Transport Equations and Multi-D Hyperbolic Conservation Laws

Lecture Notes of the Unione Matematica Italiana

5

Editorial Board

Franco Brezzi (Editor in Chief) Dipartimento di Matematica Universita di Pavia Via Ferrata I 27100 Pavia, Italy e-mail: [email protected] John M. Ball Mathematical Institute 24-29 St Giles’ Oxford OX1 3LB United Kingdom e-mail: [email protected] Alberto Bressan Department of Mathematics Penn State University University Park State College PA 16802, USA e-mail: [email protected] Fabrizio Catanese Mathematisches Institut Universitatstraße 30 95447 Bayreuth, Germany e-mail: [email protected] Carlo Cercignani Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milano, Italy e-mail: [email protected] Corrado De Concini Dipartimento di Matematica Università di Roma “La Sapienza” Piazzale Aldo Moro 2 00133 Roma, Italy e-mail: [email protected]

Persi Diaconis Department of Statistics Stanford University Stanford, CA 94305-4065, USA e-mail: [email protected], [email protected] Nicola Fusco Dipartimento di Matematica e Applicazioni Università di Napoli “Federico II”, via Cintia Complesso Universitario di Monte S. Angelo 80126 Napoli, Italy e-mail: [email protected] Carlos E. Kenig Department of Mathematics University of Chicago 1118 E 58th Street, University Avenue Chicago IL 60637, USA e-mail: [email protected] Fulvio Ricci Scuola Normale Superiore di Pisa Plazza dei Cavalieri 7 56126 Pisa, Italy e-mail: [email protected] Gerard Van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam, The Netherlands e-mail: [email protected] Cédric Villani Ecole Normale Supérieure de Lyon 46, allée d’Italie 69364 Lyon Cedex 07 France e-mail: [email protected]

The Editorial Policy can be found at the back of the volume.

Luigi Ambrosio • Gianluca Crippa Camillo De Lellis • Felix Otto Michael Westdickenberg

Transport Equations and Multi-D Hyperbolic Conservation Laws Editors Fabio Ancona Stefano Bianchini Rinaldo M. Colombo Camillo De Lellis Andrea Marson Annamaria Montanari

ABC

Authors Luigi Ambrosio

Felix Otto

[email protected]

[email protected]

Gianluca Crippa

Michael Westdickenberg

[email protected]

[email protected]

Camillo De Lellis [email protected]

Editors Fabio Ancona

Camillo De Lellis

[email protected]

[email protected]

Stefano Bianchini

Andrea Marson

[email protected]

[email protected]

Rinaldo M. Colombo

Annamaria Montanari

[email protected]

[email protected]

ISBN 978-3-540-76780-0

e-ISBN 978-3-540-76781-7

DOI 10.1007/978-3-540-76781-7 Lecture Notes of the Unione Matematica Italiana ISSN print edition: 1862-9113 ISSN electronic edition: 1862-9121 Library of Congress Control Number: 2007939405 Mathematics Subject Classification (2000): 35L45, 35L40, 35L65, 34A12, 49Q20, 28A75 c 2008 Springer-Verlag Berlin Heidelberg  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper 987654321 springer.com

Preface

This book collects the lecture notes of two courses and one mini-course held in a winter school in Bologna in January 2005. The aim of this school was to popularize techniques of geometric measure theory among researchers and PhD students in hyperbolic differential equations. Though initially developed in the context of the calculus of variations, many of these techniques have proved to be quite powerful for the treatment of some hyperbolic problems. Obviously, this point of view can be reversed: We hope that the topics of these notes will also capture the interest of some members of the elliptic community, willing to explore the links to the hyperbolic world. The courses were attended by about 70 participants (including post-doctoral and senior scientists) from institutions in Italy, Europe, and North-America. This initiative was part of a series of schools (organized by some of the people involved in the school held in Bologna) that took place in Bressanone (Bolzano) in January 2004, and in SISSA (Trieste) in June 2006. Their scope was to present problems and techniques of some of the most promising and fascinating areas of research related to nonlinear hyperbolic problems that have received new and fundamental contributions in the recent years. In particular, the school held in Bressanone offered two courses that provided an introduction to the theory of control problems for hyperbolic-like PDEs (delivered by Roberto Triggiani), and to the study of transport equations with irregular coefficients (delivered by Francois Bouchut), while the conference hosted in Trieste was organized in two courses (delivered by Laure Saint-Raymond and Cedric Villani) and in a series of invited lectures devoted to the main recent advancements in the study of Boltzmann equation. Some of the material covered by the course of Triggiani can be found in [17, 18, 20], while the main contributions of the conference on Boltzmann will be collected in a forthcoming special issue of the journal DCDS, of title “Boltzmann equations and applications”. The three contributions of the present volume gravitate all around the theory of BV functions, which play a fundamental role in the subject of hyperbolic conservation laws. However, so far in the hyperbolic community little attention has been paid to some typical problems which constitute an old topic in geometric measure

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theory: the structure and fine properties of BV functions in more than one space dimension. The lecture notes of Luigi Ambrosio and Gianluca Crippa stem from the remarkable achievement of the first author, who recently succeeded in extending the so-called DiPerna–Lions theory for transport equations to the BV setting. More precisely, consider the Cauchy problem for a transport equation with variable coefficients ⎧ ⎨ ∂t u(t, x) + b(t, x) · ∇u(t, x) = 0 , (1) ⎩ u(0, x) = u0 (x) . When b is Lipschitz, (1) can be explicitly solved via the method of characteristics: a solution u is indeed constant along the trajectories of the ODE ⎧ dΦ ⎨ dt x = b(t, Φx (t)) (2) ⎩ Φ(0, x) = x . Transport equations appear in a wealth of problems in mathematical physics, where usually the coefficient is coupled to the unknowns through some nonlinearities. This already motivates from a purely mathematical point of view the desire to develop a theory for (1) and (2) which allows for coefficients b in suitable function spaces. However, in many cases, the appearance of singularities is a well-established central fact: the development of such a theory is highly motivated from the applications themselves. In the 1980s, DiPerna and Lions developed a theory for (1) and (2) when b ∈ W 1,p (see [16]). The task of extending this theory to BV coefficients was a long-standing open question, until Luigi Ambrosio solved it in [2] with his Renormalization Theorem. Sobolev functions in W 1,p cannot jump along a hypersurface: this type of singularity is instead typical for a BV function. Therefore, not surprisingly, Ambrosio’s theorem has found immediate application to some problems in the theory of hyperbolic systems of conservation laws (see [3, 5]). Ambrosio’s result, together with some questions recently raised by Alberto Bressan, has opened the way to a series of studies on transport equations and their links with systems of conservation laws (see [4,6–13]). The notes of Ambrosio and Crippa contain an efficient introduction to the DiPerna–Lions theory, a complete proof of Ambrosio’s theorem and an overview of the further developments and open problems in the subject. The first proof of Ambrosio’s Renormalization Theorem relies on a deep result of Alberti, perhaps the deepest in the theory of BV functions (see [1]). Consider a regular open set Ω ⊂ R2 and a map u : R2 → R2 which is regular in 2 R \ ∂ Ω but jumps along the interface ∂ Ω. The distributional derivative of u is then the sum of the classical derivative (which exists in R2 \ ∂ Ω) and a singular matrixvalued radon measure ν , supported on ∂ Ω. Let μ be the nonnegative measure on R2 defined by the property that μ (A) is the length of ∂ Ω ∩ A. Moreover, denote by n the exterior unit normal to ∂ Ω and by u− and u+ , respectively, the interior and

Preface

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exterior traces of u on ∂ Ω. As a straightforward application of Gauss’ theorem, we then conclude that the measure ν is given by [(u+ − u−) ⊗ n] μ . Consider now the singular portion of the derivative of any BV vector-valued map. By elementary results in measure theory, we can always factorize it into a matrixvalued function M times a nonnegative measure μ . Alberti’s Rank-One Theorem states that the values of M are always rank-one matrices. The depth of this theorem can be appreciated if one takes into account how complicated the singular measure μ can be. Though the most recent proof of Ambrosio’s Renormalization Theorem avoids Alberti’s result, the Rank-One Theorem is a powerful tool to gain insight in subtle further questions (see for instance [6]). The notes of Camillo De Lellis is a short and self-contained introduction to Alberti’s result, where the reader can find a complete proof. As already mentioned above, the space of BV functions plays a central role in the theory of hyperbolic conservation laws. Consider for instance the Cauchy problem for a scalar conservation law ⎧ ⎨ ∂t u + divx [ f (u)] = 0 , (3) ⎩ u(0, ·) = u0 . It is a classical result of Kruzhkov that for bounded initial data u0 there exists a unique entropy solution to (3). Furthermore, if u0 is a function of bounded variation, this property is retained by the entropy solution. Scalar conservation laws typically develop discontinuities. In particular jumps along hypersurfaces, the so-called shock waves, appear in finite time, even when starting with smooth initial data. These discontinuities travel at a speed which can be computed through the so-called Rankine–Hugoniot condition. Moreover, the admissibility conditions for distributional solutions (often called entropy conditions) are in essence devised to rule out certain “non-physical” shocks. When the entropy solution has BV regularity, the structure theory for BV functions allows us to identify a jump set, where all these assertions find a suitable (measure-theoretic) interpretation. What happens if instead the initial data are merely bounded? Clearly, if f is a linear function, i.e. f  vanishes, (3) is a transport equation with constant coefficients: extremely irregular initial data are then simply preserved. When we are far from this situation, loosely speaking when the range of f  is “generic”, f is called genuinely nonlinear. In one space dimension an extensively studied case of genuine nonlinearity is that of convex fluxes f . It is then an old result of Oleinik that, under this assumption, entropy solutions are BV functions for any bounded initial data. The assumption of genuine nonlinearity implies a regularization effect for the equation. In more than one space dimension (or under milder assumptions on f ) the BV regularization no longer holds true. However, Lions, Perthame, and Tadmore gave in [19] a kinetic formulation for scalar conservation laws and applied velocity averaging methods to show regularization in fractional Sobolev spaces. The notes of Gianluca Crippa, Felix Otto, and Michael Westdickenberg start with an introduction

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to entropy solutions, genuine nonlinearity, and kinetic formulations. They then discuss the regularization effects in terms of linear function spaces for a “generalized Burgers” flux, giving optimal results. From a structural point of view, however, these estimates (even the optimal ones) are always too weak to recover the nice picture available for the BV framework, i.e. a solution which essentially has jump discontinuities behaving like shock waves. Guided by the analogy with the regularity theory developed in [14] for certain variational problems, De Lellis, Otto, and Westdickenberg in [15] showed that this picture is an outcome of an appropriate “regularity theory” for conservation laws. More precisely, the property of being an entropy solution to a scalar conservation law (with a genuinely nonlinear flux f ) allows a fairly detailed analysis of the possible singularities. The information gained by this analysis is analogous to the fine properties of a generic BV function, even when the BV estimates fail. The notes of Crippa, Otto, and Westdickenberg give an overview of the ideas and techniques used to prove this result. Many institutions have contributed funds to support the winter school of Bologna. We had a substantial financial support from the research project GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni ) – “Multidimensional problems and control problems for hyperbolic systems”; from CIRAM (Research Center of Applied Mathematics) and the Fund for International Programs of University of Bologna; and from Seminario Matematico and the Department of Mathematics of University of Brescia. We were also funded by the research project INDAM (Istituto Nazionale di Alta Matematica “F. Severi”) – “Nonlinear waves and applications to compressible and incompressible fluids”. Our deepest thanks to all these institutions which make it possible the realization of this event and as a consequence of the present volume. As a final acknowledgement, we wish to warmly thank Accademia delle Scienze di Bologna and the Department of Mathematics of Bologna for their kind hospitality and for all the help and support they have provided throughout the school. Bologna, Trieste, Brescia, Z¨urich, and Padova, September 2007

Fabio Ancona Stefano Bianchini Rinaldo M. Colombo Camillo De Lellis Andrea Marson Annamaria Montanari

References 1. A LBERTI , G. Rank-one properties for derivatives of functions with bounded variations Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 239–274. 2. A MBROSIO , L. Transport equation and Cauchy problem for BV vector fields. Invent. Math., 158 (2004), 227–260.

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3. A MBROSIO , L.; B OUCHUT, F.; D E L ELLIS , C. Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Comm. Partial Differential Equations, 29 (2004), 1635–1651. 4. A MBROSIO , L.; C RIPPA , G.; M ANIGLIA , S. Traces and fine properties of a BD class of vector fields and applications. Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 4, 527–561. 5. A MBROSIO , L.; D E L ELLIS , C. Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions. Int. Math. Res. Not. 41 (2003), 2205–2220. 6. A MBROSIO L,; D E L ELLIS , C.; M AL Y´ , J. On the chain rule for the divergence of vector fields: applications, partial results, open problems. To appear in Perspectives in Nonlinear Partial Differential Equations: in honor of Haim Brezis Preprint available at http://cvgmt.sns. it/papers/ambdel05/. 7. A MBROSIO L.; L ECUMBERRY, M.; M ANIGLIA , S. S. Lipschitz regularity and approximate differentiability of the DiPerna–Lions flow. Rend. Sem. Mat. Univ. Padova 114 (2005), 29–50. 8. B RESSAN , A. An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend. Sem. Mat. Univ. Padova 110 (2003), 103–117. 9. B RESSAN , A. A lemma and a conjecture on the cost of rearrangements. Rend. Sem. Mat. Univ. Padova 110 (2003), 97–102. 10. B RESSAN , A. Some remarks on multidimensional systems of conservation laws. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 15 (2004), 225–233. 11. C RIPPA , G.; D E L ELLIS , C. Oscillatory solutions to transport equations. Indiana Univ. Math. J. 55 (2006), 1–13. 12. C RIPPA , G.; D E L ELLIS , C. Estimates and regularity results for the DiPerna-Lions flow. To appear in J. Reine Angew. Math. Preprint available at http://cvgmt.sns.it/cgi/get.cgi/papers/ cridel06/ 13. D E L ELLIS , C. Blow-up of the BV norm in the multidimensional Keyfitz and Kranzer system. Duke Math. J. 127 (2005), 313–339. 14. D E L ELLIS , C.; OTTO , F. Structure of entropy solutions to the eikonal equation. J. Eur. Math. Soc. 5 (2003), 107–145. 15. D E L ELLIS , C.; OTTO , F.; W ESTDICKENBERG , M. Structure of entropy solutions to scalar conservation laws. Arch. Ration. Mech. Anal. 170(2) (2003), 137–184. 16. D I P ERNA , R.; L IONS , P. L. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–517. 17. L ASIECKA , I.; T RIGGIANI , R. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Discrete Contin. Dyn. Syst. (2005), suppl., 556–565. 18. L ASIECKA , I.; T RIGGIANI , R. Well-posedness and sharp uniform decay rates at the L2 (Ω)level of the Schr¨odinger equation with nonlinear boundary dissipation. J. Evol. Equ. 6 (2006), no. 3, 485–537. 19. L IONS P.-L.; P ERTHAME B.; TADMOR E. A kinetic formulation of multidimensional scalar conservation laws and related questions. J. AMS, 7 (1994) 169–191. 20. T RIGGIANI , R. Global exact controllability on HΓ10 (Ω) × L2 (Ω) of semilinear wave equations with Neumann L2 (0, T ; L2 (Γ1 ))-boundary control. In: Control theory of partial differential equations, 273–336, Lect. Notes Pure Appl. Math., 242, Chapman & Hall/CRC, Boca Raton, FL, 2005.

