| rn= o 7 (T- S, c X v )\ -h _l_ \ T: (r e X -r )lk\T, \J,(T S, Q UJ (\\ J+(T, / ^ J/c^ T, S, n n
fc = l
— DJ (T, zs, UXn)J-f (T", s, ^^j|r n =o ~l~ lj(T, s, u) — n-l-r ie 71 V t 7 /V c -r "\ — 7 (T G f — Jjj ^ / , la, LJxn ^ ' + V ) ) n / ^ — v i > n ^ / |\\\x
n
= 0
1
The lemma is proved. From Lemma 8.2 we obtain the following assertion.
D
A*
Lemma 8.3. U(r, s, xn, 0) = £ J fc (r, s, xn)ipk(r, s). k =l
Using the function U(T,s,xn,yn), for xn ^ d = diam(supp /) we have oo
u(r,s,xn)= If J+(r,s,xn -
yn)f(r,s,yn)dyn
\J
o
oo
= /
U(T,s,xn,yn)f(T,s,yn)dyn.
293
§ 8. Necessary Solvability Conditions
Taking into account this representation, from the inequality (8.4) we find oo
/ U(r,s,xn,yn)f(r,s,yn}dyn,Lp>(Wi
x {(s}an < 1}) ,Lp({xn>2d})
By the Minkowski inequality, for any a and b we have oo
J
x 7(r,0,y n )dy n ) Lp,((a, 6) x {(s)°"> < 1}) , L p ( { x n > 2 d } } oo
/ (U(T, s, xn, yn) - U(r, s, xn, 0))
x f(r,s,yn)dyn, Lp>((a, b) x {{s)a" < 1}) ,Lp({xn oo
-f
/ U(r, s,xn,Q}(f(T, s,yn]
- f ( r , Q , y n } ) d y n , Lp,((a, b) x {(s)*" < 1}) ,Lp({xn >2d}) + C ||u(t, x}, ^ ^ ( M J H = F,
c\\u(t, x),
(8-5)
We establish the estimate (8.6)
c(f) < oo.
We consider the norm of FI. By Lemma 4.2, for 0 < yn < d, 2d < xn we have
i \ U ( T t s , x n , y n ) - U ( T , s , x n , Q ) \ = yn I
Consequently,
, Lp({xn
294
4. Mixed Problems in a Quarter of Space
Similarly, n-l
x exp(-<Jx n ( S ) a -/2), MW" < U)!!, M(*n Introduce the domains ujl = {s 6 E.n-i • 2 ~ l ~ ! < (s)an < 2~'}, i = 0, 1, 2, . . . . By the above estimates, we have
EE i>0k=l
0
x exp(-(f:r n (s) "V2),
i ^ 0 k -1
i/p i>0
By (8.2), we have (1 — Q^ — \a\/p') < 0. Therefore, the following estimate holds: (8.6). By (8.5) and (8.6), we have
x / ( r , 0 , y n ) d y n i V ( ( a , 6 ) x { ( S ) a - < 1}) , Lp({xn > 2d})
By assumption, the orthogonality condition (8.3) does not hold. Therefore,
and the following two-sided estimate holds: 0 < I \ U ( T t 8 ) x n , Q ) g ( T ) , L p , ( ( a , b) x {(*)"- < <6
, Lp({xn > 2d})|| (8.7)
§ 9. Pseudoparabolic Equations
295
Introduce the notation V ( d , e ) = \ \ \ \ U ( r , S , x n j O ) g ( r ) , L p , ( ( a , 6) x { ( s ) a » < e}}\\, Lp({xn > 2cf})||, where e < 1. By Lemma 8.1 for any c > 0 we have c( 1 ~l a l/ p ^V(d^e] = V(ca»d, c~ a "£). From (8.1) and Lemma 8.3 we find U(r,s,xn,Q) £ 0, s £ Wj, i = 0, 1, 2, . . . . Therefore, the last relation can be written in the form c^-H/p') = V(ca»d, c-a»e)/V(d, e). Taking into account (8.2), for any c > 1 we obtain the estimate V(ca"d,c-a*e)/V(d,e)
^ 1.
On the other hand, in view of the inequality (8.7), we have lim V(cand,c-an£) = 0, C-K30
which leads to a contradiction. Thus, the orthogonality condition (8.3) is necessary for the solvability of the boundary- value problem (1.1) in the space Wp'i^(M^1) under the condition (8.2). ' D
§ 9. Mixed Boundary- Value Problems for Pseudoparabolic Equations In this section, we consider the mixed boundary- value problems for pseudoparabolic equations with constant coefficients £(Dt,Dx)u = f ( t , x ) , t>Q, x G E + , Bj(Dx)u\Xn=0 = Q, j = l , . . . ,/i, < > 0 , a r ' E M n _ i , D^u\t=0 = Q,
(9.1)
A: = 0 , . . . , / - ! , x € M + ,
where £(Dt,Dx} = L(Dt,Dx) + L ' ( D X ) ,
is the leading part of the operator, and the operator L ' ( D X ) consists of the lower-order terms of £(Dt, Dx). We recall the definitions given in Chapter 2, Section 2. The leading part L(Dt, Dx) of the operator £(Dt, Dx] should satisfy the following conditions.
296
4. Mixed Problems in a Quarter of Space
1. The symbol L(ir),i£) is homogeneous relative to the vector a = (a 0 ,a), QQ > 0, I/a,- 6 N, z = 1 , . . . , n , i.e., L(c a °zr?, c a z£) = cL(ir],i£), c> 0. 2. The operator LQ(DX] is quasielliptic. 3. For Re r ^ 0, £ € M n , |r| + |f | ^ 0 the following inequality holds:
We assume that the lower-order part L'(D;c} of the operator £(Dt, Dx] takes the form L'(DX] £ We indicate conditions on the boundary operators B j ( D x ) . We first note that conditions 1, 2 and the form of the operator L'(DX) means the existence of 70 > 0 such that the equation £(r,is,*A) = 0,
Rer^7o,
s e E n _i\{0},
(9.2)
has no real roots A. We assume that the number of boundary conditions in (9.1) at xn = 0 is equal to the number of roots with positive imaginary part. To simplify exposition, we assume that the boundary operators do not contain the operators D t : B j ( D x ) = D?j + ]T b j i k ( D x . ) D k X n ,
mj < l/a n ;
moreover, the symbols Bj [i^] are homogeneous relative to the vector a from condition 1 with exponents /3j, 0 0. Consider the boundary- value problem with parameters (r, s), Re r ^ 70, s6M n _!\{0}: C(r,is,DXn)v = 0, B is
D
xn > 0 ,
j( > xn}v\Xn=o = Vj> sup \v(r, s,xn)\ < oo.
J = l,---,^,
(9.3)
xn>0
We assume that this problem is uniquely solvable for any data (p\ , . . . ,ip^, i.e., it satisfies the Lopatinskii condition. Definition 9.1. We say that the mixed boundary- value problem (9.1) satisfies the Lopatinskii condition if for Re r ^> 70, s £ M n \{0} the problem (9.3) satisfies the unique solvability condition. Example 9.1 (the equation of small-amplitude oscillations of a viscous rotating fluid). We consider the first boundary- value problem for the equa-
§ 9. Pseudoparabolic Equations
297
tion of small-amplitude oscillations of a rotating viscous fluid (i/ > 0) A(A - i/A) 2 w + "2D2X3u = f ( t , x ) , 2
«U=o = A: 3 uU=o - D 3u\X3=0 =0,
t > 0, x3 > 0, t > 0, x' € M 2>
(9.4)
w|t=o = A«|t=o = 0, z G E j . Recall that for the leading part of the symbol of the differential operator, the homogeneity vector a takes the form a = (1/3,1/6,1/6,1/6). It is obvious that the equation (s2 -f s2 + A 2 )(r + vs\ + vs\ + z/A 2 ) 2 + u; 2 A 2 = 0 for s 6 M2\{0}, Re r ^ 70 ^> 0 has three roots AI(T, s), \2(r,s}, X3(r,s) in the upper half-plane and three roots in the lower half-plane, i.e., /i — 3. It is easy to verify that for s £ M2\{0}> Re T ^> 70 the following boundary- value problem satisfies the Lopatinskii condition: (-s2 - s2 + £> 2 3 )(r + vs\ + vs\ - vDlJfv + u2D2X3v = 0, ^1x3=0 = ^1,
DX3V\X3=0=2,
x3 > 0,
DX3V\X3=0 = (f>3,
sup \v(r, s,x 3 )| < oo. We formulate the main assertions about the solvability of the mixed problem (9.1). We look for a solution to the problem (9.1) in the class Wp'^M+^j), r — (1/ai, . . . , l/an), of functions having the weak derivatives , x] € <^ 1 " /cao)r (M++ 1 ), k = 0, . . . , / . Theorem 9.1. Suppose that \a\/p' + laQ > I , f ( t , x ) G L P|7 (RJ+ 1 ) n R^)), 7 > 70- Then there exists a unique solution u ( t , x ) G to the mixed problem (9.1) and the solution satisfies the estimate
where the constant c > 0 is independent of f ( t , x ) . Theorem 9.2. Let ja|/p' + /ao ^ 1, and let N be a natural number such that \a\/p' + /a 0 -f Nam-m > 1 ^ |a|/p' + /a 0 + (TV - l)a min . Then the mixed problem (9.1) has a unique solution u ( t , x ) £ Wp',^0^n+i)> 7 > 7o; /or any function f ( t , x ] £ LP|7(IR^1) swc/i
f
x
298
4. Mixed Problems in a Quarter of Space
The solution u ( t , x ) satisfies the estimate l r
where the constant c > 0 is independent of f ( t , x ) . Theorem 9.2 for |a|/p' + IO/-Q ^ 1 indicates sufficient conditions for the solvability of the mixed problem (9.1) in the spaces ^'^(IR*^). This result is similar to Theorem 4.4 in Chapter 2 about the solvability of the Cauchy problem in the space Wl'r (^^. j ) , and to Theorem 2.2 about the solvability of the mixed problems for simple Sobolev-type equations. We consider a wider scale of weight Sobolev spaces Wp'r^ ^(IR^j^), 1 < p < co, 7 > 0, 1 ^> a ^> 0, as a completion of the set of functions in C°°(IR^1) vanishing for large ( t\ + \x\) in the norm \\..l4 ™\ ixrl,r
/TTD++ Ml _
/3a-l
By definition, we have W^ Q(R++1) = W^R^) and ^S,a(KJi)-
We formulate the solvability theorem for the mixed problem (9.1) in the space Wp<^a(R++l), I < p < oo, 7 > 0, 1 ^ 1; |a|/p > 1 — /«o — \a\/p', f ( t , x ) GL p i ^(]R++ 1 )nL p i 7 (E+;L l i < 7 ( ; Q o _ 1 ) (E+)) ; 7 > 7o- T/ien t/ie mairerf problem (9.1) Aas a unique solution u ( t , x ) E VK'1^, a (M^ 1 ) ; anc? f/ie solution satisfies the estimate
4- II 1 1 / 11 4- /-r\\a(1-lao) f(tl x , -^H^n J l h ^ P . - y ^ l
where the constant c > 0 zs independent of f ( t , x )
§ 10. Pseudoparabolic Equations. Proof of Solvability
299
We illustrate Theorem 9.3 by the boundary-value problem (9.4). Since a = (1/3, 1/6,1/6,1/6), we have |a| -f Ia0 = 7/6. By Theorem 9.3, the problem (9.4) is uniquely solvable in every space Wp'^a(M.^+], I < p < oo, 7>7o,
§ 10. Sketch of Proof of Solvability of Mixed Problems for Pseudoparabolic Equations In this section, we write formulas for approximate solutions to the mixed boundary-value problem (9.1) and clarify the main difficulties appearing in the proof of estimates. We construct approximate solutions to the problem (9.1) according to the scheme described in Section 3. We write explicit formulas for these solutions. Let the right-hand side f ( t , x) of the equation in (9.1) belong to Co°(3R^1) Consider the boundary-value problem on the half-axis {xn > 0} for the ordinary differential equation with parameters (r, s ) , Re r ^ 701 s G M n -i\{0} £(r, is,DXn)u> = f ( r , s,xn), Bj(is, DXn)ui\Xn=o = 0,
xn>0,
j = 1,. .. ,fj,,
sup |co>(r, s, xn)\ < oo. xn>0
Since the Lopatinskii condition holds, this problem has a unique solution which can be explicitly written with the help of the formulas in Chapter 1, Section 9. As in Section 3, we introduce the contour integrals J+(r, s,xn) = —-
2ir
ZTT
j(T,S,Xn)
/
——rArrdA,
J L(T, is, iX) r+(T,5) J
r-(r,s)
(10.1)
Li\T, zs, zAJ
- —
where j = 1 , . . . ,/^, Re r ^ 70, s G M n _i\{0}, F + (r, s) is a contour in the complex plane surrounding all the roots X^(r,s), k = 1, ... ,/j, of the
300
4. Mixed Problems in a Quarter of Space
equation C(r,is,iX) = 0 lying in the upper half-plane and T~(r, s) is a contour surrounding all the roots A~(r, s), i = 1 , . . . , m—p, m = l/an, lying in the lower half-plane. By definition, M + (r, s , A ) = Hfe=i(^ "~ ^/c~( r > s ' ^)) and N j ( r , s, A) are polynomials in A such that 27rz
J
M+(r,.s,A)
J
'
where 61- is the Kronecker symbol. The contour integrals (10.1)~(10.3), as well as the corresponding integrals (3.4)-(3.6), are analytic functions of T, Re r > 70. We introduce analogs of the functions (3.7)-(3.9):
/
^ J+(r,s,xn - y n } f ( T , s , y n ) dyn, o oo
~J oo
3
uJ (r,s,xn) - Jj(r,s,xn)
r
/
Ij(T,s,yn)f(r,s,yn)dyn,
\)
o
where j - 1 , . . . ,p and Ij(r,s,yn) = -Bj(is, DZn) J_(r, s, zn - yn)\zn=0. Since the integrals (10.1)-(10.3) are analytic and bounded for Re r ^ 70, we can apply Theorem 5.2 in Chapter 1 to these functions. Consequently, the integrals 00
11+ (1 c T } — /Oii-^" 1 / 2 V {l,S,Xn) — v^'V
f
/ /
a(irl+~t)t/
,+ (irt i -, o 7. n \rJri \ ' '' ' ' ''
^104^ Viu'^/
f e^+^u~ (irj + 7, s, x n ) dry,
(10.5)
— oo oo
1 2
v- (t, s, *„) - (27T)- /
(i77 + 7 , s , z n ) dry,
(10.6)
where j — 1, . . . ,/j, are independent of 7 ^ 70- Furthermore, the function v(t,s,xn) = v+(t,s,xn) + v~(t,s,xn)
§ 10. Pseudoparabolic Equations. Proof of Solvability
301
is a solution to the mixed problem
/-i k=0
for t > 0, xn > 0, Bj(is,D a;n )t;| arB= o = 0 > D*v| t= o = 0 >
j = 1 , . . . ,/*,
k = Q,...,l-l,
t > 0,
xn>0,
seM n -i\{0}.
Now, we can construct a sequence of approximate solutions to the mixed problem (9.1). Since the integrals (10.4)-(10.6), as well as the corresponding integrals (3.10)-(3.12), have, in general, nonintegrable singularities at s = 0, to construct approximate solutions we use the integral representation for functions
h-1 v-la'l-i
/ h
\timesG(s}(p(y'}dsdy'
r /
r /
f
'
eX pL-^_Z_
Kn-iKn-i
dv,
(10.7)
where
1=1 As in Section 3, we define the function k
1 n /2
u+(t,x) = (2 7 r)( - )
t v~l 1//C
I
eix'sG(sva')v+(t,s,xn}dsdv,
Kn-l
/C
«;(*,x) = (2;r)( 1 - n ) /2 /t;- 1 /" e l ' x ' a G(st; 0 f ')u~(<. s » x n) ^sdt;, I/A;
In-i
fc
wi(t,x) = (2 7 r)( 1 - n )/ 2 I v~l I
eix'sG(sva')vj(t,s,xn)dsdv,
where j = 1 , . . . ,/j. Further, we assume that N in (10.7), (10.8) is such that the functions u£(t,x), u^(t,x), u j k ( t , x ) are infinitely differentiable and summable with respect to x with any power. We note that such a
302
4. Mixed Problems in a Quarter of Space
choice is always possible because the contour integrals (10.1)-(10.3) have only power-type singularities at s = 0. We consider the functions ^ uk(t, x) = u+(t} x) + u~ (t, x) + Y^ u fe(*> x )' ( 10 ' 9 ) j=i By construction, we have
B j ( D x ) u k ( t , x ' , Q ) = 0,
j = 1 , . . . , A / , t > 0, z' e M n _ i ,
where
l/k
I
By the integral representation (10.7), we have \\fk(t,x) - f ( t , x ) , £ P ,.y(M++i)|| -»0,
Ar-»oo.
Consequently, the function uk(t, x) is an approximate solution to the mixed boundary-value problem (9.1). To obtain the L p -estimates for { u k ( t , x } } , it is necessary to use analogs of the lemmas in Section 4 for the contour integrals (10.1)-(10.3). We formulate the corresponding assertions. We first recall that the equation L(r,is,i\) = Q,
Rer^O,
s 6 R n _i\{0}
(10.10)
has no real roots A. For s = 0, Re r > 0 there is a single real root (equal to zero and with multiplicity m = (1 — lao)/an), whereas the remaining (l/an — m) roots are complex. We denote by A^(r, s), . . . , A*+(r, s) and Aj~(r, s ) , . . . ,\~_(T,S] the roots lying in the upper and lower half-plane respectively for Re r > 0, s £ M n _i\{0} and vanishing at s = 0. The remaining roots A++ + j (r,s), j - 1, . . . ,(t*-p+), and \~-+k(r,s), k = 1, ... , (l/an — p, — p ~ ) , lie in the upper and lower half-plane respectively for any Re r > 0, s £ R n -i- Since all the roots of equation (10.10) are quasihomogeneous functions of the vector (GO, a') with the homogeneity exponent an, the following assertion is valid.
§ 10. Pseudoparabolic Equations. Proof of Solvability
303
Lemma 10.1. The roots of equation (10.10) satisfy the estimates S^s)"*
MO"- ^ Im A p + ++j (r,s) j = 1 , . . . , (f*-p + ), (Ji(«) a - ^ -Im Af (r,a / = !,.. . , p ~ ,
= ! , . . . , (l/a n - // - p-), (s) 2 -
1=1
^ - / a ' , (C)2 - \r\2 + (s)2 and
^1,^2 > 0 are constants. If a pseudoparabolic operator contains no lower-order terms, from Lemma 10.1 we obtain the following estimates for the contour integrals (10.1)(10.3). Lemma 10.2. For xn > 0, Re r > 0, s G M n _i\{0} the following estimates hold: \D<*nJ+(r, s, xn}\ <:
+ c 2 <s}^+i-^)«"(C} ma -- 1 ex P (-^ n <s} a -),
-^ n (O a "),
(10.11)
(10.12)
where c\, c2; 6 > 0 are constants. PROOF. By the homogeneity of the symbol L(r,is,i\) relative to the vector a = (ao, &', a n), we have °'»-1-L f ^y r+
ex
P(^»(O an £(,-/, , V|I - A
where r' = r(C)" 1 , s'k = Sfc(C)~ Q f c , k = 1, . . . , n - 1. By Lemma 10.1 the roots lying in the upper half-plane satisfy the estimates )"* <: Im A+(r'X) ^ lA^r', s')\ Im
304
4. Mixed Problems in a Quarter of Space
Similar estimates hold for the roots lying in th lower half-plane. Consequently, to estimate the contour integral Dqx J+(r, s, xn], it is necessary to consider two cases. CASE 1: S-2(s')an > 6\f2. For the contour F+ we can take the boundary of the domain {Im A > J 1 2 /4^ 2 }n{|A| < 252}. Then for xn > 0 the following inequality is obvious: / x2 \ q
u \D \ x
T ss xr "ll < r/< / V < 7 + 1 ) a n "~ 1 pxr>( ex (C\an I\\Cr J ) n)\ ^ C-\S/ P\ — — , r x rn\S,/ \ 40 2 /
CASE 2: J 2 (s') a n ^ ^i/2. The contour integral DqCnJ+(r, s} xn) can be written in the form • n ( C ) Q " A ) l J fiV,:A) n (Q*-A)
^
;
where F^~ is the boundary of the domain {Im A > 4Ji/5} H {|A| < 252} and Fy is the boundary of the domain {Im A > ^i{s / ) a "/2} fl {|A| < 5^ 2 (s') an /4}. Then for xn > 0 we obtain the inequality
The proof of (10.12) is similar.
D
We consider the pseudoparabolic operator £(Dt,Dx) = L(Dt,Dx) + a L ' ( D X ) , where L ' ( D X } = Y^ pDx- As was mentioned in Section 9, l-/a0^/9a
for sufficiently large 70 > 0 the equation £(r,is,i\) = 0, Re r ^ 70, s G ]R n _i\{0}, has no real roots A, and for s = 0 the equation has a single real root A = 0 of multiplicity m = (I — lao)/an. Consequently, for the roots of this equation an analog of Lemma 10.1 holds, which implies an analog of Lemma 10.2 for Re r ^ 70, s 6 M n _i\{0}. The following assertion is proved in the same way as Lemma 4.3. Lemma 10.3. For Re r ^ 70, s £ R n _i\{0} the following identities hold: 8k_ Dkx (J+(r, s,xn) + J-(T, s,xn))\Xn=0 = ~^~, q = l/an,
§ 10. Pseudoparabolic Equations. Proof of Solvability
305
where 6^ is the Kronecker symbol and aq(r] = OLQTI + air'"1 4- . . . + at is the coefficient at the higher-order term (i\)q of the polynomial £(r, is, iX). We consider analogs of Lemmas 4.4-4.6. The proof is similar to the proof of Lemmas 4.4-4.6, but, instead of Lemma 4.2, we should use Lemma 10.2. Lemma 10.4. For Re r ^ 70, s E M n _i\{0} the following identity holds: oo
/ e-iXXn(9(xn)J+(T,s,xn)+e(-xn)J-(T,s,xn))dxn J
= ——^—-, L(r,zs,iX)
— OO
where 0(xn) is the Heaviside function. Lemma 10.5. For Re r ^ 70, s E M n _i\{0} the following holds:
identity
r< .^ L(T,is,iX) — oo
Lemma 10.6. Let
Jj(r,s,xn} I o oo
Ij(T,8,yn)
= / DZn(Jj(r, s,xn + zn) I /j(r,s,y n + z n )<^(s,y n ) dyn)dzn,
We clarified indicated some differences at the first stage of obtaining estimates for approximate solutions (10.9). Using Lemmas 10.2-10.6 and the Lizorkin theorem about multipliers and arguing as in Sections 5-7, it is possible to establish assertions similar to the lemmas in these sections. Under the corresponding assumptions of Theorems 9.1-9.3, from the above results we obtain estimates and convergence of the sequence of approximate solutions {uk(t,x}} in the Wl*r ^(IR^j^-norms. By the construction of the sequence { u k ( t , x ) } , the limit function u ( t , x ) = lim Uk(t,x) is a solution k—too to the mixed problem (9.1) and satisfies the estimates in Theorems 9.1-9.3.
306
4. Mixed Problems in a Quarter of Space
The uniqueness of a solution to the mixed problem (9.1) is proved in the same way as the corresponding assertion for the mixed problem (1.1).
§ 1 1 . Statements of the Mixed Boundary-Value Problems for Sobolev-Type Systems In this section, we consider the mixed boundary-value problems in the quarter of the space for Sobolev-type systems K0Dtu+ + A>+ + K2D^u+ + L(Dx}u~ = f+(t, x), M(Dx}u+ = 0, +
t >0, xeM+, 1
B+u + B-D^ u-\Xn=0 = 0, w+| t = 0 = 0,
t > 0, z ' e M n - i ,
xeM+.
We consider the Sobolev-type operator (cf. Chapter 3, Section 2) 1
rtnt,L>nnX) \L(JJ , JJ — t nn n \— I - " - U ~ - ^ E ' -"-1
'~'{J-St ; -L^X ) —
L(DX
where A'o, A'i, K2 are m x m-matrices, det A'o ^ 0, L(DX] and M(DX] are m x (v — m)- and (u — m] x m-matrix operators of differentiation with respect to x with constant coefficients, and D^1 is the integration operator (D^lu(t,x)
= f u ( s , x ) d s ) . The symbol £(z>/,i£) = (^,j(^,«0) °f o operator j C ( D t , D x ) satisfies the following conditions.
tne
Assumption 1. There exist numbers si, . . . , s^ and t i , . . . , tu such that max Sk — 0, tj ^ 0, j — 1, . . . , i/, t,- ^ t i , s,- ^ s 1; z = m + 1, . . . , ix, and a vector a = (QI, . . . , a n ), a,- > 0, such that for k, j = 1,. . . , v we have lk,j(iri,it) = 0 if s f c + t j < 0 and lk,j(iri,cai£) = c8fc + '>/*,, (177, *£), c > 0 if s/c + t j ^ 0, where t^- /a, are natural numbers. Assumption 2. Th equality det(M(i0^ 1 ^(iO) = °. ^ ^ M "> and only if £ = 0.
holds if
From Assumption 1 we find ti = . . . = t m = — si = . . . = — s m . By Assumptions 1 and 2, for sufficiently large Re r > 0 we have det £(r, r~\ if) = det(rA 0 + A'i + r'1^) x det(-M(tO(r/^o + #1 + r- 1 A' 2 )- 1 L(z'0).
(11.2)
§ 11. Sobolev-Type Systems. Statements
307
Consequently, there exists 70 > 0 such that the equation det£(r,r- 1 ,z's,z'A) = 0,
Re r ^ 70,
s £ M n _i\{0}
(11.3)
has no real roots A. Let /z be the number of roots lying in the upper halfplane. In addition, we require the validity of the following assumption. Assumption 3. If the boundary operator 5_ differs from zero, then tm+l = . . . = tj, > t]..
We indicate conditions on the boundary operator in (11.1). Assumption 4. The matrices .8+ and 5_ have size // x m and // x (^ — m) respectively, and the boundary-value problem on the half-axis (rKQ + KI + T~lK2)v+ + L(is,DXn)v~ = 0, M(is,DXn)v+ = 0, +
xn>0,
l
B+v + r- B-v-\Xn=0 = v,
( 1L4)
sup \v(r}s,xn)\ < oo, Xn>0
where Re r ^ 70, s £ M n _i\{0}, satisfies the Lopatinskii condition, i.e., the problem is uniquely solvable for any vector = ( < £ > i , . . . , 0} (cf. Chapter 1, Section 10). A basis o>i(r, s, xn),... ,u; M (r, s, xn) for the space of solutions to the problem (11.4) is said to be canonical if every vector-valued function U j ( T , s , x n ) = (u+(r, s,xn),u~(r, s,xn)Y is a solution to the problem (11.4) for 71 > 70, s £ M n _i\{0) the elements ^T+l (r,s,xn), I = 1 , . . . , ^ — m, of the canonical basis of the problem (11.4) satisfy the estimate \u™+l(r,s,xn)\ <: (^(fl+Xs) 1 -*'- 1 ' +e 2 (S_))- 1 e- ( 5 < s > t t ' 1 ^, n-1
where (s)2 = ]P st-
xn ^ 0,
,
, S > 0 is a constant, £i(5+), £2(5-) are nonnegative
functions of entries of the matrices 5+ and B- respectively vanishing only for zero matrices.
308
4. Mixed Problems in a Quarter of Space We introduce Sobolev spaces of vector-valued functions similar to the m
.
spaces introduced in Chapter 3. We denote by J~] Wp3^3 (^^+1) the space j=i of vector- valued functions
For the Sobolev space with the same smoothness exponent F] W j=i we write ' ( K n i ) f°r brevity, and denote the norm as follows:
1=1 V
Further, we introduce the notation p1 = p / ( p — 1), T — £ ( 5 » ~f~ ^')> l a l = n
m+l
^
a
X] z , tmax — max{t m +i,. . . , t^}, £(r, z'^) is the conjugate to £(r, r" 1 , z^) and £+(r, z'£) is the m x m-matrix consisting of the entries of the first m rows and m columns of the matrix £(r, z'£). Theorem 11.1. Let B_ = 0, anrf let degA B+£+(iri + 7, z's, z'A) < r/a n . 7/|a|/p' > t max — ti ; then the mixed problem (11.1) /ias a unique solution U+(t, X.} G W^p1'^
(^n + l ) )
r+
= (tl/^l, • • • > tl/an),
j=m+l
(11.5)
/or any vector-valued function /+(/, x) e ^^(Kj+j) n L P , 7 (M+; L 1 (M+)) )
(11.6)
ana7 Me solution u ( t , x ) satisfies the estimate j
, ,
j=m+l
J|| + ||||/ + (^a:) ) L 1 (M+)|U P)7 ^ where the constant c > 0 zs independent of
f+(t,x).
(11.7)
§ 11. Sobolev-Type Systems. Statements
309
Theorem 11.2. Let B- = 0; and let degA B+jC+(irj -f 7, is, i\) = r/an. If \a\/p' > t max — t i + a n , then the mixed problem (11.1) has a unique solution (11.5) for any vector-valued function
and the solution u ( t , x ] satisfies the estimate v j=m+l
(11.9) where the constant c > 0 zs independent of f+(t,x). Theorem 11.3. Let B- ^ 0, one? let \a\/p' > t max - ti. // degA B+£+(ir] + 7, is, i\) < r/an> then for any right-hand side + f ( t , x ) in (11.6) the problem (11.1) zs uniquely solvable in the space (11.5) ancf £Ae solution satisfies the estimate (11.7). // degA B+C+(ir) + 7, is,i\] — r/an, then for the right-hand side + f (t,x] in (11.8) the problem (11.1) /ias a unique solution (11.5) and solution satisfies the estimate (11.9). We give examples of the mixed boundary-value problems satisfying the above conditions for the Sobolev system, the internal gravity wave system, and the gravity-gyroscopic wave system in the Boussinesq approximation. Example 11.1 (the Sobolev system). We consider the following boundaryvalue problem for the Sobolev system U
-L
t
r -4- d) —T —I-i T~7 — \/ II
— LIIU
, U/J -|-
divw~*~ = 0,
V U
.r-4- / -L
\
— l T I — JT i \L,
/ > 0, x 6 Mj,
^r&iut +6 4 / u-ds\X3=0 = 0, i=i J
t > 0, x'
(11.10)
where u = ( 0 , 0 , w ) , 6 1 , . . . ,64 are real constants. The problem (11.10) for bi = b-2 = 64 = 0, 63 = 1 is often referred to as the second boundary-value
310
4. Mixed Problems in a Quarter of Space
problem and in the case u \X3=o = 0 it is called the first boundary-value problem for the Sobolev system. In the case 61 = 62 = 63 = 0, 64 = 1, the problem (11.10) is reduced to the first boundary-value problem by differentiating the boundary condition with respect to /. As we know, the differential operator A
-W
0
w
A
0
A!
DX2
0
0
A
A; 3
i-'.r -
DT. i
JL'rr»
U
satisfies Assumptions 1, 2 and the following conditions: si = s 2 — 83 = — 1, s4 — 0, ti = t 2 = ts = 1, 14 = 2, «! = a-2 = #3 = 1, det £(r, z<^) = r 2 |£| 2 + o;2^|. Therefore, the equation det£(r, z's, z'A) — 0, Re r > 0, s £ K.2\{0} has a single root AI(T, s) = i j s l / y l + ^ 2 / T 2 lying in the upper half-plane, i.e., fi = I . We indicate conditions on the coefficients 6 1 , . . . ,64 in the boundary condition in (11.10) under which the Lopatinskii condition holds. Consider the boundary-value problem on the half-axis x3 > 0 for the system of differential equations with parameters
"r
-u
0
isl '
Ul
T
0
is-2
0
0
r
DX3
isi
is-2
DX3
0
4
= 0,
x3
(11.11) +
+
+
:
-b^v~
7
sup
, s, x3}\ + |v (r, s, z 3 )|) < oo,
where Re r ^ 70, s G M2\{0}. The first two equations of the system can be written in the form
v
- 0,
^~ = 0.
(11.12)
Therefore, the boundary-value problem (11.11) is equivalent to the following
§ 11. Sobolev-Type Systems. Statements
311
problem: DX3v~ + rv£ = 0, 2
\s\
+
'"2
-
1 LL) ••• -\I •—•
x3 > 0,
~
/
/ \ \
_,
oTT(s2 - SIU/T) }v \X3=o = Tip, /IT
/I
sup(|vj(r,s,z 3 )| + |t;~(r,s,x 3 )|) < oo, where Re r ^ 70, s € R2\{0}. A basis for the space of solutions to the system of equations DX3w~ + rw+ = 0,
is formed by the vector-valued functions
Therefore, the bounded solutions for £3 > 0 to this system have the form fw(Ttstx3y\_ceiXl(Ti8)X3f-ii(T,s)T \W
(T,8,X3)J
\
\
1
(11.14)
J
Then the unique solvability condition for the boundary-value problem (11.13) (consequently, for the boundary-value problem (11.11)) takes the form
(61 (*i + 52o;/r) + b2(s2 -
8^ IT))
+0
(11.15)
for Re r ^ 7 o , s G M 2 \{0). We give a criterion for the Lopatinskii condition (11.15). Lemma 11.1. // 64 = 0, then the Lopatinskii condition (11.15) holds if and only if b2 + 62 < co^l; where c0 = 2/(<\/l + ^ 2 /7o — !)• //6 4 ^ 0, ^/ien ^/ie Lopatinskii condition (11.15) Ao/rfs z/ and only if 63 64 ^ 0.
