c Birkh¨
auser Verlag, Basel, 1999 NoDEA Nonlinear differ. equ. appl. 6 (1999) 35 – 54 1021-9722/99/010035-20 $ 1.50+0.20/0
Nonlinear Differential Equations and Applications NoDEA
2-D Euler equation perturbed by noise Hakima BESSAIH Scuola Normale Superiore Piazza dei Cavalieri 7, I-56126 Pisa, Italia, (
[email protected]) Franco FLANDOLI Dipartimento di Matematica Applicata Via Buonanno 25B, I-56126 Pisa, Italia, (
[email protected]) Abstract A 2-dimensional Euler equations subject to a stochastic perturbation (a noise) is investigated. An existence and uniqueness result is proved with some assumptions of spatial regularity on the noise.
1
Introduction
We prove some results of well posedness for inviscid flows described by stochastic Euler equations of the following form ∂u + (u · ∇)u + ∇p = f + ∂W in (0, T ) × D ∂t , ∂t ∇ · u = 0, in (0, T ) × D (1) u · n = 0, on (0, T ) × ∂D u|t=0 = u0 . in D Here D is a regular open domain of R2 with boundary ∂D, n is the exterior normal to ∂D, u = (u1 , u2 ) is the velocity field of the fluid, p is the pressure field, u0 is the initial velocity field, f is the external force field and W is a random field. Under proper assumptions on the regularity of W in the space variable (in time it is only continuous), we prove an existence result and a uniqueness result, depending on the regularity of u0 and f . The solutions obtained here are strong solutions (in the probabilistic sense), i.e. solutions corresponding to the a priori given probability space and Brownian motion. In contrast, weak or martingale solutions, i.e. solutions defined on a suitable probability space with a suitable Brownian motion, not chosen a priori, have been constructed by [6] and [7]. We comment on the differences between these results. Two advantages of martingale solutions are: they are not restricted to the case of additive noise; they require that the Brownian motion (or continuous square
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Hakima Bessaih and Franco Flandoli
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integrable martingale) W takes values in a Sobolev space like H 1 (D). In contrast, we obtain strong solutions only for additive noise and under the assumption that W takes values in a space like H 4 (D). On the other side, the strong solutions constructed here have the following advantages, beside the fact that they are a stronger concept of solution: a) the pathwise approach developed here may be useful to study dynamical system properties (see for instance [9] for this kind of analysis, in the case of stochastic Navier-Stokes equations); b) it is not necessary that W is a martingale, and the initial random variable u0 may be anticipative (we shall not stress the later possibility and take for simplicity a deterministic initial condition); c) W can even be a single deterministic function, showing that the deterministic Euler equation is solvable also for external forces that are distributional derivatives of continuous functions. Concerning the restriction on the space regularity of W , in the last section we show a preliminary attempt to relax the H 4 (D)-condition. As far as the uniqueness is concerned, the result proved here does not have a counterpart at the level of martingale solutions yet. Note also that the difficulty indicated in the final section to weaken the H 4 (D)-condition, is similar to the difficulty arising in the proof of uniqueness of martingale solutions. In the pathwise analysis we extend certain deterministic methods. We mainly use the method sketched by [17], in contrast to other approaches that use deep regularity properties of a Stokes operator associated to equation (1); only in the preliminary stage we follow the work of [3]. The approach that we have choosen is a little bit longer, but relies only on elementary facts and is essentially selfcontained. The uniqueness is based on a method of Yudovich (which requires proper modifications in the stochastic case), revisited by [3] and [14]. It seems that Euler equations with noise or non-regular force has not been studied as far as the authors are aware, at least with the boundary condition of (1), with the exception of the two works on the martingale approach [6] and [7]. Concerning the viscous case (Navier-Stokes equations with noise or non-regular force), the literature is extensive and a lot of papers have been written on the subject; we will quote only a few of them, see for instance [4], [19], [20], [13], [2], [8], [11], [5].
1.1
Main results
Let us introduce the space H of all measurable vector fields u : D −→ R2 which are square integrable, divergence free, and tangent to the boundary n o 2 H = u ∈ L2 (D) ; ∇ · u = 0 in D, u · n = 0 on ∂D ; the meaning of the condition u · n = 0 on ∂D for such vector fields is explained for instance in [18]. The space H is a separable Hilbert space with inner product 2 of L2 (D) , denoted in the sequel by < ., . > (norm |.|). Let V be the following
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2-D Euler equation perturbed by noise
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subspace of H;
o n 2 V = u ∈ H 1 (D) ; ∇ · u = 0 in D, u · n = 0 on ∂D ;
2 The space V is a separable Hilbert space with inner product of H 1 (D) (norm k . k). Identifying H with its dual space H 0 , and H 0 with the corresponding natural subspace of the dual space V 0 , we have the standard triple V ⊂ H ⊂ V 0 with continuous dense injections. We denote the dual pairing between V and V 0 by the inner product of H. Let (Ω, F, P ) be a probability space with expectation E. The process W = W (t, ω), t ≥ 0, ω ∈ Ω, is an H-valued stochastic process defined on the probability space (for instance a Wiener process, cf. [10]), subject to the following regularity in space: for P -a.e. ω ∈ Ω, 2 (2) W (·, ω) ∈ C [0, T ]; H 4 (D) ∩ V with the mapping ω → W (·, ω) measurable in this topology, and ∇ ∧ W = 0 on (0, T ) × ∂D.