Contents

Part I Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields . . . . . . . . . . Luigi Ambrosio and Gianluca Crippa 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Continuity Equation Within the Cauchy–Lipschitz Framework . . . . . . 4 (ODE) Uniqueness Vs. (PDE) Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Flow Associated to Sobolev or BV Vector Fields . . . . . . . . . . . . . . . . . 6 Measure-Theoretic Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Differentiability of the Flow in the W 1,1 Case . . . . . . . . . . . . . . . . . . . . . . . . 8 Differentiability and Compactness of the Flow in the W1,p Case . . . . . . . . 9 Bibliographical Notes and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5 7 11 19 32 38 40 52 54

Part II A Note on Alberti’s Rank-One Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . Camillo De Lellis 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A Blow-Up Argument Leading to a Partial Result . . . . . . . . . . . . . . . . . . . . 4 The Fundamental Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Proof of Theorem 1.1 in the Planar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 63 65 66 68 74

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Contents

Part III Regularizing Effect of Nonlinearity in Multidimensional Scalar Conservation Laws . . . . . . . . . . . . . . . . . . . . . . 77 Gianluca Crippa, Felix Otto, and Michael Westdickenberg 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2 Background Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3 Entropy Solutions with BV-Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4 Structure of Entropy Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5 Kinetic Formulation, Blow-Ups and Split States . . . . . . . . . . . . . . . . . . . . . . 91 6 Classification of Split States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.1 Special Split States: No Entropy Dissipation . . . . . . . . . . . . . . . . . . . . 98 6.2 Special Split States: ν Supported on a Hyperplane . . . . . . . . . . . . . . . 101 6.3 Special Split States: ν Supported on Half a Hyperplane . . . . . . . . . . . 103 6.4 Classification of General Split States . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8 Proofs of the Regularity Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Authors Luigi Ambrosio Gianluca Crippa Scuola Normale Superiore Piazza dei Cavalieri 7 56126 Pisa, Italy E-mail: [email protected] [email protected] URL: http://cvgmt.sns.it/people/ ambrosio/

Felix Otto Institute for Applied Mathematics University of Bonn Wegelerstrae 10 53115 Bonn, Germany E-mail: [email protected] URL: http://www-mathphys.iam.unibonn.de/∼otto/

Camillo De Lellis Institut f¨ur Mathematik Universit¨at Zrich Winterthurerstrasse 190 CH-8057 Z¨urich, Switzerland E-mail: camillo.delellis@math. unizh.ch URL: http://www.math.unizh.ch/

Michael Westdickenberg School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332-0160, USA E-mail: [email protected] URL: http://www.math.gatech.edu/ ∼mwest/

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Editors Fabio Ancona Department of Mathematics and CIRAM University of Bologna Via Saragozza, 8 40123 Bologna, Italy E-mail: [email protected] URL: http://www.ciram.unibo.it/ ancona/ Stefano Bianchini SISSA-ISAS, Via Beirut, 2-4 34014 Trieste, Italy E-mail: [email protected] URL: http://people.sissa.it/ ∼bianchin/ Rinaldo M. Colombo Department of Mathematics University of Brescia Via Branze, 38 25123 Brescia, Italy E-mail: [email protected] URL: http://dm.ing.unibs.it/rinaldo/

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Camillo De Lellis Institut f¨ur Mathematik Universit¨at Z¨urich Winterthurerstrasse 190 8057 Z¨urich, Switzerland E-mail: [email protected] URL: http://www.math.unizh.ch/ Andrea Marson Department of Pure and Applied Mathematics Via Trieste, 63 35131 Padova, Italy E-mail: [email protected] URL: http://www.math.unipd.it/ ∼marson Annamaria Montanari, Department of Mathematics University of Bologna Piazza di Porta S. Donato, 5 40126 Bologna, Italy E-mail: [email protected] URL: http://www.dm.unibo.it/ ∼montanar/

Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields Luigi Ambrosio and Gianluca Crippa Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy e-mail: [email protected], [email protected] URL: http://cvgmt.sns.it/people/ambrosio/

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Continuity Equation Within the Cauchy–Lipschitz Framework . . . . . . . . . . . . . . . . . . . 4 (ODE) Uniqueness Vs. (PDE) Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Flow Associated to Sobolev or BV Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Measure-Theoretic Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Differentiability of the Flow in the W 1,1 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Differentiability and Compactness of the Flow in the W1,p Case . . . . . . . . . . . . . . . . . . . . . . 9 Bibliographical Notes and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 5 7 11 19 32 38 40 52 54

1 Introduction In these notes we discuss some recent progress on the problem of the existence, uniqueness and stability of the flow associated to a weakly differentiable (Sobolev or BV regularity with respect to the spatial variables) time-dependent vector field b(t, x) = bt (x) in Rd . Vector fields with this “low” regularity show up, for instance, in several PDEs describing the motion of fluids, and in the theory of conservation laws. We are therefore interested to the well posedness of the system of ordinary differential equations ⎧ ⎨ γ˙(t) = bt (γ (t)) (ODE) ⎩ γ (0) = x.

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4

L. Ambrosio and G. Crippa

In some situations one might hope for a “generic” uniqueness of the solutions of (ODE), i.e. for “almost every” initial datum x. But, as a matter of fact, no such uniqueness theorem is presently known in the case when b(t, ·) has a Sobolev or BV regularity (this issue is discussed in Sect. 9). An even weaker requirement is the research of a “canonical selection principle”, i.e. a strategy to select for L d -almost every x a solution X(·, x) in such a way that this selection is stable w.r.t. smooth approximations of b. In other words, we would like to know that, whenever we approximate b by smooth vector fields bh , the classical trajectories X h associated to bh satisfy lim X h (·, x) = X(·, x)

in C([0, T ]; Rd ), for L d -a.e. x.

h→∞

The following simple example, borrowed from [8], provides an illustration of the kind of phenomena that can occur. Example 1.1. Let us consider the autonomous ODE ⎧  ⎨ γ˙(t) = |γ (t)| ⎩

γ (0) = x0 .

Then, solutions of the ODE are not unique for x0 = −c2 < 0. Indeed, they reach the origin in time 2c, where they can stay for an arbitrary time T , then continuing as x(t) = 14 (t − T − 2c)2 . Let us consider for instance the Lipschitz approximation  (that could easily be made smooth) of b(γ ) = |γ | given by ⎧ ⎪ if −∞ < γ ≤ −ε 2 ; ⎪ |γ | ⎪ ⎪ ⎪ ⎪ ⎨ bε (γ ) := ε if −ε 2 ≤ γ ≤ λε − ε 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩γ − λ + 2ε 2 if λ − ε 2 ≤ γ < +∞, ε

ε

with λε − ε 2 > 0. Then, solutions of the approximating ODEs starting from −c2 reach the value −ε 2 in time tε = 2(c − ε ) and then they continue with constant speed ε until they reach λε − ε 2 , in time Tε = λε /ε . Then, they continue as λε − 2ε 2 + 14 (t − tε − Tε )2 . Choosing λε = ε T , with T > 0, by this approximation we select the solutions that don’t move, when at the origin, exactly for a time  T. Other approximations, as for instance bε (γ ) = ε + |γ |, select the solutions that move immediately away from the singularity at γ = 0. Among all possibilities, this family of solutions x(t, x0 ) is singled out by the property that x(t, ·)# L 1 is absolutely continuous with respect to L 1 , so no concentration of trajectories occurs at the origin. To see this fact, notice that we can integrate in time the identity 0 = x(t, ·)# L 1 ({0}) = L 1 ({x0 : x(t, x0 ) = 0})

Flow of Weakly Differentiable Vector Fields

5

and use Fubini’s theorem to obtain 0=



L 1 ({t : x(t, x0 ) = 0}) dx0 .

Hence, for L 1 -a.e. x0 , x(·, x0 ) does not stay at 0 for a strictly positive set of times. The theme of existence, uniqueness and stability has been trated in detail in the lectures notes [7], and more recently in [8] (where also the applications to systems of conservation laws [11, 9] and to the semi-geostrophic equation ([53]) are described), so some overlap with the content of these notes is unavoidable. Because of this fact, we decided to put here more emphasis on even more recent results [73, 15, 17, 48], relative to the differentiability properties of X(t, x) with respect to x. This is not a casual choice, as the key idea of the paper [15] was found during the Bologna school.

2 The Continuity Equation An important tool, in studying existence, uniqueness ad stability of (ODE), is the well-posedness of the Cauchy problem for the homogeneous conservative continuity equation (PDE)

d μt + Dx · (bμt ) = 0 dt

(t, x) ∈ I × Rd

and for the transport equation d wt + b · ∇wt = ct . dt We will see that there is a close link between (PDE) and (ODE), first investigated in a nonsmooth setting by DiPerna and Lions in [61]. Let us now make some basic technical remarks on the continuity equation and the transport equation: Remark 2.1 (Regularity in space of bt and μt ). (1) Since the continuity equation (PDE) is in divergence form, it makes sense without any regularity requirement on bt and/or μt , provided   I A

|bt | d|μt | dt < +∞

∀A ⊂⊂ Rd .

(1)

However, when we consider possibly singular measures μt , we must take care of the fact that the product bt μt is sensitive to modifications of bt in L d -negligible sets. In the Sobolev or BV case we will consider only measures μt = wt L d , so everything is well stated.

6

L. Ambrosio and G. Crippa

(2) On the other hand, due to the fact that the distribution bt · ∇w is defined by bt · ∇w, ϕ  := −

  I



wbt · ∇ϕ dxdt − Dx · bt , wt ϕt  dt I

∀ϕ ∈ Cc∞ (I × Rd )

(a definition consistent with the case when wt is smooth) the transport equation makes sense only if we assume that Dx · bt = div bt L d for L 1 -a.e. t ∈ I. See also [28, 29] for recent results on the transport equation when b satisfies a one-sided Lipschitz condition. Next, we consider the problem of the time continuity of t → μt and t → wt . Remark 2.2 (Regularity in time of μt ). For any test function ϕ ∈ Cc∞ (Rd ), condition (1) gives   d ϕ d μt = bt · ∇ϕ d μt ∈ L1 (I) dt Rd Rd and therefore the map t → μt , ϕ , for given ϕ , has a unique uniformly continuous representative in I. By a simple density argument we can find a unique representative μ˜ t independent of ϕ , such that t → μ˜ t , ϕ  is uniformly continuous in I for any ϕ ∈ Cc∞ (Rd ). We will always work with this representative, so that μt will be well defined for all t and even at the endpoints of I. An analogous remark applies for solutions of the transport equation. There are some other important links between the two equations: (1) The transport equation reduces to the continuity equation in the case when ct = −wt div bt . (2) Formally, one can establish a duality between the two equations via the (formal) identity d dt







d d μt wt wt d μt + dt dt    = (−bt · ∇wt + ct ) d μt + bt · ∇wt d μt = ct d μt .

wt d μt =

This duality method is a classical tool to prove uniqueness in a sufficiently smooth setting (but see also [28, 29]). (3) Finally, if we denote by Y(t, s, x) the solution of the ODE at time t, starting from x at the initial time s, i.e. d Y(t, s, x) = bt (Y(t, s, x)), dt

Y(s, s, x) = x,

then Y(t, ·, ·) are themselves solutions of the transport equation: to see this, it suffices to differentiate the semigroup identity Y(t, s, Y(s, l, x)) = Y(t, l, x) w.r.t. s to obtain, after the change of variables y = Y(s, l, x), the equation

Flow of Weakly Differentiable Vector Fields

7

d Y(t, s, y) + bs (y) · ∇Y(t, s, y) = 0. ds This property is used in a essential way in [61] to characterize the flow Y and to prove its stability properties. The approach developed here, based on [6], is based on a careful analysis of the measures transported by the flow, and ultimately on the homogeneous continuity equation only.

3 The Continuity Equation Within the Cauchy–Lipschitz Framework In this section we recall the classical representation formulas for solutions of the continuity or transport equation in the case when  b ∈ L1 [0, T ];W 1,∞ (Rd ; Rd ) . Under this assumption it is well known that solutions X(t, ·) of the ODE are unique and stable. A quantitative information can be obtained by differentiation: d |X(t, x) − X(t, y)|2 = 2bt (X(t, x)) − bt (X(t, y)), X(t, x) − X(t, y) dt ≤ 2Lip (bt )|X(t, x) − X(t, y)|2 (here Lip ( f ) denotes the least Lipschitz constant of f ), so that Gronwall lemma immediately gives

 t  Lip (X(t, ·)) ≤ exp Lip (bs ) ds . (2) 0

Turning to the continuity equation, uniqueness of measure-valued solutions can be proved by the duality method. Or, following the techniques developed in these lectures, it can be proved in a more general setting for positive measure-valued solutions (via the superposition principle) and for signed solutions μt = wt L d (via the theory of renormalized solutions). So in this section we focus only on the existence and the representation issues. The representation formula is indeed very simple: Proposition 3.1. For any initial datum μ¯ the solution of the continuity equation is given by

μt := X(t, ·)# μ¯ ,



i.e.

Rd

ϕ d μt =

 Rd

ϕ (X(t, x)) d μ¯ (x) ∀ϕ ∈ Cb (Rd ).

(3)

d dt μt

+ Dx ·

Proof. Notice first that we need only to check the distributional identity (bt μt ) = 0 on test functions of the form ψ (t)ϕ (x), so that

8

L. Ambrosio and G. Crippa

 R

ψ (t)μt , ϕ  dt +

 R

ψ (t)

 Rd

bt , ∇ϕ  d μt dt = 0.