312
4. Mixed Problems in a Quarter of Space
We denote by A' the cone { ( x , y, z] £ MS : x2 + y2 < coz2} and by K the closure of this cone. Then the lemma can be formulated as follows: Lemma 11.1 (a version). 7/64 = 0, then the Lopatinskii condition fails if and only if the point B = (61,6:2,63) does not belong to the cone K. If 64 ^ 0; then the Lopatinskii condition fails if and only if either 63 64 < 0 or the point B = ( b i , 6 2 , 63) does not belong to K. PROOF. We note that if the Lopatinskii condition fails, then it means that there exist (r, s ) , Re r ^> 70, s £ M 2 \{0}, such that Re l(r, s) = 0,
Im l(r, s} - 0.
(11.16)
We begin with the case 64 = 0. By the homogeneity with respect to s, it suffices to consider s G M 2 , \s\ — 1. Introduce the notation , ,2
d = Re
. b = Re ^/W} In the above notation, the system (11.16) can be written in the form
where ., , \si6 A(r, s); = \ v [-si
Two cases are possible: det A(r, s) — 0 or det^4(r, s) 7^ 0. Since detA(T,s) = \s\2(6b-(3d),
8 = --Q, a
(11.19)
we have det A(r, s} = 0 for s E ffi.2\{0} if and only if ft = 0; moreover, in this case, the first row of the matrix A(r, s} consists of zeros. Consequently, if the Lopatinskii condition fails for the data (61 , 6 2 , 63) and det A(r, s) = 0, then ranker, s) = rank \A(r, s) :
«3
in view of the Kronecker-Capelli
theorem. Hence 63 = 0. In the case detA(r,s) ^ 0, from (11.17) for \s\ = 1 we find (1L20) ,
( (n 21)
'
§ 11. Sobolev-Type Systems. Statements
313
where (p is the angle between the vectors I 1 and I , I . It is easy to \sij \-bJ establish that the function <3>(r) = (d2 + b2)/(Sb — (3d)2 maps the complex half-plane { r 6 C : R e r ^ 7 0 } onto the half-line {t G MI : t ^> c0}. Therefore, from (11.20) and (11.21) for 63 ^ 0 we find (6i/6 3 ) 2 + (6 2 /6 3 ) 2 ^ CQ. By the above arguments, if the Lopatinskii condition fails, then the point B — (61, 62, 63) does not belong to the cone K. To prove the inverse assertion, we assume the contrary. Let (61, 6 2 , 63) $• K, and let the Lopatinskii condition hold. Then 63 ^ 0. Otherwise, setting Im T = 0, we obtain a matrix A(T, s) in (11.18) whose first row consists of zeros. Therefore, there is s, |s| = 1, such that (11.17) holds. Taking into account the definition of the function $(r), we can write the condition (61, 6 2 , 63) $. K for 63 ^ 0 in the form (bi/b3)2 + (b2/b3)2 - $(r°), Re r° ^ 70. Then 61 = &3\/$(r 0 )cos y° and 62 = 63A/° € [0,27r). Consequently, taking s°, |s°| = 1, such that the angle between the vectors ( n ) and I • I is equal to
where TJ>I(T,S) = — Re >/l -f u» 2 /r 2 — —-|s| and ^^(r] = — Im \/l +u2/T2, 64
whereas the matrix A(r,s) is defined in (11.18). We note that Re >/l +o; 2 /r 2 > O f o r R e r > 0. If/(r°,s°) = 0 and det A(T°, s°) = 0 for given (6 1; 6 2 , 6 3 ,6 4 ), then, taking into account (11.19) and the condition r a n k e r0, *0) = rank
° 00 ) ! (r°,s
/
we find 6364 < 0. Suppose that /(r°,s°)j= 0 and det^(r°,s 0 ) ^ 0. We show that either 63 64 < 0 or (61, 62, 63) ^ K. Assume the contrary. Let 63 64 ^> 0, and let (61 , 6 2 , 63) G K. Taking into account (11.19) and (11.22), we have 61/64 - -F(r°, s°) sinfci + 2), 62/64 = -F(r°, 8°) cos(y»i + v»2),
(11.23) (11.24)
314
4. Mixed Problems in a Quarter of Space
where _ yd2 + b2 \7(^2(T°)/?/^) 2 + (^i( r °) s°)) 2 * ^ ' ^ - (Sb - (3d) |s°| o
T
0. s
o?i is the angle between the vectors ( sn ] and I b, 1, V i/ \~ J COS 0?o =
sm o>9 =
Then 6? + 6?> = 6 2 1 F 2 (r 0 ,s 0 ) > c06§, i.e., (61,62,63) £ #• Hence we arrive at a contradiction. Consequently, if the Lopatinskii condition fails, then either 6364 < 0 or (61, 6 2 , 63) £ A. We prove the converse assertion. Let 6364 < 0. We consider r° such that Im r° = 0. Then the first row of the matrix A(r°, s) in (11.18) consists of zeros and ^(r0) = 0. For s° E M 2 \{0} such that
we obtain (11.22) for j^= r°, s = s°. Let (61,62,63) ^ A'. We show that the Lopatinskii condition (11.15) fails. We assume the contrary. Let the Lopatinskii condition hold. The aforesaid means that it suffices to consider the case 63 64 ^ 0. The condition (61,62,63) ^ K can be written in the form 63
We note that F(r,s) —> +00 as |s| —>• 0, F2(r,s) > co(63/6 4 ) 2 , and for any fixed r, Re r ^ 70, the function F(r, s) monotonnically decreases as |s| increases; moreover, F(r,s) -^ Tjcvbz/b^ as |s| —>• oo. Therefore, there exist r°, Re T° ^ 70, and s° G M 2 \{0}, such that (11.23), (11.24) are satisfied; we arrive at a contradiction. Consequently, if 63 64 < 0 or (61,62,63) ^ A', then the Lopatinskii condition fails. D By Lemma 11.1, the Lopatinskii condition for the boundary- value problem for the Sobolev system (11.10) holds if the coefficients 61, . . . , 64 satisfy the following conditions: (61,63,63) 6 M 3 \{6 3 = 0}U {0,0,0}, 63600, |63| + | 6 4 | ^ 0 . (11.25)
§ 11. Sobolev-Type Systems. Statements
315
Indeed, since the opening of the cone K is CQ = 2/\/l + w 2 /7g — 1, for a given vector (61,62,63) in the domain (11.25), we can indicate a number 70 such that (61,62,63) £ K. Consequently, for Re r ^ 70, s £ M2\{0) the function l(r, s) defined by formula (11.15), does not vanish. Moreover, we have \l(r, s)\ 2 M(N - V( f c i + 6 l)/co) + |64| Re \/l + o; 2 /r 2 .
(11.26)
To prove (11.26), we introduce the notation
64 Re
b= Then
|/(r,s)| = |A(r,s)6 + /(r,s)|.
(11.27)
From (11.15) and (11.16) we find
Therefore, by Lemma 11.1 for Re r ^ 70, from (11.27) we obtain (11.26). The above criterion implies, in particular, that the first and second boundary-value problems for the Sobolev system satisfy the Lopatinskii condition. We note that the norms of the solutions to the boundary- value problem (11.11) can unboundedly increase as \s\ —> 0. Indeed, in view of (11.12)(11.15), the solution to the problem can be represented as
-i\i(r,s)r
l
For example, if 61 = 62 = 64 = 0, by (11.15), for the fourth component of the solution we have \v-(r,s,x3)\ = | e '^(^)^i _> 00>
|s| ^ o
03\S\
We illustrate Theorem 11.1-11.3 by the mixed-problem for the Sobolev system (11.10). In this case, the matrix £(r, is,i\), r = irj + 7 takes the
316
4. Mixed Problems in a Quarter of Space
form
£(r, is, iX) X )
U)X
— rsiX u>is2T — ir2 si
— TS\S2
— T S i X — U>S2X
—rs2X—ujsiX —ir2s2 — u>is\r
T(s2 + s2) —ir2X — iu2X
Therefore, B+£+(r,is, iX) = (bi 62 = (b\r(s\ + A 2 ) — b2(rsiS2 + u;A 2 } 6i(u;A — rsis2) + b2r(s2 + A 2 ) — 6 3 (rs 2 A + u>siA), -f b2(ujsiX — rs2X) + 6 3 r(s 2 -+• 5 2 ))Consequently, by Theorem 11.1, the second boundary-value problem (61 = 62 = 64 = 0) has a unique solution .,+ ^"^ \ U (+ { I , r\ X) c t lA/^-'l ''p -y /"TO> V^4 />
n~ W (+ ( I , v\ X) a t T/T/^'2/'m)++'\ K K _ -, ^"^4 )
f-t^ iil .oo\ /oj
for p > 3/2, and, by Theorem 11.3, the first boundary-value problem (61 — 62 = 63 = 0) is also uniquely solvable in the indicated spaces. If 64 = 0 and \bi\ + \b2\ ^ 0, then deg A B+£+(T, is, iX) = 2 and, by Theorem 11.2, this problem has a unique solution (11.28) for p > 3. Example 11.2 (the internal gravity wave system in the Boussinesq approximation). We consider the following boundary-value problem for the internal gravity wave system in the Boussinesq approximation: AW + + DXlu~ — f i ( t , x), t
AW* + Ac 2 w~ = f2(t, x),
w + r f s + DX3u~ = f s ( t , x ) , 0
DXluf + A 2 wJ + DX3u+ = 0,
t > 0, x E M+,
(11.29)
C
64 / u-ds\X3=0 = Q,
t> 0, x' £
0
where 6 1 , . . . ,64 are real constants. The problem (11.29) for is usually referred to as the second boundary-value problem if 61 = 62 = 64 = 0, 63 =
317
§ 11. Sobolev-Type Systems. Statements
1 and the first boundary-value problem if u \XA=Q — 0. In the case 61 = 62 = 63 = 0, 64 = 1, the problem (11.29) is reduced to the first boundaryvalue problem by differentiating the boundary condition with respect to t. The integrodifferential operator
'A
0
0 0
A 0
Dt
A,
DXl
0
satisfies conditions 1 and 2; moreover, si = 82 = 83 = —1, 84 = 0, ti = t 2 = + |£|2 + t 3 = 1, t 4 = 2, ai = a, = a3 = I , det £(r, r '1 Therefore, the equation r 2 |s| 2 + r 2 A 2 + JV 2 [s| 2 = 0, Re r ^ 70 > 0, s e R2\{0}, has a single root AI(T, s) = i>/l + A/" 2 /r 2 |s| lying in the upper half-plane, i.e., /it = 1. We indicate the conditions on the coefficients 61 , . . . , 64 in the problem (11.29) under which the Lopatinskii condition holds. Consider the boundary-value problem on the half-axis £3 > 0 for the system of differential equations with parameters
0 0 is\ ivt\ r 0 IS2 V* = 0, 4 0 'r + N2T~l DX3 oJ V / i«2 DX3 _ . + + 1^1 -r- 2^2 + 6 3 ^ + -\^v X3=0 ~ T
"r 0 0 _tsi +
x3>0,
(11.30)
oo, ar 3 >0
where Re r ^ 70 ) s € M2\{0}- The first two equations imply TV* +is\v~ = 0 and TV*. + is2v~ = 0. Therefore, the boundary-value problem (11.30) is equivalent to the following problem:
o, (11.31) sup ( \ V ^ ( T , S , x 3 )| + |v~(r, s,x 3 )|) < oo, where Re r ^ 70, s G M2\{0}. a basis for the space of solutions to the system of equations
Dx
+ (^ +
;+=0,
TDX3W+ +
=0
318
4. Mixed Problems in a Quarter of Space
is formed by the vector-valued functions n(r,s)r-1/ 1
1 i.e., the solutions to this system, bounded for x$ > 0, have the form
w~(r,s,x3) Hence the condition of the unique solvability of the boundary-value problem (11.31), and, consequently, of the problem (11.30), takes the form l(r, s} = 63
,
|S
'
, : + 64 - i(6isi + 6 2 S 2 ) + 0,
(11.32)
where Re r ^ 70, s G M 2 \{0}. We give a criterion for the Lopatinskii condition (11.32). Lemma 11.2. The Lopatinskii condition (11.32) holds if and only if [63) + 64] ^ 0 and 63 64 ^> 0. PROOF. We prove that if there exists a number 70 > 0 such that l(r, s) ^ 0 for Re r ^ 70, s G M 2 \{0}, then 63 l>4 ^ 0, |63| + |64| ^ 0. We assume the contrary. Let one of the following conditions hold: either [63) + |64| = 0 or 6364 < 0. Taking the vector ( s i , s 2 ) orthogonal to the vector (61,62) in the first case and setting s\ = —6 4 \/(Re r) 2 + A r 2 /(6 3 Re r) in the second case, we conclude that l ( r , s ) — 0 for any real part of r; we arrive at a contradiction. We prove the inverse assertion. Let 6364 ^> 0, |6s| + |64| ^ 0. It is easy to see that the expression — Im I(T, s) = 6i«i -f 62^2 ~ ^3 Im
T
— \s\
can vanish for any bj. Since Re (r/\/T 2 + A^ 2 ) > 0 for Re r > 0, we have
Re l(r, s) = 63 Re for any r, Re r > 0, s e!R 2 \{0}).
\fTZ
/
| g | + 64 / 0
+ N^
D
From the proof of the lemma it follows that if the Lopatinskii condition holds, then the following estimate holds: |/(r,s) ^ |di6s|s| -f 64 , where d\ > 0 is a constant. We note that the first and second boundary-value problems for the internal gravity wave system satisfy the Lopatinskii condition.
§11. Sobolev-Type Systems. Statements
319
Example 11.3 (the gravity-gyroscopic wave system in the Boussinesq approximation) . Similarly, we can write a criterion for the Lopatinskii condition for the following boundary-value problem for the gravity-gyroscopic wave system in the Boussinesq approximation: Dtu+ — utu^ + DXlu~ — f i ( t , x ) , Dtu% -f wuj" + Dx^u~ = f2(t, x ) , t
ds + Dtiu~ = / 3 (t,
f
t
iuf +6 4 / u~ds\X3=Q = 0,
t > 0, x' e R 2 ,
Examples 11.1-11.3 show that the solution to the boundary- value problem (11-4) can unboundedly increase as js| —> 0. Such a property of the boundary- value problems (11.4) can essentially affect the solvability conditions for the mixed problems (11.1). We trace the character of singularities at s = 0 of the solutions to the boundary- value problem (11.4) by considering two special examples of the mixed problem (11.1). Since there exists the inverse matrix to K(r] = (rA'o + A'i + r~lK-2) for Re r ^ 70, the boundary- value problem (11.4) is equivalent to the following problem:
M(is,DXn)K-l(r}L(is,DXn)v-
=0,
xn > 0,
sup \ v ~ ( r , s , x n ) \ < oo,
xn>0
where Re r ^ 70, s E M n -i\{0}. Consequently, the Lopatinskii condition can be formulated as the unique solvability of the boundary- value problem (11.33) for any vector
In the case B- — 0, the boundary condition of the boundary- value problem (11.33) takes the form B+K~l(r)L(is, DXn)v~ \Xn=o = — f . If
320
4. Mixed Problems in a Quarter of Space
v~ (r, s,xn] is a solution to the problem (11.33), then for any c > 0 the following identity holds: v- (r, s, xn) ~ c- ^T^m(c)v- (r, ca' s, C-a*xn),
(11.34)
where T M _ m (c) is a diagonal matrix with entries c tm+1 , . . . , ctv . Indeed, by Assumption 1 on the symbol of the operator jC(Dt, Dx), we have L(c a z'£) = c-^L(i£)Tv-m(c} arid M(cai£) = c t l 5 l / _ m (c)M(z'0, where 51,_m(c) is a diagonal matrix with entries cs™+1 , . , . , cs" . Since -qa
we conclude that M(f S ,D ; r JA'- 1 (r)L(zs, = c~^M(is,c-a"DZn) Tv-m (C)V-(T, ca's, zn Sv-m(c = 0,
Moreover. sup \c~tlTu-m(c)v~ (T, ca s,c~anxn}\ < co. xn>0
Consequently, the vector-valued function c~ tl T l/ _ m (c)z;""(r, ca s, c~anxn] is a solution to the boundary- value problem (11.33) for B- = 0. Since a solution is unique, we obtain (11.34). Setting c- (s}-1, (s)2 = "f] sz2/a' in (11.34), we find
and
§ 12. Sobolev-Type Systems. Approximate Solutions
321
Consequently, the jih components vm+J(t,x) of the solution to the problem (11.4) can unboundedly increase as s —> 0 with rate (s)~( tm+J ~ l l ). Therefore, for xn > 0 the following estimates hold: \V+( T u T
I
c
}\ <
— f.-5(Srn*n
\ i * j x n )\ ^ i i X
m+ \v J(T
(s)
m
'
°, _ t-, er -<5(a)° n a; T t ) +J M
— x1> • • • i " i/ — ''*) m ./, —
where the constants c, <£ > 0 are independent of T, s, xn. We consider one more example of the problem (11.1) with £?+ = 0. Let t m+1 = ... = tj,, i.e., the boundary condition in (11.33) takes the form r~lB- \Xn=Q — ¥>• If v~ (T, s, xn) is a solution to this problem, then for any c > 0 we have the identity v~ (r, s, xn) = v~ (T, ca s, c~Qnxn) which can be verified in the same way as (11.34). This identity implies that v+(r, s,xn)
v~ (r, s, xn) = v- (r, ——7, (s) a "x n ), and, consequently, for xn > 0 \v+( I ^ V Tj
6
c ^s ' x^ ^ }\/ l< ^ i—(<;}*• i N 6 / "-^p\T\
W"1"- )
6 \v~(r ^°nXn ' | y \ , ' ' ds ' J /xn / }\ I : :< ^ cce~ 'c
where the constant c > 0 is independent of r, s, xn.
§ 12. Approximate Solutions to the Mixed Problems for Sobolev-Type Systems In this section, we construct approximate solutions to the mixed boundaryvalue problem (11.1). We follow the scheme of constructing approximate solutions to the mixed problems for equations that are not solved relative to the higher-order derivative. We assume that the right-hand side f + ( t , x ) in (11.1) belongs to the space C°°(M^1). We preserve the notation for the extension of this function by zero for t < 0, xn < 0. We denote by f+(t,s,xn) the Fourier transform of the vector- valued function f+(t,x',xn) with respect to x' and by /+ (r, s, x n ), T = irj + 7, the Fourier transform of the vector- valued function e~ 7t / + (2, x',xn) with respect to (t,x'). We extend the vector-valued function f+(t,x',xn) to the whole space Kn+i- as follows: for t < 0 by zero and for x n < 0 in such a way that
322
4. Mixed Problems in a Quarter of Space
the extended vector- valued function F(t,x',xn) is sufficiently smooth and compactly supported. In Section 13, we indicate some additional conditions on F(t,x',xn). We denote by F+(t,s,xn) the Fourier transform of the vector-valued function F+(t,x',xn) with respect to x' and by F+(T, s,xn), r — irj + 7 the Fourier transform of the vector-valued function e~^tF+(t, x' , xn) with respect to ( t , x ' ) , and by F+(T],s,£,n) the Fourier transform of the vectorvalued function e~~ftF+(t, x' , xn). with respect to all variables. Consider the boundary-value problem on the half-axis {xn > 0} for the system of differential equations with parameters (r, s), Re r ^ 71 > 70,
C(r,T-l,is,DXn)u =
> > «
Xn>0
,
sup \u(r,s,xn)\ < oo,
xn>0
where, by definition, C(r r-1 ,is,D ,', n XnW L(r,r )u-The Lopatinskii condition implies the unique solvability of this problem. We write the solution to the problem (12.1) in the form UJ(T, s, xn) = WQ(T, s, xn) + v(r, s, xn),
(12.2)
where UQ(T, ^, xn) is a bounded solution to the system f + C\'r > °>
the vector-valued function v(r}s,xn) is a solution to the boundary-value problem jC(r,T-1,is,DXn)v = 0,
xn >0,
B+v+ + r-^.v-^^o = -B+u+ - T-lB-u~\Xn=Q, sup u(r, s, xn)\ < oo. a:n>0
Introduce the notation a(r,is,i\) = det C(r, r""1, is, z'A), jC(r,is,i\) is the conjugate ti £(r, r" 1 , is, z'A). For a sufficiently smooth vector-valued
§ 12, Sobolev-Type Systems. Approximate Solutions
323
function u(xn) we have £(r, r""1, is, DXn}C(r, is, DXn}ui(xn} = a(r, is, D X n ) u j ( x n ) . The equation (cf. (11.3)) a(r, is,i\) = 0, Re r ^ 71, s G M n -i\{0}, has no real roots A. Therefore, the boundary- value problem on the line a(r, is,DXn)u = g ( x n ) , sup — oo
-oo < xn < oo,
\u(r, s,xn)\ < oo,
(*• "
where Re r ^> 71, s G R n _i\{0}, has a unique solution for any bounded function g ( x n ] £ C(Ri). Consequently, using explicit formulas for the solution to this problem constructed in the same way as in Chapter 1, Section 9, for a bounded solution UQ(T, s,xn) in (12.2) we can take a vector-valued function of the form
£n +
RF (r, s,xn}= I
J+(T,s,xn-yn}F+(T,s,yn)dyn
— oo 00
4- / J-(r, s,xn - yn] F+(r, s,yn)dyn,
(12-5)
xn
where l i f \ f ex P( i;E nA) J+(T, s,xn) = — I ——:—~-d\, 2?r J a(r,is,i\) r+
(12.6)
r1
+
Furthermore, the contour F" " = F (r, s) surrounds all the roots of equation (11.3) lying in the upper half-plane and the contour F~ = F~(r,s) surrounds all the roots lying in the lower half-plane. Using the canonical basis uij(r,s,xn), j = 1, . . . ,/j,, of the boundary-value problem (11.4), we can represent the vector-valued function v(r, s,xn] in (12.2) as follows: v(r,s,xn) -
pj(r,s)uj(T,s,xn), j=i (r, s} = -B+u+(r, s, 0) - r~lB.^ (r, s, 0).
(12.8)
324
4. Mixed Problems in a Quarter of Space
From formulas (12.2), (12.4), (12.8) we obtain the following representation of a solution to boundary- value problem (12.1):
(r, s,xn) = C(r,is,DXn) where
-T-I T
yn=0
Since the vector-valued functions (12.4) and (12.8) are analytic and bounded with respect to r, Re r ^ 71, we can apply Theorem 5.2 in Chapter 1 to the vector- valued function (12.9). Consequently, the vector- valued function v(t,s,xn) = (v+(t,s,xn),v~ (t,s,xn)Y , where v+ ( t , 5, *„) - (27T)- 1 / 2
e^+^u; + (zr ? + 7, s, x n
v-(ttstxn) = (27T) is independent of 7 ^ 71 and is a solution to the mixed problem with parameter s E M n _i\{0}: K0Dtv+ + KIV+ + K2D~lv+ + L(is, DXn}v~ = f+(t, s, xn), M(is, DXn)v+ = 0, +
1
/ > 0,
B+v +B-D- v-\Xn=0 = Q) v+
t=o= °>
xn > 0, t>Q,
x
* > °-
We proceed by constructing approximate solutions to the mixed boundaryvalue problem (11.1). We can obtain a formal solution to this problem by applying the inverse Fourier operator with respect to s to the vector- valued function v(t,s,xn). However, the contour integrals (12.6), (12.7) and the components of the canonical basis of the boundary- value problem (11.4) have, generally speaking, nonintegrable singularities at s = 0. Therefore, as
§ 12. Sobolev-Type Systems. Approximate Solutions
325
in the construction of approximate solutions to the mixed boundary-value problems for simple Sobolev-type equations, it is necessary to regularize the Fourier operator. For this purpose, we use the integral representation of summable functions in Chapter 1, Section 6. We introduce the vector-valued function v~l l/k
I
eix'sG(sva')
En_i
oo f
\
/ e(ir>+^tu+(ir) + -f,s,xn)dr) \dsdv, J /
= (2ir)-n/2
f v-1
I
(12.10)
eix'sG(sva>}
(12.11) where 2N
exp(-(s}™},
s?0".
(sf
(12.12)
t=i
We choose N such that the vector-valued function is infinitely differentiable and summable with any power. By construction,
x)u(t,x}
= Q,
t>Q,
xEM+,
l
B+ u+(t, x', 0) + B.D~ u- (t, x', 0) = 0, 00, u+(Q,x) = Q,
xGR+,
where + ( t ) x ) = (27r)1-n f v-1 I/A;
f En_i
f
x'
326
4. Mixed Problems in a Quarter of Space
Since \ \ f £ ( t , x ) — f+(t,x), LP|7(IR^1)|| -» 0 as k -4 oo, the vector-valued function Uk(t, x) = (u~£ (t, x), u^(t, x)Y is a solution to the mixed boundaryvalue problem (11.1). In Section 13, we give L p -estimates for the vector- valued function Uf. (t, x) which mean the convergence of the sequence {ut-(t, x ) } to the solution of the mixed problem (11.1). To obtain estimates for the approximate solutions, we use some properties of solutions to the boundary-value problem (12.1) given below. As in Chapter 3, Section 4, for Re T ^ 7i, £ £ E n \{0} introduce the notation K(r) = rKQ + KI + T~1K2, N ( T , £ ) = Lemma 12.1. For any c > 0 for Re r ^ 71, f £ M n \{0} the following identities hold: (12.13) +
a
f) = 7V (r,c f), (12.14) where T^-m(c] and 5^_ m (c) are diagonal matrices with entries c t m + 1 , . . . , c ty ana" c S m + 1 ,. . . , cs" respectively. PROOF. From Assumption 1 on the symbol of the operator £(Dt, Dx] we have L(c a if) = c~ ^(if )7;_ m (c) and M(c Q if) = c t l 5 J / _ m (c)M(iO, where Si/_ m (c) is a diagonal matrix with entries c S m + 1 , . . . , c s ". Therefore, (12.13) holds, which implies (12.14). D Lemma 12.2. For Re r ^ 71, f £ R n fAe following identity holds:
a(r, c a if) = cra(r, if),
r = V"^ ( s i+t z -),
c > 0.
(12.15)
1=771+1
Fwr^Aermore, n
i=l
where a^ ^> a\ > 0 are constants independent of T, f. PROOF. We write (11.2) as follows: a(r, if) = det A'(r) det(—-/V(r, f)). By (12.13), we obtain (12.15). In (12.15), we set c = (f)- 1 . Taking into account that a ( r , i f ) ^ 0, Re r ^ 71, f £ M n \{0), we obtain (12.16). D We note that r Q(\ r ,; if) + f/ -> / — flo(if)' ^ \ ^/
j
k-0
a/_t-(if)T~ * n, \ ^, / >,
/ = 2m — ZAi
(12.17) v /
§ 12. Sobolev-Type Systems. Approximate Solutions
327
where ao(i£) = det A'o d e t ( — M ( i £ ) K Q L(i£)). Indeed, representing (11.2) in the form T"-2ma(T,i£) = det(A0 + r~1Kl + T~2K2) det(-M(i£)(KQ + r~lKi + r~2K2)~1L(i£)) and passing to the limit as \r\ —»• oo, we obtain (12.17). As we know, the equation a(r,is,i\) = 0, Re r ^ 71, s e M n _i\{0}, has no real roots A and for s = 0, by Lemma 12.2, has a single root with multiplicity r/an. By Lemma 12.2, all the roots A + ( r , s ) , . . . , A+(r,s), Im At(r, s) > 0, and A ] " ( r , s ) , . . . , A ~ ( r , s), Im A~(r, s) < 0, q = r/an -//, are homogeneous with respect to s relative to the vector a', i.e., A^~ (r, ca s) — c a n A^~(r, s) and \^(r,ca s} = c a n A~(r, s). Consequently, there are constants A > 6 > 0 such that all the roots satisfy the inequalities 26(s)an ^ Im At( r , s) ^ |At( r , s)| < A<s) a n , 2o\s/ ann ^ —1m Aj-- (T, sj ^ |A-- (T, sj| ^ Z\^syI n .
(12'18)
From these inequalities one can obtain estimates for the contour integrals (12.6) and (12.7). Lemma 12.3. For xn > 0 the following inequalities hold: k kJr an r l \n ^ ~ \T\~ x n ( — foi/ I ' l p'•'•"•Pv
M 9 1 Q1! ^1^6.iyj
where Re r ^ 71, 5 e M n -i\{0} one? c > 0 z's a constant independent of T and s. PROOF. We consider, for example, the integral J + (r, s,x n ). By Lemma 12.2 and the estimates (12.18), we have
2?r
J
2?r
exp(zx n A)(z'A) /e dA a(r, is, z'A)
7
ixn\)(i\)k dA, a(r,
where sj. = Sk(s) ak, k = 1 , . . . , n — 1, F+ — the boundary of the domain G+ = {A € C : |A| < 2A, Im A > S}. Using (12.16) and (12.18), we obtain (12.19). We establish (12.20) in the same way. D Lemma 12.4. Let £+(r,i£) be an m x m-matrix consisting of entries of the first m rows and m columns of the matrix JC(r, it;). Then for any
328
4. Mixed Problems in a Quarter of Space
m-dimensional vector-valued function g+(xn) 6 C^O^i) for Re r ^ 71, s £ M n _i\{0) the following identity holds: JC+(r,is,DXn)Rg+(r,s,xn) (12.21) where <7 + (£) is the Fourier transform of the vector-valued function g+(xn), and the operator R is defined by (12.5). PROOF. Consider the boundary- value problem on the real line for the system of equations jC(r, T~l,is, DXn}u =
n U
\ \U(T, s,xn}\ < oo,
sup
,
-oo < z n < oo,
/
(12.22)
— OO<Xn<00
where Re r ^ 71, s 6 M n _i\{0}. By the definition of £(r, r'1, is, DXn), this problem is equivalent to the following: Af (is, DXn) A'"1 (r)L(is, DXn)uj= M(is,DXn}K-1(r)g+(xn), sup \u>~(r,s,xn}\ < oo.
xn € MI,
(12.23)
— co<x n
Since the equation det(M(is,i\)K~1(r)L(is,i\)} = 0, Re r ^> 71, s £ Kn-i\{0}) has no real roots A, the boundary- value problem (12.23), and, consequently, the problem (12.22), is uniquely solvable. By the definition of the operator C(r,is,DXn], we have £(r, r"1, is, DXn] o jC(r, is, DXn) = a(r,is,DXn) o /. Since the solution to the problem (12.3) takes the form U(T, s,xn) = Rg(r, 5, xn], we obtain the explicit formula for the solution to the problem (12.22): + T s x
/ x rt • n \ f^9 ( , u(r,s,x n} = £ ( T , i s , D X n ) I » ^
, n}\
' 1.
By Lemma 12.3, we can apply the Fourier operator with respect to xn to vector-valued function s D'XnRg+(T,s,xn), k ^ 0. Applying the Fourier operator to (12.22), we find K(r)Z+(T}t)+L(i®Z-(Ttt)=g+(l-n),
§ 12. Sobolev-Type Systems. Approximate Solutions
329
Therefore, £+(r,£) = K~I(T}(! - N+(T,£))g+(£n). Applying the inverse Fourier operator with respect to £ n , we obtain (12.21). D Lemma 12.5. Let £~t(r, z£) be a (if — m) x m-matrix formed by entries of the last v — m rows and the first m columns of the matrix £(r,z£). Then for any m-dimensional vector-valued function g+(xn} G C^ for Re r ^ 71, s G M n -i\{0} we have
PROOF. From (12.24) it follows that
By the definition of N±(T,£), we have £-(r,£) = N±(T,£)g+(£n). fore,
There-
u;-(r,5,x n ) = (27T - 1 / 2
However, by construction, u;~(r, s,xn) = £l(r, z's, DXn)Rg+(r, s, xn).
D
Lemma 12.6. Suppose that U j ( T , s , x n ) , j — 1, . . . ,//, is the canonical basis of the boundary-value problem (11.4), g+(xn) G C^ORi) is an mdimensional vector-valued function, and V?j(r, s,x n ) = -5 + £ + (r,is, DXn)Rg+(T, s,xn) -T-1B-C^(r,is,DXn)Rg+(T,s,xn). Then for Re r ^ 71, s G M n _i\{0} oo
^j(r,s,0)wj(r, s,ar n ) = - / Dyn(ipj(T, s, yn}uj(r, s, (xn + yn))) dyn.