(3)
where
∂W2 ∂W1 − . ∂x1 ∂x2 To simplify the expression (even if some partial results require less assumptions), we impose throughout the paper the following conditions on f and u0 ∇∧W =
u0 ∈ V, f ∈ L2 (0, T ; V ).
(4)
Note that the same results are true if u0 is a V -valued random variable and f is an L2 (0, T ; V )-valued random variable. We prove the following theorems Theorem 1.1 Under the conditions (2), (3) and (4), there exists (at least) a stochastic process u(t, ω), solution of (1) in the sense that for P -a.e. ω ∈ Ω, u(., ω) ∈ C(0, T ; H) ∩ L2 ([0, T ]; V ) and
Z
t
< u(t), φ > + 0
< (u(s) · ∇)u(s), φ > ds =< u0 , φ > Z t + < f (s), φ > ds+ < W (t), φ >,
(5)
0
for every t ∈ [0, T ] and every φ ∈ V Theorem 1.2 If in addition (u0 , f ) ∈ V × L2 (0, T ; V ), ∇ ∧ u0 ∈ L∞ (D), ∇ ∧ f ∈ L∞ ([0, T ] × D), and (∆∇ ∧ W ) ∈ L∞ ([0, T ] × D) for P -a.e. ω ∈ Ω, the solution of problem (1) is unique.
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2
Approximating Navier-Stokes equations
NoDEA
The main effort to prove the previous theorem consists in the analysis of the following equation. For every ν > 0 we consider the equation of Navier-Stokes type ∂u in (0, T ) × D ∂t + (u · ∇)u + ∇p = ν∆u + f + ∂W ∂t , in (0, T ) × D ∇ · u = 0, (6) ∇ ∧ u = 0, on (0, T ) × ∂D on (0, T ) × ∂D u · n = 0, u|t=0 = u0 , in D where ∇∧u=
∂u2 ∂u1 − ∂x1 ∂x2
is the vorticity. Due to the boundary condition ∇ ∧ u = 0, this is not the classical equation for a viscous fluid in a boundary domain, but it can be studied in a similar way as we shall show below. Let us consider in each point σ0 ∈ ∂D the reference frame composed by the exterior normal n and the tangent τ . Let (xτ , xN ) the components of points of R2 and (uτ , uN ) the components of a vector with respect to this reference. If (nτ , nN ) are the components of the exterior normal n, the curvature of ∂D at the point σ0 is given by the relation ∂nτ k(σ0 ) = − . (7) ∂xτ On the other hand, the rotational of a field u will be written in the reference(τ, n) ∇∧u= The relation
∂uτ ∂uN − . ∂xτ ∂xN
(8)
Z
Z ∇u · ∇v −
a(u, v) = D
k(σ)u(σ) · v(σ)dσ, ∂D
defines a continuous and coercive bilinear form on V and Z (u · ∇)v · w, b(u, v, w) = D
defines a continuous trilinear form on V . Definition 2.1 We say that a stochastic process u(t, ω) is a weak solution of (6) if for P -a.e. ω ∈ Ω, u(., ω) ∈ C([0, T ]; H) ∩ L2 (0, T ; V )
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and Z
Z
t
t
a(u(s), φ)ds + b(u(s), u(s), φ)ds = 0 0 Z t < u0 , φ > + < f (s), φ > ds+ < W (t), φ > .
< u(t), φ > +
(9)
0
for every t ∈ [0, T ] and every φ ∈ V . Proposition 2.1 There exists a unique weak solution u(., ω) of the Navier-Stokes equation (6). Moreover, for every function ϕ(.) ∈ C([0, T ]; H) ∩ L2 (0, T ; V ) ∩ H 1 (0, T ; V 0 ) such that ϕ(T ) = 0, we have Z T ∂ϕ u(s), (s) + (u(s) · ∇)ϕ(s) ds − a(u(s), ϕ(s))ds = ∂s 0 0 Z T Z T ∂ϕ < f (s), ϕ(s) > ds + W (s), − < u0 , ϕ(0) > − (s) . ∂s 0 0 Z
T
(10)
Proof. In the following argument we initially take a given ω ∈ Ω and construct a corresponding solution; then the measurability with respect to ω is studied. Let z = u − W , then the equation (9) becomes ( < z, ˙ φ > +a(z, φ) + b(z + W, z + W, φ) =< f, φ > −a(W, φ), (11) z(0) = u0 . We will use the Faedo-Galerkin method to proof existence. Since V is separable there exists a sequence of linearly independent elements e1 , . . . , em , . . . which is complete in V . For each m, we define an approximate solution zm of (11) as follows m X zm = gjm (t)ej , j=1
and (
< z˙m , ej > +a(zm , ej ) + b(zm + W, zm + W, ej ) =< f, ej > −a(W, ej ),
(12)
zm (0) = u0m . To prove existence of solutions, we will look for a priori estimates independent of m for zm and then pass to the limit on m.