This means that we have to check that t → μt , ϕ  belongs to W 1,1 (0, T ) for any ϕ ∈ Cc∞ (Rd ) and that its distributional derivative is Rd bt , ∇ϕ  d μt . We show first that this map is absolutely continuous, and in particular W 1,1 (0, T ); then one needs only to compute the pointwise derivative. For every choice of finitely many, say n, pairwise disjoint intervals (ai , bi ) ⊂ [0, T ] we have n

∑ |ϕ (X(bi , x)) − ϕ (X(ai , x))| ≤ ∇ϕ ∞

i=1

≤ ∇ϕ ∞

 ∪i (ai ,bi )



∪i (ai ,bi )

˙ x)| dt |X(t, sup |bt | dt

and therefore an integration with respect to μ¯ gives n

∑ |μbi − μai , ϕ | ≤ μ¯ (Rd )∇ϕ ∞

i=1

 ∪i (ai ,bi )

sup |bt | dt.

The absolute continuity of the integral shows that the right hand side can be made small when ∑i (bi − ai ) is small. This proves the absolute continuity. For any x the ˙ x) = bt (X(t, x)) is fulfilled for L 1 -a.e. t ∈ [0, T ]. Then, by Fubini’s identity X(t, theorem, we know also that for L 1 -a.e. t ∈ [0, T ] the previous identity holds for μ¯ -a.e. x, and therefore d d μt , ϕ  = dt dt  =



Rd

Rd

ϕ (X(t, x)) d μ¯ (x)

∇ϕ (X(t, x)), bt (X(t, x)) d μ¯ (x)

= bt μt , ∇ϕ  for L 1 -a.e. t ∈ [0, T ].



In the case when μ¯ = ρ L d we can say something more, proving that the measures μt = X(t, ·)# μ¯ are absolutely continuous w.r.t. L d and computing explicitly their density. Let us start by recalling the classical area formula: if f : Rd → Rd is a (locally) Lipschitz map, then  A

g|J f | dx =

 Rd

∑

g(x) dy

x∈A∩ f −1 (y)

for any Borel set A ⊂ Rd and any integrable function g : A → R, where J f = det ∇ f (recall that, by Rademacher theorem, Lipschitz functions are differentiable L d a.e.). Assuming in addition that f is one to one and onto and that |J f | > 0 L d -a.e. on A we can set A = f −1 (B) and g = ρ /|J f | to obtain

Flow of Weakly Differentiable Vector Fields

 f −1 (B)

9

ρ dx =



ρ ◦ f −1 dy. B |J f |

In other words, we have got a formula for the push-forward: f# (ρ L d ) =

ρ ◦ f −1 L d . |J f |

(4)

In our case f (x) = X(t, x) is surely one to one, onto and Lipschitz. It remains to show that |JX(t, ·)| does not vanish: in fact, one can show that JX > 0 and

t 

 t  exp − [div bs ]− ∞ ds ≤ JX(t, x) ≤ exp [div bs ]+ ∞ ds (5) 0

0

for L d -a.e. x, thanks to the following fact, whose proof is left as an exercise. Exercise 3.2. If b is smooth, we have d JX(t, x) = div bt (X(t, x))JX(t, x). dt Hint: use the ODE

d dt ∇X

= ∇bt (X)∇X.

The previous exercise gives that, in the smooth case, JX(·, x) solves a linear ODE with the initial condition JX(0, x) = 1, whence the estimates on JX follow. In the general case the upper estimate on JX still holds by a smoothing argument, thanks to the lower semicontinuity of ⎧ d ⎪ ⎨Jv∞ if Jv ≥ 0 L -a.e. Φ (v) := ⎪ ⎩ +∞ otherwise with respect to the w∗ -topology of W 1,∞ (Rd ; Rd ). This is indeed the supremum of 1/p the family of Φ p , where Φ p are the polyconvex (and therefore lower semicontinuous) functionals 

Φ p (v) :=

Bp

|χ (Jv)| p dx.

Here χ (t), equal to ∞ on (−∞, 0) and equal to t on [0, +∞), is l.s.c. and convex. The lower estimate can be obtained by applying the upper one in a time reversed situation. Now we turn to the representation of solutions of the transport equation:   Proposition 3.3. If w ∈ L1loc [0, T ] × Rd solves  d wt + b · ∇wt = c ∈ L1loc [0, T ] × Rd dt

10

L. Ambrosio and G. Crippa

then, for L d -a.e. x, we have wt (X(t, x)) = w0 (x) +

t 0

cs (X(s, x)) ds

∀t ∈ [0, T ].

The (formal) proof is based on the simple observation that d d d wt ◦ X(t, x) = wt (X(t, x)) + X(t, x) · ∇wt (X(t, x)) dt dt dt d = wt (X(t, x)) + bt (X(t, x)) · ∇wt (X(t, x)) dt = ct (X(t, x)). In particular, as X(t, x) = Y(t, 0, x) = [Y(0,t, ·)]−1 (x), we get wt (y) = w0 (Y(0,t, y)) +

t 0

cs (Y(s,t, y)) ds.

We conclude this presentation of the out two simple  classical theory pointing  local variants of the assumption b ∈ L1 [0, T ];W 1,∞ (Rd ; Rd ) made throughout this section. Remark 3.4 (First local variant). The theory outlined above still works under the assumptions  1,∞ b ∈ L1 [0, T ];Wloc (Rd ; Rd ) ,

 |b| ∈ L1 [0, T ]; L∞ (Rd ) . 1 + |x|

Indeed, due to the growth condition on b, we still have pointwise uniqueness of the ODE and a uniform local control on the growth of |X(t, x)|, therefore we need only to consider a local Lipschitz condition w.r.t. x, integrable w.r.t. t. The next variant will be used in the proof of the superposition principle. 1,∞ Remark 3.5 (Second local variant). Still keeping the L1 (Wloc ) assumption, and assuming μt ≥ 0, the second growth condition on |b| can be replaced by a global, but more intrinsic, condition:

 T 0

Rd

|bt | d μt dt < +∞. 1 + |x|

(6)

Under this assumption one can show that for μ¯ -a.e. x the maximal solution X(·, x) of the ODE starting from x is defined up to t = T and still the representation μt = X(t, ·)# μ¯ holds for t ∈ [0, T ].

Flow of Weakly Differentiable Vector Fields

11

4 (ODE) Uniqueness Vs. (PDE) Uniqueness In this section we illustrate some quite general principles, whose application may depend on specific assumptions on b, relating the uniqueness for the ODE to the uniqueness for the PDE. The viewpoint adopted in this section is very close in spirit to Young’s theory [95] of generalized surfaces and controls (a theory with remarkable applications also to nonlinear PDEs [60, 88] and to Calculus of Variations [19]) and has also some connection with Brenier’s weak solutions of incompressible Euler equations [30], with Kantorovich’s viewpoint in the theory of optimal transportation [63, 85] and with Mather’s theory [80, 81, 20]: in order to study existence, uniqueness and stability with respect to perturbations of the data of solutions to the ODE, we consider suitable measures in the space of continuous maps, allowing for superposition of trajectories. Then, in some special situations we are able to show that this superposition actually does not occur, but still this “probabilistic” interpretation is very useful to understand the underlying techniques and to give an intrinsic characterization of the flow. The first very general criterion is the following. Theorem 4.1. Let A ⊂ Rd be a Borel set. The following two properties are equivalent: (a) Solutions of the ODE are unique for any x ∈ A. (b) Nonnegative measure-valued solutions of the PDE are unique for any μ¯ concentrated in A, i.e. such that μ¯ (Rd \ A) = 0. Proof. It is clear that (b) implies (a), just choosing μ¯ = δx and noticing that two ˜ of the ODE induce two different solutions of the PDE, different solutions X(t), X(t) . namely δX(t) and δX(t) ˜ The converse implication is less obvious and requires the superposition principle that we are going to describe below, and that provides the representation     ϕ d μt = ϕ (γ (t)) d η x (γ ) d μ0 (x) ∀ϕ ∈ Cb (Rd ), Rd

ΓT

Rd

with η x probability measures concentrated on the absolutely continuous integral solutions of the ODE starting from x. Therefore, when these are unique, the measures η x are unique (and are Dirac masses), so that the solutions of the PDE are unique. 

  We will use the shorter notation ΓT for the space C [0, T ]; Rd and denote by et : ΓT → Rd the evaluation maps γ → γ (t), t ∈ [0, T ]. Definition 4.2 (Superposition solutions). Let η ∈ M+ (Rd × ΓT ) be a measure concentrated on the set of pairs (x, γ ) such that γ is an absolutely continuous integral solution of the ODE with γ (0) = x. We define μtη , ϕ  :=

 Rd ×ΓT

ϕ (et (γ )) d η (x, γ )

∀ϕ ∈ Cb (Rd ).

12

L. Ambrosio and G. Crippa

By a standard approximation argument the identity defining μtη holds for any Borel function ϕ such that γ → ϕ (et (γ )) is η -integrable (or equivalently for any μtη -integrable function ϕ ). Under the (local) integrability condition  T Rd ×ΓT

0

χBR (et )|bt (et )| d η dt < +∞

∀R > 0

(7)

it is not hard to see that μtη solves the PDE with the initial condition μ¯ := (πRd )# η : indeed, let us check first that t → μtη , ϕ  is absolutely continuous for any ϕ ∈ Cc∞ (Rd ). For every choice of finitely many pairwise disjoint intervals (ai , bi ) ⊂ [0, T ] we have n

∑ |ϕ (γ (bi )) − ϕ (γ (ai))| ≤ Lip (ϕ )

i=1

 ∪i (ai ,bi )

χBR (|et (γ )|)bt (et (γ ))| dt

for η -a.e. (x, γ ), with R such that supp ϕ ⊂ BR . Therefore an integration with respect to η gives n

∑

i=1

|μbηi , ϕ  − μaηi , ϕ |

≤ Lip (ϕ )



 ∪i (ai ,bi ) Rd ×ΓT

χBR (et )|bt (et )| d η dt.

The absolute continuity of the integral shows that the right hand side can be made small when ∑i (bi − ai ) is small. This proves the absolute continuity. It remains to evaluate the time derivative of t → μtη , ϕ : we know that for η -a.e. (x, γ ) the identity γ˙(t) = bt (γ (t)) is fulfilled for L 1 -a.e. t ∈ [0, T ]. Then, by Fubini’s theorem, we know also that for L 1 -a.e. t ∈ [0, T ] the previous identity holds for η -a.e. (x, γ ), and therefore d η d μ , ϕ  = dt t dt =



 Rd ×ΓT

Rd ×ΓT

ϕ (et (γ )) d η

∇ϕ (et (γ )), bt (et (γ )) d η = bt μt , ∇ϕ 

L 1 -a.e. in [0, T ].

Remark 4.3. Actually the formula defining μtη does not contain x, and so it involves only the projection of η on ΓT . Therefore one could also consider measures σ in ΓT , concentrated on the set of solutions of the ODE (for an arbitrary initial point x). These two viewpoints are basically equivalent: given η one can build σ just by projection on ΓT , and given σ one can consider the conditional probability measures η x concentrated on the solutions of the ODE starting from x induced by the random variable γ → γ (0) in ΓT , the law μ¯ (i.e. the push forward) of the same random variable and recover η as follows:     ϕ (x, γ ) d η (x, γ ) := ϕ (x, γ ) d η x (γ ) d μ¯ (x). (8) Rd ×ΓT

Rd

ΓT

Flow of Weakly Differentiable Vector Fields

13

Our viewpoint has been chosen just for technical convenience, to avoid the use, wherever this is possible, of the conditional probability theorem. By restricting η to suitable subsets of Rd × ΓT , several manipulations with superposition solutions of the continuity equation are possible and useful, and these are not immediate to see just at the level of general solutions of the continuity equation. This is why the following result is interesting. Theorem 4.4 (Superposition principle). Let μt ∈ M+ (Rd ) be a solution of (PDE) and assume that  T |b|t (x) d μt dt < +∞. Rd 1 + |x| 0 Then μt is a superposition solution, i.e. there exists η ∈ M+ (Rd × ΓT ) such that μt = μtη for any t ∈ [0, T ]. In the proof we use the narrow convergence of positive measures, i.e. the convergence with respect to the duality with continuous and bounded functions, and the easy implication in Prokhorov compactness theorem: any tight and bounded family F in M+ (X) is (sequentially) relatively compact w.r.t. the narrow convergence. Remember that tightness means: for any ε > 0 there exists K ⊂ X compact s.t. μ (X \ K) < ε ∀μ ∈ F . A necessary and sufficient condition for tightness is the existence of a coercive functional Ψ : X → [0, ∞] such that Ψ d μ ≤ 1 for any μ ∈ F . Proof. Step 1. (Smoothing) The smoothing argument which follows has been inspired by [65]. We mollify μt w.r.t. the space variable with a kernel ρ having finite first moment M and support equal to the whole of Rd (a Gaussian, for instance), obtaining smooth and strictly positive functions μtε . We also choose a function ψ : Rd → [0, +∞) such that ψ (x) → +∞ as |x| → +∞ and  Rd

ψ (x)μ0 ∗ ρε (x) dx ≤ 1

∀ε ∈ (0, 1)

and a convex nondecreasing function Θ : R+ → R having a more than linear growth at infinity such that  T Θ(|bt |(x)) d μt dt < +∞ 1 + |x| Rd 0 (the existence of Θ is ensured by the Dunford–Pettis Theorem). Defining

μtε := μt ∗ ρε ,

btε :=

(bt μt ) ∗ ρε , μtε

it is immediate that d ε d μt + Dx · (btε μtε ) = μt ∗ ρε + Dx · (bt μt ) ∗ ρε = 0 dt dt

14

L. Ambrosio and G. Crippa

 1,∞ and that bε ∈ L1 [0, T ];Wloc (Rd ; Rd ) . Therefore Remark 3.5 can be applied and

the representation μtε = Xε (t, ·)# μ0ε still holds. Then, we define

η ε := (x, X ε (·, x))# μ0ε , so that 

ηε

Rd

ϕ d μt

= =

 Rd ×ΓT



Rd

ϕ (γ (t)) d η ε

(9)

ϕ (Xε (t, x)) d μ0ε (x) =

 Rd

ϕ d μtε .

Step 2. (Tightness) We will be using the inequality ((1 + |x|)c) ∗ ρε ≤ (1 + |x|)c ∗ ρε + ε c ∗ ρ˜ ε

(10)

for c nonnegative measure and ρ˜ (y) = |y|ρ (y), and Θ(|btε (x)|)μtε (x) ≤ (Θ(|bt |)μt ) ∗ ρε (x).