PROOF. The assertion follows from the Newton-Leibniz formula since the components of the canonical basis exponentially decay as xn —> +00. D
330
4. Mixed Problems in a Quarter of Space Lemma 12.7. For Re r ^ 71; s 6 M n _!\{0} 00
/
n'XnJ+(r, J (T s, v x r n)\ rlr e.,-iXntn u axn 0 0
PROOF. In the case / = 0, this assertion is similar to Lemma 4.4. For / = 1 we have r „
} UJ J
> *j T-t-nj
I
-n i
/
e p-»*n^n n
/
(T
S
T
} dT
±sxnj - { ' > * > , J
/ J
-oo ) I °°
nj|0
I p ~ ^ n^n /
-f e
j _ \if , A , J - n y) lI-°o o T
S
T
0
r
(T,s,xn)dxn + i^n I
e~lXn^nJ,(r,s,xn)dxn.
0
By Lemma 12.3, |J+(r, s, x n )| + | J_(r, s, — x n )| -> 0 as o?n -> +00 and, by definition, -J + (T ! S ! O) + J _ ( T l s , 0 ) = -~ / -1 . rfA, 2?r 7 a(r, zs, zA) r where the contour F surrounds all the roots X ^ ( T , S ) , X ~ ( r , s ) , i.e., -J+(r,s,0) + -/_(r,s,0) = 0 . Consequently, oo
0 lXn ^D-L^Xn J— J (T Wr fc~ \ ' it *jr ^n) UA,n
— oo
a r , zs.z'^n The case / > 1 is considered in a similar way.
D
Lemma 12.8. Elements of the operator C+(r,is, DXn) have the form aa(T}(isy' D°»,
a'a' +
(12.25)
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
331
where a a (r) are polynomials of degree 2m — v — 1. PROOF. From Assumption 1 on the symbol of the operator £(Dt,Dx] we obtain the identity a(r, cai£}£(T, i£) = a(r, if )T(c)£(r, cai£)S(c),
c > 0.
(12.26)
Since ti = . . . = t m — — si = ... = — s m , from Lemma 12.2 we have £+(T, s'f) = c~ r £ + ( r ,c a if)> which implies (12.25). The degree of polynomials a (7 (r) is obviously determined from the equality
and the representation (12.17) for a(r, z'f).
D
Lemma 12.9. For t m +i = . . . = t,, f/ie elements of £l(r, is, DXn) take the form aa(r)(is)a D°xl,
(7V + o-nan = r - t ^ + ti,
(12.27)
where aa(r) are polynomials of degree 2m — v. PROOF. By the identity (12.26) and Lemma 12.2, we have £l(r, if) = t
c "- t i- r £;t(r ) c a if) and r =
f) (st- + t,-), which implies (12.27). The j =m+l
degree of polynomials aa(r) is determined from the equality
and the representation (12.17).
D
§ 13. Solvability of Mixed Problems for Sobolev-Type Systems In Section 12, we constructed asequence of approximate solutions { u k ( t , x)} to the mixed boundary- value problems (11.1) for Sobolev-type systems K0Dtu+ + KlU+ + K2D-lu+ + L(Dx)u~ = f + ( t , x ] , M(Dx}u+ =Q,
00, x e M + ,
332
4. Mixed Problems in a Quarter of Space
In this section, we establish the Lp-estimates for the vector-valued functions {uk(t, x)} which implies the convergence Uk(t, x) -> u(t, x) as k —>• oo. The limit vector-valued function u ( t , x ) = (u+ (t , x) , u~ (t , x)Y is a solution to the problem (11.1). In accordance with formulas (12.2), (12.4), (12.8), (12.9), we write the approximate solution (12.10), (12.11) in the form
1=0
where
oo
1
/e
il>s
Q
G(sv ')(
[
i/fc
&
= (27r)-
n/2
oo
1
ix/s
/v~ fe
J
a>
G(sv )(
J
/"e \ J
l/fc
x WJ(ZT; -(- 7, s, xn}drj \dsdv, where j = 1, . . . ,/^. To obtain the Lp-estimates for vector- valued functions Mj + fc (i,;c), wj~ f c (<,x), we need some auxiliary assertions. Recall that in the construction of approximate solutions we consider the right-hand side f+(t,x',xn] of the system (11.1) in the space C°° (R^ ) under the assumption that it has compact support with respect to ( t , x ' ) and vanishes for xn ^> p. In the sequel, we will extend it to the entire space IR n +i as follows: we extend the function by zero for t < 0 and for xn < 0 in such a way that the extended vector- valued function F+(t, x' ,xn) is sufficiently smooth and compactly supported; moreover, (13.1)
<: c|| ||/+(t, x', xn), LiCM+JU, L P , 7 (R+)||, where the constant c > 0 is independent of f+(t, x' , xn).
(13.2)
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
333
For the Fourier transform we introduce the following notation: F+(t,s,xn) is the vector-valued function F+(t,x',xn) with respect to *'• ~ + F (r, s, xn)} T = ir] + 7, is the vector-valued function e~ 7 t F + (<, x1 , xn) with respect to ( t , x ' ) , F+(r),s,£n] is the vector-valued function e~'ytF+(t, x' , xn] with respect to all variables, ,s,£ n ) is the vector-valued function F+(t,x',xn) with respect to
Lemma 13.1. For the vector-valued function u£k(t,x) the following estimate holds: \\u+k(t,x), W£;+(M#i)|| ^ c \ \ f + ( t , x ) , %°;;+(^i)ll, where the constant c > 0 is independent of k and f+(t,x}. I^O.
7 > 7i, (13-3) Furthermore,
*i. ^ 2 ^ 0 0 .
(13.4)
PROOF. Using the notation in Section 12, and the extension F+(t, x', xn), we write f
r
.
ytu+ /j x\ _ ^rr)""/ 2 / i;"1 / e11 sG(sva ) °' J J l/k
(/"• 00
Introduce the notation K
K(s, n, k) = (27r)-"/
By Lemma 12.4, we have
2
/ v-1G(sva')dv. i/fc
(13.5)
334
4. Mixed Problems in a Quarter of Space
By the assumptions on the operator C(Dt, D^1 , Dx), from Lemma 12.1 it follows that the entries of the matrices (irj+j)1 K~l(ir]-\-^}(I— N+ (in+~f,l;)), I = 0, 1, are multipliers. Taking into account the definitions (12.12), (13.5) and the estimate (13.1), we obtain the inequality (13.3) and the convergence (13.4). D Lemma 13.2. For the Ith component u ™ £ l ( t , x ) of the vector-valued function
the following estimates hold:
|| J D-/+(/ > x),L p i 7 (M++ 1 )||,
(13.6)
where t m + / — ti 7i and the constant c > 0 is independent of k and f + ( t , x ) . Furthermore, /
j
I
x
0,^i V >
x
/
0,K2 ^ '
''
Pi"/V
n-f-1 / 11
V
' /
as k\, k<2 —} oo. PROOF. Using the extension F+(t, x) of the vector-valued function / + (/, x), from the definition of UQ k(t, x) for 7 > 71 we have
n/2
I v~l
! eix'sG(sva<
J
J
l/k
ln_i
~
dsdv.
Using the notation (13.5) and Lemma 12.5, we can write this expression in the following form:
(13.8)
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
335
By Lemma 12.1, the entries of the matrix itt)t
7^7i,
(13-9)
satisfy the assumptions of the Lizorkin theorem about multipliers. From (12.12), (13.5), and (13.1) we obtain (13.6) and (13.7). D Lemma 13.3. // \a\/p' > t m +j — ti, / = 1, . . . , v — m, then the following estimate holds: \\u-k(t,x)t + ||||/ + (i,x),L 1 (lR+)||,L p , 7 (E+)||),
7>7l,
where the constant c > 0 is independent of k and f+(t,x).
(13.10) Furthermore, 00.
(13.11)
PROOF. Estimating the components u™£1 (t, x) of u0 k ( t , x } , we meet two cases: tm+i — ti ^ an and tm+i — ti = ctn. CASE 1: tm+i — ti ^ cnn. Since the entries of the matrix (13.9) are multipliers, using (13.8), we obtain the following estimate for u 0,k t
n,
, 0 dt,
(13.12)
where the constant GI > 0 is independent of k and f+(t,x). Consequently, if t m + / = ti (recall that, by Assumption 1, we have tm+i ^ ti), then from the definition of K(s, n, k} and the estimate (13.1) we obtain the inequality
where the constant c > 0 is independent of k and f+(t, x}. Let tm+i > t i , t m+ / — ti ^ an. By the obvious inequalities
4. Mixed Problems in a Quarter of Space
336
where £ = (s,£n) € I^n\{0}, and the theorem about multipliers, from the inequality (13.12) we find
r[ ••*< , n,
where the constant c2 > 0 is independent of k and F+(t,x). Since ti/a,-, t m +//a; are natural, we have (t m+ / —t\}/an ^ 2. By the definitions (13.5), (12.12) and the vector-valued function F+(t, y), we can change the integrals under the sign of the L p -norm. Using the notation (13.5) and the Minkowski inequality, we find
(( J
\
1 /w"1
f(
f e^
l/k
Using the Young inequality, we find f^1 J
f J
337
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
•/•"I/-«/
I/A;
| «/
ffi n
dv [the change of variables^ = £iVa', z,- = X i V ~ a ' , i — 1 , . . . , n] K
_c2y w
/
i/fc
Taking into account the definition of G(s), the conditions \a\/p' > tm+[ — 0, and the inequalities (13.1) and (13.2), we obtain the inequality
>++ (13.13) where the constant c > 0 is independent of k and f+(t, x}. CASE 2: t m+ / — ii = an. Using the representation (13.8) and the fact that the elements of the /th row «s}a" + i£n}(N±(irj + "f,£))i, £ = (s,&»), 7 ^ 7i, are multipliers, we obtain the estimate
, n,
,0
where the constant GI > 0 is independent of k and / + (t, x). We write this
338
4. Mixed Problems in a Quarter of Space
estimate in the form .
0,k
<
elx'sK(S,n,k}
1 , L r
By the definition of K(s,n,k) and F+(t,y',yn), we can apply the Fubini theorem to the integrals under the sign of the L p -norm. Consequently,
x e
Taking into account the definition of the function A'(s, n, k) and using the Minkowski inequality and the Young inequality, we find
^ ' ( < > x ) > L p i 7 ( M + + 1 ) | | ^ c i y t;' i/fc K e -(*--y»){.)"" Q(Xn
-
yn) ds\
F+(t, y) dy,
dv
dv
dv
i/k [the change of variables Ci = £,iVa' , k
— XiV~at , i — 1, ... , n]
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems 1
r
339
an_l
i/k Since we consider the case t m +/ — ti = an, from the condition \a\/p' > tm+f — ti we obtain the estimate (13.13). By the above arguments, we also obtain the inequality (13.10). The convergence (13.11) is established in the same way. D Lemma 13. 4. Let \a\/p' > an. For the vector-valued functions u^k(t, x), j — 1, . . . ,p, the following estimate holds:
where 7 > 71 and the constant c > 0 is independent of k and f + ( t , x ) . Furthermore, fc^^-^.^.W+i)!!^ 0 '
*i,*2->oo.
(13.15)
PROOF. Using the notation introduced at the beginning of the section and (12.9), (12.10), we find
l/k
!„_!
oo
X
(II
f
/ I
vy
— oo
P
''
(D • \ 11T\ T*^ 7 \ /
an ^^ / iJ] l 7 7 1 ' \ /
\ *"V C I / J ^ «^ / ) / -^T
n
\
I <"V C 1* 1 / f T l I / V Q / Y ? ) 1^ / l ^ l ^ M / ^ ' ' * / J UO L4 C/ .
/
/I/"V <** \ t L/ Ut -O- u J .
where Uj~(irj + 7 , s , x n ) is an m-dimensional vector-valued function whose components are the first m components of the jth the vector-valued function u t (^ + T' s ' x ") f rom the canonical basis of the boundary-value problem (11.4), Pj(«? + 7, s} = -(bj,i . ..bj<m)£+(ir, + 7, is, Dyn)RF+(ir] + 7, s, yn)| y n = 0 - (ir/ -|-7)"" 1 (6j im+1 . . .bjtl,)£±(iri + i,is,Dyn)RF+(iri + 7, s, y n )| y n = 0 , (6j,i . . .6j,m) = (5 + )j, (6j, m +i • • -bjlU) = (-B_)j. Applying Lemma 12.6 and
340
4. Mixed Problems in a Quarter of Space
using the notation (13.5), we can write this representation as follows: 00
x u j , k ( >x) -
f
f
J
OO
f
J J
ffin_! -00
e
•
• '
Ms.™>
'
0
+
x ((bjti . . . bj!rn)C (irj + 7, is, Dyn)RF+(iT] + 7, s, y n ) ) x D^+iUj" (iri -f 7, s, xn + yn) dyn drjds oo
E n _ l -00
oo
0
x ((6 j(1 . . . bjim}DynC+(ir) + 7, is, Dyn)RF+(ir] 4- 7, s, y n ) ) x Z)J"u;t(z^ + 7 , s , x n + yn}dyn drjds oo
+
f J
oo
[
eir)t+ix'sK(s,n,k)(is)P'
f
\J
tJ
E n _ i -oo 0
x ((iij + 7)- 1 (fe j , m+1 . ..bjtl,}C+(irj + 7, is, Dyn)RF+(irj + 7, s, y n )) x D+1u>(ir) + 7, s, x n + ?/ n ) cfyn drjds oo
/
oo
/
x ((»»; + 7)" 1 (^,m+i • • • bjiV)DynCt(iri + 7, »«, Dyn)RF+(ir) + 7, *, yn)) x Dfcuf(iri
+ 7, «, 2?n + j/n) c?yn cf^ds
- wl(t, x} + w2k(t, x) + w3k(t, x) + wi(ttz).
(13.16)
If B+ = 0, then w l k ( t , x) = w%(t, x) = 0 and if 5_ = 0, then iyj(<, x) = w k(t,x) =0. Consider the case 5+ ^ 0 and estimate, for example, the first term w^tyx) on the right-hand of (13.16). For xn > 0, using the Heaviside function, we can write 4
oo
1
w k(t,x)= I x
j,i . . .
x D
+l
j
I
i
m
oo
j
e^+ix'sK(s,n,k)(isf9(yn
r ] +i , s , y
n
r ) +j , s , y
ui(ir) + 7 ^ - ^ n + yn] dyn dr] ds .
n
x
n
+ yn
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
341
By properties of the Fourier transform, we have w1k(t,x) =
~
~
\
x (bj,i • • • bj,m)£+(ir] + 7, is, DZn)RF+(iT} + 7, s, zn) dzn j oo
x ( f e'v^ebJDt^wfVri + i.StyJdyn} d r i f t . \ J /
(13.17)
-00
We consider two cases. CASE 1: (3n = 0. Since for B+ ^ 0 the components of the vector- valued function 00
/>
(13.18)
are multipliers, we have e^t+ix'sK(s,n,k)(isfe(xn)
and, by Lemma 12.4,
n -00
As was noted in the proof of Lemma 13.1, the entries of the matrix + 7,0),
' = 0, 1,
(13.19)
are multipliers. Taking into account the definition (13.5) of the function K(s,n,k) and the estimate (13.1), we deduce the inequality C l ||/+(t,x),
^(R+
(13-20)
342
4. Mixed Problems in a Quarter of Space
where the constant GI > 0 is independent of k and / + (i, x } . CASE 2: fln = ti /an. Since the components of the vector- valued function oo
f eiy^ D?:+lu}+ (irj + 7, *, i/ n ) dyn satisfy the assumptions of the Lizorkin theorem about multipliers, and using the definition (13.17), we obtain the inequality
x (bjt\ . . .bj}Tn)C+(ir]Jr 7,zs, DXn)RF+(ir] + 7, s,xn) dr/ds, L p (E n + i) By Lemma 12.4,
\\w1k(t,x),Lp(M++l)\\^c
and, since the entries of the matrices (13.19) are multipliers, we obtain the estimate (13.20) in the same way as in Case 2. Similarly, we can establish the inequality
where the constant 02 > 0 is independent of k and f+(t, x } . Thus, if B+ ^ 0, 5_ = 0, then
Similarly,
In fact, we obtain the inequality (13.14). The convergence (13.15) is proved in the same way. Thus, for B+ ^ 0, B_ = 0 the required assertion is proved. Suppose that B+ ^ 0 and 5_ 7^ 0. In this case, we assume that t m +i = . . . — t^ > ti and the components u~ (T, s, xn), r = irj + 7, of the canonical
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
343
basis of the boundary- value problem (11.4) satisfy the estimate
Xn
> 0,
(13.21)
where 8 > 0 is a constant, e\(B+] ^> 0, £2 (B-} ^ 0; moreover, e\(B+] and £2(6-) vanish only for 5+ = 0 and 5_ = 0 respectively. Consider the terms w%(t, x] and w%(t, x) on the right-hand of (13.16) in the case B+ ^ 0. Using Lemma 12.5 and properties of the Fourier transform, we write w%(t, x) for xn > 0 in the form w*(t,x) =
C
OO
c \ I e-iZn^e(zn)JC+(ir) + 7, is, DZn}RF+(irj + 7, s, zn] dzn J j /
— oo
We consider two cases. CASE 1: f3n = 0. By the estimate (13.21) and the relations the components of the vector- valued function oo
1
(s)* - *" I e^-f-D^wt (117 + 7, ^ yn) dyn o are multipliers. Hence
/ (13.22) By Lemma 12.9, the elements of the operator £~t(irj -f- 7, zs, DXn) have the form a a (i?7 + j}(is)ff D°^, a'a' + o-nan = r — t^ + ti, where aa(r} are
344
4. Mixed Problems in a Quarter of Space
polynomials of degree 2m — v. Therefore, to derive the estimate
\\w*k(t,x), MR+Ull ^ 'll/ + (*,*), W?;7r+(R+i)||,
(13.23)
we need the inequality oo
f e'
1
(13.24) where j3' a' 0 is independent of k and f+(t,x}. By the definition (12.5) of the operator R, we have
7,
Using the Heaviside function and properties of the Fourier transform, for the expression under the sign of the L p -norm in (13.24), we have
7, s, xn] drfds \2m~u-l
x
9(xn - yn}Dax
,s,yn) dyn
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
345
oo
+ /f 0(yn -*„)£>£;; J_(zr7 + 7, s,z n - yn] F+ (ir) + ~f , s , yn) dyn \j dr)ds J
/
where
°: J+ (ir, + 7, 8, yn ) dyn ,
,^^
As in the proof of Lemma 4.7, using Lemma 12.3, it is easy to show that the functions //+ (ir? + 7, s, £ n ) and //~ (ir) + 7, s, £ n ) satisfy the assumptions of the Lizorkin theorem. By (13.1), this implies the estimate (13.24) for /?n=0. CASE 2: j3n = ti /a n . Since the components of the vector-valued function c^-f- I?J;+1wt (^ + 7, 5, jfc ) dyn
{*)- *" 0
are multipliers, we obtain an estimate of type (13.22): oo
II^cll j
J
eW'Kfrnt
K n - l -00
x (bj>m+i . ..bjiV)£±(iri + 7, is, -Dr x
Arguing in the same way as in the case fin = 0, we obtain (13.23).
346
4. Mixed Problems in a Quarter of Space
Similarly, we obtain the inequality °'r
Thus, in the case 5+ / 0 and 5_ ^ 0, we estimate the L p -norms of all four terms in (13.16). Consequently,
where the constant c > 0 is independent of k and f + ( t , x ) . above, we find
Arguing as
\\DtU$k(t,x), Lp, 7 (M+i)|| ^ c \ \ f + ( t , x ) , LP From these inequalities we obtain the estimate (13.14). The convergence (13.15) is established in a similar way. Thus, the lemrna is proved in the case B+ / 0 and 5_ ^ 0. Consider the case B+ = 0. Then wl(t,x) = w%(t,x] = 0, i.e., we need to estimate w%(t,x) and w%(t,x) from (13.16). By the condition on the boundary operators, the estimate (13.21) also holds. Therefore, the above arguments remain valid. D Lemma 13.5. For the Ith component u™£l(t,x] of the vector-valued function u ~ i k ( t , x ] = ( u ™ £ l ( t , x ) , . . . ,u^k(t,x)Y for (3a = tm+i the following estimate holds: xj,k
where 7 > 71 and the constant c > 0 is independent of k and f + ( t , x ) Furthermore, +l
k (t (l> XT\ ) ~
r)'x3u?/km +'/'f T K('TO++ —^. Un (l> XT} )> LP,l( n + i ) MI \ \ ~^
n "^ Zb9fi (16.
JJ
as ki, k-2 —> co.
PROOF. Like (13.16), we have 6-1*0% ujik(t,x)=
f J
t I' J
^t+lx'sK(s}n,k}(isf
J
K n _ i -oo 0 /
_
\
x ( (6^! . . . bjim)£+ (ir, + 7, is, Dyn)RF+(ir, + 1,s,yn)\
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
347
x D^+lu~ (it] + 7, s, xn +yn] dyn drjds CO
+ j
OO
e^t+ix'sK(s,n,k)(is)^
j j
n_i
-oo 0
x x D U } ~ (ir] + 7, s, xn + y n ) dyn dry ds oo
+ j
oo
eir}t+ix'sK(s,n,k}(isf'
j j
x
x D^+1u;~ (irj + 7, s, xn + y n ) rfyn drjds oo
n_!
-00
oo
0
/
\
x ( (irJ + 7)"1 (bj,m+i • • • bj,v)Dyn£±(ir) + 7, is, Dyn)RF+(ir) + 7, «, yn) J x D^u~ (irj + 7, s, xn + y n ) dyn - tiJjU*, x) + w2k(t, x} + w3k(t, x} + w4k(t, x } .
(13.27)
Fordefiniteness, we estimate the first components w\ k(t,x),i = 1, . . . ,4, of the vector-valued function wlk(t,x}. Consider the case B+ ^ 0, 5_ = 0. Then w%(t, x} = ^(^, ^) = 0- As in the case (13.17), we have 00
I i/
x
/ eiyn*ne(yn)DPn+luT+ (ir) + 7, s, yn) dyn } drjd£.
(13.28)
— 00
Let (3n = 0. Since the function oo
^tm+1-ti
iyn e
^DynuJi+1(ir] + ^ts,yn)dyn
(13.29)
348
4. Mixed Problems in a Quarter of Space
is a multiplier and fi'a1 = tm+i , arguing in the same way as in the proof of the estimate (13.20), we find
\ \ w l i k ( t , x ) , Ip(M+i)ll ^ c|l/ + (^)> W?; 7 r ( R nJi)ll. +
where the constant c > 0 is independent of k and f then the function
(13.30)
(t, x). If (3n — t m +i /an,
satisfies the assumptions of the Lizorkin theorem. Since /3' — 0, we again obtain the estimate (13.30). The following inequality is proved in a similar way:
\\wl>k(t,x), MR+WH **c\\f+(t,x), ^ p °; 7 r (M+i)||.
(13.31)
The convergence l ^0,
i = 1,2,
(13.32)
as fci, A:2 —> co is established in the same way. Thus, in the case J3+ 7^ 0 and 5_ = 0, we proved (13.25) and (13.26) for / = 1. Consider the case B+ ^ 0, 5_ ^ 0. It is obvious that (13.29), (13.31), (13.32) are obtained as above. Therefore, we can consider only the functions w^k(t,x) and w^k(t,x}. As in the proof of the estimate (13.23), we first write an explicit expression for MI k(t, x) in the case xn > 0:
*,*) = f f
r
*/
/ e-li^e(zn)Ct(ir} + f,i8,D
Taking into account the estimate (13.21), we can show that the function
7, s, yn
yn
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
349
is a multiplier. Therefore, for /?n = 0, (3'a1 = t m + i, repeating the arguments of the proof of the estimate (13.23), we arrive at the inequality
K fc (*,*), MKJi)ll ^ c|l/ + (*,*). W^R+i)!!,
(13.33)
where the constant c > 0 is independent of A; and f+(t, x). If 0n = t m +i fan, then it is obvious that the function
is a multiplier. Since /?'«' = 0, we again obtain the inequality (13.33). Arguing in a similar way, we prove the inequality l , k > >
and the convergence \\^Q,
i = 3,4,
*i, k2 -> oo.
Thus, in the case B+ ^ 0, B- ^ 0, the inequality (13.25) and the convergence (13.26) are proved for / = 1. It is easy to consider the last case. 5+ = 0, B- ^ 0. D Lemma 13.6. Let B_ = 0, and let deg A (6j i i . . .bjiTn}jC+(ir] + 7, is, z'A) < r/an. If \ot\lp1 > tm+i—ti, then the Ith component u™£l(t,x) of the vector-valued function u~k(t,x) satisfies the estimate }\\ ^ c ( \ \ f + ( t , x ) , +
(t,x), L^RJ)!!, L P , 7 (M+)||),
7 > 7i,
where the constant c > 0 is independent of k and f+(t,x). \\->Q,
klt
(13-34) Furthermore,
fc2->oo.
(13.35)
PROOF. To prove Lemma 13.5, we use the representation (13.27). We consider this representation for J3 = 0, i.e., for B- — 0 we have e~~*tu~k(t, x) = wk(t,x) + w ^ t j X ) . For defmiteness, we estimate the first components w\ k(t, x), i — 1,2, of these terms.
4. Mixed Problems in a Quarter of Space
350
Taking into account the representation of the function w\ k ( t , x ] in the form (13.28) with /3 = 0 and using the multiplier (13.29), we find
ei t+ix K 8
n
" '' ( > >
' tm+1
o C+(irj + 7, is, DXn)RF+(irj + 7, s, xn) dr) ds, Lp(R++l By Lemma 12.8, the elements of the operator (6^1 . . . 6j im )£ + (277+7, is, DXn ) have the form aa(irj + -y)(is}a D°^, a'a!-\-anan — r, where a a ( r ] are polynomials of degree 2m — v—\. By the assumptions of the lemma, an ^ r/an — I . Therefore, by the definition (12.5) of the operator R, we have
X
f
^
/ Daj:l J+ (ir] + 7,5, xn - yn)F+(ir] + 7, s,yn)dyn
,s,yn}dyn \ J drjds,
^n + 1,
/
By properties of the Fourier transform and Lemma 12.7, this expression can be written in the form >++
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
351
ir
>t+ixtK(s, n + 1, k ) ( s ) t l ~ t m +> (isy'aa(iri + 7)
Since cr'a' ~?> an, By Lemma 12.2, the function
is a multiplier. Consequently,
>, n + 1. A)
The last inequality can be written as follows:
f
- t m + 1 +a n I I
\
xn r I /
J
(s)an(xn-yn)
s ?.
'
\flll '
) J, r Pi
Arguing as in the proof of Lemma 13.3, under the condition tm+i — t i , we find
M),
where the constant c > 0 is independent of k and In the same way, we prove the estimate
)!!, I Pl7
, x).
and the convergence
Estimating the remaining components of wl(t,x), w%(t,x] in a similar way, we obtain (13.34) and (13.35). D
352
4. Mixed Problems in a Quarter of Space
Lemma 13.7. Let B- = 0, and let deg A (6 Ji i . . .bj:m)£+(ir/ + 7, is, z'A) — r/an. If \oc\/p' > tm+i — ti -\-oLn, then the. Ith component u™£l(t, x} of the vector-valued function ujk(t,x) satisfies the estimate
(t, x), i|))
7>7l,
where the constant c > 0 is independent of k and f+(t,x). the convergence takes place: (13.35).
(13.36) Furthermore,
PROOF. Recall that t eix>sK(s,n,
u-k(t,x)=
where
l«/n=l
As was already mentioned, the elements of the operator (bjt\ .. . bjiTn)£+(ir/+ •y,is,Dyn) take the form aa(ir) -f l)(is)° Dy^, a1 a1 + crnan — r. By the assumptions of the lemmas, there are elements such that 0,7(^77 -f j)Dynan . By the proof of Lemma 13.6, it suffices to estimate a vector-valued function of the form OO
vk (*,*) =
ix>s
f e J
K(s,n,k}(
f e^aa(in + 7} \
J
ds,
where Fq(ir/ + 7 , s , y n ) is the qth component of F+(ir/ + 7 , s , y n ) . By the definition (12.5) of the operator /?, we can write the vector- valued function
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
353
Vk(t, x) in the form oo
eirit+ix>sK(s, n,k}aa(irj + 7) V1^ + 7)^(117 + 7, s,y n )
vk(t,x) =
+ / Dry/n°lnJ+(iri + i,s,yn - zn)Fq(irj + ^,s,zn) dzn — oo oo
\ Dryl"n J_ (IT; + 7, s, yn - zn}Fq(ir) + 7, s, zn) dzn j \yn=Q / / Vn.
x wj (117 + 7, *, *n) ^ c?s = vj (t, x) + uj(t, x) + vj(t, i), are
(13.37) r ari
where 6(177 + T) ^he coefficients at the leading term (z'A) / of the polynomial 0(177 + 7, is, iA). We estimate the first term f ^ ( t , x ) on the right-hand of (13.37). For definiteness, we consider the first component v\ k ( t , x ) . Using the NewtonLeibniz formula, we can write it in the form oo
J l -00 0
77 + 7,s,x n + y n ) d y n d r j d s
ln-1 -00 0
x b~i(irj + -f)Fq(iT] + ^,s,yn)Dynuj1l+1(ir} + j,s,xn + yn)dyndr)ds We consider the first term w i ^ f t j x ) on the right-hand side of the last equality. As in the case (13.17), for xn > 0 we write it in the form wlik (t,x) = J J c E n -oo oo
/ f f / e-l \ J
354
4. Mixed Problems in a Quarter of Space
By the properties of the components of the vector-valued function w • (irj 7, s, xn), the function
satisfies the assumptions of the Lizorkin theorem about multipliers. Therefore,
, n,
/ " ~r t^n
where , 0 - (27r)-"/ 2 / e-"*D,Jq(t, z) dz. J
Arguing as in the proof of Lemmas 13.3 and 13.6, under the condition |a|/p' > t m + i — ti +a n , we obtain the inequality fr*), I P (K+i)|| ^ c(\\DxJq(t,x), I Pl7 (K+i)|| + III Dxjq(t, x ) , ^ ( K + J H , I Pl7 (M+)|| ),
(13.38)
+
where the constant c > 0 is independent of A; and / (2, x). The term wz^ft, x] is estimated in a similar way. However, in this case, to obtain the estimate \ \ w 2 i k ( t , x ) , LpORj (13.39) the condition aj/p' > t m +i — ti is sufficient. Indeed, as in the case we write W2tk(t, x) for xn > 0 in the form
x I
e
lZniin
6(zn}Fq(ir]-\-^}s}zn}dzri
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
355
By the properties of the components of the vector- valued function u • (i 7 , s , x n ) , the function
satisfies the assumptions of the Lizorkin theorem about multipliers. Therefore,
, n, fc)
where
Comparing with the proof of Lemma 13.6, we obtain (13.39) under the condition that Ictj/p' > tm+i — ti. From the estimates (13.38) and (13,39) we find
We consider the second and third terms v%(t, x) and v%(t, x] on the righthand side of (13.37). By the definition of the contour integrals (12.6) and (12.7), we have r/an-l
m=0
where r/an —
356
4. Mixed Problems in a Quarter of Space
Consequently, the vector- valued function v%(t,x) + v%(t,x) can be represented in the form «?
/ m=0
J
I J*+ixtK(8tnt J
x aa (ir) + 7)6m (177 + 7, s}D™n RFq(iri + s, yn)\yn=QUj
("7 + 7, «, z
We note that bm(irj -\- 7, ca s) — cr~manbm(ir] + 7, s), c > 0, and 0 ^ m
where the constant c > 0 is independent of k and f+(t,x). The above arguments imply the estimate (13.36). The convergence (13.35) is established in the same way. CD Lemma 13.8. Let 5_ ^ 0, B+ = 0. // \a\fpf > tm+i -ti, then the Ith component u"^'(/,x) of the vector-valued function u~k(t,x) satisfies the estimate (13.34) and the convergence (13.35). PROOF. In the proof of Lemma 13.5, we wrote the representation (13.27). Using this representation, for ft = 0 and B+ = 0 we find e~'ytu~k(t,x) = w%(t,x) + w % ( t , x ) . We consider the vector- valued function w % ( t , x ) . As in the proof of Lemma 13.5, for the /th component for xn > 0 we have
J
iz
f
e-
^e(zn)C+(i
oo X ( — oo
Since the function
iT, + 7, «, y n ) dyn
357
§ 13. Sobolev-Type Systems. Solvability of Mixed Problems
is a multiplier, the following estimate holds:
**+**'•
\\u>i,k(t>*)>'
K(8,n,k)e(xn)(bj,m+l...bjil,)
o Ct(irj + 7, is, DXn)RF+(ir1 + 7, s, xn) d-qds, Lp(Rn+l) Using Lemma 12.5, this expression can be written in the form 3
Hi*) f c ((iI , XT\) , ITv p^11$++ r \\Wi ^ M ^ i j jMl l l
r II J
oo
r
I 011*+" 16 J
ll
Recall that the entries of the matrix ( (cf. (13.9)) are multipliers. Consequently,
N*(ir) + 7,0) 7 ^ 7i
Repeating the proof of Lemma 13.3, for |a|/p' > tm+i — ti we find
(t,x), where the constant c > 0 is independent of k and /"*"(<, x). In the same way, we establish the inequality
^ c(\\f+(t,x), and the convergence 1
i = 3,4,
oo.