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A priori estimates We multiply the equation (11) by gjm (t) and add these equations for j = 1, . . . , m. We get < z˙m , zm > +a(zm , zm ) + b(zm + W, zm + W, zm ) =< f, zm > −a(W, zm ). < z˙m , zm >=
1 d |zm |2H . 2 dt
By the incompressibility condition b(zm + W, zm + W, zm ) = −b(zm + W, zm , zm + W ) = −b(zm + W, zm , zm ) − b(zm + W, zm , W ) = −b(zm + W, zm , W ). By integration by part
Z (zm + W ) · ∇zm · W Z zm .∇zm · W + W · ∇zm · W = D D Z Z 1 2 zm ∇·W −2 zm · ∇W · W. = − 2 D D
b(zm + W, zm , W ) =
ZD
Consequently |b(zm + W, zm , W )| ≤ C |∇W |[L∞ (D)]2 |zm |2H + |zm |H |∇W |[L2 (D)]2 |W |[L∞ (D)]2 ≤ C |W |[H 3 ]2 |zm |2H + |zm |H |W |[H 1 ]2 |W |[H 2 ]2 . Because of ∇ ∧ W |∂D = 0
Z
Z ∇W ∇zm − ν
a(W, zm ) = ν Z
k(σ)W zm Z
D
∂D
∇W ∇zm − ν
= ν Z
D
= −ν
∂D
∂W zm ∂n
∆W zm = −ν < ∆W, zm > . D
For arbitrary > 0 (see [16]) Z k(σ)|zm |2 ≤ |∇zm |2 + C()|zm |2 . ∂D
ν being an arbitrary constant in (0, ν0 ), we have d |zm |2H + ν|zm |2V dt
≤ C(ν0 )(|zm |2H (1 + |W |[H 3 ]2 + |W |2[H 1 ]2 ) + |f |2[L2 (D)]2 + |W |2[H 2 ]2 ).
(13)
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We integrate between (0, t) and then we apply Gronwall’s Lemma, we obtain Z t 2 2 2 2 |zm (0)|H + C(ν0 ) (|f |[L2 (D)]2 + |W |[H 2 ]2 ) sup |zm (t)|H ≤ 0
0
Z t (1 + |W |[H 3 ]2 + |W |2[H 1 ]2 ) , exp c(ν0 )
(14)
0
∀t ∈ [0, T ]. which yields that zm is bounded in L∞ (0, T ; H) uniformly in ν and m. Now integrate (13) between (0, t); we obtain Z Z t ν t |zm (s)|2V ≤ |zm (0)|2H + C(ν0 )(|zm |L∞ (0,t;H) (1 + |W |[H 3 ]2 + |W |2[H 1 ]2 ) 2 0 0 Z t + (|f |2[L2 ]2 + |W |2[H 2 ]2 )). (15) 0
∀t ∈ [0, T ]. Now, from (12) and the bounds already proved, it follows that zm is uniformly bounded in W 1,2 (0, T ; V 0 ). Passage to the limit and conclusion We recall that the previous estimates have been proved for a given arbitrary ω ∈ Ω. By classical compactness arguments, we can extract a subsequence (thus depending on ω), still denoted by zm , which converges to a function z ∈ L2 (0, T ; V ) ∩ L∞ (0, T ; H) ∩ W 1,2 (0, T ; V 0 ) in the topologies zm −→ z in L2 (0, T ; V ) weakly, zm −→ z in L2 (0, T ; H) strongly, and
zm −→ z in W 1,2 (0, T ; V 0 ) weakly, .
Moreover, z ∈ C([0, T ]; H) by the following Lemma (see [17]) Lemma 2.1 Let V, H, V 0 be three Hilbert spaces, each space included in the following one, V 0 being the dual of V . If a function u belongs to L2 (0, T ; V ) and its derivative u0 belongs to L2 (0, T ; V 0 ) then u is almost everywhere equal to a a function continuous from [0, T ] into H. We prove now that equations (9) and (10) are satisfied. The proof is similar, so we give only that of (10) which is more difficult. From (12), it follows Z T Z T ∂ϕ zm (s), (s) + < zm (s) + W (s), (zm (s) + W (s)) · ∇ϕ(s) > ds ∂s 0 0 Z T Z T − a(zm (s) + W (s), ϕ(s))ds = − < u0 , ϕ(0) > − < f (s), ϕ(s) > ds 0
0
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Hakima Bessaih and Franco Flandoli
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forall ϕ ∈ W 1,2 (0, T ; Hm ), where Hm = span {e1 , . . . , em }. Here, we have used the identity b(zm + W, zm + W.ϕ) = −b(zm + W, ϕ, zm + W ). Using the convergence of zm to z in the various topologies, it is easy to obtain Z T ∂ϕ z(s), < z(s) + W (s), (z(s) + W (s)) · ∇ϕ(s) > ds (s) + ∂s 0 0 Z T Z T − a(z(s) + W (s), ϕ(s))ds = − < u0 , ϕ(0) > − < f (s), ϕ > ds
Z
T
0
(16)
0