(11)

The proof of the first one is elementary, while the proof of the second one follows by applying Jensen’s inequality with the convex l.s.c. function (z,t) → Θ(|z|/t)t (set equal to +∞ if t < 0, or t = 0 and z = 0, and equal to 0 if z = t = 0) and with the measure ρε (x − ·)L d . Let us introduce the functional Ψ(x, γ ) := ψ (x) +

 T Θ(|γ˙|)

1 + |γ |

0

dt,

set equal to +∞ on ΓT \ AC([0, T ]; Rd ). Using Ascoli–Arzel´a theorem, it is not hard to show that Ψ is coercive (it suffices to show that max |γ | is bounded on the sublevels {Ψ ≤ t}). Since 

 T Θ(|γ˙|)

≤

0

 T



and

 T

1 + |γ | Θ(|bt |(x)) d μt dt (1 + ε M) 1 + |x| Rd 0 Rd ×ΓT

(10),(11)

dt d η ε (x, γ ) =

Rd ×ΓT

ψ (x) d η ε (x, γ ) =

 Rd

0

Rd

Θ(|btε |) ε d μt dt 1 + |x|

ψ (x) d μ0ε ≤ 1

we obtain that Ψ d η ε is uniformly bounded for ε ∈ (0, 1), and therefore Prokhorov compactness theorem tells us that the family η ε is narrowly sequentially relatively compact as ε ↓ 0. If η is any limit point we can pass to the limit in (9) to obtain that μt = μtη . Step 3. (η is concentrated on solutions of the ODE) It suffices to show that

Flow of Weakly Differentiable Vector Fields

15

  γ (t) − x − t bs (γ (s)) ds



0

1 + max |γ |

Rd ×ΓT

dη = 0

(12)

[0,T ]

for any t ∈ [0, T ]. The technical difficulty is that this test function, due to the lack of regularity of b, is not continuous. To this aim, we prove first that     T γ (t) − x − t cs (γ (s)) ds |bs − cs | 0 dη ≤ d μs ds (13) 1 + max |γ | 0 Rd ×ΓT Rd 1 + |x| [0,T ]

for any continuous function c with compact support. Then, choosing a sequence (cn ) converging to b in L1 (ν ; Rd ), with 

ϕ (s, x) d ν (s, x) :=

 T 0

Rd

ϕ (s, x) d μs (x) ds 1 + |x|

and noticing that 

 T |bs (γ (s)) − cns (γ (s))|

Rd ×ΓT

0

1 + |γ (s)|

dsd η =

 T 0

Rd

|bs − cns | d μs ds → 0, 1 + |x|

we can pass to the limit in (13) with c = cn to obtain (12). It remains to show (13). This is a limiting argument based on the fact that (12) holds for bε , η ε :    γ (t) − x − t cs (γ (s)) ds 0 dη ε 1 + max |γ | Rd ×ΓT [0,T ]    ε X (t, x) − x − 0t cs (X ε (s, x)) ds ε d μ0 (x) = 1 + max |X ε (·, x)| Rd [0,T ]    t ε ε  t ε  |bεs − cs | ε 0 bs (X (s, x)) − cs (X (s, x)) ds ε d d μs ds μ (x) ≤ = 0 1 + max |Xε (·, x)| Rd 0 Rd 1 + |x| ≤ ≤

[0,T ]

 t 0

Rd

0

Rd

 t

 

t |bεs − cεs | ε |cεs − cs | ε d μs ds + d μs ds 1 + |x| 0 Rd 1 + |x|  t |bs − cs | |cεs − cs | ε d μs ds + d μs ds. 1 + |x| 0 Rd 1 + |x|

In the last inequalities we added and subtracted ctε := (ct μt ) ∗ ρε /μtε . Since ctε → ct uniformly as ε ↓ 0 thanks to the uniform continuity of c, passing to the limit in the chain of inequalities above we obtain (13). 

The applicability of Theorem 4.1 is strongly limited by the fact that, on one hand, pointwise uniqueness properties for the ODE are known only in very special situations, for instance when there is a Lipschitz or a one-sided Lipschitz (or

16

L. Ambrosio and G. Crippa

log-Lipschitz, Osgood. . . ) condition on b. On the other hand, also uniqueness for general measure-valued solutions is known only in special situations. It turns out that in many cases uniqueness of the PDE can only be proved in smaller classes L of solutions, and it is natural to think that this should reflect into a weaker uniqueness condition at the level of the ODE. We will see indeed that there is uniqueness in the “selection sense”. In order to illustrate this concept, in the following we consider a convex class Lb of measurevalued solutions μt ∈ M+ (Rd ) of the continuity equation relative to b, satisfying the following monotonicity property: 0 ≤ μt ≤ μt ∈ Lb

=⇒

μt ∈ Lb

(14)

whenever μt still solves the continuity equation relative to b, and satisfies the integrability condition  T 0

Rd

|bt (x)|

d μ (x)dt < +∞. 1 + |x| t

The typical application will be with absolutely continuous measures μt = wt L d , whose densities satisfy   some quantitative and possibly time-depending bound e.g. L∞ (L1 ) ∩ L∞ (L∞ ) . Definition 4.5 (Lb -Lagrangian flows). Given the class Lb , we say that X(t, x) is a Lb -Lagrangian flow starting from μ¯ ∈ M+ (Rd ) (at time 0) if the following two properties hold: (a) X(·, x) is absolutely continuous in [0, T ] and satisfies X(t, x) = x +

t 0

bs (X(s, x)) ds

∀t ∈ [0, T ]

for μ¯ -a.e. x; (b) μt := X(t, ·)# μ¯ ∈ Lb . Heuristically Lb -Lagrangian flows can be thought as suitable selections of the (possibly non unique) solutions of the ODE, made in such a way to produce a density in Lb . See Example 1.1 for an illustration of this concept. We will show that the Lb -Lagrangian flow starting from μ¯ is unique, modulo ¯μ -negligible sets, whenever uniqueness1 for the PDE holds in the class Lb , i.e.

μ0 = μ0

=⇒

μt = μt ∀t ∈ [0, T ]

whenever μt and μt belong to Lb . Before stating and proving the uniqueness theorem for Lb -Lagrangian flows, we state two elementary but useful results. The first one is a simple exercise: 1

We thank A. Figalli and F. Flandoli for pointing out that the argument works with minor variants when only uniqueness is imposed at the level of the PDE, and not necessarily the comparison principle.

Flow of Weakly Differentiable Vector Fields

17

Exercise 4.6. Let σ ∈ M+ (ΓT ) and let D ⊂ [0, T ] be a dense set. Show that σ is a Dirac mass in ΓT iff its projections (e(t))# σ , t ∈ D, are Dirac masses in Rd . The second one is concerned with a family of measures η x : Lemma 4.7. Let η x be a measurable family of positive finite measures in ΓT with the following property: for any t ∈ [0, T ] and any pair of disjoint Borel sets E, E ⊂ Rd we have   η x ({γ : γ (t) ∈ E}) η x {γ : γ (t) ∈ E } = 0 μ¯ -a.e. in Rd . (15) Then η x is a Dirac mass for μ¯ -a.e. x. Proof. Taking into account Exercise 4.6, for a fixed t ∈ (0, T ] it suffices to check that the measures λx := γ (t)# η x are Dirac masses for μ¯ -a.e. x. Then (15) gives λx (E)λx (E ) = 0 μ¯ -a.e. for any pair of disjoint Borel sets E, E ⊂ Rd . Let δ > 0 and let us consider a partition of Rd in countably many Borel sets Ri having a diameter less then δ . Then, as λx (Ri )λx (R j ) = 0 μ¯ -a.e. whenever i = j, we have a corresponding decomposition of μ¯ -almost all of Rd in Borel sets Ai such that supp λx ⊂ Ri for any x ∈ Ai (just take {λx (Ri ) > 0} and subtract from it all other sets {λx (R j ) > 0}, j = i). Since δ is arbitrary the statement is proved. 

Theorem 4.8 (Uniqueness of Lb -Lagrangian flows). Assume that the PDE has the uniqueness property in Lb . Then the Lb -Lagrangian flow starting from μ¯ is unique, i.e. two different selections X1 (t, x) and X2 (t, x) of solutions of the ODE inducing solutions of the continuity equation in Lb satisfy X1 (·, x) = X2 (·, x)

in ΓT , for μ¯ -a.e. x.

More generally, if η 1 , η 2 ∈ M+ (Rd × ΓT ) are concentrated on the pairs (x, γ ) with

γ absolutely continuous solution of the ODE, and if μtη = μtη ∈ Lb , then η 1 = η 2 and η i are concentrated on the graph of a map x → X(·, x) which is the unique Lb -Lagrangian flow. 1

2

Proof. If the first statement were false we could produce a measure η not concentrated on a graph inducing a solution μtη ∈ Lb of the PDE. This is not possible, thanks to the next result (Theorem 4.10). The measure η can be built as follows: 1 1 η := (η 1 + η 2 ) = [(x, X 1 (·, x))# μ¯ + (x, X2 (·, x))# μ¯ ] . 2 2 Since Lb is convex we still have μtη = 12 (μtη + μtη ) ∈ Lb . In a similar way, one can use Theorem 4.10 first to show that η i are concentrated on graphs, and then the previous combination argument to show that η 1 = η 2 . 

1

2

Remark 4.9. In the same vein, one can also show that X1 (·, x) = X 2 (·, x)

in ΓT for μ¯ 1 ∧ μ¯ 2 -a.e. x

whenever X 1 , X 2 are Lb -Lagrangian flows starting respectively from μ¯ 1 and μ¯ 2 .

18

L. Ambrosio and G. Crippa

We used the following basic result, having some analogy with Kantorovich’s and Mather’s theories. Theorem 4.10. Assume that the PDE has the uniqueness property in Lb . Let η ∈ M+ (Rd × ΓT ) be concentrated on the pairs (x, γ ) with γ absolutely continuous solution of the ODE starting from x, and assume that μtη ∈ Lb . Then η is concentrated on a graph, i.e. there exists a function x → X(·, x) ∈ ΓT such that   η = x, X(·, x) # μ¯ , with μ¯ := (πRd )# η = μ0η . Proof. We use the representation (8) of η , given by the disintegration theorem, the criterion stated in Lemma 4.7 and argue by contradiction. If the thesis is false then η x is not a Dirac mass in a set of positive μ¯ measure and we can find t ∈ (0, T ], disjoint Borel sets E, E ⊂ Rd and a Borel set C with μ¯ (C) > 0 such that     η x {γ : γ (t) ∈ E} η x {γ : γ (t) ∈ E } > 0 ∀x ∈ C. Possibly passing to a smaller set having still strictly positive μ¯ measure we can assume that 0 < η x ({γ : γ (t) ∈ E}) ≤ M η x ({γ : γ (t) ∈ E })

∀x ∈ C

(16)

for some constant M. We define measures η 1 , η 2 whose disintegrations η 1x , η 2x are given by

η 1x := χC (x)η x {γ : γ (t) ∈ E},

η 2x := M χC (x)η x {γ : γ (t) ∈ E }

and denote by μsi , s ∈ [0,t], the (superposition) solutions of the continuity equation induced by η i . Then

μ01 = η x ({γ : γ (t) ∈ E})μ¯ C,

μ02 = M η x ({γ : γ (t) ∈ E })μ¯ C,

so that (16) yields μ01 ≤ μ02 . On the other hand, μt1 is orthogonal to μt2 : precisely, denoting by η tx the image of η x under the map γ → γ (t), we have

μt1 =

 C

η tx E d μ (x) ⊥ M

 C

η tx E d μ (x) = μt2 .

In order to conclude, let f : Rd → [0, 1] be the density of μ01 with respect to μ02 and set η˜ 2x := M f (x)χC (x)η x {γ : γ (t) ∈ E }. We define the measure η˜ 2 whose disintegration is given by η˜ 2x and denote by μ˜ s2 , s ∈ [0,t], the (superposition) solution of the continuity equation induced by η˜ 2 . Notice also that μsi ≤ μs and so the monotonicity assumption (14) on Lb gives μsi ∈ Lb , and since η˜ 2 ≤ η 2 we obtain that μ˜ s2 ∈ Lb as well. By construction μ01 = μ˜ 02 , while μt1 is orthogonal to μt2 , a measure larger than μ˜ t2 . We have thus built two different solutions of the PDE with the same initial condition. 

Flow of Weakly Differentiable Vector Fields

19

Now we come to the existence of Lb -Lagrangian flows. Theorem 4.11 (Existence of Lb -Lagrangian flows). Assume that the PDE has the uniqueness property in Lb and that for some μ¯ ∈ M+ (Rd ) there exists a solution μt ∈ Lb with μ0 = μ¯ . Then there exists a (unique) Lb -Lagrangian flow starting from μ¯ . Proof. By the superposition principle we can represent μt as (et )# η for some η ∈ M+ (Rd × ΓT ) concentrated on pairs (x, γ ) solutions of the ODE. Then, Theorem 4.10 tells us that η is concentrated on a graph, i.e. there exists a function x → X(·, x) ∈ ΓT such that   x, X(·, x) # μ¯ = η . Pushing both sides via et we obtain X(t, ·)# μ¯ = (et )# η = μt ∈ Lb , and therefore X is a Lb -Lagrangian flow.



5 The Flow Associated to Sobolev or BV Vector Fields Here we discuss the well-posedness of the continuity or transport equations assuming that bt (·) has a Sobolev regularity, following [61]. Then, the general theory previously developed provides existence and uniqueness of the L -Lagrangian flow, with L := L∞ (L1 ) ∩ L∞ (L∞ ). We denote by I ⊂ R an open interval.   Definition 5.1 (Renormalized solutions). Let b ∈ L1loc I; L1loc (Rd ; Rd ) be such that D · bt = div bt L d for L 1 -a.e. t ∈ I, with  div bt ∈ L1loc I; L1loc (Rd ) .  ∞  d Let w ∈ L∞ loc I; Lloc (R ) and assume that c :=

d w + b · ∇w ∈ L1loc (I × Rd ). dt

(17)

Then, we say that w is a renormalized solution of (17) if d β (w) + b · ∇β (w) = cβ (w) dt

∀β ∈ C1 (R).

Equivalently, recalling the definition of the distribution b · ∇w, the definition could be given in a conservative form, writing d β (w) + Dx · (bβ (w)) = cβ (w) + div bt β (w). dt

20

L. Ambrosio and G. Crippa

Notice also that the concept makes sense, choosing properly the class of “test” functions β , also for w not satisfying (17), or not even locally integrable. This is particularly relevant in connection with DiPerna–Lions’s existence theorem for the Boltzmann equation [62], or with the case when w is the characteristic of an unbounded vector field b. This concept is also reminiscent of Kruzkhov’s concept of entropy solution for a scalar conservation law d u + Dx · (f (u)) = 0 dt

u : (0, +∞) × Rd → R.