Arguing int he same way as in the proof of Lemmas 13.6-13.8, one can establish the following assertion. Lemma 13.9. Let B- ^ 0, B+ ^ 0, and let \a\/p' > t m+ / — ti . Suppose that deg A (6j_i . . .bj>m)£+(ir) + f,is,i\) < r/an. Then the Ith component
358
4. Mixed Problems in a Quarter of Space
of the vector-valued function u~k(t,x) satisfies (13.34) and (13.35). Let deg A (6j i i . . .bjtTn}£+(in+~/, is, i\) — r/an. Then (13.36) and (13.35) hold. PROOF OF THEOREMS 11.1-11.3. From Lemmas 13.1-13.9 it follows that the sequence {iik(t,x)}, where U k ( t , x ) = (u^(t, x ) , u^(t, x ) ) 1 , is a Cauchy sequence in the space W*£+ (R*^) x W^'^^), 7 > 71. Since the Sobolev spaces are complete, the limit vector-valued function u(t,x) exists and u+(t,x) = lim ut(t,x), u ~ ( t , x ] = lim u^(t,x); moreover, the k—too
k-*oo
estimates indicated in the theorem are valid. By the construction of the sequence { u ^ ( t , x ) } , we conclude that u ( t , x ) is a solution to the problem (11.1). Since C^JE^+J is dense in Lp^Rj+j) n Lp^(R+; L i ( M + ) ) , the problem (11.1) is solvable for any right-hand sides indicated in the theorems. The uniqueness is proved in the same way as for the mixed boundaryvalue problems for Sobolev-type equations. D
§ 14. Mixed Boundary- Value Problems for Pseudoparabolic Systems In this section, we consider the mixed problems in the quarter of the space for pseudoparabolic systems of the form K0Dtu+ + Kl(Dx}u+ + K2(Dx}u+ + L(Dx)u- = f + ( t , x ) , M(Dx}u+ = 0, +
Bu \Xn=0 = Q,
/ > 0 , zeMj, t>Q, x ' e M n - i ,
We consider a pseudoparabolic operator (cf. Chapter 3, Section 2)
0 where Ki(Dx), L ( D X ) , M(DX] — are matrix differential operators with respect to x with constant coefficients of size m x m, m x (y — ra) , (y — m] x m respectively and A'o is a nonsingular number matrix. Definition 14.1. A matrix operator £o(A, Ac) is said to be pseudoparabolic if the symbol of this operator satisfies the following conditions. Assumption 1. Let s\ = . . . = sm = Q, t\ = . . . = im = I . Suppose that there are numbers s m + i , . . . ,8^, t m + i , . . . , t^ such that 0 ^ Sj ^ — 1, tj ^ 0, j — m + 1, . . . , is, and a vector a — (a\} . . . , an), cti > 0, i = 1, . . . , n, such
§ 14. Pseudoparabolic Systems
359
that for k, j = 1, . . . , v lk,j(iTl, i£] ~ 0,
Sfc + tj < 0,
lkij(cir],cait} = cSk +t>lktj(ir),it),
c>0,
s f c + t j ^ 0,
where tj /on are natural numbers. Assumption 2. Let det(rA 0 + Ai(z£)) ^ 0, Re r ^ 0, ^ E M n , |r| + |f|^ 0; moreover, det(M(is}(rKQ+Ki(i^))~1 L(is}) =Q,s G R n if and only if s = 0. In addition, we assume that for the symbol A'2(i£) = (^/,j( z O) °f the operator of the lower-order terms K-2(DX] the following condition holds (for the sake of convenience, we continue the enumeration of the assumptions in Definition 14.1). Assumption 3. There exist numbers qij, 0 ^ qij < I such that kij (cai^} c '-ikij(it),c>Q. q
From Assumption 2 it follow that for Re r ^ 0, £ G M n , |r| + |^| ^ 0 det£ 0 (r, z'O = det(rA' 0 + tfi(i'O) det(-M(z'0(rA' 0 -f A 1 (zO)~ 1 L(iO).
(14.2)
r
Consequently, the equation det £o(' , is, z'A) = 0, Re r ^ 0, s G M n -i\{0} has no real roots A. By Assumptions 1 and 3 and Definition 14.1, for a sufficiently large 70 > 0 the equation det£(r,is,«A) = 0 ,
Re 7 - ^ 7 0 ,
s e M n _i\{0},
(14.3)
where
has no real roots A either. Let // be the number of roots lying in the upper half-plane. We formulate the conditions on the boundary // x m-matrix B in (14.1). We assume that the boundary-valuer problem on the half-axis £(T, is,DXn)v = 0,
xn > 0,
+
Bv \Xn=0 = (f>,
(14.4)
sup \v(r,s,xn)\ < oo,
xn>0
where Re r ^ 70 > 0, s £ IR n _i\{0}, satisfies the Lopatinskii condition, i.e., the problem is uniquely solvable for any vector (p.
360
4. Mixed Problems in a Quarter of Space
We recall (cf. Chapter 1, Section 10) that, by the Lopatinskii condition, the number of rows of the matrix B should be equal to the number of roots of equation (14.3) lying in the upper half-plane {Im A > 0}. Example 14.1 (the linearized Navier-Stokes equations). We consider the boundary-value problem for the linearized Navier-Stokes equations - i / w - [u+,uj] + Vu~ = f+(t,x),
u
divu+ = f ~ ( t , x ) , +
u \X3=0 = 0,
t > 0, x £R+,
(14.5)
t > 0, x' € M 2 ,
Let us show that this problem satisfies the above conditions. For the sake of simplicity, we set jx = 1, |tJ| = 0. The operators defining the system have the form
'1 0 0
L(DX) =
-A
0 0' 1 0 0 1
(Dx DX2 \D,
0 A 0
0 0 -A
M(DX) = (DXl
moreover, c*i = a-2 — a 3 = 1/2, 84 = —1/2, t 4 = 1/2, det£(r, zf) = (r + |^| 2 ) 2 |^| 2 . Therefore, the equation det£(r, is, iA) - 0, Re r ^ 0, s e M 2 \{0) has three roots \i(r, s) = i s\, \I(T, s] — Aa(r, s] — i\/r -\- \s\2 lying in the upper half-plane. We verify the Lopatinskii condition. Consider the boundary-valued problem in the half-axis £3 > 0 for the system of differential equations with parameters
0 0
0
r + M2T+
,s
2
- D23
DX3
S\lp(\V (T,S,X3)\
4 4
\v-y
(14.6)
*3=0 = +
is i /«f\ is-i = 0, z3 > 0, DX3 0 _
+ \V~(T,S,X3)\)
< 00,
where Re r > 0, s E E2\{0}. It is easy to verify that the basis for the space of bounded solutions to the system (13 > 0) £(T, is, DX3]w — 0,
x3 > 0
(14.7)
§ 14. Pseudoparabolic Systems
361
is formed by the vector-valued functions
/ is\ \
\ Consequently, the general formulas for the solutions to the system (14.7) that are bounded for x3 > 0 takes the form w(r, s, x 3 ) = Y, ckwk(r, s, x3). Then the validity of the Lopatinskii condition for the problem (14.6) is equivalent to the unique solvability of the system
- \s However, the determinant of this matrix H\A" + I S I 2 (| S | — \/r + |s|2) is different from zero for Re r > 0, s G R2\{0}) i- e -> ^ ne Lopatinskii condition is satisfied. We note that the norms of solutions to the boundary-value problem (14.6) can unboundedly increase as |s| —*• 0. It is easy to see if we write explicit formulas for the basis solutions to the boundary-value problem. Indeed, if
I W°(T,8,X3)
=
362
4. Mixed Problems in a Quarter of Space
where p = -\fr + |s|2, f p\
/ isi \ is-} ei(r,s) , ~\s
r
\-
e 2 (r, s) =
/
0 isi
,
/O \ P e 3 (r,s) = IS-2
w
v°/
If (pi = 0, tf-2 — 1, (f>z = 0, then the solution takes the form
If (fi = 0, (f>2 — 0, 993 = 1, then the solution takes the form '»>
|s|(|s|-p) vr
Consequently, for the fourth component of the third vector-valued function we have '.^T-J^L-reoo,
|s| -> 0.
We formulate the results for the mixed problem (14.1). Theorem 14.1. Let f + ( t , x ] 6 L^OR^) D L P , 7 (M+; Li(E+)), 7 > 7o// |a'|/p' > max{t m + i,... , U,}, ^Aen t/ie problem (14.1) /ifls a unique solution u+(t, x)
j=m+l
and the following estimate holds:
where the constant c > 0 is independent of f+(t,x}.
§ 15. Pseudoparabolic Systems. Approximate Solutions
363
Theorem 14.2. Let f+(t,x) G Lp^M^), 7 > 70- Then the problem (14.1) has a solution u ( t , x ) such that u+(t,x) G ^^(M^), D^ui(t,x) G Lp^(l&n+i)> / = m + 1 , . . . , v, (3la. = ii, and the following estimate holds:
where the constant c > 0 is independent of f+(t,x}. Remark 14.1. For the Cauchy problem jC(Dt,Dx)u — f ( t , x ) , ( t , x ) G M+ +1 , u+ t_Q= 0 we have an assertion similar to Theorem 14.1 provided that \a\/p' > max{t m + i,... , t,,} (cf. Chapter 3, Section 7).
§ 15. Approximate Solutions to Mixed Problems for Pseudoparabolic Systems In this section, we present formulas for approximate solutions to the mixed boundary- valued problem (14.1). Here, we follow the scheme described in Sections 3, 10, 12. Let the right-hand side /+(/, x) of (14.1) belong to C§°(M+i). As above, we extend it for t < 0, xn < 0 by zero and preserve the same notation for the extended function. We denote by f+(t,s,xn) the Fourier transform of f+(t, x', xn) with respect to x' and by f+(r, s, xn), r = ir) + 7 the Fourier transform of e ~ ' y t f + ( t , x' , xn) with respect to ( t , x ' ) . Consider the boundary- valued problem in the half- axis {xn > 0} for the system of differential equations with parameters (r, s), Re r ^ 70, s G
sup \u(r,s,xn)\ < oo.
xn>0
By the Lopatinskii condition, the problem is uniquely solvable. We write its solution in the form u(r, s, xn) = UQ(T, s, xn) + v(r, s, x n ),
(15.2)
364
4. Mixed Problems in a Quarter of Space
where U>Q(T, s, xn) is a bounded solution to the system
r(r zs, ?•? L> n Xn)u> }/ >——i/ If +J / V\ >
xrn >•>> u,n
and the vector-valued function v(r, s, xn) is a solution to the boundary-value problem of the form (14.4): £(T, is, DXn)v — 0,
xn > 0,
B^L^-^.^osup |u(r,5,x n )| < oo. rn>0
We introduce the notation a(r, is, z'A) = det £(r, is, z'A), where £(r, is, z'A) is conjugate to £(r, z's,z'A). As in Section 12, for a solution U>O(T, s,x n ) we can take a vector-valued function of the form u
V
/
(15.3)
where ,s,xn) = If J+(r,s,xn -
yn)f+(T,s,yn)dyn
o oo
/ J-(T,s,xn-yn)f+(r,s,yn)dyn,
(15.4)
r+
r / \ ! f exp(z'x n A) J _ ( r ) s , x n ) = -— / \ ./ d\; 2?r 7 a(r, zs, zA) r-
15.6
here, the contour F+ = F + (r, s) surrounds all the roots of equation (14.3), i.e., det £(r, z's, z'A) = 0, Re r ^ 70, s e M n _i\{0}, lying in the upper halfplane and the contour F~ — F~(r, s) surrounds the roots lying in the lower half-plane. We indicate explicitly an expression for V(T, s,xn) in (15.2).
§ 15. Pseudoparabolic Systems. Approximate Solutions
365
Let WJ(T, s,xn), j = 1, . . . ,/j, be a canonical basis of the boundaryvalue problem (14.4), i.e., WJ(T, s,xn) is a solution to the boundary-value problem
£(r, is,DXn)uj = 0, xn > 0, ^ + L = o= e ;> sup |u;t(r,s,ar n )| < oo, *n>0
J
where Re r ^ 70, s G M n _i\{0} and ej is the unit vector whose jth component is equal to 1. The elements of the canonical basis for xn > 0 satisfy the inequalities
|u^(r, s,xn)\ ^ c(exp(-Sxn(s)an) + exp(-6xn(()an)),
i = 1 , . . . ,m,
*0""" exp(-^ n ( S ) Q ») + (O^*"-where / = m + 1, . . . , i/ and c, S > 0 are absolute constants. Using the canonical basis, we can write the vector-valued function u(r, s,xn) in the form V(T, s,xn) =
j(r,s)wj(r,s,z n ),
y(r, s) = -5wJ(r, s, 0).
(15.7)
Taking into account (15.2), (15.3), (15.7), we write the solution to the boundary- value problem (15.1) in the form
Rf+(T,8,xn) 0 (15.8)
where
y n =0
Since the vector-valued functions (15.3) and (15.7) are analytic and bounded with respect to r, Re r ^ 70, we can apply Theorem 5.2 in Chapter
366
4. Mixed Problems in a Quarter of Space
1 to the vector-valued function (15.8). Then the vector- valued functions v+(t,s,xn) = (27T)- 1 / 2 / e «/
are independent of 7 ^ 70, moreover, they form a solution to the mixed problem with parameter s £ M n _i\{0| K0Dtv+ + Ki(is,DXn)v+ + K2(is,DXn)v+ + L(is,DXn)v~ — J?+(/( L i s*i rx n1j j —
M(is}DXn)v+ =Q,
V
t>0,
xn>0,
=o=°' ^ > ° ' = 0, xn>0. t-O
We construct a sequence of approximate solutions using the integral representation of summable functions presented in Chapter 1, Section 6. We introduce the vector-valued function Uk(t,x) — (u£(t, x } , u^(t, x ) Y , where v~l
elxsG(sva
J
J
l/k
En-i
(15.9)
l/k
Kn-i
(15.10)
where (s}2N),
(s}2 = Vs, 2 / a '. t=i
(15.11)
§ 15. Pseudoparabolic Systems. Approximate Solutions
367
We choose TV such that U k ( t , x ) is infinitely differentiable and summable with respect to x with any power. By construction, KQDtu+(t, x) + Ki(Dx)u%(t,x) + K2(Dx)u+(t, x) + L(Dx}u~ (t, x) = / fc + M), M(Dx)u+(t, x) = Q,
t > 0, x e M+,
where k 1 n
f+(t,x) = (27r) -
j v-1
I
j
ei(xl-»l)'G(sva')f+(t,i/txn)d8di/dv.
l/k
Since \ \ f £ ( t , x ] — f+(t,x], Lpi7(]R^1)|| —>• 0 as &.'—>• oo, the vector-valued function U k ( t , x ] is an approximate solution to the mixed boundary- value problem (14.1). In Section 16, we establish the Lp-estimates for U k ( t , x ) , which imply that the sequence {uk(t,x)} converges and the limit function is a solution to the mixed problem (14.1). For the sake of simplicity, we consider pseudoparabolic systems without lower-order terms, i.e., K-2(DX] — 0. To estimate approximate solutions, we essentially use some properties of solutions to the boundary- value problem (15.1) (cf. below). As in Chapter 3, Sections 9 and 10, for Re r ^ 0, £ G M n \{0} we introduce the notation K(r,£) - rA'0 + A'i (z£), N(T,£) = M(i£)K-l( Lemma 15.1. For any numbers c 1 ; C2 > 0 for Re T ^ 0, £ G M n \{0} the following identities hold:
where T^-m(c) is a diagonal matrix with entries c tm+1 , . . . , ctv . PROOF. By Assumption 1 on the symbol of the operator £o(Dt, Dx), Kl(cai^ = cK1(i^},
(15.12)
368
4. Mixed Problems in a Quarter of Space m(c), a
M(c z'0 = c5 l/ _ m (c)M(»0,
c > 0,
(15.13) (15.14)
where S,,_m(c) is a diagonal matrix with entries c Sm+1 , . . . ,c s ". Hence we obtain the required identities for N*(T,£) and N+(T, £). D Lemma 15.2. The determinant det £o(ir), z'£) is a homogeneous function relative to the vector (I, a] with homogeneity exponent r = ^ (s,- + t,-) ; 1=1 i.e., detjCo(ciT],cai^) = crdetJC,o(irj,i^}. Furthermore, for any (T,£), Re r ^ 0, £ £ M n ; /Ae following estimate holds: ai(O r ° r - r ° ^ |det£ 0 (r,iOI ^ a2(O r o (C> r - r °, n
(15-15)
.
where (£) 2 = ^ ^, f o t l , (C) 2 = |T|2 + (C) 2 > ro = r — Zm+v and the constants i=l
a-2 ^ ai > 0 are independent of r, £. PROOF. The homogeneity of det £o(^,^) relative to the vector ( l , a ) follows from (15.12)-(15.14). We prove (15.15). From (14.2) it follows that det£ 0 (r, «0 = c 2m -^c r 2 ° det(q lrKQ + K^c^it)) x det(-M(cJ a zO(cr 1 rA' 0 + /^(cpz'Or^cJ 0 ^)), c
i , C2 > 0- We set GI = (^), c% = (^). By Assumption 2 on the operator £o(Dt,Dx), there exist constants 02 ^ ai > 0 such that
which implies the estimate (15.15).
D
Leinina 15.3. The following relation holds: i-\ det£ 0 (r,iO = a0(i^rl + ^a,_ f c (iOr f c ,
/ = 2m - i/,
(15.16)
Ai=0
where Re r ^ 0, <^ € M n , |T| + |^| 7^ 0, OQ(^) zs a polynomial homogeneous relative to the vector a with homogeneity exponent ro = r — 2m + v , V
r = £Xs,- + t,-), such that a 0 (zf) = 0, £ € M n; «/ ancf on/y if £ = 0. f=i
§ 15. Pseudoparabolic Systems. Approximate Solutions
369
PROOF. By (14.2) and (15.12)-(15.14), r"~ 2m detC0(r, i£) = det(K0 + r- K(i£)) det(-M(if)(#o + r- 1 A'(^))- 1 L(z^)). Passing to the limit as \r\ —>• oo, we obtain (15.16). Furthermore, l
By Assumption 2, we have QO(Z'£) = 0 for £ £ IRn if and only if £ = 0.
D
As was mentioned in Section 14, the equation det CQ(T, is,iA) = 0, Re r ^ 0, s £ R n -i\{0} has no real roots A. If s = 0, then, by Lemmas 15.2 and 15.3, there is a single real root (equal to zero and with multiplicity m = ro/a n ), whereas the remaining (r/an — m) roots are complex. We denote by Af(r,s), . . . , A++(r,s), A^(r,s), . . . , A~_(r,s), p+ +p~ = m, the roots lying in the upper and lower half-planes respectively for Re r ^ 0, s £ M n _i\{0} and equal to zero at s = 0. The remaining roots A++ .(r, s), .7 = 1 , . . . , ( ^ - p + ) , and A~_ + / c (r,s), k=l,... ,(r/an - p - p~) lie in the upper and lower half-plane respectively for any Re r > 0, s £ M n _ i . Since all the roots are quasihomogeneous functions relative to the vector (I, a'} with homogeneity exponent an, the following assertion holds. Lemma 15.4. The following estimates hold: S^s)"" ^ Im \t(r,s) ^ \\t(r,s}\ <: 62(s)a» ,
^i(C) an ^ I m \t+ + j ( r > s ) ^ j = 1,...
,(lA-p+),
1 = 1 , . . . ,p~,
n-1
2
^1,^2 > 0 are constants. This lemma is similar to Lemma 10.1. Repeating the proof of Lemma 10.2, we obtain an assertion similar to this lemma. Lemma 15.5. For xn > 0, Re r > 0, s £ R n _i\{0} for the contour
370
4. Mixed Problems in a Quarter of Space
integrals (15.5) and (15.6) the following estimates hold:
where 01,03,6 > 0 are constants. Due to this lemma, we can establish the following analogs of Lemmas 12.4-12.6. Lemma 15.6. Let jC+(r,i^) be an m x m-matrix consisting of entries of the first m rows and m columns of the matrix £(r, z£). Then for Re r ^ 0; s 6 M n _!\{0} the following identity holds:
= (27r)
,
'
/•
/ e t X n * n K ~ l ( T , £ ) ( I - N+(r,£))f+(n,s,£n} J
d£ n ,
(15.17)
— oo
where /^"(T/,^) 25 t/ie Fourier transform of the vector-valued function PROOF. Consider the boundary-value problem for the system of differential equations with parameters Re r ^ 0, s G M n -i\{0)
( sup
— oo<x n
,. \ T,S,X ) n
_
n
U
j
(15.18)
|^(T, s, xn}\ < oo.
Since the equation det £O(T, is, iX) = 0, Re r ^ 0, s G M n -i\{0} has no real roots A, this problem has a unique solution which can be represented in the form (15.3). By Lemma 15.5, we can apply the Fourier operator with respect to xn to the solution. Applying the Fourier operator to the system (15.18), we find K(r, t)2+ (r, 0 + L(iO£- (r, 0 = /+ (17, 0 ,
§ 16. Convergence of Approximate Solutions
Consequently, w+(r,£) - ff-^r.OC/ - N+(T,t))f + ( ] , £ ) . inverse Fourier operator, we obtain (15.17).
371
Applying the D
Lemma 15.7. Let £J!(r, z'£) be a (y — m) x m-matrix formed by entries of the last v — m rows and the first m columns of the matrix £(r, z'£). Then for Re r ^ 0, s £ M n _i\{0} we have
= (27T)- 1 / 2
e"-^(r,0fo,0d&,.
(15-20)
PROOF. By (15.19), we have tD~(r, £) = N * ( T , £ ) f + ( r ) , £ ) . Applying the inverse Fourier operator and using the representation for the solution to the problem (15.18), we obtain (15.20). D Lemma 15.8. For Re r ^ 0, s G !!!.„_ i\{0} the following tions hold:
representa-
oo
r S
^•( , )UJ(T, s,xn) = / Dyn o x WJ(T,S, (xn +yn)))dyn, where j = 1, . . . , / / anrf (6^1 . . .&j, m ) Z5 ^e j^A row of the matrix B. PROOF. The assertion follows from the Newton-Leibniz formula since the components of the canonical basis exponentially decay as xn —> +00. D
§ 16. Convergence of Approximate Solutions In this section, we prove the Lp -estimates for the approximate solutions (15.9), (15.10) which imply the existence of a solution to the mixed problem (14.1). To simplify the exposition, we consider pseudoparabolic systems without lower-order terms. By formulas (15.2), (15.3), (15.7), (15.8), we can write the approxin mate solution (15.9), (15.10) in the form u£(t, ar) = j] w nc(^> ^)> uk (*> x ) = /=o '
372
4. Mixed Problems in a Quarter of Space
A*
u
(
k
U II u
(t
x)\
L
T 1 /
-' I/ T,
f
x ) = (/TT)
Q,k\ > ^l/
f
i
I e
I v
I
I
J l/k
J !„_!
7, s,x n )rf7/ J d s d u , /
?/ u
I/
T*
jk\Li-L
l/k
xf
i
f . / e^rj+^t(pj(irj
\
\ + 7, s}ujj(ir) + 7, s , x n ) d r j \dsdv,
^
X
— OO
where j = 1, . . . ,//. We estimate the vector- valued functions u f k ( t , x ) and
Lemma 16.1. For Me vector-valued function v . Q k ( t , x ) the following estimate holds: I < c\\ f (t x] L
}\\
flR +
where the constant c > 0 is independent of k and f (t,x).
"Y > 0
(16 1)
Furthermore,
PROOF. Using the notation in Section 15, we write k t +
n
cf>~1 ii ^ • * n t -(fl * iT} "'/ — — (9Tr}~ I^ / /
2
f
f
II n~l
//
»/
J
l/k
En-l
,
i
a plx sCl(sv \/ ^-^ \
OO
( f i t~
\J
~ •
'
— oo
\
J
Introduce the notation k
K(s, n, k) = (27r)- n / 2 / v-1G(sva')dv. i/k
(16.3)
§ 16. Convergence of Approximate Solutions
373
By Lemma 15.6, we have CO
e 7t
~ «?,*(<.*) = J J
By conditions on the pseudoparabolic operator Co(Dt, Dx) and Lemma 15.1, the entries of the matrices (^+7) K~1 (z'fy+7, £)(!— N+ (zr/-f 7, £)), ^ K~l (117 7,^)(/ — N+(irj + 7,£)), where 0 ^ /?a ^ 1, satisfy the assumptions of the Lizorkin theorem about multipliers. Therefore,
Jf Jf e *(«'-¥'
n 4- 1 ,Tt-i-i,
,-T*
where the constant c > 0 is independent of k and f+(t,x). From the definitions (16.3) and (15.11) we obtain the inequality (16.1). Similarly, we obtain (16.2). D Lemma 16.2. Let fi = ($,... ,/3ln), I = m + 1, . . . ,i/, 0la = tt. Then the components o f v , Q k ( t , x ) — (u™£1 (t , x) , . . . , U Q k ( t , x } Y satisfy the estimate i)ll, where the constant c > 0 is independent of k and f+(t,x}.
7 > 0,
Furthermore,
II -* 0 i, -z —>• co. PROOF. By definition,
(/"
~ 7, is, D X n ) R f + ( i r ) + 7, s, z n
(16.4)
(16.5)
4. Mixed Problems in a Quarter of Space
374
Using the notation (16.3) and Lemma 15.7, we can write this expression as follows:
(16.6) From the conditions on the pseudoparabolic operator £o(Dt , Ac) and Lemma 15.1 it follows that the entries of the matrix (Slj^'}N^(ir] + 7,$), 7 > 0, satisfy the assumptions of the Lizorkin theorem. Therefore, for any multiindex (3l we have \\D?lul0ik(t,x),Lpt,(R++l}\<:C
x f+(t, y', xn] dy1 ds, LP(1R^ where the constant c > 0 is independent of k and f+(t, x } . By the definitions (16.3) and (15.11), we obtain the inequality (16.4). Similarly, we prove (16.5). D Lemma 16.3. // \ot\/p' > t/ ; then for the component u l Q k ( t , x ] the vector-valued function U Q k ( t , x ) the following estimate holds:
)||),
7 > 0,
where the constant c > 0 is independent of k and f+(t,x). 0>
(16.7) Furthermore,
klt ^ 2 - ^ 0 0 .
(16.8)
PROOF. By the conditions on the pseudoparabolic operator Co(Dt, Dx), from Lemma 15.1 it follows that the entries of the matrix T,,_ m ((£})./V+ (iry + 7 i O > 7 > 0) satisfy the assumptions of the Lizorkin theorem. Taking into account (16.6), for each component u'0 k ( t , x) we obtain the estimate
§ 16. Convergence of Approximate Solutions
375
where F+(t,£) is the Fourier transform with respect to x of the vectorvalued function /"*"(<,#), and the constant c > 0 is independent of k and f+(t,x). Consequently, if \a\fp' > t / , then, repeating similar arguments from the proof of Lemma 13.3, we obtain the inequality
from which we obtain (16.7). Similarly, we establish (16.8). D
Lemma 16.4. The vector-valued functions u^k(t,x), j = 1 , . . . , [i satisfy the estimate (Rj+JII ^ c\\f+(t,*), W^n+i)ll,
Kfc(M),
7 > 0,
(16.9)
where the constant c > 0 25 independent of k and f+(t, x). Furthermore, \\^Q,
*i, *2-*oo.
(16.10)
PROOF. Using Lemma 15.8, we find oo
oo
f -00ffin_!
0
+
o £+(T, is, Dyn}Rf (r, s, yn}}Dyn^(r', s' , (C>°-(xn + yn)) dyn dsdr, OO
f -ooffin_1 0
x /^^ s,yn))wt(r',s',(C) a n (zn + yn)} dyn dsdn = wlk(t,x) + w k ( t , x ) , where r = ir, + 7, r' = r^)' 1 , (C)2 = |r|2 + <s} 2 , s'q = sq(t)-a<*, q = 1 , . . . , n — 1. The vector-valued functions wk(t, x ) , if|(t, x) are estimated by the same scheme. We consider, for example, wk(t,x). Using the Heaviside function, we can write wk(t, x) for xn > 0 as follows: CO
I ~l(t,x)= *.
w
OO
j
j
e^+ix>sK(s,n,k)e(yn)((bj>l...bj>m}
-00 K n _ ! -00
Mo C+(r, is, Dyn}Rf+(r, s, yn}}6(xn + y n )
x Dynu}^(T',s',(()an(xn
+ y n )) dyn dsdr)
376
4. Mixed Problems in a Quarter of Space
and, by the formula of the Fourier transform of convolution, we have t(t,x}=
I -oo E n
oo
x ( j \— Joo
e-iz^9(zn}(bJil...bJim)£+(T,iS,DZn)Rf+(T,S,zn)dzn} /
oo
x ( ( ei»*t~0(yn)DyHu}(T',8l,(Qa'-yn)dyn}dtdT)t \ J /
£ = (*,£„)
— oo
It is easy to show that the components of the vector-valued function
0
are multipliers. Consequently, CO
j
j
ei^+^>K(s,n,k}8(xn]
r = ir) + 7. Repeating the proof of Lemma 16.1, we obtain the estimate
where the constant GI > 0 is independent of k and f+(t,x}. estimate is established in the same way:
The following
\\w2k(t,x), W These estimates imply (16.9). We establish (16.10) in a similar way.
D
Lemma 16.5. Let j3l = ( # , . . . ,/?{,), / = m + 1, . . . , v, pl a = t / . Then the components of u~k(t,x}, Lp.^M+i)!! ^ c||/+(i,x), LP.^M+J!)!!,
7 > 0, (16.11)
§ 16. Convergence of Approximate Solutions
377
where the constant c > 0 is independent of k and f+(t,x).
Furthermore,
l
l ki(t,x)-D^u k2(t,x),Lp^(R++l}\\^Q
(16.12)
as ATI, ki —>• oo. PROOF. Using Lemma 15.8 and arguing in the same way as in the proof of Lemma 16.4, we find 00
'
j,k
OO
f
J
o £+(r, is, Dyn)Rf+(r, s, yn))DVnuJ (r', s',
+ j
oo
j
j
e^t+ix'sK(S,n,k}((bjtl...bj>m)
o DynC+(r, is, Dyn)Rf+(r, s, yn))u~ (T', s', (Qan(xn + y n )) dyn dsdr) 4 / j \ ) +, w k(t, x).
/1 ^? i o\
(lo.lo)
Estimates for the corresponding derivatives of the components of the vectorvalued functions w%(t,x) and wk(t,x) are proved by the same scheme. For definiteness, we consider the first vector- valued function. Introduce the diagonal (i/ — m) x (v — m)-matrix operator T>P with differentiation operators D%m , . . . , D% on the diagonal. We denote by (5^) the symbol of this operator. For the sake of simplicity, we set /3ln = 0. Using the Heaviside function and the formula for the Fourier transform of convolution in the same way as in the proof of Lemma 16.4, we have
~
~
\
x (fcj-,1 . . . bj>m)£+(T, is, DZn)Rf+(r, s, zn] dzn J oo
ft
el'^-%n)Dymw(r/l«/>(C>a"yn)dyn
x (E )
d^dr,,
xn > 0.
— 00
We can show that the components of the vector- valued function oo
1
(C)- feiv*e~Tl,-m((8))DVnuj(T)8tyn)dyn)
Re r > 0
(16.14)
378
4. Mixed Problems in a Quarter of Space
are multipliers in the space L p (M n + i). Since the functions s@ (s} multipliers in L p (lR n _i), the following estimate holds: m w k
t(
are
++
x(Q(bjil...bJtm)£+(T,iS,DXn}Rf+(T,s,xn)dsdrl,Lp(R++1) T — ir/ + 7. Repeating the proof of Lemma 16.1, we derive the estimate T \\S^ flU"^"'" \\\ <" r,\\f^~(-t k3(t ^ f c j T^ X J , Ljp j V ) r\/ ' T P ri i i J I I 5^ f-l
where the constant GI > 0 is independent of k and f + ( t , x ) . We establish the estimate
c2\\f+(t,x), in the same way. These estimates imply (16.11). We can establish (16.12) in a similar way. D Lemma 16.6. // \&'\/p' > t/, then the components ul- k ( t , x ) , I = m + 1, . . . , v, of the vector-valued functions u~k(t,x), j — 1, . . . , /j,, satisfy the estimate \ \ u l j i k ( t , x ) , Lp.^M+JOII ^ c ( \ \ f + ( t , x ) , + ||||/+(<1x)>L1(M+)|>Lpi7(M+)||)>
7>0,
where the constant c > 0 is independent of k and f+(t,x). \\uljlkl(t,x)-Uljik3(ttx),Lpti(R++l)\\^Qt
(16.15) Furthermore,
ATI, ^2-^00.