forall ϕ ∈ W 1,2 (0,T ;Hm ), forall m ∈ N . Now, given ϕ ∈ C([0, T ]; H)∩L2 (0, T ; V )∩ L2 (0, T ; V 0 ), there exists a sequence ϕn ∈ W 1,2 (0, T ; Hn ) which converges to ϕ in the following topologies ϕn −→ ϕ weakly in L2 (0, T ; V ), and W 1,2 (0, T ; V 0 ) and ϕn (0) −→ ϕ(0) in H. (This fact can be proved by solving by Galerkin method a linear parabolic equation satisfied by ϕ). Then, by density (16) implies (10), where we have used u(t) := z(t) + W (t). Up to now we have proved all the statements of the proposition for any given ω. We have omitted the proof of uniqueness which is classical and similar to the energy estimates given above. In this way we know that the solution u corresponding to a given ω is the limit (in the appropriate sense) of the entire sequence of Galerkin approximations (thus we do not need to work with subsequences depending on ω). Since these approximations are measurable in ω, the limit function ω → u is also measurable. This completes the proof. Proposition 2.2 Let β = ∇ ∧ u, where u is the solution of the Navier-Stokes equation (6) (hence in particular β ∈ L2 ((0, T ) × D)). For every function ψ(.) ∈ C([0, T ]; H01 (D)) ∩ L2 (0, T ; H 2 (D)) ∩ H 1 (0, T ; L2 (D)) such that ψ(T ) = 0, we have Z T ∂ψ (s) + ∆ψ(s) + (u(s) · ∇)ψ(s) ds = − < ∇ ∧ u0 , ψ(0) > β(s), ∂s 0 Z T Z T ∂ψ − < ∇ ∧ f (s), ψ > ds + ∇ ∧ W (s), (s) . ∂s 0 0
(17)
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Proof. We plug in particular ϕ = ∇⊥ ψ, where ∇⊥ = (D2 , −D1 ) in (10); we obtain Z
∂∇⊥ ψ < u, >+ ∂s 0 Z − < u0 , ∇⊥ ψ0 > − T
Z
T
Z
⊥
T
< u, u · ∇∇ ψ > − 0 T
< f, ∇⊥ ψ > +
0
Z
a(u, ∇⊥ ψ) =
0 T
< W, 0
∂∇⊥ ψ >. ∂s
(18)
Using the fact ψ|∂D = 0 and the integration by part for the first term on the left hand side of the above inequality we obtain < u,
∂ψ ∂∇⊥ ψ >= − < ∇ ∧ u, >. ∂s ∂s
We apply an integration by part for the second term on the left hand side of (18); we have Z < u, u · ∇∇⊥ ψ > = ui uj Dj (∇⊥ ψ)i ZD = ui uj Dj Di⊥ ψ D Z Z ⊥ Di (ui uj )Dj ψ + ui uj Dj ψn⊥ = − i D ∂D Z Z = − (Di⊥ ui )uj Dj ψ − ui (Dj⊥ uj Dj ψ D Z D ⊥ ui uj )Dj ψni (19) + ∂D
Another integration by part for the second term on the right hand side of (19) yields that Z Z Z ui (Dj⊥ uj )Dj ψ = − Dj (ui (Di⊥ uj ))ψ + ui (Di⊥ uj )ψnj D D ∂D Z Z ⊥ = − Dj ui (Di uj )ψ − ui (Dj Di⊥ uj )ψ D D Z ui (Di⊥ uj )ψnj . (20) + ∂D
Because of ∇ · u = 0, the first and the second term on the right hand side of (20) are equal to zero and we obtain Z Z ⊥ ui (Dj uj )Dj ψ = ui (Di⊥ uj )ψnj . D
∂D
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Hakima Bessaih and Franco Flandoli
So that
Z
⊥
< u, u · ∇∇ ψ >= − < β, u · ∇ψ > −
NoDEA
ui (Di⊥ uj )ψnj − ui uj Dj Di⊥ ψn⊥ i .
∂D
Since we have the following hypothesis on the boundary ∂D u · n|∂D = 0, and ∇⊥ ψ · n|∂D = 0, we have Z
ui (Di⊥ uj )ψnj − ui uj Dj Di⊥ ψn⊥ = i
∂D
Z u · n (u1 D2 ψ − D1 ψu2 ) = 0, ∂D
which implies that < u, u · ∇∇⊥ ψ >= − < β, u · ∇ψ > . We apply the integration by part twice for the third term on the left hand side of (18), we get Z Z Di uj Di Dj ⊥ψ − ku · ∇⊥ ψ a(u, ∇⊥ ψ) = D ∂D Z Z 2 uj Di Dj ⊥ψ + uj Di Dj⊥ ψni − ku · ∇⊥ ψ = − ∂D Z D ⊥ 2 Dj uj Di ψ = D Z ⊥ uj Di Dj⊥ ψni − uj Di2 ψn⊥ (21) + j − kuj Dj ψ . ∂D
Since u · n = 0 on ∂D, we have: ∂ (nτ uτ + nN uN ) = 0. ∂xτ We have also (DN ψ)nτ − (Dτ ψ)nN = 0. Using (7) and the above boundary conditions, the last boundary integral in (21) is equal to zero. It remains to apply the integration by part for the integrals in the right hand side of (18) to have the result. When a function β ∈ L2 (0, T )×D) satisfies the previous variational equation, we call it a generalized solution of the following equation ∂β in (0, T ) × D ∂t + (u · ∇)β = ν∆β + ∇ ∧ f + ∂∇∧W ∂t , (22) β = 0, on (0, T ) × ∂D β|t=0 = ∇ ∧ u0 , in D We have the following uniqueness result
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Proposition 2.3 There exists a unique generalized solution β ∈ L2 ((0, T ) × D) of the previous equation. Proof. Assume that β 0 and β 00 are generalized solutions, and set β = β 0 − β 00 . Then β is a generalized solution with data equal to zero, i.e. it satisfies Z 0
T
∂ψ β(s), (s) + ν∆ψ(s) + (u(s) · ∇)ψ(s) ds = 0, ∂s