In this case only a distributional one-sided inequality is required: d η (u) + Dx · (q(u)) ≤ 0 dt for any convex entropy–entropy flux pair (η , q) (i.e. η is convex and η f = q ). Remark 5.2 (Time continuity). Using the fact that both t → wt and t → β (wt ) have uniformly continuous representatives (w.r.t. the w∗ −L∞ ¯ t , σt , we obtain loc topology) w that t → w¯ t is continuous with respect to the strong L1loc topology for L 1 -a.e. t, and precisely for any t such that σt = β (w¯ t ). The proof follows by a classical weakstrong convergence argument: fn  f ,

β ( fn )  β ( f )

=⇒

fn → f

provided β is strictly convex. In the case of scalar conservation laws there are analogous and more precise results [92, 82]. We remark the fact that, in general, a renormalized solution does not need to have a representative which is strongly continuous for every t. This can be seen using a variation of an example given by Depauw [59] (see Remark 2.7 of [26]). Depauw’s example provides a divergence free vector field a ∈ L∞ ([0, 1] × R2 ; R2 ), with a(t, ·) ∈ BVloc (R2 ; R2 ) for L 1 -a.e. t ∈ [0, 1] (but a ∈ L1 ([0, 1]; BVloc )) such that the Cauchy problem  ∂t u + a · ∇u = 0 u(0, ·) = 0 has a nontrivial solution, with |u| ¯ = 1 L 3 -a.e. in [0, 1] × R2 and with the property that u(t, ¯ ·)  0 as t ↓ 0, but this convergence is not strong. Now consider a vector field b on [−1, 1] × R2 defined as Depauw’s vector field for t > 0, and set b(t, x) = −a(−t, x) for t < 0. It is simple to check (as only affine functions β˜ (t) = a + bt need to be checked, because for any β there exists an affine β˜ such that β˜ (±1) = β (±1)) that the function  u(t, ¯ x) if t > 0 w(t, ¯ x) = u(−t, ¯ x) if t < 0 is a renormalized solution of ∂t w + b · ∇w = 0, but this solution is not strongly continuous at t = 0.

Flow of Weakly Differentiable Vector Fields

21

Remark 5.3. A new insight in the theory of renormalized solutions has been obtained in the recent paper [26]. In particular, it is proved that for a vector field b ∈ L∞ ([0, T ] × Rd ; Rd ) with zero divergence (and without any regularity assumption) the following two conditions are equivalent (the L2 framework has been considered just for simplicity): (i) b has the uniqueness property for weak solutions in C([0, T ]; w − L2 (Rd )) for both the forward and the backward Cauchy problems starting respectively from 0 and T , i.e. the only solutions in C([0, T ]; w − L2 (Rd )) to the problems   ∂t uF + b · ∇uF = 0 ∂t uB + b · ∇uB = 0 and uF (0, ·) = 0 uB (T, ·) = 0 are uF ≡ 0 and uB ≡ 0; (ii) Every weak solution in C([0, T ]; w − L2 (Rd )) of ∂t u + b · ∇u = 0 is strongly continuous (i.e. lies in C([0, T ]; s − L2 (Rd ))) and is a renormalized solution. The proof of this equivalence is obtained through the study of the approximation properties of the solution of the transport equation, with respect to the norm of the graph of the transport operator (see Theorem 2.1 of [26] for the details). Using the concept of renormalized solution we can prove a comparison principle in the following natural class L :    (18) L := w ∈ L∞ [0, T ]; L1 (Rd ) ∩ L∞ [0, T ]; L∞ (Rd ) :   w ∈ C [0, T ]; w∗ − L∞ (Rd ) . Theorem 5.4 (Comparison principle). Assume that   |b| ∈ L1 [0, T ]; L∞ (Rd ) + L1 [0, T ]; L1 (Rd ) , 1 + |x| that D · bt = div bt L d for L 1 -a.e. t ∈ [0, T ], and that  [div bt ]− ∈ L1loc [0, T ) × Rd .

(19)

(20)

Setting bt ≡ 0 for t < 0, assume in addition that any solution of (17) in (−∞, T )× Rd is renormalized. Then the comparison principle for the continuity equation holds in the class L . Proof. By the linearity of the equation, it suffices to show that w ∈ L and w0 ≤ 0 implies wt ≤ 0 for any t ∈ [0, T ]. We extend first the PDE to negative times, setting wt = w0 . Then, fix a cut-off function ϕ ∈ Cc∞ (Rd ) with supp ϕ ⊂ B2 (0) and ϕ ≡ 1 on B1 (0), and the renormalization functions  βε (t) := ε 2 + (t +)2 − ε ∈ C1 (R).

22

L. Ambrosio and G. Crippa

Notice that

βε (t) ↑ t +

t βε (t) − βε (t) ∈ [0, ε ].

as ε ↓ 0,

(21)

We know that d βε (wt ) + Dx · (bβε (wt )) = div bt (βε (wt ) − wt βε (wt )) dt in the sense of distributions in (−∞, T ) × Rd . Plugging ϕR (·) := ϕ (·/R), with R ≥ 1, into the PDE we obtain d dt

 Rd

ϕR βε (wt ) dx =

 Rd

βε (wt )bt , ∇ϕR  dx+

 Rd

ϕR div bt (βε (wt )−wt βε (wt )) dx.

Splitting b as b1 + b2 , with  |b1 | ∈ L1 [0, T ]; L∞ (Rd ) 1 + |x|

and

 |b2 | ∈ L1 [0, T ]; L1 (Rd ) 1 + |x|

and using the inequality 1 3 χ ≤ χ R {R≤|x|≤2R} 1 + |x| {R≤|x|} we can estimate the first integral in the right hand side with     b1t   |b1t |  dx. 3∇ϕ ∞  |w | dx + 3∇ ϕ  w  ∞ t ∞  1 + |x|  {|x|≥R} t 1 + |x| {|x|≥R} ∞ The second integral can be estimated with

ε

 Rd

ϕR [div bt ]− dx.

Passing to the limit first as ε ↓ 0 and then as R → +∞ and using the integrability assumptions on b and w we get d dt

 Rd

wt+ dx ≤ 0

in the distribution sense in R. Since the function vanishes for negative times, this suffices to conclude using Gronwall lemma. 

DiPerna and Lions proved that all distributional solutions are renormalized when there is a Sobolev regularity with respect to the spatial variables.  1,1 d Theorem 5.5. Let b ∈ L1loc I;Wloc (Rd ; Rd ) and let w ∈ L∞ loc (I × R ) be a distributional solution of (17). Then w is a renormalized solution.

Flow of Weakly Differentiable Vector Fields

23

Proof. We mollify with respect to the spatial variables and we set rε := (b · ∇w) ∗ ρε − b · (∇(w ∗ ρε )), to obtain

wε := w ∗ ρε

d ε w + b · ∇wε = c ∗ ρε − rε . dt

By the smoothness of wε w.r.t. x, the PDE above tells us that dtd wtε ∈ L1loc , therefore 1,1 (I × Rd ) and we can apply the standard chain rule in Sobolev spaces, wε ∈ Wloc getting d β (wε ) + b · ∇β (wε ) = β (wε )c ∗ ρε − β (wε )rε . dt When we let ε ↓ 0, the convergence in the distribution sense of all terms in the identity above is trivial, with the exception of the last one. To ensure its convergence to zero, it seems necessary to show that rε → 0 strongly in L1loc , or at least that rε are equi-integrable (as this would imply their weak L1 convergence to 0; remember also that β (wε ) is locally equibounded w.r.t. ε ). This is indeed the case, and it is exactly here that the Sobolev regularity plays a role. 

∞ d Proposition  5.6 (Strong convergence of commutators). If w ∈ Lloc (I × R ) and 1,1 b ∈ L1loc I;Wloc (Rd ; Rd ) we have

L1loc - lim (b · ∇w) ∗ ρε − b · (∇(w ∗ ρε )) = 0. ε ↓0

Proof. Playing with the definitions of b · ∇w and of the convolution product of a distribution and a smooth function, one proves first the identity rε (t, x) =

 Rd

w(t, x − ε y)

(bt (x − ε y) − bt (x)) · ∇ρ (y) dy − (wdiv bt ) ∗ ρε (x). (22) ε

Introducing the commutators in the (easier) conservative form Rε := (Dx · (bw)) ∗ ρε − Dx · (bwε ) (here we set again wε := w ∗ ρε ) it suffices to show that Rε = Lε − wε div bt , where Lε (t, x) :=

 Rd

w(t, z)(bt (x) − bt (z)) · ∇ρε (z − x) dz.

Indeed, for any test function ϕ , we have that Rε , ϕ  is given by − =−

  I

wb · ∇ρε ∗ ϕ dydt −

   I

  I

ϕ b · ∇ρε ∗ wdxdt −

wt (y)bt (y) · ∇ρε (y − x)ϕ (x) dxdydt

  I

wε ϕ div bt dxdt

24

L. Ambrosio and G. Crippa

− =

   I

  I

bt (x)∇ρε (x − y)wt (y)ϕ (x)dydxdt −

Lε ϕ dxdt −

  I

  I

wε ϕ div bt dxdt

wε ϕ div bt dxdt

(in the last equality we used the fact that ∇ρ is odd). Then, one uses the strong convergence of translations in L p and the strong convergence of the difference quotients (a property that characterizes functions in Sobolev spaces) u(x + ε z) − u(x) → ∇u(x)z ε

1,1 strongly in L1loc , for u ∈ Wloc

to obtain that rε strongly converge in L1loc (I × Rd ) to −w(t, x) The elementary identity

 Rd

∇bt (x)y, ∇ρ (y) dy − w(t, x)div bt (x).  Rd

yi

∂ρ dy = −δi j ∂yj

then shows that the limit is 0 (this can also be derived by the fact that, in any case, the limit of rε in the distribution sense should be 0). 

In this context, given μ¯ = ρ L d with ρ ∈ L1 ∩L∞ , the L -Lagrangian flow starting from μ¯ (at time 0) is defined by the following two properties: (a) X(·, x) is absolutely continuous in [0, T ] and satisfies X(t, x) = x +

t 0

bs (X(s, x)) ds

∀t ∈ [0, T ]

for μ¯ -a.e. x; (b) X(t, ·)# μ¯ ≤ CL d for all t ∈ [0, T ], with C independent of t. Summing up what we obtained so far, the general theory provides us with the following existence and uniqueness result. 1 Theorem 5.7 (Existence  and uniqueness of L -Lagrangian flows). Let b ∈ L 1,1 [0, T ];Wloc (Rd ; Rd ) be satisfying

  |b| ∈ L1 [0, T ]; L1 (Rd ) + L1 [0, T ]; L∞ (Rd ) ; 1 + |x|   (ii) [div bt ]− ∈ L1 [0, T ]; L∞ (Rd ) .

(i)

Then the L -Lagrangian flow relative to b exists and is unique. Proof. By the previous results, the comparison principle holds for the continuity equation relative to b. Therefore the general theory previously developed applies, and Theorem 4.8 provides uniqueness of the L -Lagrangian flow.

Flow of Weakly Differentiable Vector Fields

25

As for the existence, still the general theory (Theorem 4.11) tells us that it can be achieved provided we are able to solve, within L , the continuity equation d w + Dx · (bw) = 0 dt

(23)

for any nonnegative initial datum w0 ∈ L1 ∩ L∞ . The existence of these solutions can be immediately achieved by a smoothing argument: we approximate b in L1loc by smooth bh with a uniform bound in L1 (L∞ ) for [div bth ]− . This bound, in turn, provides a uniform lower bound on JXh and finally a uniform upper bound on wth = (w0 /JXth ) ◦ (Xth )−1 , solving d h w + Dx · (bh wh ) = 0. dt 

Therefore, any weak limit of wh solves (23).

Notice also that, choosing for instance a Gaussian, we obtain that the L Lagrangian flow is well defined up to L d -negligible sets (and independent of μ¯  L d , thanks to Remark 4.9). It is interesting to compare our characterization of Lagrangian flows with the one given in [61]. Heuristically, while the DiPerna–Lions characterization is based on the semigroup of transformations x → X(t, x), our characterization is based on the properties of the map x → X(·, x). Remark 5.8. The definition of the flow in [61] is based on the following three properties:

∂Y (t, s, x) = b (t, Y(t, s, x)) and Y(s, s, x) = x in the distribution sense in (0, T ) × ∂t d R ; (b) The image λt of L d under Y(t, s, ·) satisfies

(a)

1 d L ≤ λt ≤ CL d C

for some constant C > 0;

(c) For all s, s ,t ∈ [0, T ] we have   Y t, s, Y(s, s , x) = Y(t, s , x)

for L d -a.e. x.

Then, Y(t, s, x) corresponds, in our notation, to the flow Xs (t, x) starting at time s (well defined even for t < s if one has two-sided L∞ bounds on the divergence). In our setting condition (c) can be recovered as a consequence of the following argument: assume to fix the ideas that s ≤ s ≤ T and define ⎧

s ⎪ if t ∈ [s , s]; ⎪ ⎨X (t, x) ˜ x) := X(t,  ⎪ ⎪ ⎩X s t, X s (s, x) if t ∈ [s, T ].

26

L. Ambrosio and G. Crippa

˜ x) is an integral solution of the ODE in [s , T ] for It is immediate to check that X(·, ˜ ·)# μ¯ is bounded by C2 L d . Then, Theorem 5.7 (with s as L d -a.e. x and that X(t, ˜ x) = X(·, s , x) in [s , T ] for L d -a.e. x, whence (c) follows. initial time) gives X(·, Let us now discuss the stability properties of L -Lagrangian flows, in the special case when L is defined as in (18). We need the following lemma. Lemma 5.9. Assume that bh : [0, T ] × Rd → Rd satisfy   bh ∈ L1 [0, T ]; L1 (Rd ; Rd ) + L1 [0, T ]; L∞ (Rd ; Rd ) 1 + |x|

(24)

and that we can write bh = b1h + b2h , with  bounded and equi-integrable in L1 [0, T ] × Rd ,

|b1h | 1 + |x|

   |b2 (t, ·)|   h  sup   ∈ L1 (0, T ),   1 + |x| h

(25)

(26)

∞

b1h → b1 ,

b2h → b2

L d+1 -a.e. in (0, T ) × Rd .

(27)

Then, setting b = b + b , we have 1

2

 T

lim

Rd

h→∞ 0

|bh − b| dxdt = 0. 1 + |x|d+2

(28)

Proof. Without loss of generality we can assume that b1 = b2 = 0, hence b = 0. The convergence to 0 of  T |b2h | dxdt d+2 Rd 1 + |x| 0 follows by the standard dominated convergence theorem. The convergence to 0 of  T 0

Rd

|b1h | dxdt 1 + |x|

(and a fortiori of the integrals with the factor 1 + |x|d+2) follows by the Vitali dominated convergence theorem. 