(16.16)
PROOF. To prove Lemma 16.5, we write the representation (cf. (16.13)) e-^uj^t, x) = wl(t, x) + wl(t, x), 7 > 0. We set wi(t,x) = (w>k'm+l(t,x),... ,w^(t,x))\
i = 3,4.
Then, under the condition \a'\/p' > t / , it is possible to prove the estimates
(16.17)
§ 16. Convergence of Approximate Solutions
379
where i = 3,4 and the constant c > 0 is independent of k and Indeed, for example, for i = 3 we have
f+(t,x).
wl(t,x)= -ooffi n
~ ~ \ x (6j,i . . . &j, m )£ + (r, is, DZn)Rf+(r,s,zn)dzn J oo
(
I eiy^e(yn)Dynu-(T',s',((:)a-yn)dyn}
\ J
d^dr),
xn > 0.
/
Since the components of the vector-valued function (16.14) are multipliers in L p (R n +i), the following estimate holds: \\wl'l(t,x),Lp(R+^}\\^c
j
I
e^t+ix'sK(s,n,k)e(xn)
Taking into account the condition |a'|/V > ti and repeating the corresponding arguments of the proof of Lemmas 13.3 and 13.6, we obtain (16.17). From the inequalities (16.17) we directly obtain (16.15). The proof of (16.16) is similar. D PROOF OF THEOREM 14.1. Under the assumptions Theorem 14.1, from Lemmas 16.1-16.6 we find
where the constant c > 0 is independent of k and / + (t, x). Moreover,
l-»o, || -»0 as &i, A?2 —>• oo. Since the Sobolev spaces are complete, there exists u(t, x) — lim uit(t,x), such that u(t,x) = (u+(t, x), u~(t, x)Y, where u+(t,x) G +J1)) w ~ ( t , a r ) € ^^'(Mj^), and it is a solution to the mixed problem (14.1) for f+(t,x) G ^(K+Jj). Since the space ^(Mj+J is
380
4. Mixed Problems in a Quarter of Space
dense in the space L P)7 (IR^ 1 ) H L p i 7 (M^; Li(M^)), a solution exists for f+(t,x) G £ Pl7 (Rj+i) nl p i 7 (Mj";Li(IR+)). The uniqueness of a solution is proved in the same way as in the case of the mixed boundary- value problems for equations. D The proof of Theorem 14.2 immediately follows from Lemmas 16.1, 16.2, 16.4, 16.5. The proof of the theorem in the case K2(DX) ^ 0 is similar.
Chapter 5 Qualitative Properties of Solutions to Sobolev-Type Equations In this chapter, we consider some aspects of the qualitative theory of Sobolevtype equations. In Section 1, we introduce the Sobolev-Wiener spaces and establish embedding theorems. These theorems are used in Sections 2, 3 in the study of asymptotic properties of solutions to the mixed boundary-value problems in cylindrical domains. In Section 4, we study the asymptotic behavior of algebraic moments of the solution to the first boundary-value problem for the Sobolev equation as t —> oo. We restrict ourselves to the first boundary-value problem for the Sobolev equation in the three-dimensional space. However, the results can be generalized to the n-dimensional case and the Sobolev system. Similar results can be obtained for the internal wave equation, the gravity-gyroscopic wave equation, the Rossby wave equation, and so on. In Section 5, we study asymptotic properties of solutions the Cauchy problem for one equation appearing in the study of small-amplitude oscillations of a rotating compressible fluid. This equation is solved relative to the higher-order time-derivative. The asymptotic behavior as t —>• oo of solutions to this equation essentially depends on the presence of the zero moments of the initial data. A similar situation holds for solutions to the Sobolev equation, the internal wave equations, and the Rossby wave equation (cf. S. V. Uspenskii, G. V. Demidenko, and V. G. Perepelkin [1] and S. V. Uspenskii and G. V. Demidenko [2]).
§ 1. Sobolev-Wiener Spaces We define function spaces introduced by S. V. Uspenskii on the basis of the Sobolev spaces and the Paley-Wiener ones. The embedding theorems allow us to give the qualitative characteristic of the function u ( t , x ) with 381
382
5. Qualitative Properties of Solutions to Sobolev-Type Equations
respect to a distinguished variable as t —> oo. Let G be a bounded domain in M rl , and let Q = { ( t , x) : t > 0, x £ G}. We set 00
= f e-Tu(t,x)dt.
(1.1)
Let /o ^ 1) r ^ 0, ^ — ( ^ o , ^ i ) • • • i In), 1 ^ p < oo. We say that a function u(t,x) belongs to the Sobolev- Wiener space Wlp>r ' (Q) [ f u ( t , x ) belongs to the weight Sobolev space equipped with the norm
1/p
0 G
where N > 0, and the norm sup k=0
1/p
k=Qi=l^
is finite, where 6 > 0 is some real number. We say that the function u(t) defined in (0,oo) is oscillating if in any neighborhood of infinity it change its sign. Theorem 1.1. Let u(t,x) £ Wp n. If 0 = 1 - pa — \a\/P > 0) then, at any interior point x° £ G, the function DlDpxu(t, x°), regarded as a function of the variable t either is summable in (0,oo) and monotonnically tends to zero as t —> oo or Dl+lD?u(t, x°) is oscillating in (0, oo). PROOF. Since for u(t. x) the norm (1.2) is finite, the function Drt+ku(j,x) is defined in the domain G for all 7 > 0 and belongs to the Sobolev space Wp(G), 1 = ( / i , . . . , l n ) - We denote by Ge C G a compact set such that the distance between any point of this set and the boundary dG is greater than
§ 1. Sobolev- Wiener Spaces
383
e. Setting k = 0, 1 and applying the embedding Sobolev theorem for the compact set Ge , we obtain the estimate sup \Dl^Dftxu(^x}\ ^ c\\u(t,x), W^(Q)\\.
(1.3)
This estimate is valid for all 7 E (0,5] with constant c > 0 independent of 7. Then for any point x° £ Gs we have sup | z f t i ( 7 , z°)KCl,
0<7<<5
sup p ^ ( 7 J x 0 ) U c 1 ,
(1.4)
0<7<<5
We prove the estimates 0
(1.5)
,«°)|A^c2,
(1.6)
)\dt^c2,
where NI = N if N > 1 and JV^ = 1 + cr if N ^ 1. Since for u(t, x) the norm (1.2) is finite, for any t £ [0, oo) and k = 0, 1 the following norm is finite: f
=
f
/ \Drt+kDl^u(t,x)\rdx+ / \Drt+ku(t,x}\ G G
By the Sobolev embedding theorem, for x 6 GE we have
r
\ !/P
/ |Dr+*u(< > a r)|"dx) J /
._
'- G
,
(1.7)
where k = 0, 1 and the constant c is independent of it(t, x) and t. Multiplying the inequalities (1.7) by (l + tf)"^1 and integrating with respect to t, we find
(
C
f
(1 + *)~M£ / \Drt+kD^u(t,x)\^dx+ V »• — 1 *^
I \Drt+ku(t, J
384
5. Qualitative Properties of Solutions to Sobolev-Type Equations
Applying the Holder inequality, we find
<: c 0
G
(1.8) Setting x = x° in (1.8), we obtain the estimates (1.5) and (1.6). To complete the proof, it suffices to apply Lemma 1.2 (below) to the function Drt D?u(t, x°). D To prove Lemma 1.2 used in the proof of Theorem 1.1, we need the following assertion. Lemma 1.1. Let a function v(t) be defined in [0,oo) ; and let t)~N\v(t)\dt
sup
(1.9)
0<7<<5
where N > 0; 8 > 0 are real numbers. Then v(t) is either oscillating in (0,oo) or summable in (0,oo) and does not change the sign in some neighborhood of oo. PROOF. Since the function v(t) satisfies the condition (1.9), the Laplace transform
exists for any 7 > 0. Therefore, either v(t) is oscillating in (0,oo) or there exists R > 0 such that for t > R the function v(t] does not change its sign. We assume that v(t] is not oscillating. We prove that it is summable in (0, oo). We have < c for any
7 6 (Q,S).
Therefore, R
f(l+trN\v(t)\dt\ J
/
§ 1. Sobolev-Wiener Spaces
385
where the constant c\ depends only on R. Without loss of generality, we assume that v(t) ^ 0 in (R, oo). Then for any RI > R and 0 < 7 < 6 we have [ e-^v(t) dt ^ ci (l + 1(1 + t)-N\v(t)\ dt J
\
R
J
0
or, by the Fatou lemma,
RI RI RI I v(t] dt = lim e-^v(t) dt <£ sup / e~^v J J ^->° o< 7 <<57
R
R
dt
R
oo
[(l+t)-N\v(t)\dt\. J
/
Since the function v(t) has the same sign on (R, oo) and RI > R is arbitrary, we find oo
v(t) dt ^ ci (l + I (I + tYN\v(t)\dt\ \
J
/
or oo
/KOI dt < c?.
D
Lemma 1.2. Let the function v(t] be defined in [0,oo) and have the weak derivative vt(t). Furthermore,
sup
dt < c,
sup
I/ e j| o
dt
where N > 0 and 8 > 0 are some real numbers. Then either v(t) is summable in (0,oo) and monotonnically tends to zero as t —> oo or vt(t) is oscillating in (0,oo) ; i.e., it changes the sign in any neighborhood of infinity.
386
5. Qualitative Properties of Solutions to Sobolev-Type Equations
PROOF. The assumptions of Lemma 1.1 are satisfied by v(t) and by v t ( t ) . Therefore, v t ( t ) is either oscillating or summable in (0,oo) and does not change the sign in some neighborhood of oo. If the derivative Vt(t) is a summable function of constant sign in some neighborhood of oo, then v(t) cannot be an oscillating function. By Lemma 1.1, the function v(t) is summable in (0,oo). Since the sign of Vt(t) does not change in some neighborhood of oo, the function v(t) monotonnically tends to zero. D Theorem 1.1 can be regarded as a generalization of the Bochner theorem about almost periodicity (cf., for example, B. M. Levitan and V. V. Zhikov [1]) to the case of oscillating functions. We establish some sufficient conditions for the oscillation of functions (the monotone convergence to zero as t —> oo) under the additional assumptions that u(t, x) is an entire function oft at each interior point x G G. Theorem 1.2. Let u ( t , x ) £ Wl'r(Q] satisfy the assumptions of Theorem 1.1. We assume that the derivative D%.u(t,x°) at any point x° G G is an entire function o f t ; moreover, in any ball {\t\ < R} of the complex plane, the derivative Drt D%u(t, x°] is represented as a uniformly 00
convergent series Drt Dpxu(t, z°) - £ ak(x°)tk. Let Drt+kD?u(t, x°) £ 0 k=0 n
for k = 0, I , and let Pn(t,x°) = ^ ak(x°)tk.
If for any large number
k =0
N > 0 there are numbers a > N, b > N and a sequence of polynomials Pnu(t,x°], nv —>• oo, such that each polynomial in the interval (a, 6) has zero, then the function Drt+k D?.u(t, x°] regarded as a function oft is an oscillating function in (0,oo). PROOF. Since the sequence of polynomials Pnv(t,x°) uniformly converges to an analytic function DrtD^.u(t,x°} in any ball {\t\ < R} of the complex plane, from the Hurwitz theorem (cf., for example, E. C. Titchmarsh [1]) it follows that each limit point of zeros of the polynomials Pnv(t,x°) is a zero of the function D[D^.u(t, x ° ) . Then, in any neighborhood of co, the function DrtD^.u(t} x°) has a finite real zero. Consequently, the function Dl+lD?u(t, x°] is oscillating in (0,oo). D Theorem 1.3. Let u ( t , x ) 6 Wlp'T'(Q) satisfy the assumptions of Theorem 1.1. We assume that, in any point x° G G, the derivative D£u(t,x°) is an entire function oft; moreover, in any ball {\t\ < R}, the derivative °) is represented as a uniformly convergent series
k=0
§ 2. Mixed Problems
387
We assume that Dl+lDxu(t, x°] 3= 0. Introduce the notation Pn(t,x°) = n
Y^ ^k(z°)tk • If there exists a real number ko > 0 and a sequence of k-O
polynomials Pnv(t,x°), nv —>• oo, such that for all zeros {z^v} of the polynomials Pn^(t,x°) we have the estimate Re z^v ^ k0, then the function Drt Dxu(t, x°] is summable in (0,oo) and monotonnically tends to zero as t —> oo. PROOF. Since the sequence of Pnv(t, x°) uniformly converges in any ball {\t\ < R} of the complex plane to the analytic function D[ +1 D£w(t, x ° ] , from the Hurwitz theorem it follows that the entire function Dl+1Dxu(t, x°) has no zeros in Re t > &o- Hence the function Dl+iDxu(t, x°] has no real zeros in (&o,oo) and, consequently, cannot be an oscillating function. By Theorem 1.1, D^Dxu(t, x°) regarded as a function of/ is summable in (0, oo) and monotonnically tends to zero as t —>• oo. D
§ 2. Mixed Problems for Sobolev-Type Equations in Cylindrical Domains We consider the general mixed boundary- value problem for simple Sobolevtype equations 1-1 LQ(x- Dx}D[u + Li-k (x; Dx}Dktu = F(t, x), t > 0, x 6 G,
. = $j(t,x'),
j = l,...,m,t>Q, x'edG,
u = 0, t< 0 in a cylindrical domain Q — {t > Q, x € G c M n } . The class of such problems contains, in particular, boundary-value problems for the Sobolev equation, the internal wave equations, the gravity-gyroscopic wave equation, the Rossby wave equation, the Boussinesq equations and so on. We further assume that G C M n is a bounded domain with (n — l)-dimensional smooth boundary dG. We formulate the assumptions on the operator /-i L ( x ; Dt,Dx) = L0(x; Dx)Dlt + ^ Li-k (*; DX)D* . k=Q
Assumption 1. The operator LQ(X\DX] is an elliptic operator of order 2m. In the case n = 2, for any linearly independent vectors £' and £" the equation LQ(X;£' + A£") = 0 has exactly m roots A in the half-plane Im A > 0.
388
5. Qualitative Properties of Solutions to Sobolev-Type Equations Assumption 2. The order of each operator Li-k(x\ Dx] is at most 2m.
We assume that the coefficients of the operator L(x;Dt,Dx) are sufficiently smooth. We formulate the assumptions on the boundary operators in (2.1) Assumption 3. The operators Bj(x; Dt, Dx) have the form Bj(x-DttDt) = bj(x-Dx)Dlt'
+
the order of the operator b j ( x ; , Dx) is equal to rrij
L0(x;Dx)v = g(x),
x£G, l
bj(x;Dx)v
QG
= il>j(x ),
j=l,...,m
satisfies the Lopatinskii condition ( cf., for example, J. L. Lions and E. Magenes [1], H. Triebel [1], L. Hormander [2]). We define the weight Sobolev space Wl^(Q], /, r 6 N, 7 > 0, of functions u(t,x) with finite norm \e-*u(t, x), L 2 ((0, oo); W?(G}}\\ , L2((0,oo)xG)|| such that D f w | t _ 0 = 0, k = 0, . . . , / - 1. Let V"2' r (C 7 x G ) , C 7 = {r = (T + irj eC, Re r > 7). A function g(r, x) belongs to the class V 2 ' r (C 7 x G) if for almost all x G G the function g(r, x} is analytic with respect to r E C 7 , has the weak derivatives \/3\ <; r, in the domain G, and the following norm is finite: 00
,
-, x), V2' (C-y x G)|| = sup
(
F
(
/
Re r > 7 \ J — 00
+
7'
"'" '
'
°"
\ 1/2
Using the Paley-Wiener theorem (cf. Chapter 1, Section 5), one can show that the integral Laplace operator C maps the space W2' rr~(Q) onto the
§ 2. Mixed Problems
389
space V^'^C-y x G) in a one-to-one and mutually continuous manner. Such assertions are usually referred to as "generalized Paley-Wiener theorems" (cf., for example, M. S. Agranovich and M. I. Vishik [I] and S. V. Uspenskii, G. V. Demidenko, and V. G. Perepelkin [1]). We introduce the space K 2 ' r (C 7 x dG). A function y?(r, #') belongs to K 2 ' r (C 7 x dG) if for almost all x' G dG the function ip(T,x') is analytic with respect to r G C 7 , for every r G C 7 the function <^(r, x') belongs to W% (dG) , and the following norm is finite:
= sup Re r>7
— oo 1/2
Theorem 2.1. Le£ F(*, x) G W2;2 r,x' G V -'-4m-m'Then there exists 70 > 0 suc/i that the question on the solvability of the mixed problem (2.1) in the space W?.^4m(Q), 7 > 70, is reduced to the solution to the Fredholm equation of the second kind. Theorem 2.2. If for any g(x) G L2(G), ^(x1) G W 2 2m ~ mj ~ 1/2 (<9G), j = I , . . . ,m, the boundary-value problem (2.2) is uniquely solvable in the space W%m(G}, then there exists 70 > 0 such that the mixed problem (2.1) has a unique solution u(t,x) G W$lfm(Q), 7 > 70, for any F(t,x] G w£m(Q), £[^](r,x') G y22'-i-4m-^->1/2(C7 x dG), j=l,...,m, and the solution satisfies the estimate c(\\F(t}x),W%m '~~'j; i4m—rrij'~ 1/2//pi
£)/^MI \ (y—y x c/GJII ),
/o Q^
(^-3j
where the constant c > 0 25 independent of F(t,x) and $j(t,x'). From Theorem 2.2 it follows, for example, the unique solvability of the
390
5. Qualitative Properties of Solutions to Sobolev-Type Equations
first boundary-value problem
for any e J> 0. In particular, this is valid for the Boussinesq equations, the Sobolev equation, the Rossby wave equation, the gravity-gyroscopic wave equation, and so on. PROOF OF THEOREM 2.1. By the generalized Paley-Wiener theorem, the study of the solvability of the mixed problem (2.1) in the space Wj,^ m' (Q} ls reduced to the study of the boundary-value problem with parameter r E C 7 for the stationary equation /-i fc= o
(2.4)
j — 1,. . . , m. Assuming that Re r > 7 > 0, we can write this problem in the form i-\ L Q ( X ] L)X)V
-f- y ^ T k
L[^k(X]
IJxjV
— T
9\T> X ) ,
X E G,
(2.5)
=°
bj(x;Dx)v+
^
rfc~/J&fe|/3
j = 1, . . . ,m. As is known, the boundary-value problems for elliptic operators satisfying the Lopatinskii condition are reduced to the Fredholm equation of the second kind by constructing the right and left regularizes (cf., for example, L. Hormander [2]). We assume that these regularizers are already constructed and apply them to our problem. We consider the right regularizer -Bright for the operator A | -T> • J j
\ — J T „ i o-» • ij
\
t* O 01 I T* * \J
i
S O 0
where S is the trace operator on dG. By definition,
f "7*" L)
Ir
§ 2. Mixed Problems
391
where q ^ 0, TJ = 2m — mj — 1/2, is a linear continuous operator; moreover,
where the operator m
m
T : W|(G) x U W?+q(dG) -> W|(G) x JJ W^ j=i j=i is completely continuous. For brevity, we introduce the notation m
Mq = Wj(G) x JJ W? j=i and set
We look for a solution to the problem (2.4) in the form V(T,X) = Bright { / , ¥ > ! , . . . , ^ m } ( T , x ) .
Using the definition of -R r i g ht, from (2.5) we find f ( r , x) + T 0 { f , i,... ,v>m}(r,x)
= — g(r, x) = G(r,x),
x G G,
or, in the vector notation, » i > . . . > ¥ ) m } = {G>*i>...>*m}.
(2.6)
392
5. Qualitative Properties of Solutions to Sobolev-Type Equations
Let us show that there exists a sufficiently large 70 > 0 such that for any fixed r, Re r > 70, the norm of the operator M(r) is strictly less than 1. By the properties of the operator /i r i g ht, for any fixed T, Re r ^> 7 > 1, we have * ^ , v ll&k >HB\.£}T // -j " ' \/3\^2m,k^l-l
n Ml/,
Similarly,
, W^(dG)\\}moreover, the constants c' and c" are independent of r. Consequently, for sufficiently large 70 > 1 we have }
Rer;>7o.
Therefore, for any fixed r, Re r ^ 70, the operator equation (2.6) can be written in the form (/ + T o (7 + M(r))- 1 ){/ ) ft,... , £m} = {G, * i , . . . , *m};
(2-7)
where {/,v?i, . . . ,<£ m } = (/ + Af(r)){/, v?i, - - . ,Vm}- The operator (7 + M(r))" 1 is bounded in A^9 for any fixed T, Re r ^ 70; moreover, ||(/ + M(r))~ 1 || ^ c(7o), and 0(70) is independent of r. The operator T is completely continuous in Mq and is independent of r. Therefore, the operator T o (/ + M ( r } } ~ 1 is also completely continuous in Mq\ moreover, sup
\\To(I + M(r)rl\\
r:Re T^7 0
Therefore, for any fixed T, Re r ^ 70, the operator equation (2.7) is the Fredholm equation of the second kind. Thus, the question on the existence of a solution to the boundary-value problem (2.5) has been reduced to the question on finding {/, m} from equation (2.7).
§ 2. Mixed Problems
393
By the Fredholm alternative, either a solution { / , £ > i , . . . ,£>m} exists for any {G, W i , . . . , \I>m} 6 .M9 (in this case, the solution is unique) or {G, ^i, . . . , ^m} satisfies some orthogonality conditions. In the last case, the solution {/, i, . . . ,£>m} is orthogonal to all solutions to the homogeneous equation, such a solution {/,£>i, . . . ,£>m} is unique. From the Fredholm theorems it follows that for any fixed r, Re r ^> 70 a unique solution to equation (2.7) satisfies the following estimate c\\T
^c(7o
By the boundedness of the operator (/ + M(r))" 1 , we have (2.8)
Since the solution to the problem (2.4) can be found in the form v — Bright!/, V?i, • • • , ¥>m}, from the properties of Hright and the estimate (2.8) for fixed r, Re r ^ 70 we find c(7o)||{G, *i, . . . , *TO Therefore, for 7 > 70 we have Mr,*), V 2 2/ ' 4m (C, x G)|| ^ c ( 7 o ) | | G ( r , x ) , Vf '2m(C, x G)||
Taking into account that G(r, x) = r~lg(r,x), ^ J ( T , X ' ) = T~IJI[}J(T}X'}, we find
(2.9) By the generalized Paley-Wiener theorem, u(t,x) = C ~ l [ v ] ( t , x ) is a solution to the mixed problem (2.1) satisfying the estimate (2.3). Similarly, the question on the uniqueness of a solution can be reduced to the solution of the Fredholm equation of the second kind by using the left regularizer .Rieft of the vector operator A(x; Dx). O
394
5. Qualitative Properties of Solutions to Sobolev-Type Equations
PROOF OF THEOREM 2.2. By the assumptions in the problem (2.2), the regularizer -R r j g ht is inverse to the operator A(x;Dx). Consequently, equation (2.6) can be written in the form
Since the norm of the operator M(r] is less than 1 for Re r ;> 70, where 70 > 0 is a sufficiently large number, we have
Hence the function u(t,x) = ^"^Hrightj/, <{>i, • • • ,pm})(t,x) is a solution to the mixed problem (2.1). The uniqueness of a solution is proved in a similar way. By the Paley- Wiener theorem, the required estimate follows from (2.9). D The following assertion directly follows from Theorems 2.2 and 1.1. Corollary. Let the assumptions of Theorem 2.2 hold, and let the solution to the problem (2.1) belong to the Sobolev-Wiener space W^'r(Q}, I - ( / 0 , / i , • • • , / n ) - // 1 - pot > H/2, r > IQ, a = (a 1} . . . ,an), otj = l / l j , p = ( p i , . . . ,pn), Pj ^ 0 are integer, then at each interior point x° 6 G either DlD^.u(t^xQ] is summable in (0,co) and monotonmcally tends to zero as t —>• co or DrtJrlD^.u(t, x°) is an oscillating function in (0,oo).
§ 3. Properties of Solutions to the First Boundary- Value Problem for the Sobolev Equation We consider the first boundary-value problem for the Sobolev equation u + Dl3u = 0, =0,
t > 0, x E G, (3.1)
00,
We assume that G C MS is a bounded domain with sufficiently smooth boundary dG and ^i(cc),
>6},
u£(x°] = { x 6 M 3 : | x - x ° | < e}, Sex° = x£R3: x - x ° = e ,
§ 3. Properties of Solutions to the First Boundary- Value Problems
395
3
(Vu(t, x),Vv(t, x)) = ^ DX]u(t, x } D X ] v ( t , x),
oo
U(T, x) = —== I e~Ttu(t,x}dt,
Re r > 0.
The solution to the Dirichlet problem AU = f ( x ) , x 6 G, v QG= 0 is denoted by v(x) = &-1f(x). Theorem 3.1. At any interior point x° £ G, the solution to the boundary-value problem (3.1) possesses the following property: either D^u(t,x°} is summable with respect to t on (0, oo) and monotone decreases to zero as t —>• oo or D f u ( t , x ° ) is an oscillating function. We divide the proof of Theorem into several lemmas. Lemma 3.1 (the energy identity). Let u ( t , x ] be a solution to the problem (3.1), and let
Then Ek(t} = Ek(0), k = 1,2. Lemma 3.2. The Laplace transform (3.1) satisfies the estimate
c
of the solution to the problem
t (W^)!2 + |Av> 2 (z)| 2 ) dx, J \ /
Im r = 0,
(3.2)
G
where the constant c > 0 is independent of 2(x). PROOF. Since the solution to the problem (3.1) can have at most power growth as t —> oo (cf. S. V. Uspenskii and G. V. Demidenko [1]), for the Laplace transform we have , x) + D^3u(r, x) = -^=(Ay>2 + rAy?i),
Re r > 0.
396
5. Qualitative Properties of Solutions to Sobolev-Type Equations
Multiplying by w(r, x) and integrating by parts for Im r — 0, we find
f
~
f
r I \(Vu(r,x), Vv?!(z)) \dx+ I \u(r, x)A
J
G
G
\ 1//2 / r
f
T2\Vu(T,x)\2dx)
/ |Vpi(*)| 2 \J G x 1/2 x y/2
J
/
U(T, x)\2 dx \
G
( / JA<^>2( 2; )| 2 dx ) G
By the Steklov inequality f\u(r,x)\2dx<:c(G) J
f\DX3u(r,x)\2dx,
(3.3)
J
G
G
we have /('r 2 |Vw(r,x)| 2 ^+(^ 3 ^r ! x)) 2 ^x J
G
\
/
, I/ (-r^l^JTil-r I / V t X l / j tT L iMl
2
x 1/2 21 4D X 3?/Yr /7r1 ^ II ^ (I ^ \ ) T-^ / / !yt-*
«/
/
G
/ r
x f / N*'
G
which implies (3.2).
D
We consider a function r<j(x) e C*o°(G) such that r<j(x) = 1 for x £ G$ and r,j(x) = 0 for x G G\G$/2- Then the function U(5(i,x) = u(t,x)rs(x) is a solution to the problem u6 + D2zu6 = f ( t , x ] ,
us
t=Q=
Dtus t^0=(p2ts(x), (3.4)
where f ( t , x) = 1(\7Dtu(t, x), Vrj(x)) + 2A ? 3 u(t, x)D l 3 r<5(x) + u(t, x)D23rs(x),
(3.5)
§ 3. Properties of Solutions to the First Boundary-Value Problems
397
- X°)2 + ( X 2 - Z°) 2 + ( X3 - *°)2.
S. L. Sobolev [7] proved that for x ^ 0 the function $(t,x,x°) satisfies the equation AD%$ + D%3$ — 0- Using this function, it is possible to write an explicit formula for the solution to the Cauchy problem for the above equation (cf. formula (158) in [7]). We write the Sobolev formula for the solution to the Cauchy problem (3.4) at an arbitrary point x° 6 MS: 1 f \ us(t,x°) = -— I -J0(tp/r)A(pi!s(x}dx 47T J r 13
I
tp/r
ri/ f
~ T~ 4?r JI ~\ p V JI 13
0
t
~ TI I ~( 47T J J p \
M*)** /( s - a:)
dxds
-
Lemma 3.3. The Laplace transform of a function f ( t , x ] defined in (3.5) satisfies the estimate sup / |/(T, x)| 2 dx ^ c, ^ G
(3-7)
where the constant c > 0 depends only on x and G. PROOF. From the estimates for the solution to the boundary-value problem (3.1)
398
5. Qualitative Properties of Solutions to Sobolev-Type Equations
(cf. S. V. Uspenskii and G. V. Demidenko [1]) and Lemma 3.1 we have c2tM+c2t
|i/|£l,
(3.9)
where the constants c i , C 2 > 0 are independent of i. Therefore, the function f ( r , x) for Re r > 0 is defined; moreover,
/(r, x) - 2r 2 (Vw(r, x), Vr,j(x)) -.2 ' u(r,x}Dlr 6(x]
^
1
DX3u(r, x)DX3r6(x) + r 2 u(r,
(2r(Vu(0,z), Vrs(x)}
27T
, x,
ru(0,
Taking into account the inequalities (3.2) and (3.3), for Im r — 0 we obtain the estimate (3.7). D Lemma 3.4. For any point x° £ G$ the following estimates hold: 00
sup 0
/ e-TtD?u(t,x°)dt
< 00,
J
(3.10) sup
C 2 < 00.
0
PROOF. From the expression for the Laplace transform of the derivative of a function we have sup
£[D?u](T,x0)\^\D?u(Q,x°)\+
0
sup
r\jC[D^u](r,x°)\.
0
Consequently, it suffices to prove (3.10). For this purpose, we use the Sobolev formula (3.6). Differentiating (3.6) twice with respect to t and taking into account the definition of the functions <£(£, x, x°), ipjts(x), f ( t , x ) , we find at x° £ G$: D?u(t,x°) = --— I 47T J G
Dl
4?r 1
— s,x,x
4> 0 G
)f(s,x)dxds.
(3.11)
§ 3. Properties of Solutions to the First Boundary- Value Problems
399
Applying the Laplace transform, we find }(r, x°) = ~
L\D\*\(T, x, x°)A
l
- -1
-f(r, x)dx-
JC[$](T, x, x°)f(T, x) dx. G
By the equalities (0, Z, z 00) = 1/r, Dt$(Q,x,x°} ° = 0 and the formula — r
(cf., for example, I. S. Gradshteyn and I. M. Ryzhik [1]), we have
|£[D?u](r>x°)|^c(/'l X''
G
X
(/' /
I / I *
/
\ iO
i
*
/
\ iO
i < * /
\ i O\
«
By Lemma 3.3 we obtain the inequality (3.10).
I
*/ 2
D
PROOF OF THEOREM 3.1. The assertion immediately follows from Lemmas 1.1, 1.2, 3.4 and the inequality (3.8). The following theorem yields estimates for the amplitude of oscillations of the derivatives of the solution to the boundary- value problem (3.1). Theorem 3. 2. At any interior point x° G G, the solution to the problem (3.1) satisfies the estimates \DtD»u(t, x°)\ ^ c ( * + i 2 + 1),
|,,| ^ 0,
(3.12)
where the constant c > 0 is independent of t. PROOF. Making the change of variables y — x — or 0 , we write formula
400
5. Qualitative Properties of Solutions to Sobolev-Type Equations
(3.11) in the form D*u(tt x°) = --L
D**(t, y, 0)Ap M (y + *°) dy
G'
t
-~ 47T
0 G'
Then
1/2
f
2
•||D£/(t,x), L 2 (G)||
/r
/ \D»f(s,y + x°)l2
\J
0
G'
G'
ds. /
Since i,6(x),
\ 1/2
/ /•
f / \Dt$(t-S}y,0)\2dy} / \v / 0
ds,
(3.13)
G'
where the constant GI > 0 is independent o f t . We estimate the integral 1/2
f f f /(*)= / f / J
T
\v/
G'
1
Assuming that G C {y G ^3 '• \y\ ^ R} and making the spherical change, in view of the definition of the function 3>(i, y, 0), we find
}(} \1/2 \ I ( t } \ ^c(R) I I / sin 2 ^(Jo(7-sin^)) 2 ^) dr. T
0
Further, we choose T > 1 such that J\(t] ^ 1/V^, t ^ T. Using the equality ./o(0 — ~^i(^) and representing the integration interval in the
§ 4. Algebraic Moments
401
form (0,7r) = II(T) U 7 2 (r), where /i(r) = {9 : sin6 > Tr'1} and / 2 (r) = {9 :sine ^ Tr~1}, we find 1/2
f I si
dr
J
(T)
(R)
/•/ r
1/2
i I sin^ •/ \ •/
T
/ 2 (r)
T
J 2 (r)
By (3.13), the inequality (3.12) holds.