for all ψ(.) as in proposition 2.4. When ψ(.) varies, the expression ∂ψ ∂s (s)+ν∆ψ(s)+ (u(s) · ∇)ψ(s) describes a dense set in L2 ((0, T ) × D) (in fact the whole space), by the lemma below. Hence β = 0. Lemma 2.2 For v0 ∈ H01 (D) and g ∈ L2 ((0, T ) × D), the following equation ∂v in (0, T ) × D ∂t + (u · ∇)v = ν∆v + g, (23) v = 0, on (0, T ) × ∂D v|t=0 = v0 , in D has a unique solution v ∈ C([0, T ]; H01 (D)) ∩ L2 (0, T ; H 2 (D)) ∩ H 1 (0, T ; L2 (D)). Assuming only v0 ∈ L2 (D) and g ∈ L2 (0, T ; H −1 (D)), it has a unique solution v ∈ C([0, T ]; L2 (D)) ∩ L2 (0, T ; H01 (D)) ∩ H 1 (0, T ; H −1 (D)). Step 1. We have the following a priori estimates: Z 1 d 2 = (ν∆v + g − (u · ∇)v)v |v| 2 2 dt L (D) D ν ≤ −ν|v|2H 1 (D) + |v|2H 1 (D) + C0 |g|2H −1 (D) 4 +C1 |v|H 1 (D) |v|L4 (D) |u|[L4 (D)]2 and |v|H 1 (D) |v|L4 (D) |u|[L4 (D)]2
ν 2 |v| 1 4 H (D) +C2 |v|L2 (D) |v|H 1 (D) |u|[L2 (D)]2 |u|[H 1 (D)]2 ν 2 ≤ + C3 |v|2L2 (D) |u|[L2 (D)]2 |u|[H 1 (D)]2 |v| 1 2 H (D) ≤
so that ν 1 d 2 |v|L2 (D) − |v|2H 1 (D) ≤ C0 |g|2H −1 (D) + C4 |v|2L2 (D) |u|[L2 (D)]2 |u|[H 1 (D)]2 . 2 dt 4
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Hakima Bessaih and Franco Flandoli
NoDEA
Whence (by Gronwall lemma and again by the same inequality, using the regularity of u which implies that |u|[L2 (D)]2 |u|[H 1 (D)]2 ∈ L1 (0, T )) Z T sup |v(t)|2L2 (D) < ∞, |v(s)|2H 1 (D) ds < ∞. t∈[0,T ]
0
Proving these estimates for classical Galerkin approximation and passing to the limit in the classical way, we prove that there exists a solution v ∈ C([0, T ]; L2 (D)) ∩ L2 (0, T ; H01 (D)) ∩ H 1 (0, T ; H −1 (D)). The uniqueness is proved by the very similar estimates. Step 2. We have the following additional a priori estimate Z 1 d |∇v|2L2 (D) = − (ν∆v + g − (u · ∇)v)∆v 2 dt D ν ≤ −ν|∆v|2L2 (D) + |∆v|2L2 (D) + C5 |g|2L2 (D) 4 + C6 |∆v|L2 (D) |v|W 1,4 (D) |u|[L4 (D)]2 and |∆v||v|W 1,4 (D) |u|[L4 (D)]2
ν |∆v|2L2 (D) 4 + C7 |v|H 1 (D) |v|H 2 (D) |u|[L2 (D)]2 |u|[H 1 (D)]2 ν ≤ |∆v|2L2 (D) + C8 |v|H 1 (D) |u|[L2 (D)]2 |u|[H 1 (D)]2 2 ≤
so that ν 1 d |∇v|2L2 (D) − |∆v|2L2 (D) ≤ C5 |g|2L2 (D) + C9 |∇v|L2 (D) |u|[L2 (D)]2 |u|[H 1 (D)]2 . 2 dt 4 Whence (as in step 1) Z sup |∇v|2[L2 (D)]2 < ∞,
[0,T ]
T
|∆v|2L2 (D) ds < ∞. 0
Proving these estimates for classical Galerkin approximations and passing to the limit in the classical way, we prove that the solution of step 1 satisfies the regularity required by the lemma. Proposition 2.4 The solution β given by proposition 2.4 satisfies β ∈ C [0, T ]; L2 (D) ∩ L2 0, T ; H01 (D) . Moreover, it satisfies k β kC([0,T ];L2 (D)) ≤ C, where the constant is independent of ν (it depends on ω).
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Proof. We prove that there exists a solution β of equation (22) with such regularity. Since it is automatically a generalized solution (as it can be verified as proposition 2.1), it coincides with the generalized solution given by proposition 2.4. Setting formally z = β − ∇ ∧ W we get the equation ∂z in (0, T ) × D ∂t + (u · ∇)z = ν∆z + g, (25) z = 0, on (0, T ) × ∂D z|t=0 = v0 , in D where g = ∇ ∧ f − (u · ∇)(∇ ∧ W ) + ν∆(∇ ∧ W ). Since ∇ ∧ u0 ∈ L2 (D) and g ∈ L2 (0, T ; H −1 (D)), there exists a unique solution z ∈ C [0, T ]; L2 (D) ∩ L2 0, T ; H01 (D) ∩ H 1 0, T ; H0−1 (D) . By the lemma 2.6, it is straightforward to see that β := z+∇∧W is a solution of the first equation of R(23) and has the regularity required by the previous proposition. Moreover, since D ((u · ∇)z)z = 0 Z 1 d 2 (ν∆z + g − (u · ∇)z)∆z |z|L2 (D) = 2 dt D = −ν|z|2H 1 (D) + C|z|L2 (D) |g|L2 (D) . Hence for all t ∈ [0, T ], |z(t)|2L2 (D)
Z ≤ |∇ ∧ u0 |2L2 (D) + C(
Z
T
|z(s)|2L2 (D) ds +
0
T
(|∇ ∧ f (s)|2L2 (D) 0
+ |∇∇ ∧ W (s)|L∞ (D) |u(s)|L2 (D) + |∆∇ ∧ W (s)|2L2 (D) )ds). Using Gronwall’s Lemma and the estimate (14) the desired bound is proved.