Theorem 5.10. Let Xh be L -Lagrangian flows relative to vector fields bh , starting from μ¯ = ρ¯ L d and satisfying: (a) Xh (t, ·)# μ¯ ≤ CL d , with C independent of h and t ∈ [0, T ]; (b) bh = b1h + b2h with bih satisfying (25), (26) and (27). Assume that the continuity equation with b = b1 + b2 as vector field satisfies the uniqueness property in L . Then there exists a unique L -Lagrangian flow X relative to b and x → Xh (·, x) converge to x → X(·, x) in μ¯ -measure, i.e.

Flow of Weakly Differentiable Vector Fields



lim

h→∞ Rd

27

1 ∧ max |Xh (t, x) − X(t, x)| d μ¯ (x) = 0. [0,T ]

Remark 5.11 (Stability with weak convergence in time). We remark the fact that the hypothesis of strong convergence in both time and space of the vector fields in the stability theorem is not natural in view of the applications to the theory of fluid mechanics (see Theorem II.7 in [61] and [75], in particular Theorem 2.5), and it is in contrast with the general philosophy that time regularity is less important than spatial regolarity (only summability with respect to time is necessary for the renormalization property to hold). For instance, this form of the stability theorem does not include the case of weakly converging vector fields which depend only on the time variable, while it is clear that in this case the convergence of the flows holds. However, as a consequence of the quantitative estimates presented in Sect. 8, it is possible 1,p ) for some p > 1 (and for simto show that, under uniform bounds in L∞ ([0, T ];Wloc ∞ plicity under uniform bounds in L ), the following form of weak convergence with respect to the time is sufficient to get the thesis:  T 0

bh (t, x)η (t) dt −→

 T 0

b(t, x)η (t) dt

in L1loc (Rd ) for every η ∈ Cc∞ (0, T ).

Indeed, fix a parameter ε > 0 and regularize with respect to the spatial variables only using a standard convolution kernel ρε . We can rewrite the difference Xh (t, x)− X(t, x) as   Xh (t, x) − X(t, x) = X h (t, x) − Xεh (t, x) + X εh (t, x) − Xε (t, x)  + X ε (t, x) − X(t, x) , where Xε and Xεh are the flows relative to the spatial regularizations bε and bεh respectively. Now, it is simple to check that • The last term goes to zero with ε , by the stability theorem stated above; • The first term goes to zero with ε , uniformly with respect to h: this is due to the fact that the difference bεh − bh goes to zero in L1loc ([0, T ] × Rd ) uniformly with respect to h (thanks to the uniform control in W 1,p of the vector fields bh ), hence we can apply the quantitative version of the stability theorem (see Theorem 8.12), and we get the desired convergence; • The second term goes to zero for h → ∞ when ε is kept fixed, because we are dealing with flows relative to vector fields which are smooth with respect to the space variable, uniformly in time, and weak convergence with respect to the time is enough to get the stability. Then, in order to conclude, it is enough to let first h → ∞, to eliminate the second term, and then ε → 0. It is not clear to us whether this result extends to the W 1,1 or the BV case: the quantitative estimates are available only for W 1,p vector fields, where p is strictly greater than 1, hence the strategy described before does not extend to that case.

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L. Ambrosio and G. Crippa

Proof of Theorem 5.10. We define η h as the push forward of μ¯ under the map x → (x, X h (·, x)) and argue as in the proof of Theorem 4.4. Step 1. (Tightness of η h ) We claim that the family η h is tight: indeed, by the remarks made after the statement of Theorem 4.4, it suffices to find a coercive functional Ψ : Rd × ΓT → [0, ∞) whose integrals w.r.t. all measures η h are uniformly bounded. Since μ¯ has finite mass we can find a function ϕ : Rd → [0, ∞) such that ϕ ∈ L1 (μ¯ ) and ϕ (x) → ∞ as |x| → ∞. Analogously, we can find a function ψ : Rd → [0, ∞) such that  T ψ (b) dxdt < ∞ 0 Rd 1 + |x| and ψ (x) → ∞ as |x| → ∞. Then, we define Ψ(x, γ ) := ϕ (x) + ϕ (γ (0)) +

 T ψ (γ˙) 0

1 + |γ |

dt

and notice that the coercivity of Ψ follows immediately from Lemma 5.12 below. Then, using assumption (a), we obtain:      T h ψ (X˙ (t, x)) Ψ(x, γ ) d η h = 2ϕ (x) + dt d μ¯ (x) h 0 1 + |X (t, x)| Rd ×ΓT Rd 

 T

ψ (bh (t, X h (t, x))) d μ¯ (x)dt 1 + |Xh (t, x)| 0 Rd Rd   T ψ (bh ) dy dt. ϕ d μ¯ + C ≤2 Rd Rd 1 + |y| 0 =2

ϕ d μ¯ +

Therefore the integrals of Ψ are uniformly bounded. Step 2. (The limit flow belongs to L ) Let now η be a narrow limit point of η h along some subsequence that, for notational simplicity, will not be relabelled. Let us show first that μtη = (et )# η is representable as wt L d with wt belonging to L . Indeed, assumption (a) gives η

μt h = X h (t, ·)# μ¯ = wth L d

with wth ∞ ≤ C.

η

(29)

Therefore, as μtη is the narrow limit of μt h , the same property is preserved in the limit for some w ∈ L∞ . Moreover, the narrow continuity of t → μtη immediately yields the w∗ -continuity of t → wt and this proves that wt ∈ L . Step 3. (η is concentrated on solutions of the ODE) Next we show that η is con∞ d centrated on the class of solutions   of the ODE. Let t¯ ∈ [0, dT ], χ ∈1Cc (R ) with 1 ∞ d 0 ≤ χ ≤ 1, c ∈ L [0, t¯]; L (R ) , with c(t, ·) continuous in R for L -a.e. t ∈ [0, t¯], and define   t¯  ¯  γ (t ) − x − 0 c(s, γ (s)) ds t¯ Φc (x, γ ) := χ (x) . 1 + sup |γ |d+2 [0,t¯]

It is immediate to check that Φct¯ ∈ Cb (Rd × ΓT ), so that

Flow of Weakly Differentiable Vector Fields

 Rd ×ΓT

Φct¯ d η = lim

29



Φct¯ d ηh     t¯   0 bh (s, X h (s, x)) − c(s, X h (s, x)) ds = lim χ (x) d μ¯ (x) h→∞ Rd 1 + sup |Xh (·, x)|d+2 [0,t¯]    t¯  h bh (s, X (s, x)) − c(s, X h (s, x)) ≤ lim sup dsd μ¯ (x) 1 + |Xh (s, x)|d+2 Rd 0 h→∞  t¯  |bh (s, y) − c(s, y)| dsdy ≤ C lim sup d 1 + |y|d+2 0 R h→∞  t¯  |b(s, y) − c(s, y)| dsdy. =C d 1 + |y|d+2 0 R h→∞ Rd ×ΓT

Now, as (by lower semicontinuity)   |b| ∈ L1 [0, T ]; L1 (Rd ) + L∞ [0, T ]; L∞ (Rd ) , 1 + |x| we can find a sequence of vector fields ch (t, x) continuous with respect to x and satisfying the assumptions of Lemma 5.9. Indeed, writing b = b1 + b2 = (f 1 + f 2 )(1 + |x|), with   |f 1 | ∈ L1 [0, T ]; L1 (Rd ) , |f 2 | ∈ L1 [0, T ]; L∞ (Rd ) , we define c = (ch )1 + (ch )2 , with (ch )i := ((1 + |x|)f i ) ∗ ρεh

i = 1, 2

with εh ↓ 0. It is not hard to show that |(ch )i | ≤ 2(1 + |x|)(|f i | ∗ ρεh )

i = 1, 2

if the support of the convolution kernel ρ is contained in the unit ball. Therefore  |(ch )1 | is bounded and equi-integrable in L1 [0, T ] × Rd , 1 + |x|    |(ch )2 (t, ·)|  1  sup   1 + |x|  ∈ L (0, T ). h ∞

(30)

(31)

In the previous estimate we can now choose c = ch and use Lemma 5.9 again to obtain   t¯   ¯  γ (t ) − x − 0 ch (s, γ (s)) ds χ (x) d η = 0. (32) lim h→∞ Rd ×ΓT 1 + sup |γ |d+2 [0,t¯]

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L. Ambrosio and G. Crippa

Now, using the upper bound (29) and Lemma 5.9 once more we get t¯



lim sup

Rd ×ΓT

h→∞

≤ lim sup

0 |ch (s, γ (s)) − b(s, γ (s))| ds 1 + sup |γ |d+2

h→∞

|ch (s, γ (s)) − c(s, γ (s))| d η ds 1 + |γ (s)|d+2 h − b| dxds = 0. 1 + |x|d+2

Rd ×ΓT T |c

0

≤ C lim sup

(33)

[0,t¯]

 T

h→∞

dη

 0

Rd

Hence, from (32) and (33) and Fatou’s lemma we infer that for χη -a.e. (x, γ ) there is a subsequence εi(l) such that    t¯  t¯   ε i(l)  lim γ (t¯) − x − b (s, γ (s)) ds + |bεi(l) (s, γ (s)) − b(s, γ (s)| ds = 0, l→∞ 0 0 so that

γ (t¯) = x +

 t¯ 0

b(s, γ (s)) ds.

Choosing a sequence of cut-off functions χR and letting t vary in Q∩[0, T ] we obtain that (x, γ ) solve the ODE in [0, T ] for η -a.e. (x, γ ). Step 4. (Conclusion) As we are assuming that the uniqueness property holds in L for the continuity equation relative to b, we are now in the position of applying Theorem 4.10, which says that under these conditions necessarily

η = (x, X(·, x))# μ¯ for a suitable map x → X(·, x). Clearly, by the concentration property of η , X(·, x) has to be a solution of the ODE for μ¯ -a.e. x. This proves that X is the L -Lagrangian flow relative to b. The convergence in measure of X h to X follows by a general principle, stated in Lemma 5.13 below. 

Lemma 5.12 (A coercive functional in ΓT ). Let ϕ , ψ : Rd → R and let

Φ (γ ) := ϕ (γ (0)) +

 T ψ (γ˙) 0

1 + |γ |

dt

be defined on the subspace of ΓT made by absolutely continuous maps, and set equal to +∞ outside. If ϕ (x), ψ (x) → +∞ as |x| → ∞ then all sublevel sets {Φ ≤ c}, c ∈ R, are compact in ΓT . Proof. Let γn be such that Φ (γn ) is bounded and notice that necessarily |γn (0)| is bounded, by the assumption on ϕ . By integration of the ODE d γ (t) γ˙(t) ln(1 + |γ (t)|) = · dt |γ (t)| 1 + |γ (t)|

Flow of Weakly Differentiable Vector Fields

31

one obtains that also sup[0,T ] |γn | is uniformly bounded. As a consequence the factor 1/(1 + |γn|) inside the integral part of Φ can be uniformly estimated from below, and therefore (due to the more than linear growth at infinity of ψ ) the sequence |γ˙n | is equi-integrable in L1 (0, T ); Rd . As a consequence the sequence (γn ) is relatively 

compact in ΓT . Lemma 5.13 (Narrow convergence and convergence in measure). Let vh , v : X → Y be Borel maps and let μ¯ ∈ M+ (X). Then vh → v in μ¯ -measure iff (x, vh (x))# μ¯ converges to (x, v(x))# μ¯ narrowly in M+ (X × Y ). Proof. If vh → v in μ¯ -measure then ϕ (x, vh (x)) converges in L1 (μ¯ ) to ϕ (x, v(x)), and we immediately obtain the convergence of the push-forward measures. Conversely, let δ > 0 and, for any ε > 0, let w ∈ Cb (X;Y ) be such that μ¯ ({v = w}) ≤ ε . We define dY (y, w(x)) ϕ (x, y) := 1 ∧ ∈ Cb (X × Y ) δ and notice that

μ¯ ({v = w}) +

 X×Y

 X×Y

ϕ d(x, vh (x))# μ¯ ≥ μ¯ ({dY (v, vh ) > δ }),

ϕ d(x, v(x))# μ¯ ≤ μ¯ ({w = v}).

Taking into account the narrow convergence of the push-forward we obtain that lim sup μ¯ ({dY (v, vh ) > δ }) ≤ 2μ¯ ({w = v}) ≤ 2ε ; h→∞

since ε is arbitrary the proof is achieved.



The renormalization Theorem 5.5 has been extended in [6] to the case of a BV dependence w.r.t. the spatial variables, but still assuming that D · bt  L d for L 1 -a.e. t ∈ (0, T ). (34)   Theorem 5.14. Let b ∈ L1loc (0, T ); BVloc (Rd ; Rd ) be satisfying (34). Then any   d of distributional solution w ∈ L∞ loc (0, T ) × R  d w + Dx · (bw) = c ∈ L1loc (0, T ) × Rd dt is a renormalized solution. A self contained proof of this result, slightly simpler than the one given in the original paper [6], is given in [7] and [8]. The original argument in [6] was indeed based on deep result of G. Alberti [2], saying that for a BVloc function u : Rd → Rm the matrix M(x) in the polar decomposition Du = M|Du| has rank 1 for |Ds u|-a.e. x, i.e. there exist unit vectors ξ (x) ∈ Rd and η (x) ∈ Rm such that

32

L. Ambrosio and G. Crippa

M(x)z = η (x)z, ξ (x). However, we observe that in the application of this result to the Keyfitz–Kranzer system [11, 9] the vector field b is of the form f (ρ ) with ρ scalar and f ∈ C1 vectorial, so the rank-one structure of the distributional derivative (as a whole) is easy to check. Analogously, in the case of the semi-geostrophic system considered in [53], the vector field is a monotone map, and for this class of BV functions a much simpler proof of the rank-one property is available [3]. As in the Sobolev case we can now obtain from the general theory given in Sect. 3 existence and uniqueness of L -Lagrangian flows, with L = L∞ (L1 ) ∩ L∞ (L∞ ): we just replace of Theorem 5.7 the assumption b ∈ in the statement   1,1 1 d d 1 L [0, T ];Wloc (R ; R ) with b ∈ L [0, T ]; BVloc (Rd ; Rd ) , assuming as usual that D · bt  L d for L 1 -a.e. t ∈ [0, T ]. Analogously, by applying Theorem 5.10 we obtain stability of L -Lagrangian flows when b is as in Theorem 5.14.