D
Remark 3.1. Estimates of the form (3.12) are valid for the derivatives of the form D?+lD»u(t, z°), / > 0, |i/| ^ 0. Remark 3.2. Analogs of Theorems 3.1 and 3.2 are valid for the Sobolev system.
§ 4. Algebraic Moments of Solutions to the First Boundary- Value Problem for the Sobolev Equation Properties of functions by the method of algebraic moments were studied by many authors. This question was first considered in the classical works of P. L. Chebyshev and A. A. Markov. Various aspects concerning the study of algebraic moments are contained in the monograph of M. G. Krein and A. A. Nudel'man [1]. In this section, we study the behavior of the algebraic moments of the solution to the first boundary-value problem for the Sobolev equation in a cylindrical domain Q = { ( t , x ] : t > 0, x G G c MS) as t —>• oo. We consider a special class of domains G on the boundary of which some polynomial P(x) vanishes. This class contains, in particular, a ball, a half-ball, a quarter of a ball, a paraboloid, a cone, We consider the case of a ball in detail. We prove that, if the average of the initial data 2(x] vanish, then the average of the solution to the problem (3.1) also vanishes. We define the function class C 2 ' 2 ([0,T] x G). We say that a function u(t, x) belongs to the space C 2 ' 2 ([0, T] x G) if u(t, x) is defined in [0, T] x G,
402
5. Qualitative Properties of Solutions to Sobolev-Type Equations
has continuous partial derivatives with respect to Xk, k — 1, 2, 3, up to the second order, and D^D^.u(t,x) are continuous on [0,T] x G for v — 1,2, |/?| ^ 2. Let G be the ball with radius R: x\ + x\ + x\ ^ R2. Lemma 4.1. Let u ( t , x ) E G 2|2 ([0,T] x G) be a solution to the problem (3.1). Then the zeroth-order moment of the solution u ( t , x ) has the form
r /
G
r
u(t, xj dx — v 3 sin^t / v o) / <^2(2')^^~(~ cos(t / v o) / y ? i ( x ) u x .
(4.1)
G
PROOF. Consider the function ui(t - r , x ) = (R2 - \x2 ) sin(ai(< - r)),
(4.2)
where a\ > 0 is some constant. This function vanishes on the boundary of the ball G. We multiply the identity L(DT, Dx)u(r, x) = AD2u(r, x) + D23u(r, x} = 0 by u\(t — r, x) and integrate:
t r r 0= / / L(DT, DX)U(T, x)ui(t - T, x) dx dr J
J
o G t
U(T, x ) L ( D T , Dx)ui(t — T, x} dx dr 0
G
+ If Dru(r, x)Aui(t — r, x) * dx — f u(r, x}Dr&.u\(t — r, x) dx. (4.3) J
n
J
We choose the constant a\ in (4.2) in such a way that L(DT, Dx}u\(t — r,x) = 0, i.e., (6af - 2)sinai(< - r) = 0. Then ai = 1/v^, and (4.3) implies (4.1). D Corollary. If the mean of the initial data 2(x) vanish, then the mean of the solution u ( t , x ) to the problem (3.1) also vanishes for any t. Lemma 4.2. Let u ( t , x ] 6 C 2|2 ([0,T] x G) be a solution to the problem (3.1) for any T > 0. Then the first-order moments of the solution u ( t , x)
§ 4. Algebraic Moments
403
have the following form:
I u(t, x)xj dx G /
f 2(x)xj dx + cos(t/v5) / t p i ( x ) x j d x ,
(4-4)
•J
G
G
j U(t,
3 \ f
3
5 / GJ
I /3 \ f
V V s yG J
1
where j = 1,2. PROOF. We consider the functions Uj+i(t — r, x) = (/?2 — |x| 2 )xj sin(aj+i(i — r)),
j = 1, 2, 3,
where c*2, t*s, <^4 are negative constants. We note that wJ7 -+i \ ^ — 0. ro •" ' ^' *-* i |^(J O^lvf The constants &j+i are chosen in such a way that L(D T , £) x )itj + i(< — r, x) = 0. Then Q 2 = a3 = l/\/5 and a 4 = ^3/5. Replacing MI(/,O;) in (4.3) by ^2(^,2:), u z ( t , x ] , u ^ ( t , x ] , we obtain (4.4) and (4.5). D )
T
Corollary. // the first-order moments of the initial data i(x), ^f>i(x] vanish, then the first-order moments of the solution u(t,x) to the problem (3.1) also vanish for any t. We consider an algorithm for obtaining moments of arbitrary order / of the solution u(t,x) to the problem (3.1) in terms of moments of order at most / of the initial data
(4.6)
bN
where /?7, a are constants, 7 = (71,72,73), 7i ^ 0, i — 1,2,3. We denote by NI the number of different combinations of 71, 72, 73. Then NI — (/ + l)(/ + 2)/2. We have L(DT,Dx)v(t - r , x ) = AD2.^/ - T,X) + D%3v(t - T,X)
404
5. Qualitative Properties of Solutions to Sobolev-Type Equations
where Ay(/? 7 ,a 2 ), j97(/?7, a 2 ) are linear functions of /?7 and a 2 . We consider the expression ^x^(-\x\2) sin(a(* -
= L(DT ,DX) - -IfT
and compute Dl
1) - /? 7l+2 , 72 , 73 _ 2 (7 1 + 2) (71 +^71+2,72-2,73 +^ 7 l +2, 7 2 , 7 3-2)(7l + 3 =
We note that if some index 7 contains the component 7,- — 2 < 0, i = 1,2,3, then the corresponding /?7 vanishes. Similarly, we can compute the derivatives
n2 |7|=/
Taking into account the above computations, we find 5 7 (/? 7 ,a 2 ) = a 2 (07 1 ,7 2 ,73 4- 0 7l+2 , 72 _ 2|73 +/? 7l+2|72i73 _ 2 )(7 2 + 871 + 2) + (071,72,73 + 071+2,72-2,73 + 07i +2,72,73-2) (l\ + 372 + 2) + (071,72,73 + 071+2,72-2,73 + 071 +2,72,73 -2) (73 + 373 + 2) = /
j
Pa,i(a )0<
\<7\=l
For B-y = 0, J7J = / we obtain the homogeneous linear system of order JV/:
We find a 2 from the following condition. Assumption A: A1 = d e t ( p a , 7 ( a 2 ) ) = 0 .
(4.8)
§ 4. Algebraic Moments
405
Then we have AT/ roots af , . . . , aj^ . From (4.7) we find 0a^ , . . . , 0a,Ni which depend, at least, on one arbitrary constant and, consequently, do not vanish identically. Substituting the obtained values of a\ , . . . , aj^ and AT.I) • • • j fla.Nt m (4-6), we find the functions - r)),
vk(t
We note that v^
.
f
k=l,...,NL (4.9)
= 0. For each function Vk(t — r, x) we have
r 0 = / / L(DT, DX)U(T, x} Vk(t — r, x) dx dr 0 G t
= I I u(r, x} L(Dr, Dx}vk(t — T, x) dxdr 0 G
+ / DTU(T, x)Af/c(t — r, x) dx — u(r, x)DTAvfc(< — r, x) G ° G
dx,
which implies
/
w(r, x}DTAvk(t — r, x)|\T-t _ dx
G t
= I I u(r, x)L(DT,Dx)vk(t - r, x} dx dr 0 G T
G
~
+ j u(r, x)Dr&vk(t - T, x)| T=f) dx.
(4.10)
We assume that we already obtain moments of order / — 2 (/ ^ 2) of the solution u(t,x) to the problem (3.1). We make computations for the moments of order /. Taking into account the definition (4.9) of the functions vk(t — r, x),
406
5. Qualitative Properties of Solutions to Sobolev-Type Equations
k = 1,. . . , Ni, from (4.10) for all k = I , . . . , NI we find li^T, XJ LJ-j- L\Vk (t — T, £ J
UX
"•' f
fsina
T)s[n(akt
7,m J
.
\
f
+ Pa,m I cos(a m r) sin(a/c(^ — T}} dr j — / (f2(x)/\vk(t (pi(x)DTAvk(t
— r, x)
- r,x)\T_Qdx,
G
where HaiTn, Pa,m are linear combinations of moments of order |\(x), -2.(x}. We consider the integrals. Assume that am ^ ak and a m / — a/c, we have t / sin(a m r) sin(ak(t - T))' ' dr = -= —^-5sin(a m i) --r\/^„ r\>^ r\i **
/
a
J 0
a
a
k ~ m
k
m _
a f\^
sin(aA;^),
m
t /
\
• /
/•
\\ j
®k
/
,\
®m
/
,\
cos(amT)sm(ak(t - r ) ) dr = —=-r-cos(a m ^j -- 5-^-cos(a/ct). a
a
a
k ~ m
0
In the case a m = t f • , \ • i u I sm(a m r) sm(o;/ c (t J o t / \ • / / cos(a m r)sin(o; fc (/ / o
a
k~ m
\\j sin(a t) tcos(akt) r)) dr - — k --- -^a/c ^ xx , rjjrfr =
In the case a m = — f • i \ • i u \\j tcos(akt) I sm(amT) sm(ak(t - r}) dr -
J 0
£
t
r / cos(a m T) sin(a/c (t — T}} dr —
sm(akt) A&k
§ 4. Algebraic Moments
407
Thus, for am = ak and am — — ak the relation (4.10) includes terms admitting the linear growth with respect to t. However, these terms are cancelled since, as is known (cf. R. A. Aleksandryan [1] and R. T. Denchev [1]), the solution to the problem (3.1) in a ball is an almost periodic function and, consequently, is bounded with respect to t. The aforesaid means that j u(r, x)DTAvk(t - r, x)\r=t dx = $k(t),
(4.11)
G
where =
E
E |7|=i-2 m=l
9^ sin(cM) + P*k cos(akt}}.
(4.12)
We consider
where
E 9(^,*)^
hi=/
- - E
[(^1,72,73,* + /^71+2,72-2,73,/e + #n+2,7 3 ,7s-2,fc)(7l -f 871 + 2)
M=i + (/^7i,72,73,* + ^7i+2,-Y2-2,73,/c + ^71+2,72,73 -2,/c) (l\ + 3^2 + 2) , 7 2 l 7 3 -2,fc) (73 + 373 + 2)]x 7 .
Setting k = 1,. . . , NI in (4.11), we obtain the system of equations
r J
_
408
5. Qualitative Properties of Solutions to Sobolev-Type Equations
where $*k(t) is a function of the form (4.12). The system (4.13) is uniquely solvable if the following condition holds. Assumption B:
Then the moments fu(t,x)x^dx of order |7J = /, / ^ 2, of the solution G u(t, x} to the problem (3.1) are determined in terms of the moments of order at most / of the initial data 2(x}. Theorem 4.1. Let u(t, x) £ C* 2 ' 2 ([0, T] x G) be a solution to the problem (3.1) for any T > 0. We assume that condition is satisfied. Then the moments of order I ^> 2 of the solution u(t,x) admits the following representation: Na
f
(
f
I u(t,x}x'1dx — ^P ^ f aa
V G
r \ s(a ai/c /) / -2(x] vanish, then the moments of order I of the solution u ( t , x ] to the problem (3.1) also vanish for any t. We introduce the norm
1/9
= ([ [(\D?Vu(t,x)\2+\DtVu(t,x)\2+\u(t,x)\2)dxdt} \«/
J
0 G
. /
(4.14)
§ 4. Algebraic Moments
409
Definition 4.1. A function u(t,x) belongs to wl'1((Q,T}xG),ifu(t,x) belongs to the closure in the norm (4.14) of functions in C 2 ' 2 ([0,T] x G) with compact support in a neighborhood of the boundary with compact support in a neighborhood of the boundary ([0,T] x dG). We denote by GS a subdomain of the bounded domain G whose points x satisfy the condition p(x,dG) > <5. We define a weak solution to the problem (3.1) for piecewise-smooth domains G. Definition 4.2. A function u(t,x) € G 2 ' 2 ([0,T] x G«j)n W 2ll ((0,T) x G) is a weak solution to the problem (3.1) if L(Dt,Dx)u(t, x} = D2&u(t, x} + D2X3u(t, x) = 0, u
=< 1 x
DtU
=
2 x
t=o ^ ( ^ \t=o ^ ( )' for any T > 0 and S > 0.
(*, x} 6 ((0, T) x G5),
X G
^ S
Lemma 4. 3. Let G = {\x\ < R, x\ > 0}. Then the first- order moments with respect to x\ of the weak solution to the problem (3.1) takes the form I u(t, x}x\ dx G
/ z(x)xi dx + cos(
(4.15)
PROOF. Consider the function u2(t -T,X) = (R2 - \x\2}xl sin((< It is obvious that L(DT, Dx)u2(t — T, x) = 0, u2 rQ ^ QG = 0. We introduce a the sequence of functions un(t,x) G C 2 ' 2 ([0,T] x G) with compact support in a neighborhood of the boundary [0,T] x dG that converges in the W2'l((Q, T) x G)-norm to the solution u(t, x). We also introduce a sequence of compactly supported functions U2,n(^> x) € C 2 ' 2 ([0, T] x G), in a neighborhood of the boundary [0, T] x<9G that converges in the W2<1((Q, T) xG)-norm to the solution u 2 (<,x). Then t 0= /
L(DT,Dx}u(T,x}u2(t-T,x)dxdr
0 G
t
= lim / / L(DT,DX)U(T, x) u2 n(t - T, x) dx dr. "-»-°0 J
J
0 G
410
5. Qualitative Properties of Solutions to Sobolev-Type Equations
Integrating by parts the rignt-hand side and taking into account the properties of the solution u(t, x) and the fact that U2,n(t, x) have compact support, we obtain the equality
10 f if>2(x)xi dx -- — I u(t, x)xi dx V5 J /
G
H
10
•= coss(//\/5)
V5
G
which implies (4.15).
D
Lemma 4.4. Let G = {\x\ < R, xi > 0, x 2 > 0}. Then the secondorder moment with respect to x\x<2 of the weak solution takes the form
r
j
j \JJ\L. x ]x i x 2 ax
G
= v7sin(£/v7) / tp2(x}xiX-2 dx + cos(//\/7) / iPi(x}xix2 dx. J J G
G
PROOF. Consider the function u4(t — r, x) = (R2 — \x\2)xiX2sm((t — r)/\/7). We note that L(Dr, Dx)u4(t — r,x) = 0, u4 r 0 o o N x S G = 0. Arguing as in the proof of Lemma 4.3, we obtain the required assertion. D Lemma 4.5. Let G = {xl + x^ < x3}; and let u(t, x) e C 2 ' 2 ([0, T] x G) be a solution to the problem (3.1). We assume that \D"D"u(t, x}\ = o(|x|~ 5 ) as |x| —> c«o for \a\ ^ 2; 0 ^ t/ ^ 2. T/ien f/ie zeroth-order moment of the weak solution u(t,x) takes the form f f f I u(t,x)dx = t I 2(x) dx + / G
G
G
PROOF. Consider the function
u5(t - T,X) = (x3 - x\- x\}(t - r}. We note that L(DT,Dx)u5(t - r,x) = 0, u5 0=
t r r / / 0 G
QG
= 0. Integrating by parts
L(DT,Di;)u(T,x)u5(t-T,x)dxdT
§ 4. Algebraic Moments
411
and using the conditions on the solution at infinity, we find 0 = 4t I y>2(x) dx — 4 / u(t,x) dx + 4 /
D
Thus, if the domain G is the interior of the paraboloid, then the zerothorder moment of the weak solution u(t, x) is not bounded with respect to /. Lemma 4.6. Let G = [x\ + x\ < z§}, and let u(t,x) e C 2 ' 2 ([0,T] x G^)n wl'l((Q,T) x G) be the weak solution to the problem (3.1). Let \D%D%u(t,x)\ = o(|x|- 5 ) as |z| -»• oo for \a\ ^ 2, 0 ^ v ^ 2. Then the zeroth- and first- order moments of the solution u ( t , x ] have the form I u(t,x}dx = sht I y>2(x)dx + cht I tf>i(x)dx, G
G
(4.16)
G
/ u(t, x)x$dx — l/v3sin(v3t) / ^2(^)^3 dx + cos(v32) / (pi(x}xsdx, (4.17) PROOF. Consider the function — r,x) — (x3 — x 1 — x 2 )sh(< — r). We note that L(DT , Dx)u6(t — T,X) — 0 and UQ parts the equality
9G
= 0. Integrating by
t 0 = / / L(DT , Dx}u(r, x) u6(t - T, x) dx dr 0
G
and taking into account the conditions at infinity, we find 0 = 4sht / 2(%) dx — 4 / u(t, x) dx + 4ch< /
412
5. Qualitative Properties of Solutions to Sobolev-Type Equations
Therefore,
t 0 = / / L(DT, DX)U(T, x) ui(t — T, x) dxdr = — sin(v32) / 2if>2(x G
0 G
+ \/3 / 2u(r, x)x3dx - \/3cos(\/3*) / 2pi(x G
which implies (4.17).
D
Thus, if G is the interior of a cone, then the zeroth-order moment of the solution u(t,x) exponentially grows as t —> oo, whereas the first-order moment with respect to x3 is a periodic function. The above results can be generalized to the Sobolev equations.
§ 5. Asymptotic Behavior of Solutions to Some Problems in Hydrodynamics In this section, we study the asymptotic behavior as t —>• oo of the solutions to the Cauchy problem for an equation of the form L(Dt , Dx}u = C(A3D?u + FD2X3u) - a(D*u + (F :Q,
xGM3)
(5.1)
where a, C, F, V , W are nonnegative numbers, A n is the Laplace operator in M n . Such equations appear in the study of small-amplitude oscillations of a rotating stratificated fluid if we take into account the compressibility (cf., for example, S. A. Gabov and A. T. Sveshnikov [1]). For a = 0, F = a;2 equation (5.1) is the Sobolev equation. We consider the Cauchy problem for equation (5.1)
We assume that 0 < a ^ 1 and F > W. Definition 5.1. A function y>(x) belongs to W^q(R3) if (x) £ and , W^(R3)\\ =
11(1 + WDlrtx), LiCRg)!! < oo.
§ 5. Asymptotic Behavior of Solutions
413
Theorem 5.1. Let ^(x) G W^5p(R3), i = 1 , . . . ,4, p ^ 2, TI > 6p, and
let j(x'}svi(x',x3} dx' = 0, n&2
0 ^ \a\ <: 2p - 1,
(5.3)
[(x3)kpi(x', x3) dx3 = Q, 0
sup K*, x)| ^ c(K)tll*-'
\\VM> ^ r ^ l i ,
* -> oo,
(5.5)
where c(K) is a constant depending on diam/i'. Theorem 5.2. Let 4, and let
MIy
[
>xa
Ax1 x)dx3 = Q
' ^z^
Kt<4
(56)
Then for the solution to the Cauchy problem (5.2) on any compact set K C Ma we have 4
where c(K) is a constant depending on diamA'. To prove the theorems, we write formulas for the solution to the Cauchy problem (5.2). It is easy to verify that the solution can be represented as
= ui(t,x) + w 2 (^,ar) + u3(t,x) + « 4 ( < , x ) ,
(5.7)
414
5. Qualitative Properties of Solutions to Sobolev-Type Equations
where £,-(£) = (27r)~ 3 / 2 / e-iy^i(y)dy,
I ^ i ^ 4,
D = (C\£\2 + a(F + VC))2 - ^[aWC^l + & + FCQ + aVCF]. We consider the first term u i ( x . t ) on the right-hand of (5.7). Lemma 5.1. Let the assumptions of Theorem 5.1 hold. Then on any compact set K C M 3 sup x£K
^ c(K}t^2-P(\\^(x), Wr^p(R3)\\ + |^ 3 (x), W^5p(R3)\\)
(5.8)
as t —> oo, where c(K) is a constant depending on diamA'. PROOF. We represent u i ( t , x ) in the form
It is obvious that
(27r
3 2
/
We prove an estimate of the form (5.8) for u\(t, x ) . Using the conditions
§ 5. Asymptotic Behavior of Solutions
415
(5.3) and (5.4), we find tot 1
1
x / / J
/ exp(-zA 2 p A 2 p_i . . . .
0
la
0
In the spherical coordinates £1 = p sin ^ cos ^, £2 = p s i n ^ s i n ^ , ^3 = pcosfl, the integral u\(t, x] can be written in the form oo TT 2rr ,ix£(p,0,ifj)
1
2
X
£ (0
0 0
„
tiz
i1 X
2-2r+4p
f
J/
i1
/
. / exp(-iA 2p . . . X \ y ' £ ' ( p , 0 , i j } -i^2P
0
0
x ylP'
We consider the integral
..«,*) o i
52
(0
i
x / . . . / exp(-iA 2p . . . 0
0
2 2 \ \2 \2P-1 P ~ 1 / // \ X /\ 2 p_ 1^2p — 2 • • • / ^ i /^2p — Ip 2 p — 2 • • • "l " ^2p • • •
and show that for p ^ 1 the following estimate holds: sup |J| ^ c(/C)i 1 / 2 -"(pP|y|" + I)/,
* -> oo.
(5.9)
416
5. Qualitative Properties of Solutions to Sobolev-Type Equations
Using the explicit expression for u\(t,x] in the spherical coordinates and the estimate (5.9), we find sup \u\(t,x)\ ^ c( Since
we have
x / t-l — J P o
sin(A(£)t)D0[sin2p9cos2p-1
where
i
i r . .. I exp(-z'A 2p
o
o
Do [sin2" tf cos2"'1 tfT^, ^ V, y)#
Using the equality
we find
(C(aW - F}}
o •0 - (2p - 1) sin2p 9 cos 2p ~ 3 9}TB'
§ 5. Asymptotic Behavior of Solutions
417
Applying (5.10), (5.11) and integrating by parts p—l times (p > 1), we find
J=
(C(aW-
where Q
is a polynomial in B*(t)A(Q and — A2(£)) and their derivatives with respect to 9 of order at most We represent the integral J as the sum of three integrals l/V l/V/7
J
==/
7T-1/-S/F
+
1=1 The inequality
|/!| + |/3| ^ c^/ 2 -P((|x| + |y|)p- V-1 + I)/, follows from the relations |cos0#(A(f)*)Pa,/j(cos0,sin0)| ^ c, P 2 sin0cos0 />
66
2
sin0cos0
c(aw_n
c(aw_n
(5.12)
5. Qualitative Properties of Solutions to Sobolev-Type Equations
418
The last inequality is valid because
a sin 6
sin 2
Cp -aF + aVC)2 + 4Cp2(F - aW} sin2 9 asin# -aW)
We estimate the term J 2 . For this purpose, we introduce the notation M ( 9 , p ] = y^(si a,ft
Using (5.10), (5.11) and assuming that p— I is odd for definiteness, we find
sin «=i Since for large t we have |sin0| ^ 1/(2V?), 5 £ following estimate holds: I/24
We represent the integral /f as follows: 5
7T-<5
-K-\l\fl
(C(aW - F))P cos 0
TT - 1/V^], the
§ 5. Asymptotic Behavior of Solutions
419
where S > 0 is a sufficiently small fixed number such that |sin0| > 9/2, 6e [1/vM- Then \ I % ' 1 \ ^ C 2 ( ( \ x \ + \y\)pPp + l)ppi~p(Vi-8-1)- Similarly, K23>3| ^ c 3 ((|z| + \y\)PpP + l)pPt-P(Vi- S-1). Since sin2 S' the above arguments lead to the inequality ",
i —> oo,
which implies (5.9) in view of (5.12). As was mentioned above, (5.9) implies an estimate of the form (5.8) for the function u\(t, x). In the case p < 1, we obtain the estimate (5.9) by similar computations. Using the inequality (5.9), we obtain an estimate of the form (5.8) for the function u\(t, x ) . The second term u 2 (^, x) is estimated in the same way. D PROOF OF THEOREM 5.1. As in Lemma 5.1, we can estimate the terms u i ( x , t ) , I = 2 , 3 , 4 , on the right-hand side of (5.7). Finally, we obtain the inequalities = 2,3,4,
which imply the required estimate (5.5) in Theorem 5.1.
/->«>,
D
Theorem 5.2 is obtained from the following lemma which is proved as above. Lemma 5.2. Let the assumptions of Theorem 5.2 hold. Then on any compact set K C Ma 4
^e^
L
\ui(t,x)\ ^ c(K)t-l/2 V||v?i(z), ^[^(Ms)!!,
l^/^4,
<->oo.
Remark 5.1. Suppose that the initial data of the Cauchy problem (5.2) satisfy the assumptions of Theorems 5.1 and have compact support. By the Sobolev theorem (cf. S. L. Sobolev [10, p. 82]), we can represent them in the divergence form <£>;(x) = ]P D^F^(x), where F^(x) is a compactly |a|=2p
supported function in the space W[ 1+2p (M3). From the estimate (5.5) we
420
5. Qualitative Properties of Solutions to Sobolev-Type Equations
have
,
-+oo.
t|4T2p Remark 5.2. If the initial data of the Cauchy problem (5.2) do not satisfy the orthogonality conditions (5.3), (5.4), we can construct a solution that does not decrease as t —> oo. Indeed, let s(x) = — VCv(x), (f>2(x] — ¥>4(z) = 0, where v(x) ^ 0 does not satisfy the conditions (5.3) and (5.4). We assume that VC > W, F > VC. We look for a solution to the Cauchy problem (5.2) in the form u ( t , x ) — cos(\fVCt)v(x). Substituting u ( t , x ] into equation (5.1), we find d2
= 0.
(5.13)
We write the equation of characteristic directions A = (Ai, A2, AS), |Aj = 1 for equation (5.13): VC\X\2 - aW(\\ + \\] - FA§ = 0. We determine the characteristic directions satisfying this equation: VC-aW(l-Xl)-F\l
= 0,
By the assumptions on the constants F, V , C, and W , we see that A§ is limited 0 to 1. Since |A| = 1, we conclude that AI and \2 satisfy the relation A' + +A A '_- l1 - AA^ --1 1 - VC-aW A! 2 3
F-VC
If \\ runs form 0 to 1, we obtain the corresponding values of \\. For example, for \\ = 0 we have \\ — -=-— and A| = —-=-=-. Then the function F-VC
VC-aW F-aW
is a solution to equation (5.13). Then we obtain the following oscillating solution to the Cauchy problem (5.2): , x u(t,x) =
F-VC
IVC-aW
Bibliographical Comments The theory of the Cauchy problem for equations and systems not solved with respect to the higher-order derivative (cf. Preface) i-i
was created by M. I. Vishik [1], A. I. Vol'pert and S. I. Hudyaev [1], S. A. Galpern [1, 2], G. V. Demidenko [4, 5, 13-15], Yu. A. Dubinskii [2], A. S. Kalashnikov [1], A. G. Kostychenko and G. I. Eskin [1], S. G. Krein [1], 0. A. Ladyzhenskaya [1], A. A. Lokshin [1], V. N. Maslennikova [1], I.I.Matveeva [1], A. L. Pavlov [2, 3], G. A. Sviridyuk [1], G. I. Eskin [1], S. I. Yanov [1], V. R. Gopala Rao and T. W. Ting [1], T. V. Gramchev [1], H. A. Levine [1, 2], W. Rundell [1], A. Favini and A. Yagi [1], and others. Boundary-value problems for classes of equations and systems of the form (0.1) were studied by M. I. Vishik [1], K. K. Golovkin [1, 2], A. A. Dezin [1], G. V. Demidenko [1, 2, 11], G. V. Demidenko and I. I. Matveeva [2], Yu. A. Dubinskii [1], A. A. losiph'yan [1], A. I. Kozhanov [1], L. I. Komarnyts'ka and B. I. Ptashnyk [1], 0. A. Ladyzhenskaya [1, 2], V. N. Maslennikova [2], 1.1. Matveeva [2, 3], I. Sh. Mogilevskii [1], A. P. Oskolkov [1], A. L. Pavlov [1, 2], V. K. Romanko [1, 2], F. G. Selezneva and S. D. Eidel'man [1], V. A. Solonnikov [1, 3], S. V. Uspenskii and G. V. Demidenko [1], J. E. Lagnese [1, 2], T. W. Ting [1], R. E. Showalter [1-3], R. E. Showalter and T. W. Ting [1], and others. Some aspects of the general theory of boundary-value problems for Sobolev-type equations are presented in the monographs H. Gajewski, K. Groger, and K. Zacharias [1], R. W. Carroll and R. E. Showalter [1], S. V. Uspenskii, G. V. Demidenko, and V. G. Perepelkin [1]. Among equations of the form (0.1), the most popular equations are the Sobolev equation and the internal wave equation and the gravity-gyroscopic wave equation, and among systems of the form (0.1) the linearized NavierStokes system and the Sobolev system are mostly known. Asymptotic properties of solutions as t —>• oo, spectral problems and other statements of problems for the Sobolev equations were studied by R. A. Aleksandryan [1], N. N. Vakhaniya [1], G. V. Virabyan [1], S. A. Galpern [3], R. T. Denchev [1], T. I. Zelenyak [1-4] and his former students (Yu. N. Grigor'ev [1], B. V. Kapitonov [1], V. V. Skazka [1], M. V. Fokin [1, 2]), 421
422
Bibliographical Comments
and A. Garadzhaev [1], G. V. Demidenko [3], V. G. Lezhnev [1], V. P. Maslov [1], S. G. Ovsepyan [1], S. V. Uspenskii and E. N. Vasil'eva [2-5], and others. Various properties of solutions to the system of small oscillations of a rotating fluid were studied by V. N. Maslennikova [1-3] and her former students (V. N. Maslennikova and M. E. Bogovskii [1, 2], V. N. Maslennikova and A. I. Giniatullin [1], V. N. Maslennikova and P. P. Kumar [1], A. V. Glushko [1], I. M. Petunin [1]), and H. P. Greenspan [1], G. V. Demidenko and I. I. Matveeva [1], A. S. Kalashnikov [1], V. I. Lebedev [1], R. F. Parada and C. A. Fraguela [1], S. V. Uspenskii and E. N. Vasil'eva [1], S. V. Uspenskii, E. N. Vasil'eva, and S. I. Yanov [1], S. V. Uspenskii and S. I. Yanov [1], S. I. Yanov [1-3], and others. Detailed bibliography can be found in the surveys R. A. Aleksandryan, Yu. M. Berezanskii, V. A. Il'in, and A. G. Kostyuchenko [1], A. A. Dezin, T. I. Zelenyak, and V. N. Maslennikova [1], A. A. Dezin and V. N. Maslennikova [1], T. I. Zelenyak and V. P. Mikhailov [1]. The qualitative properties of solutions to the internal wave equation and the gravity-gyroscopic wave equation, as well as the Ross by waves were studied by S. A. Gabov [1, 2], S. A. Gabov and G. Yu. Malysheva [1], A. M. Il'in [1], A. V. Krasnozhon and Yu. D. Pletner [1], S. Ya. SekerzhZen'kovich [1], S. V. Uspenskii and G. V. Demidenko [1, 2], and others. More detailed references can be found in the monographs S. A. Gabov and A. G. Sveshrnkov [1, 2], J. Lighthill [1], P. H. LeBlond and L. A. Mysak [1], Yu. Z. Miropol'skn [1]. Systematical representation of results on dynamics of incompressible fluids and large bibliography on the Navier-Stokes equations and the Sobolev system are contained, for example, in the monographs S. M. Belonosov and K. A. Chernous [1], N. D. Kopachevsky, S. G. Krein, and Ngo Zuy Can [1], O. A. Ladyzhenskaya [1], J. Serrin [1], R. Temam [lj.
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1. Mathematical Principles of Classical Fluid Mechanics. In: S. Fliigge, Handbuch der Physik, Bd 8, Berlin: Springer-Verlag, 1959. V. V. Skazka
1. Asymptotics as t —>• oo of solutions of a problem of mathematical physics. Math. USSR Sb. 54:1-37, 1986. S. L. Sobolev
1. Methode nouvelle a resoudre le probleme de Cauchy pour les equations lineaires hyperboliques normales. Mat. Sb. 1:39-72, 1936. 2. On a theorem in functional analysis. Mat. Sb. 4:471-497, 1938. 3. Some Applications of Functional Analysis in Mathematical Physics. 3rd ed. Translations of Mathematical Monographs, 90. Providence, RI: Amer. Math. Soc., 1991. 4. On a new problem for systems of partial differential equations [in Russian] Dokl. Akad. Nauk SSSR, 81(No. 6):1007-1009, 1951. 5. The Cauchy problem for a special case of systems that are not of Kovalevskaya type [in Russian]. Dokl. Akad. Nauk SSSR, 82(No. 2):205208, 1952. 6. On a new problem in mathematical physics [in Russian]. Uspekhi Mat. Nauk, 7(No. 1):139-140, 1952. 7. On a new problem in mathematical physics [in Russian]. Izvest. Akad. Nauk SSSR, 18(No 1):3-50, 1954. 8. On a motion of a symmetric gyroscop with cavity occupied by a fluid [in Russian] Prikl. Mekhanika Tekhn. Fiz. 3:20-55, 1960. 9. Sur une classe des problemes de physique mathematique. In: 48 Riunione della Societa Italiana per il progresse della Sienze. Roma, 1965, pp. 192208. 10. Selected Problems of the Theory of Function Spaces and Distributions. Moscow: Nauka, 1989. V. A. Solonnikov
1. On differential properties of a solution to the first boundary-value problem for nonstationary Navier-Stokes eqautions, Proc. Steklovlnst. Math. 73:221-291, 1964. 2. On boundary-value problems for linear parabolic systems of differential equations of general form. Proc. Steklovlnst. Math. 83, (1965), p. 164.