Proposition 2.5 The solution u given by proposition 2.1 satisfies u ∈ C [0, T ]; H 1 (D) ∩ L2 0, T ; H 2 (D) . Moreover, it satisfies k u kC([0,T ];H 1 (D)) ≤ C(ω) where the constant is independent of ν. Proof. Observe that since ∇ · u = 0, u satisfies the following elliptic system ∆u = −∇⊥ β, β|∂D = 0, (26) u · n|∂D = 0.
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We multiply the first equation of (26) by u and integrate over D. Since β|∂D = 0, we obtain Z 2 |∇u|L2 (D) = ∇u · u · n+ < β, ∇ ∧ u > . (27) ∂D
For an arbitrary > 0 we have (see [17]) Z ∇u · u · n ≤ |∇u|2L2 (D) + C()|u|2L2 (D) , ∂D
Which yields that |∇u|2L2 (D) ≤ C |β|2L2 (D) + |u|2L2 (D) .
The result follows from (14), (24) and (28).
3
(28)
Proof of Theorem 1.1 and Theorem 1.2
Using the estimates of section 2, we are able to proof Theorem 1.1 and Theorem 1.2.
3.1
Proof of Theorem 1.1 (Existence)
From Proposition 2.7, we have that uν , weak solution of Navier Stokes equations 6 is uniformly bounded, for any given ω ∈ Ω, in L∞ (0, T ; [H 1 (D)]2 ) so in d L∞ (0, T ; V ). Since uν satisfies (9), dt uν remains bounded in L2 (0, T ; V 0 ). The embedding V ⊂ H being compact, this implies that we can extract from uν a subsequence (also called uν ) which converge weakly in L2 (0, T ; V ) and strongly in 2 L2 ([0, T ] × D) (see [11] pp 58). The subsequence depends on ω, a priori. From Lemma 2.3, u ∈ C([0, T ]; H) and verifies that u(0) = u0 .
(29)
From strong convergence of uν in L2 (Q) we deduce that lim < (uν · ∇)uν , φ >=< (u · ∇)u, φ >
ν→0
∀φ ∈ V.
(30)
It follows from (29) and (30) that the limit u is solution of (5). We have proved theorem 1.1 except for the measurability of the mapping ω → u. The measurability of u follows from the following measurable selection theorem Lemma 3.1 Let X and Y be two separable Banach spaces and Λ a multiple-valued mapping from X to the set of nonempty closed subsets of Y , the graph of Λ being closed.
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49
Then Λ admits a universally Radon measurable selection, i.e., there exists a mapping L from X to Y , such that L(x) ∈ Λ(x)
∀x ∈ X,
and L is measurable for any Radon measure defined on the Borel sets of X.
To apply this result we take o n 2 X = x ∈ C [0, T ]; H 4 (D) ∩ V | ∇ ∧ x = 0 on (0, T ) × ∂D , Y = C ([0, T ]; H) ∩ L2 (0, T ; V ) with the natural topologies. We have proved that for every x ∈ X there exists at least one solution u ∈ Y (indeed, we can take a trivial probability space and te deterministic process W = x), so Λ(x) 6= ∅. If xn → x in X and un ∈ Λ(xn ) converges to some u ∈ Y , it is easy to pass to the limit in the Euler equation and see that u ∈ Λ(x), proving that Λ is closed. Thus there is a measurable selection x → U (x). Now, given the original probability space and process W , the composition ω ∈ Ω → W (·, ω) ∈ X → u := U (W (·, ω)) ∈ Y gives us a measurable solution u(t, ω) of the Euler equation. The proof is complete.
3.2
Proof of Theorem 1.2 (Uniqueness)
Lemma 3.2 Under the assumptions of theorem 1.2, uν and its limit u are in Xp = L∞ (0, T ; (W 1,p (D))2 ) (1 ≤ p < ∞), for P -a.e. ω ∈ Ω. Besides, there exists a random constant C(ω) such that sup |u(., ω)|Xp , |uν (., ω)|Xp ≤ C(ω)p(|∇ ∧ u0 |L∞ (D) + |u0 |L∞ (0,T ;V ) + |∇ ∧ f |L∞ (Q) (31) + |∆∇ ∧ W (., ω)|L∞ (Q) + |f |L∞ (0,T ;V ) ). Proof. Since uν ∈ L∞ (0, T ; V ) (Proposition 2.8), by Sobolev embedding theorem uν remains bounded in L∞ (0, T ; [L4 (D)]2 ). On the other hand zν = βν − ∇ ∧ W is solution of (25), where βν = ∇ ∧ uν . We multiply the first equation of (25) by |zν |2 zν and integrate over D to obtain Z Z Z Z 1 d |zν |4 + ν (∇zν )2 |zν |2 = (∇ ∧ f )|zν |2 zν + ν (∆∇ ∧ W )|zν |2 zν 4 dt D D D Z D Z 2 + (uν · ∇)zν |zν | zν − (uν · ∇)(∇ ∧ W )|zν |2 zν (32) D
D
Using H¨ older inequality for the terms in the right hand side of (32) and then the Gronwall Lemma we obtain that sup |zν (t)|4L4 (D) ≤ (|z0 |4L4 (D)
0
|∇ ∧ f |4L4 (D) + |∆∇ ∧ W |4L4 (D) + |∇∇ ∧ W |4L∞ (D) |uν |4L4 (D) )eT .