6 Measure-Theoretic Differentials In this section we introduce some weak differentiability notions, based on measuretheoretic limits of difference quotients, and we compare them (all results of this section are taken from [17]). An important remark is that none of these concepts gives additional informations on the derivative in the sense of distributions. Conversely, whenever the derivative in the sense of distributions is a measure with locally finite variation, the map is L d -a.e. approximately differentiable. We recall that a sequence of measurable maps ( fk ) defined on an open set Ω ⊂ Rd is said to converge locally in measure to a measurable map f if lim L d ({x ∈ K : | fk (x) − f (x)| > ε }) = 0

k→∞

for any compact set K ⊂ A and any ε > 0. The following simple lemma will be used in many occasions. Lemma 6.1. Let fk , f : Ω × Rd → R be measurable functions. Suppose that, for each y ∈ Rd , fk (·, y) → f (·, y) locally in measure in Ω as k → ∞. Then fk → f locally in measure in Ω × Rd as k → ∞. Proof. Let K ⊂ Ω be a compact set. We fix γ > 0 and set gk (y) = L d ({x ∈ K : | fk (x, y) − f (x, y)| ≥ γ }) . Then from the dominated convergence theorem we infer that gk → 0

in L1loc (Rd ).   In particular, Fubini’s theorem gives that L 2d (K × B) ∩ {| fk − f | > γ } → 0 for any ball B ⊂ Rd . 

Flow of Weakly Differentiable Vector Fields

33

As a consequence of the Lusin theorem, it is easy to check that for any measurable function f : Ω → R the functions f (x + h) converge to f locally in measure as h → 0. Therefore it is natural to study the behaviour, still with respect to local convergence in measure, of the difference quotients. This leads to the following definition. Definition 6.2 (Fr´echet and Gˆateaux derivative in measure). Let f : Ω ⊂ Rd → R be a measurable function. We say that g : Ω → Rd is the (Fr´echet) derivative in measure of f if lim

h→0

f (x + h) − f (x) − g(x) · h =0 |h|

locally in measure in Ω.

Analogously, g is called Gˆateaux derivative in measure if the difference quotients above tend to 0 locally in measure in Ω along all lines passing through the origin. Another differentiability condition, that we call directional differentiability in measure, involves an averaging procedure also on the direction. It appeared first in [73] in connection with the differentiability properties of the flow associated to Sobolev vector fields (see the next section) and it can be stated as follows. Definition 6.3 (Directional differentiability in measure). We say that f : Ω → R is directionally differentiable in measure if there exists W : Ω × Rd → R such that f (x + ry) − f (x) − rW(x, y) →0 r

locally in measure in Ω × Rd as h → 0.

The following result surprisingly shows that these three concepts are equivalent. Theorem 6.4. Let f : Ω ⊂ Rd → R be a measurable function. Then the following assertions are equivalent: (i) f is (Fr´echet) differentiable in measure in Ω; (ii) f is Gˆateaux differentiable in measure in Ω; (iii) f is directionally differentiable in measure in Ω. Moreover, the derivative in measure g is linked to the directional derivative in measure W by g(x) · y = W (x, y)

for L 2d -a.e. (x, y) ∈ Ω × Rd .

Proof. We start with some preliminary remarks. It is obvious that if g is the derivative in measure of f then g is also the Gˆateaux derivative in measure of f , and that these derivatives in measure are unique, up to L d -negligible sets. Moreover, Gˆateaux differentiability in measure implies directional differentiability in measure, with W (x, y) = g(x)y: this fact is an immediate consequence of Lemma 6.1. The harder implication is the one from directional differentiability in measure to Fr´echet differentiability in measure. We need the following lemma (see [71, 56]): let f : Rd → R be a measurable function such that

34

L. Ambrosio and G. Crippa

f (y + z) = f (y) + f (z) for L 2d -a.e. (y, z) ∈ Rd × Rd .

Then there exists a ∈ Rd such that f (y) = a · y for L d -a.e. y ∈ Rd . Step 1. Using the above mentioned lemma, we show the “a.e. linearity” of W (x, ·). We define f (x + δ y) − f (x) − δ W (x, y) , δ f (x + δ y + δ z) − f (x + δ y) − δ W(x + δ y, z) Gδ (x, y, z) = , δ − f (x + δ y + δ z) + f (x) + δ W(x, y + z) Hδ (x, y, z) = . δ Fδ (x, y, z) =

Then Fδ → 0 as δ → 0, Gδ → 0 as δ → 0,

(35) (36)

as δ → 0

(37)

Hδ → 0

locally in measure with respect to (x, y, z). Indeed, (35) and (37) are easy applications of Lemma 6.1. To verify (36) we need also the following shift argument. Let Ω

⊂⊂ Ω ⊂⊂ Ω. For small δ we obtain   L 3d (x, y, z) ∈ Ω

: Gδ (x, y, z)) ≥ γ   = L 3d (x, y, z) ∈ Ω

: | f (x+δ y+δ z)− f (x+δ y)−δ W (x+δ y, z)| ≥ γδ   →0 ≤ L 3d (x , y, z) ∈ Ω : | f (x +δ z) − f (x ) − δ W (x , z)| ≥ γδ where we used the change of variables x = x + δ y. This proves (36). From Lemma 6.1 we infer that W (x + δ y, z) − W (x, z) → 0

as δ → 0.

Adding the three identities we obtain that W (x, y + z) − W (x, y) − W (x, z) = 0 for L 3d -a.e. (x, y, z) ∈ Ω × Rd × Rd . Therefore, for L d -a.e. x, W (x, ·) is representable as a linear function. This concludes Step 1. Step 2. Let α d be the measure of the unit ball in Rd . We fix a set Ω ⊂⊂ Ω and ε , γ > 0. We find δ1 > 0 such that B(x, 2δ1 ) ⊂ Ω, Consider the sets

x ∈ Ω .

Flow of Weakly Differentiable Vector Fields

35

Aδ (x) = {y ∈ B(0, 2) : | f (x + δ y) − f (x) − δ g(x) · y| ≥ γδ }, 0 < δ < δ1 , x ∈ Ω . Since (x, y) → g(x) · y is a directional derivative of f in measure, by the Fubini theorem,  L d (Aδ (x)) dx → 0 as δ → 0. Ω

Therefore there exists δ2 > 0 such that for 0 < δ < δ2 we have  α d  L d x ∈ Ω : L d (Aδ (x)) ≥ ≤ ε. 4 By Lusin theorem, there exists a compact set K ⊂ Ω such that L d (Ω \ K) < ε and g|K is continuous. By the uniform continuity, there exists δ3 > 0 such that for 0 < δ < δ3 we have x, x ∈ K, |x − x | < δ

=⇒

|g(x) − g(x )| < γ .

Let h ∈ Rd , |h| = r < 12 min{δ1 , δ2 , δ3 }. Consider the sets E := {x ∈ Ω : L d (Ar (x)) ≥ 14 α d }, E := {x ∈ Ω : L d (Ar (x + h)) ≥ 14 α d }, F := Ω \ K, F := {x ∈ Ω : x + h ∈ / K}. Then

|E ∪ E ∪ F ∪ F | ≤ 4ε .

Let x ∈ Ω \ (E ∪ E ∪ F ∪ F ). Then both x and x + h belong to K and L d (B(0, 1) ∩ Ar (x)) ≤ 14 α d ,   L d {y ∈ B(0, 1) : y − hr ∈ Ar (x + h)} ≤ 14 α d . Thus there exists y ∈ B(0, 1) \ Ar (x) such that y := y − hr ∈ / Ar (x + h). Since |y | < 2, the fact that y ∈ / Ar (x + h) means that x + h + ry ∈ Ω and | f (x + h + ry ) − f (x + h) − rg(x + h) · y | ≤ γ r. Since h + ry = ry, we have | f (x+h)− f (x)−g(x) · h| ≤ | f (x+h)− f (x+h+ry )+rg(x+h) · y | + | f (x+ry)− f (x)−rg(x) · y| + |r(g(x) − g(x + h)) · y | ≤ γ r + γ r + γ r|y | ≤ 4|h|γ .

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L. Ambrosio and G. Crippa

This shows that

f (x + h) − f (x) − g(x) · h →0 |h|

in measure w.r.t. x as h → 0.



Now we introduce another more classical weak differentiability property (extensively studied, for instance, in [68]). It has still a measure-theoretic character, but unlike differentiability in measure it has a pointwise meaning. Definition 6.5 (Approximate differentiability). Let f : Ω → R be a measurable function and let x ∈ Ω. We say that a ∈ Rd is an approximate derivative of f at x if   | f (x + h) − f (x) − a · h| has zero Lebesgue density at 0 for >ε h: any ε > 0. |h| As we are concerned here with convergence in measure, it is worth mentioning that approximate differentiability at x is equivalent to the convergence lim

δ →0

f (x + δ y) − f (x) − δ a · y = 0 locally in measure w.r.t. y ∈ Rd . δ

The following proposition shows that functions which are approximately differentiable on a measurable set A essentially coincide with functions that can be approximated, in the Lusin sense, by Lipschitz maps. Theorem 6.6 (Lusin theorem for approximately differentiable maps). Let f : Ω → R. Assume that there exists a sequence of measurable sets An ⊂ Ω such that L d (Ω \ ∪n An ) = 0 and f |An is Lipschitz for any n. Then f is approximately differentiable at L d -a.e. x ∈ Ω. Conversely, if f is approximately differentiable at all points of Ω ⊂ Ω, we can write Ω as a countable union of sets An such that f |An is Lipschitz for any n (up to a redefinition in a L d -negligible set). Proof. With no loss of generality we can assume that the sets An are pairwise disjoint. By Mc Shane Lipschitz extension theorem we can find Lipschitz functions gn : Rd → R extending f |An . By Lebesgue differentiation theorem and Rademacher theorem, L d -a.e. x ∈ An is both a point of density 1 of An and a differentiability point of gn . We claim that at any of these points x the function f is approximately differentiable, with approximate differentiable equal to ∇gn (x). Indeed, it suffices to notice that the difference quotients of f and of gn may differ only on Ω \ An, that has zero density at x. Clearly, as L d -a.e. x ∈ Ω has this property for some n, so this proves the L d -a.e. approximate differentiability of f . The converse statement is proved in Theorem 3.1.16 of [68]. 

The following theorem shows that, among all the differentiability properties considered in this section, L d -a.e. approximate differentiability is the stronger one.

Flow of Weakly Differentiable Vector Fields

37

Even on the real line, an example built in [17] shows that differentiability in measure does not imply L 1 -a.e. approximate differentiability (in fact, the function built in [17] is nowhere approximately differentiable). Theorem 6.7. Suppose that f : Ω → R is approximately differentiable L d -a.e. in Ω. Then f is differentiable in measure. Proof. Let g be the approximate derivative of f . By the previous result there exist Lipschitz functions f j and pairwise disjoint measurable sets A j ⊂ Ω such that f j = f on A j , ∇ f j = g on A j ,

  d L Ω \ A j = 0. j

We fix Ω ⊂⊂ Ω and γ > 0 and find δ > 0 such that B(x, δ ) ⊂ Ω for each x ∈ Ω . We set E(h) := {x ∈ Ω : | f (x + h) − f (x) − g(x)h| ≥ γ |h|}, Then E(h) ⊂ N ∪

h ∈ B(0, δ ).



(E j (h) ∪ Fj (h)),

j

where L d (N) = 0 and   E j (h) = x ∈ Ω ∩ A j : x + h ∈ / Aj ,  Fj (h) = x ∈ Ω ∩ A j : | f j (x + h) − f j (x) − ∇ f j (x)h| ≥ γ |h|}. Then L d (E j (h)) → 0 as h → 0 by the L1loc continuity of translations. Using the differentiability of f j on A j , we obtain that also L d (Fj (h)) → 0 as h → 0. Choose ε > 0. Since E j (h) ∪ Fj (h) ⊂ A j and ∑ j L d (A j ) ≤ L d (Ω) < ∞, we can find an index m independent of h such that ∞

∑

L d (E j (h) ∪ Fj (h)) ≤ ε .

j=m+1

Then lim sup L d (E(h)) ≤ h→0

m

∑ limh→0sup L d (E j (h) ∪ Fj (h)) + ε = ε .

j=1

It follows that L d (E(h)) → 0 as required.



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L. Ambrosio and G. Crippa

7 Differentiability of the Flow in the W 1,1 Case In this section we discuss the differentiability properties of the L -Lagrangian flow associated to W 1,1 vector fields, briefly describing the results obtained in [73]. Notice that no differentiability property is presently known in the BV case; on the other hand, in the W 1,p case, with p > 1, much stronger results are available [48], and we will present them in the next section. The main theorem in [73] is the following:  1,1 (Rd ; Rd ) be satisfying Theorem 7.1. Let b ∈ L1 [0, T ];Wloc   |b| ∈ L1 [0, T ]; L1 (Rd ) + L1 [0, T ]; L∞ (Rd ) ; 1 + |x|   (ii) [div bt ] ∈ L1 [0, T ]; L∞ (Rd ) ;

(i)

and let X(t, x) be the corresponding L -Lagrangian flow, given by Theorem 5.7. Then for all t ∈ [0, T ] there exists a measurable function Wt : Rd × Rd → Rd such that X(t, x + ε y) − X(t, x) − ε Wt (x, y) →0 ε

locally in measure in Rdx × Rdy

as ε ↓ 0. The result actually stated in [73] is slightly stronger, as the convergence above is also shown to be uniform with respect to time. Having in mind the terminology and the results of the previous section, the result can also be rephrased as follows: for all t ∈ [0, T ] there exists a matrix-valued measurable function Gt : Rd → M d×d (i.e. the derivative in measure) such that X(t, x + h) − X(t, x) − Gt (x)h →0 |h|

locally in measure in Rdx as h → 0.