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3. Estimates for a solution to an initial-boundary-value problem for the linear nonstationary Navier-Stokes equations, Zap. Nauch. Sem. LOMI, 59:178-254,1976. E. M. Stein 1. Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton University Press, 1970. 2. Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton: Princeton University Press, 1993. R. Temam 1. Navier-Stokes Equations. Theory and Numerical Analysis. Providence, RI: Amer. Math. Soc., 2001. T. W. Ting 1. Parabolic and pseudo-parabolic partial differential equations. J. Math. Soc. Japan. 21(No. 3):440-453, 1969. E. C. Titchmarsh 1. The Theory of Functions. 2nd ed. London: Oxford University Press, 1975. H. Triebel 1. Interpolation Theory, Function Spaces, Differential Operators. 2nd ed. Leipzig: Barth, 1995. C. Truesdell 1. A First Course in Rational Continuum Mechanics. 2nd ed Boston: Academic Press, 1991. S. V. Uspenskii 1. On the representation of functions defined by a class of hypoelliptic operators. Proc. Steklov Inst. Math. 117:343-352, 1974. 2. On differential properties of solutions of a class of pseudodifferential equations on infinity. 1,11. Sib. Math. J. 13:457-466; 629-633, 1972. S. V. Uspenskii and E. N. Vasil'eva 1. On the behavior at infinity of a solution of a problem in hydrodynamics. Proc. Steklov Inst. Math. 192:235-245, 1992. 2. Qualitative analysis of the solution to one of S. L. Sobolev's problems for t -» oo. Proc. Steklov Inst. Math. 210:205-212, 1995. 3. Local properties of solutions to the Sobolev equation [in Russian]. Vestn. Ross. Univ. Druzh. Nar. 3, No.1:143-155, 1996. 4. Algebraic moments of the solution of the first initial-boundary value problem for the Sobolev equation for special domains. Proc. Steklov Inst. Math. 214:279-290, 1996. 5. Limit investigation at infinity of the Sobolev-Wiener function classes in tube domains. Proc. Steklov Inst. Math. 232:319-326, 2001. S. V. Uspenskii, E. N. Vasil'eva, and S. I. Yanov 1. Differential properties of the solution to the first mixed boundary value problem for the Sobolev system. Proc. Steklov Inst. Math. 227:305-313, 1999.
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S. V. Uspenskii and G. V. Demidenko 1. On mixed boundary value problems for a class of equations not solved with respect to the highest-order derivative[in Russian]. Tr. Semin. S. L. Soboleva 2:92-115, 1980. 2. On the behavior of the solutions of some problems of hydrodynamics as t tends to infinity. Sov. Math. Dokl. 31:218-220, 1985. S. V. Uspenskii, G. V. Demidenko, and V. G. Perepelkin 1. Embedding Theorems and Applications to Differential Equations. Novosibirsk: Nauka, 1984. S. V. Uspenskii and B. N. Chistyakov 1. The transition to polynomials upon convergence of \x\ —>• oo for solutions of a class of pseudodifferential equations. Sib. Math. J. 16:806-819, 1975. 2. On solutions of some class of pseudo-differential equations becoming a polynomial as \x\ —> oo [in Russian]. Tr. Semin. S. L. Soboleva 1:119135, 1979. S. V. Uspenskii and S. I. Yanov 1. On the algebraic moments of a solution to the first initial-boundary value problem for the Sobolev system in the case of special domains. Sib. Math. J. 40(No.l):165-173, 1999. N. N. Vakhaniya 1. On a boundary-valued problem with prescribed values on the boundary for a hyperbolic system equivalent to the equation of string vibration. Dokl. Akad. Nauk SSSR, 116(No. 6):906-909, 1957. G. V. Virabyan 1. On spectral properties of operators generated by systems of differential Sobolev-type equations of higher order. Dokl. Akad. Nauk SSSR, 150 (No. 1):13-16, 1963. M. I. Vishik 1. The Cauchy problem for equations with operator coefficients; mixed boundary value problem for systems of differential equations and approximations methods for their solutions. Am. Math. Soc. Transl. II. 24:173-278,1963. A. I. Vol'pert and S. I. Hudyaev 1. On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR Sb. 16:517-544, 1973. G. Whitham 1. Linear and Nonlinear Waves. New York: Wiley, 1999. S. I. Yanov 1. On the Cauchy problem for a class of systems of not Cauchy-Kovalevskaya type [in Russian]. In: Functional Analysis and Mathematical Physics. Novosibirsk: Inst. Math. Akad. Nauk SSSR, 1987, pp. 125-139 2. On existence of a solution periodic in t to the first exterior mixed problem for the Sobolev system. Sib. Math. J. 34( No.l):189-198 (1993).
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3. The behavior as t —> oo of the solution of the first exterior mixed boundary-value problem for the Sobolev system. Differ. Equations 34(No.4):554-558, 1998. T. I. Zelenyak 1. On generalized eigenvalues of the operator connected with a S. L. Sobolev problem [in Russian]. Sib. Mat. Zh. 9(No. 5):1075-1092, 1968. 2. On the behavior at infinity of solutions of a mixed problem [in Russian]. Differnts. Uravn. 5(No. 9):1676-1689, 1969. 3. Selected Problems in Qualitative Partial Differential Equations [in Russian]. Novosibirsk: Novosibirsk Gos. Univers, 1970. 4. On some properties of solutions to the equations of small oscillations of a rotating fluid [in Russian]. Dinamika Sploshn. Sredy, 107:58-64, 1993. T. I. Zelenyak and V. P. Mikhailov 1. Asymptotic behavior of solutions to some boundary-value problems in mathematical physics as t —> oo [in Russian]. In: Differentsial'nye Uravneniya. Moscow: Nauka, 1970. pp. 96-118.
Appendix S. L. Sobolev On a New Problem in Mathematical Physics Izvestiya Akad. Nauk SSSR, 18 (1954), 3-50 In this paper, we consider a system of partial differential equation that is not a Kovalevskaya system. We consider the Cauchy problem and the mixed problem in arbitrary smooth domain. We prove the existence of a solution in a Hilbert space H and the continuous dependence on the initial data. The Cauchy problem in an unbounded space is solved explicitly.
§ 1. Statement of the Problem
The following system of partial differential equation appears in some problems in mathematical physics and mechanics: dv * _ ~W~
fl
°P , +p fa *"
y
^y.--v -?P + F Q,
dt dvz
—
"X
n
dy
I
dvy Q oy
r
yi
(L)
dp
+Fz
-dT-~d'z dvx ~Q ox
I
'
I
dvz __ Q oz —
9>
where vx, vy, vz are components of some vector and p is a scalar function in a domain f2 of the three-dimensions space. The boundary of fi is denoted by S. The coordinates of points of the domain Q, are denoted by x, y, z. Depending on the physical problem, some conditions can be imposed on the boundary of the domain, for example, P\s = Q or
[vxcosnx + Vycosny + vzcosnz]s = 0.
(2a) ] \ (2b) J
To determine the solution to this problem, it is necessary to prescribe the 437
438
S. L. Sobolev
initial values of v by the condition vx\t=o = v(x°'(x,y, z), I vy\t=0 = v ( y 0 ) ( x , y , z ) J > v f \ t = o = v{°\x,y, z } . }
(3)
In some other cases, the boundary conditions can be more complicated. In addition to this main problem, we also look for solutions to the system (1) with the initial conditions (3) in an unbounded space. Furthermore, the boundary conditions (2) disappear and should be replaced with some condition at infinity. Sometimes, it is convenient to write the system (1) and the conditions (2) and (3) in the vector notation. Denote by i, j, k the unit vectors parallel to the coordinate axes and represent the system (1) in the form N(v,p) = — - [ i T x k ] + gradp = F,
div v = g. The boundary condition (2b) takes the form ^n|s = 0,
(2*)
and the initial condition (3) is written as the equality v\t=0 = ^(x,y,z).
(3*)
We consider the case where v are given in a bounded domain Q, as well as the case where v is given in the entire space. However, in the first case, we restrict ourselves to the simplest qualitative investigation of solutions to the system (1) with the conditions (2a) or (2b), without consideration, for example, of the behavior of these solutions for large values o f / which requires the study of delicate spectral properties of the corresponding operators. The main problem is to study solutions to the system (1) with the initial conditions (3) in an unbounded space with the natural conditions at infinity. We explicitly write the solution to this problem, which allows us to make a number of qualitative conclusions about the behavior of solutions to our problem and explicitly indicate the solution to the problem in some special case of a bounded medium. § 2. The Main Equations in a Function Space We consider a Hilbert space H of complex vectors v given in a domain £7. We assume that the components of the vectors are square-integrable over Q. We define the inner product in H by the formula
We will consider two cases: the case where £7 coincides with the whole space and the case of a bounded domain Q homeomorphic to a ball. The condition
On a New Problem in Mathematical Physics
439
on the topology of the domain is not essential and is introduced only for the sake of simplicity. In the space H there is a linear manifold G\ of vectors vi = grad if>,
(5)
where is a function having continuous derivatives of any order inside the domain. Vectors of the form (5) have, in turn, continuous derivatives of any order and satisfy the equations rotui=0. (6) Vectors satisfying (6) are usually said to be potential As is known, for any unboundedly differentiable vector the condition (6) is necessary and sufficient for it can be represented in the form (5). Another linear manifold J\ C H consists of vectors of the form v-2 = rot$,
(7)
where the vector $ has continuous derivatives of any order. Vectors of the form (7) satisfy the equation div i 7 2 = 0 .
(8)
Vectors satisfying (8) are usually said to be solenoidal. As is known, for any unboundedly differentiable vector the condition (8) is necessary and sufficient for it can be represented in the form (7). Let HQ be a linear manifold of smooth vectors v each of which vanishes outside some (corresponding to this vector) finite domain C$ lying, together with its boundary, inside f2 and has derivatives of any order. We say that such vectors are cut-off vectors. We denote by Jo the manifold of smooth solenoidal cut-off vectors JQ C HQ and by Go the manifold of smooth potential cut-off vectors GO C HQ. Lemma I. // tt is the entire space, any element v of H orthogonal to all elements of GO and JQ can be equal only to zero. PROOF. First of all, we note that the orthogonality of v to all elements of GO and JQ implies that the vector v is orthogonal to the image of any vector u in HQ, under the action of the Laplace operator, i.e., the image of any smooth vector vanishing outside some finite interior subdomain Gj . Indeed, Au> = grad div u — rot rot uJ. (9) However, -^
u>! = grad divtJ e Go, uj-2 = rot rot u 6 JQ , Hence
(tr,w 1 )-(i? 1 w 2 ) = o.
(n)
440
S. L. Sobolev
Thus, each component v is orthogonal to all functions of the form A^, where ^ is a smooth function vanishing outside some domain C^ . In other words, + 00
,3).
(12)
As usual, we conclude [cf. (*)] that u,- is a harmonic function and, consequently, admits the representation oo
^ = £>"yn('-)(0)¥,),
(is)
n=0
where Yn (d,) are some spherical Laplace functions. In addition, the function Vi should be square-integrable over the whole space: + 00
(14) — CO
We introduce the notation
Then
If there is a nonzero number |&n | 2 , then the sum on the right-hand of (15) is unboundedly increases if A increases. Since this contradicts (14), all the \bn | 2 's vanish. Hence each vi, t^, ^3 vanishes, which is required. Lemma II. The manifold Go is orthogonal to J\. PROOF. Let vl G G0, v2 € Ji. Then, by (7),
+00
+00
+ OO
+OO
3.
(16)
The first integral on the right-hand of (16) is reduced to the integral over a finite domain f l V l , whereas the second integral vanishes in view of
On a New Problem in Mathematical Physics
441
(6). Hence + 00
= -
f f f d i v [ vi x
where 51,/! is the surface bounding the volume £lVl . The last integral vanishes since vi vanishes on SVl. Hence + 00
Q
(17)
and the lemma is proved. Lemma HI. The manifold G\ is orthogonal to JQ. PROOF. Let v\ e GI, v-z e Jo. Then + 00
+00
= ///(grad v, — oo + 00
(18)
The second integral on the right-hand of (18) vanishes in view of (8), and the first integral is reduced to the integral over the finite domain £1V2 . We have + 00
n dS.
(19)
But v%n vanishes on SV3. Hence + 00
The lemma is proved. From Lemmas II and III we obtain the following assertion. Corollary. The manifolds JQ and GQ are orthogonal. We denote by Go, Jo, GI, and J\ the closures GO, Jo, G\, and J\ for brevity. The following assertion holds. Theorem. // Q is the entire space, it is possible to represent the Hllbert space H as H = J®G,
442
S. L. Sobolev
where J = J0 = Jl and G — GO — G\. PROOF. Indeed, Jo and Go are orthogonal because are the closures of two orthogonal manifolds. Moreover, H does not contain any element orthogonal to Go and JQ simultaneously. Consequently, H = Jo ® GOHowever, J\ D Jo and J\ is orthogonal to Go- Hence J\ coincides with JQ. In the same way, G\ D GO and GI is orthogonal to JQ. Consequently, GI coincides with GO- The theorem is proved. D We proceed by considering finite domains. Lemma IV. // v in H is orthogonal to both manifolds JQ and GO, then v is a harmonic vector. In other words, the curl, as well as the divergence, of this vector vanishes. Arguing as in the proof of Lemma I, we conclude that all the components of the vector v, i.e., the functions vx, vy, and vz, are harmonic functions of variables x, y, z and have continuous derivatives of any order. Further, let v^ = grad (p\ 6 GO, and let (p\ satisfy the condition ipi = 0 outside
V\ C 0.
(20)
By the assumptions of the lemma, we find — 0 = / / / (v, grad (p\)
if>ivndSSi
vidivvdft.
(21)
Vi
The first term on the right-hand side of (21) vanishes since
= 0.
This is possible only if diviJ^O. Let V2 £ Jo, and let v^
=
(22)
rot ^2, where
^2 = 0 outside
V2 C
fl.
(23)
On a New Problem in Mathematical Physics
443
Then f f f ( v , h}dn = Q = f f f ( v , rot 4? 2 ) dtt n v2
=
£ div[V xv\dtt + £ V2
($z,Totv)dti =
V2
x v] -n)dS + fff($z,Totv)dn.
(24)
The first term on the right-hand side of (24) vanishes since $2 = 0 on 52Consequently, for any ^2 satisfying (23) we have
This is possible only if
rot if =0.
(25)
Consequently, any vector v orthogonal to J\ and GO simultaneously is a harmonic vector. The lemma is proved. This lemma has two importance consequences. Lemma V. // a vector v is orthogonal to GO and J\ simultaneously, then it is equal to zero identically. Indeed, since v is orthogonal to GO and Jo, it is harmonic and, consequently, div v = 0. Hence it admits the representation v = rot^ and, consequently, v £ J\. Since v is orthogonal to Ji, we have v = 0. Lemma VI. // v is orthogonal to G\ and JQ simultaneously, then it is equal to zero identically. Indeed, v is orthogonal to GI and Jo, it is harmonic and, consequently, admits the representation v = grad
If the vector v is orthogonal to GO, Jo and 7, then it is identically equal to zero. Indeed, since this vector is orthogonal to GO, it belongs to J\. Since
444
S. L. Sobolev
it is orthogonal to JQ, it belongs to G*I. Consequently, this vector belongs to / and, by the orthogonality to /, is equal to z^ro identically, which is required. We note that / consists of harmonic vectors, which follows from Lemma IV. Returning to our system of equations, we reduce it to a more convenient form. We construct a vector v* satisfying the condition div v* = g and (for the problem (2b)) the additional condition
This can be done if we set v* — grad v,
Aw = g ,
—— dn
— 0. s
Making the change of unknown functions by the formula v = v* + vi , we obtain for vi the same system of equations but with the condition div v^ = 0. Thus, we can restrict ourselves to the case g — 0. We study the system (1) in a Hilbert space H . For the unknown we take an element of the Hilbert space v. By the equation diviT=0
(!'),
the vector v is solenoidal. Our next goal is to find solutions v such that
t>n|s = 0.
(2b)
If the boundary S is smooth, for smooth functions v we have ) = 0,
(26)
where is an arbitrary function having infinitely many derivatives. We consider weak solutions to the problem. For this purpose, we replace the condition (26) by the requirement that v is an arbitrary element of JQ. If v is a sufficiently smooth vector and has the limit value vn = 0 on the sufficiently smooth surface S, then the fact that v belongs to Jo implies the condition (2b), as well as and equation (I'). Indeed, the right-hand side (26) vanishes for any
vn\s = 0.
dv To find — from equation (1*), we should subtract the vector grad p
from the vector [v x k] + F in such a way that the result vector belongs to JoThe vector grad p is some element of G\ . We define grad p in the weak sense as an arbitrary element v\ of G\ .
On a New Problem in Mathematical Physics
445
From (1*) it follows that the vector v\ satisfying our condition is defined in a unique way by the formula t?i = />„*{[« xk] + F},
(27)
where PQ is the projection from H into G\. Furthermore, k] + F}l
(28)
where Pois the projection from H into JQ. Thus, equations (1*), together with (2b), can be written as a single vector equation (28). We consider the first problem about integration of equations (1*) under the condition (2a). To generalize the statement of this problem, we can regard v as an arbitrary element of «/i since no boundary conditions are imposed on this vector. dv -* To compute — , we should subtract from [v x k] + F a potential vector grad p such that p vanishes on the boundary and after subtracting we obtain a solenoidal vector. It is easy to see that for sufficiently smooth p and smooth boundary S the vector grad p is orthogonal to any element v-^ in «/i- Indeed,
> grad p) dto = jjj n
&
(P div t72) dtl = - II ' pv?n dS - f f f (p div v2) dtt.
n
(29)
s
Both terms on the right-hand side vanish and, consequently, 2,grad p}d$l = 0,
if v-z C -A- Therefore, to generalize the problem, it is natural to replace grad p with an arbitrary vector v\ in Go- In this statement, the computation dv -> of — is possible for any v 6 H and F 6 H and leads to a unique result. As it follows from (1*), it suffices to take
where Pj" is the projection from H into GO; moreover, we find ^ = Pi{[t;xk] + F} > where P! is the projection from H into Ji.
(30)
446
S. L. Sobolev
Thus, equation (1*), together with the condition (2a), can be written as a single equation (30). If the above expression P*{[v x k] + F} = grad p is actually the gradient of a smooth function, then we can assume that this function vanishes on the boundary. We write the orthogonality condition for grad p to any element of J\. We transform this expression according to formula (29). For v?n we can take any function with zero mean. Then the right-hand side of (29) can be identically equal to zero only if p — const. As is known ( * ) , for grad p 6 G\ the function p always exists. Finally, we consider the last case where the space Q is unbounded. To dv obtain — from [v x k] + F, we should subtract grad p G G. In order to generalize the result we can, as above, to write the problem in the form
where P is the projection from // into J. § 3. Representation of Solution as Power Series Using the representation of the Hilbert space, we can easily construct a solution as a power series. In this section, we consider only equation (31) since equations (28) and (30) can be treated in the same way. We start with the homogeneous equation. We denote by Av the operator P[v x k]. It is obvious that Consequently, the norm of the operator does not exceed 1. Moreover, the equation — = Av
(32)
t t2 v — eAtvQ = v0 + -AvQ -\—-A VQ + • • • . 1 2 !
(33)
has a solution
Indeed, the series (33) converges uniformly with respect to t since the norm of its nth term satisfies the inequality /" ^
It is obvious that dv dt
= AVn -\
t 2~ A Vn 1
-\
^2 . 2!
i i _i
ii
On a New Problem in Mathematical Physics
447
moreover, the series of the derivatives uniformly converges. Consequently,
dv — = Av. dt Moreover, v\t=o = VQ.
(34)
Thus, the problem is solved. We establish that it is well posed. For this purpose, we need to show the continuous dependence of the solution on the initial data. Let \\VQ — VQ\\ < e. We consider the vectors v = etAvg and 7 = e M vJ. Then
Consequently, the solution in the space H continuously depends on VQ which is also given in the space H. Hence our problem is well posed. We can find a solution to a nonhomogeneous equation in a similar way. We write this equation in the form
and obtain the solution to this problem according to the general formula for the ordinary linear equation with constant coefficients, namely, t f
(36)
Indeed, the integral on the right-hand of (36) has, obviously, meaning because the norm of the integrand does not exceed H-F^i)!)^*"* 1 !. Differentiating both sides of (36) with respect to /, we find
— = AeAtvQ + A t e^-^PFfti) dt J o
dt, + PF(t),
which implies that this formula expresses a solution to the problem. It is obvious that the constructed solution satisfies the initial conditions. The well-posedness of this problem is obvious. § 4. Potential Functions for Solution Multiplying the second equation in (1) by ±i and adding with the first equality, we can write the system (1) in the form
448
S. L. Sobolev
= -( — + i—
A _ -A
di(Vl~tVv)~ dvzL ,
Q dt
— —
dx
dy
dp L. 4.
Q
dz
i
dvz
,d_ dn
d_ dx
9-
(37) Using formulas (37), we can find a solution to the system (1) in a simple form. Introducing the notation _ dx vx — ivy — w, vx + v = Fx + iFy = U, Fx- iFy = U
(38)
for brevity, we find
dt
(39)
dvz _
dp
~Q
Q
—
dw
'
dw\
z
'
dv
The solution to the system (39) can be represented in the form
w = w1 + w
+w
,
— _ —7 I —77 , —777
where iy j , tw^, v[ is a special solution to the system
(40)
On a New Problem in Mathematical Physics
449
w11, w , vjzf, p11 is a special solution to the system d_ dt d_ dt
(41)
dp ii dz '
dt
dvj dz ' J
2V
and, finally, w111, w111, vfzn, pin is a solution to the corresponding homogeneous system
. ni — + z w111 = _
_„ •)/)•* J —
dp111 ^-,
dp "1
dt (42)
dp III dz ' dwin\
+ —=
dz
2V
= 0.
It is easy to construct a solution to the system (40) because it is a system of ordinary differential equations. The first three equations in (41) connect each unknown function w11, 11 w , v™ with the unknown p11; moreover, all the operators
d_ dC
d
d_ <9C'
d_ dt'
d_ dz'
(43)
in these equations commutate. Therefore, we can look for a special solution to equation (41) in the form
P
=
w11 =
w11 =
Ji
where $7/ is some potential. The operator Mp is the least common multiple (product) of the operators on the left-hand sides of the first three equations in (41), and each of the operators Mw, M w , Mv is the product of the operator on the right-hand of the corresponding equation and the completion of the operator on the left-hand side of the same equation to the operator
450
S. L. Sobolev
Mp. Thus,
d (3 dt
--
--
d
\( d
L 7 7 I
__
7
dt
\d d
--
7
--
7,
dt
(44) L
--
<
dt
d
\d
T - * / ^OT 02-
If we write formulas (44) in detail, we obtain the following expressions for vx, vy, vz, p:
<W
dxdf*
+
dydt '
dydt dzdt2
p» =
(45)
+
d3®11
W
dt
Substituting the expressions (44) into the last equation in (41), we obtain the following equation for the potential $ 7/ :
dz<
=
^A +
<9z2
(46)
Later we will show how to find a special solution to equation (46). It is natural to try to represent functions w111, w111, v[n, pni, which are a general solution to equation (42), in the same form as in the case of
451
On a New Problem in Mathematical Physics
equations (41). Namely,
P
///
d
d \ l 757 ot ~ J
(d
= -;a7 ot 757 dt
+*
=
d ^j,j * >
d_ dz
„/// _
(47) ./// -
r)3
r)2 °
'^ dz
where the function <&IIf is some solution to homogeneous equation (48)
We show that such a representation is always valid. We preliminarily establish that the vectors v and p satisfy the equations Lp = 0,
(49) (50)
Instead of equation (50), it suffices to consider the equation
(51) Lvz = 0.
(52)
To prove (51) and (52), we apply some of th operators M p , Mw, Mw, Mz to the equation 1 fdw111 2 ( <9C
+
dwni \ dv1/1 _ df ) + dz
452
S. L. Sobolev
For example, d d
d
\ rrr
Id
We find 1 a2
d f d
dt
'
~
' Idt
dt
d f d
or, using the equations 111 UI
dz
z
dp111
dz '
i d2 \d f d A d f d .M d2 fu . .! , v .! ,
7//
i.e.,
It,777 - 0. The remaining equations (51), (52), (49) are proved in a similar way. We consider the system (47) as a system of equations with respect to a single unknown function <£ //7 assuming that the functions vx , vy , vz , p are given and satisfy the system (42). Let us show that equations (48) and (47) are compatible and determine a function $ up to a harmonic function x(x> y) °f two variables (we omit the notation /// for brevity). Indeed, the general solution to the first equation in (47) is as follows: —— = Cz(x, y, z)cost + Cs(x,y, z)sin£ C/I
t
- /sin^-JiMx.y,*,*!)^!.
(53)
453
On a New Problem in Mathematical Physics
We compute L--. We have
t — I sm(t —
02p(x,y,z,t1} d2 f d
dy22
* ~ \adx 22
sint —
aay 22
t
c
fsin(t-
f d2 - I s i n r & — p ( x , y,z,t- r) dr -
dz2
• = -smt -57 ot
ay2
,
-cost,
+
Choosing C2 and C3 in a suitable way, we can reach that L-%- vanishes. It is obvious that the solution to the following equations exists: C/V
<92C3
d
A
,
(54)
Choosing C2 and C3, we obtain the value — up to two arbitrary functions Xi(x> y> z) We have
an
C/6
d X2(x> V> z) that are harmonic with respect to x and y.
—
,yt z} cost- X 2 ( x , y , z ) sin t,
where Ci denotes all the terms on the right-hand side of (53) except for terms containing xi anc* X2For $ we obtain the equality
t I
Q = Ci(x,y,z)
/ £ l ( x , y , z , t - r)dr.
= Ci(x,y,z) Computing L$, we find t f
7X 2
J
+
t=o
t=0
S. L. Sobolev
454
Choosing C\ form the solvable equation dz2
dt
we obtain for <J> the final expression $ = $0 + D0(x, y) + zDi(x, y) + xi(x, y,
t + x
where DQ and D\ are arbitrary functions and xi( ,y,z) and X 2 ( x , y , z ) are arbitrary harmonic functions of x and y. Furthermore, o is a solution to the equation L$ = 0. Let us show that under a suitable choice of these functions, all the remaining equations of the system (47) are also satisfied. Indeed, consider the differences dzdt2
dz (55)
dydt dydt2
dxdt
We have dp
dt3
dt
is independent of /. Consequently, Lipz = 0 In view of (50). Consequently,
On the other hand,
= o,
(56)
Consider the expressions
d^°} _ ^(o) Oi
rv
)
dt
By equations (1), (47) and (55), we find dxdt3
dt
" + dt ' ~ dydt3 Consequently, Yx
dxdt
dp dx
dt ,
_
= 0,
d dv.
^
(57)
On a New Problem in Mathematical Physics
455
Hence •0J. ' = BI(X, y, z} cost + B2(x, y, z) sint, V4 = -Bi(x,y,z)smt + B2(x, y, z) cost.
(58)
On the other hand, using (48) and the homogeneous system (1), we find
dx
dy
(59)
dz
= 0,
(60)
= 0.
(61)
From (60) and (61) we find that dx2
dy2
dx2
(62)
dy2
Equation (59), together with (56) and (58), yields
fdB-2 --TT-J- ) s i n * = ' \ dx dy J
'dBi
dx
' dy J
Hence
0.
(63)
A : ( x , y ) = 0, 1 dx
dy
dB2 dx
dBj dy
(64)
= 0.
We see that
(65)
_ du dx'
du dy'
(66)
where u(x, y, z) is a harmonic function of x and y. We consider the expressions for ipx, tpy, and ^ z : d3® 2
dxdt
52$ + -—
VT. —
+, cos t^
y
d3® dydt2
$Xi dy
(67)
dxdt
,(o) . . .( dxi - ibi, + sin t ( - -z—
dzft?
dxi dx J'
4--S--W, = V ' i ° ) + I > i ( a r ) y ) . dz
dy
dx J '
456
S. L. Sobolev
Choosing D\} Xi> X2 such that to satisfy the equations A0(x,y)+Di(x,y) = 0, ~dx~
+
~d^
= B2
'
},
(68)
ET~ = BI,
~^7
for example, setting xi — 0, X2 = u, we find tfrx = tjjy = ipz = 0. Consequently, formula (47) is proved. § 5. Green-Type Integral Formulas We consider two systems of functions vx, vy, vz, p and wx, wy, wz, q. Consider the expression
vy. j dvz.z\\ 4. [fdwx _ ,,, _i_ [/ dvx±. j' dv i _.,L iI, ,,^X ^ _L n T ^ I Q ^ Q Q, y T^ Q V dx dy dz JI 1 > \\ dt dx ' W
Q
dwy _ dt
x
_ 5g^ dy J
yy
, (dwz _ <9g^ -- - - z \ dt dz J
, /3w x , 5^ — - — — \ dx dy dz
It is easy to see that Z is represented as r\
f\
Z = -Q-(VXWX + VyWy + vzwz) + -^-(pwx + qvx) + r\
r\
(pwz+qvz).
(69)
Integrating this equality over some four-dimensional cylinder (^3, 0 ^ t ^ to) with axis parallel to the /-axis and using the Ostrogradski formula, we find
dvx dt
~oT"
dp \ dx J I X W
Vy y ~T ~Z
(dv
dp\
i
dp\
(dv
dv
dv \
y z x z + {-^7-+vx + ^-}wy+(— r + — }wz+(——+-^+-^-)q + dt d y j y \ d t dz J \dx dy dz J d™* ... , 9q\ __ , f d w y , _ _ _ , dq\ __ , fdwz , dq —^w T -7— t/ T I —77T Wr T ^^ I vy T I —XT T ^~ OT dx J \ dt dy J \ dt dz y
O atu-r dx
Q
au;,, Q au;^2 dy dz
x
^
/»/•/»
tC
+ v y w y -(- i^ui*,
to
- / 0
{(pwx + qvx)cosnx + (pwy + qvy)cosny S3
(pwz + qvz)cosnz} dS3dt,
(70)
On a New Problem in Mathematical Physics
457
where n denotes the inward normal to the surface 63 bounding ^3. Consider the case where the family of functions wx, wy, wz, q satisfies homogeneous system (1). In this case, we find *„ dv
o
n
* t
'
..y , dP\ , ...y ffov ... , dP y dxj y \ dt * dy
dv dp\ fdvx dvv dv ^-z + jZ)+q(-^ + —2+ z^ dt dz J \dx dy dz to
\to
= III (vxwx +vywy + vzwz) dtt(pwn+qvn)dSdt, (71) n ° o s where vn and wn are the normal components of v\ and w. We will use the Green formula in the form (71). If vx, vy, vz, p satisfy, in turn, the homogeneous system (1), then formula (71) becomes as follows: to (VXWX
+VyWy
+VZWZ)
(pwn+qvn)dS}dt.
(72)
Formulas (71) and (72) are obtained for a bounded domain Q. We prove that if the domain ft contains infinity, but the functions vx, vy, vz, grad p and wx, wy, wz, grad q are square-integrable over this domain, then these formulas remain valid. It suffices to establish these formulas for the exterior of sufficiently large ball because any domain can be represented as the sum of a bounded domain and its exterior. The formulas for the sum of domains can be obtain by adding the formulas for each term. Remark. Any solenoidal vector v outside some ball f2 can be continuously extended to the whole space such that the extended vector is also solenoidal and has continuous derivatives. Indeed, such a solenoidal vector {Tcan be represented outside the ball fi by the formula v = rot A. Extending A to the entire space such that the second-order derivative of the extended function remain continuous, we obtain the required assertion. Let the vector grad p be square-integrable outside the domain f2. Extending the function p by continuity to the entire space and taking into account the above arguments (cf. Section 2), we have + 00
Consequently,
(v, grad p)dtt = - fjj(v, grad p] rffl. oo-n
458
S. L. Sobolev
However,
= - [fvn.pdS,
(IT]
where n* is the inward normal to f2. Replacing vn* — —vn, where n is the normal to oo — £7, we find
(v, grad p)d£l = - 11 vnp dS. This formula immediately implies the validity of (71) and (72) for the exterior of the ball fi and, consequently, for any domain if we recall the proof of this formula and use the relation (72*). § 6. Special Solutions to the Main Equation (48)
In this section, we indicate some special solutions to the main equation (48), using which, we can explicitly construct the general solution to the problem. We set (73) where
p2 = (x - x0)2 + (y - y0}2 , r 2 = (x - x0)2 + (y- yo)2 + (z - z0)2 , T — I — tQ.