+ 0
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Under the hypothesis of the Lemma, it yields βν ∈ L∞ (0, T ; L4 (D)). Using the system (26) which is elliptic, we deduce that uν ∈ L∞ (0, T ; W 1,4 (D)) which implies by Sobolev theorem that uν ∈ L∞ (Q). Now we apply the maximum principle for zν , solution of (25); we get |zν |L∞ (Q)
≤ C(|∇ ∧ u0 |L∞ (D) + |∇ ∧ f |L∞ (Q) +|∆∇ ∧ W |L∞ (Q) + |(uν · ∇)∇ ∧ W |L∞ (Q) ).
(33)
As in [3], using the system 26 and in virtue of 33, we obtain the following estimate |uν |Xp
≤ Cp(|∇ ∧ u0 |L∞ (D) + |∇ ∧ f |L∞ (Q) + |∆∇ ∧ β|L∞ (Q) +|(uν · ∇)∇ ∧ W |L∞ (Q) + |uν |[Lp (Q)]2 ),
(34)
where C = C(ω) is a constant independent of ν. According to the hypothesis of the Lemma and following the argument of [3], we obtain (31) for uν . Passing to the limit on ν, we obtain the same estimate for u. Lemma 3.3 There exists a constant C > 0, independent of p, such that for every function u ∈ H 1 (D) we have |u|Lp (D) ≤ Cp1/2 |u|H 1 (D) , for 2 ≤ p < ∞.
(35)
Proof. See [14]. Let us assume that u0 and u00 are generalized solutions of (1) with the same initial data and the same external body force and set u = u0 − u00 , then u is a generalized solution with data equal to zero i.e. it satisfies < u0 , ϕ > + < (u. · ∇)u, ϕ >= 0, ∀ϕ ∈ V (36) u(0) = 0. In particular for ϕ = u, we have d 2 |u| 2 2 dt [L (D)]
= −2 < (u. · ∇)u, u > = < (u. · ∇)u0 , ϕ > Z = −2 (u. · ∇)u · u. D
By H¨older’s inequality we have 1 1 d 2 |u|[L2 (D)]2 ≤ 2|∇u0 |[Lp (D)]2 |u|2[L2p0 (D)]2 , + 0 = 1. dt p p Using the estimate (31), we obtain (the constants will depend on ω in the sequel) d 2 2 . |u| 2 2 ≤ Cp|u| [L2p0 (D)]2 dt [L (D)]
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51
Now using an interpolation result we obtain |u|[L2p0 (D)]2 ≤ |u|1−λ |u|λ[Lp (D)]2 , λ =
1 ≤ 1, 3 ≤ p < ∞. p−2
Hence, we have d 2λ |u| 2 2 dt [L (D)]
d 2 |u| 2 2 dt [L (D)] 2 ≤ λ|u|2λ−2 [L2 (D)]2 Cp|u|[L2p0 (D)]2 = λ|u|2λ−2 [L2 (D)]2
≤ λCp|u|2λ [L2 (D)]2 . Since λ =
1 p−2
(37)
and u(0) = 0, integration of 37 leads the estimate |u(t)|2[L2 (D)]2 ≤ (Ct)
p−2 2
(
1 p p−2 ) 2 (Cp) 2 . p−2
(38)
Suppose now that t is so small that Ct < 1 (thus it depends on ω). If we let p−2 p−2 1 p p → ∞, then (Ct) 2 (Cp) 2 → 0 while ( p−2 ) 2 remains bounded. Hence u(t) = 0 if Ct < 1. We repeat the argument a finite number of times (depending on ω) to cover the whole [0, T ], so that the uniqueness is proved.
4
Towards existence and uniqueness results under more general noise
The result of existence of martingale solutions proved in [6] requires only that the noise belongs to a Sobolev space like H 1 , while the existence result of strong solutions proved here, requires that the noise belongs to a Sobolev space like H 4 . To obtain strong solutions under more general noise seems to be a very difficult problem. In this section we assume that W is a Brownian motion and we indicate a line of proof to get an existence result under more general conditions on the noise. However, the method is based on an estimate which has not been proved yet (it is true in the deterministic case). Thus the reason to express the following conditional theorem is just to indicate a possible direction for further investigations. The same unproved estimate (even a weaker version) would give a uniqueness result for martingale solutions. Let W (t) be a Wiener process on a probability space √ (Ω, F, P ) with values in H, and covariance Q in H. We may think of it as QB(t), where B(t) is a cylindrical Wiener process in H (cf. [10]). Let Qn be a sequence bounded √ of linear √ selfadjoint non-negative operators in H, nuclear, such that | Q − Q| →0 n L 2 (H) √ the regularity required by theorem and assume also that W (t) = Qn B(t) has √ 1.1. Let un be the solutions corresponding to Qn B(t) given by theorem 1.1, and assume to have proved that they are adapted to the Brownian motion.
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Theorem 4.1 Assume that there exists C0 (R) such that ! Z T
|∇un (t)|L∞ (D) > R
P
≤ C0 (R),
limR→∞ C0 (R) = 0.