We compute the operator L$. For the sake of simplicity, we first set £o = yo — zo — ^o = 0.
Using the cylindrical coordinates, we find
where
— —,
s)(m + s + l ) z 2 - (m [(2m + 2s + 2)z 2 - p*]W(t) + z*e
2 2 T}/1 44- 9}r 4-"^ 9s i^ fjs n^ ) f)r ]$>"(f\ J V > / 4* ^ I\(9m \
On a New Problem in Mathematical Physics
459
- 2(m + 3)(m + s + 2)p 2 r 2 + (m + s + 2)(m + s + [(2m + 5)z 2 r 2 - (2m + 2s + 7)p2z2]£V'"(t) + Thus,
{[(m + s)(m + s
- m —4 (2m + 2s + 2)£*'(0 + (£2 + (m + 2)2)*"(0 + (2m + 5)^'"(0 -f
As we see, L$ can vanish only if the following equalities hold simultaneously: AI = £*"' + [m - (s + l)(s - 2)]*" + £*' + (m + «)* = 0, ] A 2 = «e 2 ^( /v ) + (2m + 5)^/// + [,e2 + (m + 2) 2 ]^ / + I (74) m + 5 m + s + l * = 0.
A direct computation shows that A2 - ^^ - (m + s + 1)A! = (s - 1)2[^'" + (m d£ It is easy to see that the second equation in (74) follows from the first equation for s — 1. Thus, for s = 1 we obtain for the unknown function the ordinary equation A! = £9'" + (m + 2)*" + £*' + (m + 1)* = 0. (75) The solutions to equation (75) form some class of solutions to the equation L$ = 0. Equation (75) can be solved in a finite form for any m with the help of the Lommel function or Bessel function. We are interested in some special solutions to this equation. For m = 0 we have A! = £V'" + 2#" + &' + * = 0.
This equation can be written in the form A! = £ IV" + i*w +i-±
*' + V + i*' +
where N = $" -I— ^' + ^. Consequently, solutions to this equation are solutions to the Bessel equation *" + i*' + * = 0
460
S. L. Sobolev
or solutions to the equation
y» + itf' + y - I that are the Lommel function 5_i ? o(^)Thus, the following function is a solution to (48): «o=ijof^V r \rj
(76)
For m — — 1 we obtain a solution in the form € (77)
Po Indeed, in this case, equation (75) takes the form
(78)
^'" + $" + ^' = 0, and is obviously a solution to the last equation ^'(^) = Jo(0For the sake of convenience, we introduce the notation
(79)
As is known, the function S(£) is expressed in terms of the Lommel function, but it is not essential for our purpose. Consider the case m — —2. In this case, we obtain a solution to the problem of the form * = ^[5(0 + ./o(0],
(80)
which can be easily verified by substitution. § 7. Another Class of Special Solutions
We set $ = zp m r- m -'*(0.
(81)
We compute the operator L<2>: <92 dt?
~
ftl
= *A
m
r — m —s —2
On a New Problem in Mathematical Physics
461
-(2m + 2s + 4)p m+2 r- m -*- 4 ]$"(0 + 2* V»- m -—5*
s - 1)2[£*"' + (m + s + 2)tf"] + (2m + 4)*" [Ai - 2s*"] i = x2pmr-m-'-*L
J
I
+(m + s + l)(Ai + 2s*") + ((s - I) 2 + 2 -
") + (m
"' + (m + s + 2)*"] - p2[A!
We see that for s = 1 and s = 3 two equations for * follows each from other and, consequently, =0 the function (81) satisfies the equation L$ = 0. We could obtain the second solution solving the equations
m
*
=0
^'
for any s. The equation for * takes the form
£*'" + (m + 3s + 2 - s2)*" + £*' + (m + s)* = 0.
(83)
We do not consider equation (82) in detail, but consider equation (83) for s = 1 and s = 3. For s = 1 from (83) we find
£*"' + (m + 4)*" + £*' + (m + 1)* = 0.
(84)
For s = 3 from (83) we find
£*"' + (m + 2)*" + W + (m + 3)* = 0.
(85)
462
S. L. Sobolev
The integration of these equations is again reduced to the Lommel function and the Bessel function. We indicate an important solution to (85) for m = — 2:
£tf" / + £tf' + tf = 0.
(86)
The following function is a solution to this equation:
which can be easily verified by direct differentiation. Replacing x with y , we obtain additional solutions to equation (48). Thus, «=
.
^
=
4
^
,
(87)
Using the constructed special solutions to (48), we can start to construct the general solution to our problem. Differentiating the solution to (80) with respect to z, we obtain one more important solution
*=
(0 + UK>] +
(?) + UK)]-^ =
Jl
. (89)
For m = —3, s = 3 we obtain the equation £ $ » ' _ $ " + £ $ ' = 0,
(90)
whose solution is, in particular, the function
Indeed,
which implies the required assertion. Using the obtained solutions, we can construct the system of functions
*/,
*//,
*//', Q,
On a New Problem in Mathematical Physics
463
where
_ X-X
- -fc
^12
(nA *• — Mr
—
0
p(t-t
0
)
t-]
;—-w—
y — yo p(t — to) , f t — t0 ' / - 1 r\
-
o
/-)O
t/Q Ip
«
i
J
«
$21 _ y - y o p ( * ~ ^ ° ) j ^ ^ ~ ^ Q \ p2 r
r
r
, 22 _ X — XQ p(t —to)
Z-Z — p2rT
0
p(t-t -r
0
^
nO
_ / />
>—'IM x
*.
'
t — to
)
y - y0_,, t - to,
X — XQ „ . ^ — to ,
t0\P r
J
It is obvious that each of these functions satisfies the equations
(92) where LQ denotes the operators obtained from the operator L by replacing the variables x, y, z, t with the variables XQ, yo, ZQ, to. Differentiating, we see that the functions (91) satisfy the equations dQ } - -*u — — — — , dxQ
dt
, 1
Q) _
7
—
~
dQ
(93)
_—___
dyo'
dQ dt0
dx0
+ -^^ +
dz0
= 0.
(94)
§ 8. Three Special Solutions to Equation (1 Using the potentials <£/, $//, $///, Q, it is possible to construct three special solutions to the homogeneous system (1*) using formula (47). We
464
S. L. Sobolev
find
(95)
where ,111 _
122
,111 _
.112
w}22 = -
121
dxdt ' ,111 _
,123
113
dzdt* ' ,111 _
dt3 '
113
dt '
l21
' ^ rt!23 dt3
'
q
~
dt
Similarly,
(96) 7
where ,211 _
.211 y
wll2 = 333, 21
212 = _.
dydt*
,211 _
221
dxdt 213
dt3 '
221
33^ 22 U *
2
dydt ,21
dz
,211 _
W
222
^22 W222 = -
.223
'
q™3 = -. dt
^
'
dt
465
On a New Problem in Mathematical Physics Finally, _ ,,,3i
w.///
,32
(97) = W*1 + W33
where ,31 _
,32 _
dxdfl _ W,,31 y =
W3l =
dydt ' ,..32
_
y
dydt"2
---
dxdt '
,,33 _
dz '
dzdt*
<731 = -
q33 = -
dt3
dt
We will use the solutions (95), (96) and (97). § 9. Computations of Some Auxiliary Integrals
For our purpose, we need to compute some definite integrals. For the sake of completeness, we recall their computation. We consider (98)
x*(t) =
v - e'
w.(0 =
(99)
We show how some of these integrals are expressed in terms of other integrals. First of all, we make the change of variables £ = t£:
jQ(tt)d(; -C 2 , , /y\ _ y2s 2s
/ C ^o(C /C^cOdC
"•W- 1 /^7T^ c:
(100)
(101)
Differentiating the first equation with respect to t, we find d_
dt
x,(t)
t2s
(102)
466
S. L. Sobolev
Moreover. (103)
d
Using the equality —[£.7,5(0] = ^"(O + -7u(0 = -f-7o(0 and differentiatd£ irig both sides of (103) with respect to t, we find
(104)
Thus, the computation of all Xs aild w s is reduced to the computation of some of them. We compute the integral (105)
Fig. 1 Note that Xi(t)
can
be represented as an integral in the complex plane £:
SP-? where Cf is an open contour such that its endpoints are located at the point £ = 0 on two sheets of the Riernannian surface of the function \/t'2 — £ 2 surrounding the point £ = t. We find
i ftf
2
A function
1
vt^e #»u . l
_ £2
WO-
satisfies the equation
1
d-
1
/^-e i yt^-e oe e ^
On a New Problem in Mathematical Physics
467
or
Hence
d
I
-77^
r c
- o2 MJ ' C
+
Consequently,
+ xi (0 = o.
ui-
Thus, Xi(/) = a cost + bsin/, We note that ;
(106)
vanishes at / = 0. Hence
To determine the constant 6, we note that
o
Consequently,
= 1.
t->o
(107)
Thus,
Xi(t) — sin/. Using (102) and (104), we write a number of equalities t
.(«) = /
Xi('^
o
(108)
468
S. L. Sobolev
d /sin A
3i at \ -rr /
t
dt = t2 sin<
cos. =
1 --
+ 9) sin i + (2*3 - 9t) cost.
o vA^
(109)
Further, cost-I t
It is obvious that d(tuQ) = — xi — — s i n / , dt
= cost — 1.
It is convenient to write the integrals which follow from the above formulas:
s'mt t '
cos^-1
o
j
o
i
t
o v/l-C
sin t
cos t
x-2
\/l - C n\.
f.*~)
v/l-C
= * --*
.
3 5
o
/i
(110)
»^- ' smt + — j" cost,j
2
•J
2
1 __3_
7 t3 9
2
9
/-o
>/l-C
We will use these formulas in the what follows. § 10. Computation of vx We pass to the solution of our problem. We consider the system (1*), where F is an arbitrary vector of exterior forces. We assume that it is
469
On a New Problem in Mathematical Physics
square-integrable over the entire space. We represent the vector F as the sum of two terms F = F\ + F%, where F\ is a potential vector and FI is a solenoidal vector; moreover, each of them is square-integrable. Let F\ = grad ty. We set p' = p — ty. Then the system (1*) is written in the for; dv -* — = (r7xk)-gradp'-F2. Hence, without loss of generality, we can assume that F in (1*) is a solenoidal vector. We cut off the cylinder
and apply formula (71) to the volume fi/^, obtained by this method. Let v and p are unknown functions satisfying (1*), and let w and q be w1 and q1. We find fff(vtw')
fff(vtwr)
rfn-
JJJ
t=t
dflt=Q
•>•>•>
to
fJJ[(wIF)+qI9}dSl
sh,r,
dt. (112)
&h,r,
We pass to the limit as 17 —>• 0 and find
lim and
dt. o
We note that the components of the vector w1 at t = to take the values
1
1 r
../_
1 .../_ rr_ >}I _ z dxdz
(U3)
Indeed, at t = to we have = 0,
dt
= 0,
dt
A = 0.
Using (95) and (96), we obtain the required assertion. By (113), we find
,1 ///(,-,*<, fih.r;
U ^h,1
i*'***
470
S. L. Sobolev
s1-
vn ~ dS, ox
(114)
where n is the inward normal to the surface Sh,n bounding flh,rj- The last integral is written as follows:
1
27r
r_ dx
My
y- yo\ x - XQ -
I
__,
T)
-h 0
+
X - XQ 0
0
-pdpdif.
p dp d(p + / / v
' '"
0
0
The limits of the last two terms in this formula are zero since each of these terms does not exceed the quantity X - XQ v
z\ 0
pdpd(p —
0
where u' is the solid angle under which one, being at the point XQ, yo, z0 (u/ , can observe the bottom of a sufficiently narrow cylinder and both integrals are as small as desired). Thus, d
Hm
~
vn7^ dx + /J 27T
F f = - hm / / v v J J
}2
(y - y 0 ) ( x - XQ}\
— + vy-=-- \ d z d i f ) . r )
-h 0
We can verify that 27T
lim I I v
dz d(p — 0.
77-+0
Indeed, (y-yo)(x -
-h 0 since the function
(y- yo}(x -
On a New Problem in Mathematical Physics
471
is odd. In the same time, we have +h 2?r
jm / / (vy - u y U 0 i y 0 l Z o .
(y-
dz dip = 0
TJ-»-0
-h 0
since the integrand 77 sin <£> cos (f>
tends to zero everywhere except for a neighborhood of z — ZQ and the integral over this neighborhood does not exceed the quantity + 00
dz
max \v,, —
— 27rmax|t; y
/ *
and, consequently, is also small as desired. In the same way, we prove that 27T
f t (x-xtf lim / / v -- dzd(f> — v 3 J J
im lim f Ir,->o ^J J h (
-h 0
/
+h
271
dz
— vx(xQ,yQ,ZQ,tQ}r)2 I
= v x ( x Q , y Q , z Q , t 0 ) 2 T r lim / TJ-fO J
OO
/
J,-
/
,
where z — ZQ = rj£. We have
rfc
= 1.
Finally, ^0///("'
= 27rw r (xo, yo, ^o, ^o)-
(115)
t=ta
We note that it was important in our process that we dealt with a cylinder letting its radius to zero. Taking other surfaces or using other methods of limit passage, we can obtain quite different results.
472
S. L. Sobolev
§ 11. Computation of vx and vy. Continuation We compute the limit
0
.
We set
// pwHk dS = K^k ,
// qiikvx cos nx dS = L^k ,
Sh,r,
Sh,n
// q^k vy cos ny dS = UJk ,
// qijkvz cos nz dS = Li/k. Sh,n
Sh,r,
Introducing the notation lim K^k = k^k,
lim Djy k = l\
rj->.0
r;-+0
k k lim L*/ = iy , X X
lim Uj Z
from (95) we find t
lim
T)-+0
o 7121 / 1 2 3 \ _L C/ 1 1 1 _L 7 113 7121 /1 'r ~ ^ ) + ('y + 'y ~ 'y ~ ly
+ (/!" + /I" _ /121 _ /123JJ rfl +
f( 0
Now, we compute Jb'J*. l^ , l*fk, iyk. We begin with lijk. We note that lim Ljk = lim and
UmLJ/* = lim // Wy|x 0 ,yo,*o9 i j ' f c cosny dS, Sh,t)
limL'/ f c = lim ff vz\Xo>y0tZoq^k cosnzdS. 1-*°
Indeed, for example, lim / / (v (vx x- Vrlrn. vx\Xo>yo!Zo )qtju p d z d c p - 0 U n. Z n)<7 77—>•() yy y Sh,rj
On a New Problem in Mathematical Physics
473
because everywhere, except for the point z = ZQ, the integrand q^kp tends to zero and, in a neighborhood of this point, the integral is so small as desired because vx — vx\XOiy0iZo is small. Taking this fact into account and noting that g111 and int he cylindrical coordinates, we concluder that, among the integrals /'• 7/c , only the integrals /£ n , /* 13 , /* 2 1 , l^23 do not vanish and that /121 _ l/121 _ l/123 _ l /123 _ l /111 _l /113 _l /111 _l /113 _ n x — z — x — z — y — y — z — z ~ u-
l
It is obvious that
r r ^$12 = -Vy(xQ,y0,z0,t) lim / / -^-rjcosnydz dp -h o
;121
l
y
r r #3$i2 = -Vy(x0,y0,z0,t) lim / / •q cos ny dz dip -h 0
+h 2n
/;
r r 5$u — — v x ( x 0 , y o , Z Q , t ) lim / / — 7 ; — T J C O S H X d z d p ri->OJ
J
Ot
-h 0
+h
r
r Q3$n
= -vx(xQ,y0,z0,t) lim / /
rjcosnx dzdp
-h o
where r = ^ — ^Q- It is obvious that 03 (r) = a'^r); moreover, 04 (r) = a3' (r). Differentiating OI(T) with respect to r and replacing the variables x and y, which means that the value of this integral remains unchanged, we find
a a (r) = ai'(r), a 4 (r) = a' 2 (r) = a'{'(r). J
(121)
474
S. L. Sobolev
Thus, it suffices to compute + h 27T
a
i
r
=
at
//
77 cos ny dz dtp =
-h 0
+ h 2vr
= lim / / (»-»)'"('-«.).,, (q ?7->-o /
/
o^r
r
Vr /
rf2 ^
(122)
-/i 0
Making the change of integration variables in the last integral and setting — = £, we find r z
2
*'
p
-c
1
dz
C'
P
(123)
2
which implies OI(T) = 2?r(cosr —
(124)
in view G: (110). Hence ;123 /„
=
^(cosr — 1), '
— l>7.
2/0,
(125)
-vx(xo,yo,z0,t)2irsinr, (126) u/c
We compute the integrals. Ar . For this purpose, we slightly transform them. We set dp
+ (x -
XQ)
dx
(y-yo) (x - XQ) -
dp I (ydy ^
\ - -~yo) oz
(127)
We prove that the second term of this formulas tends to zero as 77 —> 0. Indeed, in a neighborhood of the point XQ, yo, ZQ the integrand does not exceed r6(r/), where S(rj) tends to zero as rj —>• 0. On the other hand, (wl^kp) tends to zero everywhere outside this neighborhood so that it does not exceed — anywhere. r
475
On a New Problem in Mathematical Physics
Consequently, this integral over the neighborhood of XQ, j/o, ZQ does not exceed the quantity
whereas over the remaining part it is as small as desired. Thus, it remains to compute the first term in (127). It is easy to verify that
ljkdS = 0.
=p
II
(128)
This follows from the fact that all w^k are odd functions of one of the variables x — XQ, y — yo, z — ZQ and have the common odd exponent with respect to all these variables. Let
(129)
dp d~z
l
L'kdS.
As above, lim .
rj->0
lim
rj->-0
T?->-0
Then we have ijk 1
_ dp ~ 'dx
Lijk _ SP_ y ~ dy
ijk
where Wk = lim f f ( x - xQ)wHkdS, 1-K>JJ SH,V
bf=limJJ'(y-y0)wirikdS, Sh,rj
b^k = rlim f f ( z - zQ}w^kdS. ?->'0 J J On the areas z = ±/i of the surface of the cylinder Sh,r) the limits of the integrals of w]/k vanish. Hence we need to compute the integrals 27T
-h 0
476
S. L. Sobolev
We have +h ITT
-h 0 +h 2?r
-/i 0 2?r
-h 0 2?r
12
= lim
dxdt
dydt
dz d(p.
-/i 0
Hence
i f
27T
=
+00
7 7
r
0 -oo 2vr +00
0
-oo
2?r +00
f
f
1
r5
J
J
0
-oo
r
c/z V
Applying a transformation as above, we find i
i
'1-C 2 and, in view of (110), = 27rcosr. 12
(130)
Further, 6^ = 0 since we have an odd function of y — j/o under the integral sign. In the same way, we have 6^21 = 0. We compute 2T+/» All
= 1—>0 lim J/ J/
o -h
2
TO
T~ (\ — } JG (^ T '
477
On a New Problem in Mathematical Physics 27T
00
= '//&
= 27TT f ^p^ =
J
J]^C
It is easy to see that fe^13 and 6^23 vanish because the integrands vanish. Thus, .111
(132)
All k\ik also vanish. This fact is true because in the computation of the corresponding b*/k the integrand is an odd function of (z — ZQ). In by11 and ^ 22 , the integrand is an odd function of y — y^. Hence these integrals also vanish. It remains to compute the integrals by12 and by21. For by12 we find 27T
-fc 0 OO
ZJT
x 0 ) 2 ( y - y0) pr*
pr*
=// 0
- t/ 0 ) :
-co
(133)
Further, +h 2?r
= lim / l[(x - x0)(y - y0)wlx21 + (y - y^w™} dzd? = i-+° J J -h o
27T
0
+00
-oo
pr T (pT\ — Jo I — 1 r \ r / 27T
+00
2 (^-^o) 2 (y-yo) fpr\2 4 I— ) pr
\ r /
pr
(PT\\ j_j J7lQ ( — ) f dz dip = \ r /J
//?r\
—,y 0 i — i _ \ r/ r 0 —oo
pr^
478
S. L. Sobolev
,3
*>
P \
(PT\
c
\ r /
R I JQ ( — ) dz dip =
r /
4
-r ^
+
s or, in view of (110), l21
(134)
— — 27rsinr,
i.e., 7,112 _ Ky
,121 Ky
_
Q
~
U .
(135)
Combining the above arguments, we have ux
Returning to formula (112) and introducing the notation lim
= P. V.
we find 27rvx(x0, y0, z0,t0) - 2?r to
(136)
P.V.c.
= P. V. o
n
We use the first equation in this system. We see that dp
dx
d
dt
Fx.
Substituting this expression into (136) and making simple transformations, we find 1 1 f f f vx(zo,yo,zo,to) - -vx(x0,yo,z0,Q) + P. V.— /// (v, w1) ^
47T 7 J J
n
_L 4?r'
£ =0
(137)
On a New Problem in Mathematical Physics
479
To compute the value VV(XQ, yo, zo,/o), we argue as in the case of w11. It is easy to see that the same result is obtained from (137) with y replaced by — y and vy replaced by —vy. Moreover, exchanging the roles of the variables x and y, we arrive at the required result. Finally, we have vy(xo, 2/o, zo, to) = ^vy(x0,yo, *o, 0) + P.V.— / / / (v} wn)\t=Q dtt + n to (
r i
+ — P.V.c. HI
[(wnF) + qng^ dtt 1 dt. (138)
§ 12. Computation of vz We proceed by computing vz. For this purpose, we again apply formula (71) to the unknown solution and the function w111 in the same domain fi . We find
v, wln)\t=Q dn in
F) + q'"g] d
(139)
We pass to the limit as /i -» 0 and find lim (v,w"<)\t=todn, -+° J J J
(140)
h
to
lim I I I (pwn h-+o J0 [J J sk,«
+a
(141)
vn) dS\ dt. J
We note that for t = to the components of the vector w111 take the values
d2-
d2-
^-
( 142 )
By formula (142), we find
rrr I
= jjj
ni-.n
9-\
Ugrad-
V
/
I] •S'h.rj
vUnn -j-r dS, ~c
480
S. L. Sobolev
where n is the inward normal to the surface Sh^ bounding fi/j^. The last integral is written as follows:
a1vn —— dS - dz
vx
x - x0
hV
y ~yo
-r]dz dip—
-h 0
T)
27T
—p dp d
o o
— p dp d
The first term in this formula tends to zero because the integrand is bounded as h —> 0 and the integration domain is eliminated. The second and third terms are represented the form: 2?r
0
T)
27T
Xo,yo,z0,t
0
0
Q
27T r?
0
0 2?r
In the last two integrals, the integrand is so small as desired because the function vz is continuous. The first two integrals are the product
where u; is the spatial angle subtended at the origin by the upper and lower bases of the cylinder. Hence we can conclude that
lim
vn-~-dS - dz
(143)
In this case, the limit is twice as large as that in the case where we pass to the limit with respect to 77. We compute the limit of the integral (141). We use the above notation. As above, we have;
lim
h—)• 0
7/
+ qII!vn} dS dt =
(/f + /f + fc31 + k33} dt, (144)
On a New Problem in Mathematical Physics
481
where I3j = lim
q3jvzcosnzdS, •5/l.FJ
k3j - lim IJpw3jcosnzdS.
It is obvious that the limit of the integrals Pj and l\j is zero: on the upper and lower bases vx and vv do not participate, whereas the integrals over the lateral surfaces tend to zero. Thus, we need to compute /31, I32, For /33 we have 2ir T]
d
r r I =\imjj J [Vz\ 33
o o
„ ,? , \\z-zQ\p(t-t (p(t-tQ)\\ , Q) 1 - -47T hm / v, \Xo,yo
dp.
JQ
J 0
We make the change of variables by setting p/r = £:
pdp= Furthermore,
(145) It is obvious that
=
2 +/i + v z
fe y y ^ ' ° * ' °-^ ~w~p dp d(p = 0
<93 $m
0
rf2/33= 4Trv (x ,y ,z ,t)sinT
—r-
z
0
0
0
(146)
and, consequently, / 3 1 +/ 3 3 = 0.
(147)
482
S. L. Sobolev
It remains to compute A;31 and A;33. We set
k33 = lim
+ (X - XQ)
dp dx
wndS
dp dy
( y - yo) -
dp
dp
p-p
dp dx
(x - XQ) -
(z - z0)wndS.
~d~z
The limit of the second integral is zero since the integrand contains a factor in a neighborhood of the point XQ, yo, zo that does not greater that r6(h), where 6(h) —> 0 and the values of wn at z — ZQ = h and z — ZQ = —h differ by only the sign. In the first integral, only one term can have a nonzero limit:
( - ZQ)-—-W \ (Z n The remaining terms vanish since the values of wn at z = ±/i different by only the sign and the integrals over the lateral surface vanish because this function multiplied by (x — XQ) and (y — yo) is odd. We compute
After simple transformations, we find 33 2
= 47P
®P dI
where z pr r,t ~ I\ — — /i [i — '_ J/'. '/ pr ' C( T
(i-C2)2
-f- T
On a New Problem in Mathematical Physics
483
By (110), we find = 0.
Further,
d2c(r)
= 47T
= 0.
Consequently, Jim im y \//(Pwnn + 9in» n )
(148)
- 0.
Returning to formula (139), we introduce the notation. We set (149)
lim////>*, = P. V. Passing to the limit in (139), we find ^vz(x0,y0,zQ,t0)-P.V.d.fff(v,win} t=o
qnig] dtl dt or, finally,
Vz(xo,yQ,z0,t0) = —P.V.d. / / / (u,
t=o I
.V.d. jjj[(wni,F} + qIug} dSl\ dt. (150) o
n
Thereby our problem is completely solved. § 13. The Cauchy Problem for a Fourth- Order Equation
The equation
is of independent interest. We consider the Cauchy problem for this equation in an unbounded medium, i.e., we look for a solution to this equation such that du (152)
t=o
484
S. L. Sobolev
The problem has a number of interesting properties, and we indicate how this problem can be solved explicitly. For this purpose, we construct a formula similar to the Green formula: d2u\
fd2&w d2w\ _ + W~ !hi) =
d \ dAu
. dw]
dt \
—c
dt
d \dud2w
1
r\r*
r r» f) } d du d w dy [dy dt2
d Idu fd2w
d3w
d w dydt2
\
d .~ -
i,
+W +W + 7rbr \^7T )-uJTdz (-JZT dz \ d z \ dt2 J \ dt2 I• L
\
/
•
^
•
y
v
x
/
,„,.,..
l53
j
Integrating this formula over the volume fi bounded by the surface S, we have
at =
i i i i
(
-I {II 0
r
o o9
a a u; ,,, u n _ „2 L 5n <^
I S
A
dt
[" 5Ait JJJffln r ^ t
ixi-ji»
OW
dt
t-tQ
dw~\ U
dt\
t=o
( dw <92w 5u — w-^— } cosnz\ dS > dt. 2 \ ~Q ' ^ 5n ' \ dz dzj \ I U
W
(154) Formula (154) was derived under the assumption that the domain Q is bounded. This formula is also valid if Q contains infinity, if, for example, 1
r.%
the functions u. w. -r-rdecay at infinity as —, the first-order derivatives dt2 R as —-, and the second-order derivatives as —-. This formula can be proved by the simple limit passage. Let u be the required solution to equation (151). We set (155)
and apply formula (154) to the volume fi/^ obtained by eliminating form the space the cylinder 5/,iT, with radius 77 and height 2/i around the point £0) yo, ZQ. Passing to the limit as S^,^ tends to the point XQ, j/o, ZQ, i.e., as TJ —» 0 for example, we find /////isf^WUJ
}JJJ P
V
r
J
[
On a New Problem in Mathematical Physics
485
ld
t-t0
^f
d~t
dt
dfit=t0
_-*«_l- a l,.
t=o
p
Lr
\
/ J
r
\
/
x
J
J
In this case, the limit is independent of the method of contracting the surface S*/,^ since all the integrals exist not only in the sense of the principal value. We compute each term on the right-hand side of the above equality. It is obvious that for t = to
1= (V"M [IE (V-M1 = Ijr0(o) = -r' = o' A p"\ r ) dt [p~ \ r )\ Thus, the first term on the right-hand side of (156) takes the form
n'
Au dtl = 4Tru(xQ,y0,ZQ,t0}. r
The second term can be written in the form
We show that the limit of the last integral is equal to zero. It is obvious that all the terms containing cosnz vanish under the limit passage since cos nz differs from zero only on the small wall of the cylinder. It is easy to see that
(157)
S = 0. du . This is true because -r— is bounded and an
is of order -. We compute the limit of the remaining term. It is simple to r verify that
..hm ///< IftI u-^-^-o d a2 r-=•u / P t-t0\\ dS>dt \
n-x>J [JJ
dndt2 [p \
r )\
}
=
486
S. L. Sobolev to
= /f u ( x 0 , y Q , z 0 , t ) lim /f /f co
Q3
ri
/
/ —/
-S ( p
_p
sh,n
\
r
Moreover, _ _
^
,
,.
d
fff f
d [1
hm / / ——— -c,, p - dSc = hm — — / / ~- 4 4 -J - Q (^-} ^ dS = 2
dpdt
[p \
r J\
= ^d [ /Wfr-K & 1 70 v/T^ I V
n^o dt J J dp [r
Vr
0
which is required. After these remarks, we can pass to the limit in formula (156). Transferring the known terms to one side and dividing by 4n, we obtain the finial result + 00
-
f 0, U(X
y0, ZQ,
l
-
to f f !l r0 f P— \ A ^ "•=• l~fpto — iIII /i _ i f i
(158)
Formula (158) answers the question. It remains to verify that we actually obtain a solution to the problem. § 14. Verification of the Obtained Solution We first note that it suffices to verify the existence of a solution under the homogeneous conditions. Indeed, it is obvious that the function v = U — UQ —tui, satisfies the new nonhomogeneous equation and homogeneous boundary conditions. Under some natural restrictions on <J>, the function u defined by formula (158) has continuous spatial derivatives, as well as time-derivative of order up to the spatial coordinates and time up to the required order. We prove that the function u satisfies equation (151). Let t p ( x , y , z , t ) be an arbitrary infinitely differentiable function vanishing outside V0. This function satisfies the identity
(159)
On a New Problem in Mathematical Physics
487
provided that any point of the domain V^ has the coordinate t less than T. This identity can be obtained from formula (158) if we set t± — T — t. Multiplying both sides of (159) by $(#0, 2/o> ^0,^0) and integrating over the entire space XQ, t/o, ZQ, to, we find ////
T>t>t0
We transform this integral in the double integral by integration with respect to XQ, t/o, ZQ, to inside the domain and performing the exterior integration with respect to x, y, z, t. We find / / / / il>(xQ,yQ,ZQ,t0)<&(xQ,yQ,zQ,tQ)dSlodtQ = = ffff
Lil>(x,y,z,t)u(x,y>z,t)dadt
or, integrating by parts, (xo,y0,z0,t0)[Lu-3>}dnQdt0 = Q.
(160)
This integral identity holds for any ip, which implies
Lu = $, which is required. § 15. Some Qualitative Consequence of the Obtained Formulas As we see, the solution of all our problems is connected with the function V=-JQ(t-} = -J0(t8me). r \ r/ r
(161)
Using these functions, as well as some other functions of the same type, we have constructed the general solution to all these problems. We trace how the function V changes with time. We consider a sphere with constant radius. On this sphere, the function V at each time moment, depends only on the polar angle 6. The argument of the Bessel function Jo changes from 0 to t. on this sphere. If the time grows, more and more waves caused by the maxima and minima of the Bessel function will be placed in the interval between the pole and the equator of the sphere. Waves appear at the equator and move towards the pole so that they accumulate on the sphere but not disappear, on the sphere. Thus, more and more short waves will be formed from long waves. This phenomenon is of interest.
488
S. L. Sobolev
REFERENCES 1
Sobolev S. L. Some applications of functional analysis to mathematical physics, Leningrad, Leningr. Gos. Univ., 1950. Submitted 14.VII.1953
Subject Index
A
H
Average
Holder space Hormander theorem Hausdorff-Young theorem
Basis canonical Besov space Boussinesq equation
307 38 50
Cone condition
2 14 13
Laplace operator, integral 18 Laplace transform 18 Lizorkin theorem 14 Lopatinskii condition 39, 296
35
M D
Mikhlin theorem
DuBois-Reymond lemma
14
7 N
E
Equation — Sobolev-type simple — pseudohyperbolic strictly — pseudoparabolic
Fourier multiplier Fourier operator — inverse Fourier transform — inverse Function oscillating
Nikol'skii space O
48, 51 48, 51 51, 120 48, 511
Operator, averaging P
Paley-Wiener theorem Plancherel theorem
23
11
R
13 7 9 7 9 382
Riemann-Lebesgue theorem 9 Rossby wave equation 49
Sobolev equation 489
49
Subject Index
490
Sobolev space — anisotropic Sobolev system Sobolev-type equation — simple Sobolev-type system
34 34 154 49 48, 51 156
Theorem on extension Theorem on trace Trace
36 37 37
/-horn condition — strong
35 35