0
Then un converges in probability to a process u in L2 (0, T ; H), i.e. Z limn→∞ P (
T
|un (t) − u(t)|2H dt > ) = 0 for all > 0,
0
and u is a solution of equation (1) corresponding to the noise W (t), in the following sense: for P -a.e. ω, u(., ω) ∈ L2 (0, T ; H), and the equation (10) is satisfied (without the bilinear coercive term). Proof. By Ito formula, and integration by parts, with the use of the boundary conditions, p p d |un (t) − um (t)|2H = 2 < un − um , d(un − um ) >H +| Qn − Q|2L2 (H) dt = −2 < un − um , (un − um ) · ∇un + um · ∇(un − um ) >H D p E p p p + un − um , Qn − Q dB(t) + | Qn − Q|2L2 (H) dt H
≤ C|∇un (t)|L∞ (D) |un (t) − um (t)|2H D p E p p p + un − um , Qn − Q dB(t) + | Qn − Q|2L2 (H) dt. H
Hence by Ito formula again, Rt − C|∇un (s)|L∞ (D) ds 0 |un (t) − um (t)|2H d e Rt − C|∇un (s)|L∞ (D) ds = −C|∇un (t)|L∞ (D) e 0 |un (t) − um (t)|2H Rt − C|∇un (s)|L∞ (D) ds + e 0 d|un (t) − um (t)|2H Rt − C|∇un (s)|L∞ (D) ds ≤ e 0 × p E D p p p Qn − Q dB(t) + | Qn − Q|2L2 (H) dt . 2 un (t) − um (t), H
By classical stopping time argument we get Rt p p − C|∇un (s)|L∞ (D) ds 2 |un (t) − um (t)|H ≤ | Qn − Q|2L2 (H) T. E e 0 Hence also E
−
e
RT 0
C|∇un (s)|L∞ (D) ds
Z
T
! |un (t) − um (t)|2H dt
0
p p ≤ | Qn − Q|2L2 (H) T 2 .
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2-D Euler equation perturbed by noise
53
If a, b, and R are positive numbers, the inequality a > implies either ae−b > e−R or b > R. Hence Z T Z T |un (t) − um (t)|2H dt > ) ≤ P ( C|∇un (s)|L∞ (D) ds > R) P( 0
0
+P
−
RT
e
0
C|∇un (s)|L∞ (D) ds
Z
!
T
|un (t) −
um (t)|2H dt
−R
>e
0
p p ≤ C0 (R/C) + (eR /)| Qn − Q|2L2 (H) T 2 . Similarly, for each t ∈ [0, T ], P (|un (t) −
um (t)|2H
Z t > ) ≤ P ( C|∇un (s)|L∞ (D) ds > R) 0 Rt − C|∇un (s)|L∞ (D) ds |un (t) − um (t)|2H > e−R +P e 0 p p ≤ C0 (R/C) + (eR /)| Qn − Q|2L2 (H) T.
It follows that the sequence un (.) is a Cauchy sequence in L2 (0, T ; H) in probability, hence it converges in probability to an L2 (0, T ; H)-valued random variable u(.); an the same is true for every t ∈ [0, T ]. There exists a subsequence that converges P -a.s. in L2 (0, T ; H) to u, and such convergence allows us to pass to the limit in the weak formulation of the Euler equation, to show that u is a solution. The proof is complete.
References [1] R.A. ADAMS, Sobolev spaces, Academic press, 1975. [2] S. ALBEVERIO, A.B. CRUZEIRO, Global flow and invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids, Comm. Math. Phys. 129, 431–444 (1990). ´ [3] C. BARDOS, Existence et unicit´e de la solution de l’´equation d’Euler en dimensions deux, Jour. Math. Anal. Appl. 40, 769–780 (1972). [4] A. BENSOUSSAN, R. TEMAM, Equations stochastiques du type NavierStokes, J. Funct. Anal, 13, 195–222 (1973). [5] H. BESSAIH, Semi-linearized compressible Navier-Stokes perturbed by a White noise, Scuola Normale Superiore Pisa, Preprint, 1997. [6] H. BESSAIH, Martingale solutions for stochastic Euler equations, Scuola Normale Superiore Pisa, Preprint n.22, July 1997.
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[7] Z. BRZERNIAK, S. PESZAT, work in preparation. [8] M. CAPINSKI, D. GATAREK, Stochastic equations in Hilbert space with application to Navier-Stokes equations in any dimension, J. Func. Anal 126, 26–35 (1994). [9] H. CRAUEL, F. FLANDOLI, Attractors for random dynamical systems, Prob. Th. and Related Fields 100, 365–393 (1994). [10] G. DA PRATO, J. ZABCZYK, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. [11] F. FLANDOLI, D. GATAREK, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory relat. Fields 102, 367– 391 (1995). [12] F. FLANDOLI, Irreducibility of the 3-D Stochastic Navier-Stokes equation, to appear in J. Func. Anal. ´ [13] H. FUJITA-YASHIMA, Equations de Navier-Stokes stochastiques non Homog`enes et applications, Scuola Normale Superiore, Pisa, 1992. [14] T. KATO, A remark on a theorem of C. Bardos on the 2D-Euler equation, Preprint, 1992. [15] J.L. LIONS, Quelques M´ethodes de R´esolution de Probl`emes aux limites, Dunod Paris 1961. ´ [16] J.L. LIONS, Equations Diff´erentielles Op´erationelles et Probl`emes aux limites, Springer-Verlag, Berlin, 1961. [17] P.L. LIONS, Mathematical Topics in Fluid Mechanics, Vol 1, Incompressible Models, Oxford Sci. Publ, Oxford, 1996. [18] R. TEMAM, Navier-Stokes equations, North-Holland, 1984. ´ [19] M. VIOT, Solutions faibles d’Equations aux D´eriv´ees Partielles Stochastiques non Lin´eaires, Th`ese de Doctorat, Paris VI, 1976. [20] M.J. VISHIK, A.V. FURSIKOV, Mathematical Problems of Statistical Hydromechanics, Kluwer, Dordrecht, 1980.
Received October 7, 1997