The link between Gt and Wt is given by Gt (x)(y) = Wt (x, y) for L 2d -a.e. (x, y). So, Gt can be interpreted as the derivative of the flow map X(t, ·). The strategy of the proof in [73] is to look at the behaviour of the 2d-dimensional flows Y ε arising from the difference quotients of X: Y ε (t, x, y) :=

X(t, x + ε y) − X(t, x) . ε

It is immediate to check (see also [15]) that Y ε are L -Lagrangian flows relative to the vector fields Bε defined by

 b(t, x + ε y) − b(t, x) ε B (t, x, y) = b(t, x), . ε Therefore, it is natural to expect that their limit Y (if any) should be a flow relative to the limit vector field

Flow of Weakly Differentiable Vector Fields

39

B(t, x, y) = (b(t, x), ∇x b(t, x)y) . Notice that div Bt (x, y) = 2divbt (x), and therefore  div Bt ∈ L1 [0, T ]; L∞ (R2d ) . On the other hand, the last d components of B have no regularity with respect to the x-variable. However, in this special case, an anisotropic smoothing argument (by a regularization in the x variable much faster than the one in the y variable), based on the special structure of this vector field (see also [25, 74]), still guarantees that bounded solutions to the continuity equation with velocity field B are renormalizable. Moreover, using the renormalization property, a variant of the argument used in Theorem 5.4 shows that the continuity equation with B as vector field satisfies the comparison principle in the class  d 1 d L ∗ := L ∩ L∞ [0, T ]; L∞ loc (Rx ; L (Ry )) , where L is defined as in (18) (in 2d space dimensions). The reason for this restriction to the smaller space L ∗ is the fact that |B|/(1 + |x| + |y|) in general does not belong to   L1 [0, T ]; L1 (Rdx × Rdy ) + L1 [0, T ]; L∞ (Rdx × Rdy ) , because the last d components do not tend to 0 as |y| → ∞ while x is kept fixed (and their limit is possibly unbounded as a function of x). If b satisfies condition (ii) above, the last d components B2 of B satisfy instead  |B2 | ∈ L1 [0, T ]; L1loc (Rdx , L1 (Rdy ) + L∞ (Rdy )) . 1 + |y| For this reason, a weaker growth condition on B turns into a stronger growth condition on w. Then, the renormalization property ensures the well posedness of the continuity equation and therefore (much like as in Theorem 4.8 and Theorem 5.4) that the L ∗ -Lagrangian flow relative to B is unique. A smoothing argument (see [73] for details) proves its existence even within the smaller class L ∗ . So, denoting by Y the L ∗ -Lagrangian flow relative to B, we can represent it as Y(t, x, y) = (X(t, x), W(t, x, y)) for some map W. Finally, the same argument used for the existence of Y shows that Y is the limit, with respect to local convergence in measure (uniform w.r.t. time) of Y ε : this is due to the fact that Bε have properties analogous to the ones of B (in particular they have uniformly bounded divergences) and converge to B. This leads to the proof of Theorem 7.1.

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L. Ambrosio and G. Crippa

8 Differentiability and Compactness of the Flow in the W 1,p Case In this section we present some recent results, obtained in [48], relative to the approximate differentiability and to the Lipschitz properties of regular Lagrangian flows associated to W 1,p vector fields, with p > 1. The first results relative to the approximative differentiability of the flow have been obtained in [15]. An important fact that we want to remark from the beginning is that the approach of [15] and [48] is completely different from the one of [73] described in the previous section: these two papers are not based on the theory of renormalized solutions, but new kind of estimates are introduced. The general idea, which we are going to explain with all the details in the following, is trying to find estimates for the spatial gradient of the flow in terms of bounds on the derivative of the vector field. We start by recalling some basic facts about the theory of maximal functions. These tools will be used throughout all this section. We begin with the definition of the local maximal function. Definition 8.1 (Maximal function). Let μ be a (vector-valued) measure with locally finite total variation. For every λ > 0, we define the (λ -local) maximal function of μ as |μ |(Br (x)) Mλ μ (x) = sup x ∈ Rd . d (B (x)) L r 0 1; however, this estimate is false in the case p = 1. The second lemma will be used to estimate the difference quotients of the vector field by means of the maximal function of the spatial derivative of the vector field. Lemma 8.2. The local maximal function of μ is finite for L d -a.e. x ∈ Rd and, for every R > 0, we have  BR (0)

Mλ f (y) dy ≤ cd,R + cd

 BR+λ (0)

| f (y)| log(2 + | f (y)|) dy ∀λ > 0.

For p > 1 we have  BR (0)

(Mλ f (y)) p dy ≤ cd,p

 BR+λ (0)

| f (y)| p dy

∀λ > 0.

Lemma 8.3. If u ∈ BV (Rd ; Rm ) then there exists an L d -negligible set N ⊂ Rd such that

Flow of Weakly Differentiable Vector Fields

41

|u(x) − u(y)| ≤ cd |x − y| (Mλ Du(x) + Mλ Du(y)) for x, y ∈ Rd \ N with |x − y| ≤ λ . We also recall Chebyshev inequality for a measurable function f : Ω ⊂ Rd → R:  1/q   1 L d {| f | > t} L {| f | > t} ≤  f L p (Ω) , | f (x)| dx ≤ t {| f |>t} t d

which immediately implies  1/p  f L p (Ω) L d {| f | > t} . ≤ t

(38)

In all this section we will make the following assumptions on the vector field b : [0, T ] × Rd → Rd : (A) b ∈ L1 ([0, T ];W 1,p (Rd ; Rd )) for some p > 1; (B) b ∈ L∞ ([0, T ] × Rd ; Rd ); (C) [divx b]− ∈ L1 ([0, T ]; L∞ (Rd )). We remark that condition (B) could be relaxed to |b(t, x)| ∈ L1 ([0, T ]; L1 (Rd )) + L1 ([0, T ]; L∞ (Rd )), 1 + |x| getting some slightly weaker results (due to the possible unboundedness of the velocity field), but in this introductory presentation we prefer to avoid these technicalities and focus on the case of a uniformly bounded vector field. We also notice that, under this assumption, the global W 1,p hypothesis is not essential: thanks to the finite speed of propagation, we could truncate our vector field out of a ball, then 1,p hypothesis. However, we prefer to assume getting the same results with just a Wloc this global condition, mainly in order to simplify typographically some estimates. We refer to [48] for these more general hypotheses, the main modification being an estimate of the superlevels of the flow. We recall that in this context the results of Sect. 5 read as follow. For every vector field b satisfying assumptions (A), (B) and (C) there exists a unique regular Lagrangian flow X, that is a measurable map X : [0, T ] × Rd → Rd which satisfies the following two properties: (i) For L d -a.e. x ∈ Rd the map t → X(t, x) is an absolutely continuous solution of γ˙(t) = b(t, γ (t)) with γ (0) = x (the ODE is fulfilled in the integral sense or, equivalently, in the a.e. sense); (ii) There exists a constant C independent of t such that X(t, ·)# L d ≤ CL d for every t ∈ [0, T ]. For every vector field satisfying assumption (C), the quantity

 T  C = exp [divx b(t, ·)]− L∞ (Rd ) dt 0

42

L. Ambrosio and G. Crippa

satisfies the inequality in the second property of the regular Lagrangian flow. However, we remark the fact that all our estimates will depend on the compressibility constant L of X, i.e. the best constant C for which (ii) holds, rather than on the L1 (L∞ ) norm of [divx b]− . This is an important remark in the context of the compactness theorem (see Theorem 8.7): some compactness results under bounds on the divergence were already available in [61], while one of the merits of this new approach is this weaker requirement. The starting point of the estimates given in [48] is already present in [15] (and, at least in a formal way, in [61]): in a smooth context, we can control the time derivative dtd log (|∇X|) with |∇b|(X). The strategy of [15] allows to make this remark rigorous: it is possible to consider some integral quantities which contain a discretization of the space gradient of the flow and to prove some estimates along the flow. Then, the application of Egorov theorem allows the passage from integral estimates to pointwise estimates on big sets, and from this it is possible to recover Lipschitz regularity on big sets. However, the application of Egorov theorem implies a loss of quantitative informations: this strategy does not allow a control of the Lipschitz constant in terms of the size of the “neglected” set. Starting from this results, the main point of [48] is a modification of the estimates in such a way that quantitative informations are not lost. We define (for p > 1 and R > 0) the following integral quantity:



sup sup A p (R, X) := BR (0)

0≤t≤T 0
 Br (x)

log



  p 1/p |X(t, x) − X(t, y)| + 1 dy dx . r

(39)

Heuristically, one can view |X(t, x) − X(t, y)|/r as a “discrete gradient” of the flow X. Then the quantity A p (R, X) is constructed averaging this discrete gradient over balls of radius r, asking some uniformity with respect to t and r and finally integrating on a bounded set with respect to the second variable. We are now going to give some quantitative estimates of the quantity A p (R, X) in the case when the map X is the regular Lagrangian flow associated to a vector field b satisfying assumptions (A), (B) and (C). The estimate will depend only on the L1 (L p ) norm of the derivative of b and on the compressibility constant L. In all the following computations we will denote by cq1 ,...,qn universal constants which depend only on the parameters q1 , . . . , qn and which can change from line to line. To simplify the notation we will also denote by L p and L1 (L p ) the global spaces L p (Rd ) and L1 ([0, T ]; L p (Rd )), respectively. Out of the smooth setting, all the computations are easily justified by condition (i) in the definition of regular Lagrangian flow; however, the reader could check the estimates in the smooth case, and then obtain the general case simply by an approximation procedure (based on Theorem 5.10). Proposition 8.4. Let b be a vector field satisfying assumptions (A), (B) and (C). Denote by X its regular Lagrangian flow and let L be the compressibility constant of the flow. Then we have

Flow of Weakly Differentiable Vector Fields

43





A p (R, X) ≤ C R, L, Dx bL1 (L p ) . Proof. For 0 ≤ t ≤ T , 0 < r < 2R and x ∈ BR (0) define 

 |X(t, x) − X(t, y)| Q(t, x, r) := + 1 dy. log r Br (x) With some easy computations we get      dX dQ dX  (t, x, r) ≤ (t, x) − (t, y) (|X(t, x) − X(t, y)| + r)−1 dy  dt dt dt Br (x) =



|b(t, X(t, x)) − b(t, X(t, y))| dy. |X(t, x) − X(t, y)| + r Br (x)

(40)

˜ We now set R˜ = 4R + 2T b∞ . Since we clearly have |X(t, x) − X(t, y)| ≤ R, applying Lemma 8.3 we can estimate dQ (t, x, r) dt ≤ cd



Br (x)

(MR˜ Db(t, X(t, x)) + MR˜ Db(t, X(t, y)))

≤ cd MR˜ Db(t, X(t, x)) + cd

 Br (x)

|X(t, x) − X(t, y)| dy |X(t, x) − X(t, y)| + r

MR˜ Db(t, X(t, y)) dy.

Integrating with respect to the time, we estimate sup Q(t, x, r) ≤ c + cd 0≤t≤T

 T 0

MR˜ Db(t, X(t, x)) dt + cd

 T 0

Br (x)

MR˜ Db(t, X(t, y)) dydt. (41)

Passing to the supremum for 0 < r < 2R we obtain sup

sup Q(t, x, r)

0≤t≤T 0
≤ c + cd

0

MR˜ Db(t, X(t, x)) dt + cd

 T



sup

Br (x)

0 0
MR˜ Db(t, X(t, y)) dydt.

Taking the L p norm over BR (0) we get   

   |X(t, x) − X(t, y)|  + 1 dy log A p (R, X) =  sup sup  p r 0≤t≤T 0
 T  + cd   0 sup

0
p

Lx (BR (0))

  MR˜ Db(t, X(t, y)) dydt   B (x)



r

p

Lx (BR (0))

.

44

L. Ambrosio and G. Crippa

Recalling Lemma 8.2 and the definition of the compressibility constant L, the first integral can be estimated with  T

cd

0

MR˜ Db(t, X(t, x))Lxp (BR (0)) dt ≤ cd L1/p

 T 0

≤ cd,p L1/p

MR˜ Db(t, x)Lxp (BR+T b

 T 0

Db(t, x)Lxp (B

The second integral can be estimated in a similar way with   T     sup  cd [(M Db) ◦ (t, X(t, ·))] (y) dy R˜   0

= cd

 T

≤ cd,p = cd,p

0

0
Br (x)

∞ (0))

˜ R+R+T b∞ (0))

dt dt.

dt p

Lx (BR (0))

M2R [(MR˜ Db) ◦ (t, X(t, ·))] (x)Lxp (BR (0)) dt

 T 0

[(MR˜ Db) ◦ (t, X(t, ·))] (x)Lxp (B3R (0)) dt

0

(MR˜ Db) ◦ (t, X(t, x))Lxp (B3R (0)) dt

 T

≤ cd,p L1/p ≤ cd,p L1/p

 T 0

 T 0

MR˜ Db(t, x)Lxp (B3R+T b Db(t, x)Lxp (B

∞ (0))

3R+T b∞ +R˜ (0))

dt

dt.

Then we obtain the desired estimate for A p (R, X).



This estimate implies in an easy way a Lusin-type approximation of the flow with Lipschitz maps. As we observed before the proof of the proposition, the main point is that the estimate is now quantitative, and this will imply a precise control of the Lipschitz constant, in terms of the measure of the set we are “neglecting” (this explicit control was one of the open problems stated in [15]). This will be the key point in the proof of the compactness theorem. Theorem 8.5 (Lipschitz estimates). Let b be a vector field satisfying assumptions (A), (B) and (C) and denote by X its regular Lagrangian flow. Then, for every ε , R > 0, we can find a set K ⊂ BR (0) such that • L d (BR (0) \ K) ≤ ε ; • For any 0 ≤ t ≤ T we have Lip (X(t, ·)|K ) ≤ exp

cd A p (R, X) , ε 1/p

with A p (R, X) satisfying the estimate of Proposition 8.4. Proof. Fix ε > 0 and R > 0. We apply Proposition 8.4 and equation (38) to obtain a constant

Flow of Weakly Differentiable Vector Fields

45

M = M(ε , p, A p (R, X)) =

A p (R, X) ε 1/p

and a set K ⊂ BR (0) with L d (BR (0) \ K) ≤ ε and 

 |X(t, x) − X(t, y)| + 1 dy ≤ M log sup sup r 0≤t≤T 0
∀x ∈ K.

This clearly means that

 Br (x)

log

 |X(t, x) − X(t, y)| + 1 dy ≤ M r

for every x ∈ K, t ∈ [0, T ] and r ∈]0, 2R[. Now fix x, y ∈ K. Clearly |x − y| < 2R. Set r = |x − y| and compute

 

 |X(t, x) − X(t, y)| |X(t, x) − X(t, y)| log +1 = + 1 dz log r r Br (x)∩Br (y)



  |X(t, x) − X(t, z)| |X(t, y) − X(t, z)| ≤ log + 1 + log + 1 dz r r Br (x)∩Br (y) 

 |X(t, x) − X(t, z)| + 1 dz ≤ cd log r Br (x) 

 |X(t, y) − X(t, z)| + cd + 1 dz log r Br (y) cd A p (R, X) ≤ cd M = . ε 1/p This implies that

 cd A p (R, X) |X(t, x) − X(t, y)| ≤ exp |x − y| ε 1/p Therefore Lip(X(t, ·)|K ) ≤ exp

for every x, y ∈ K.

cd A p (R, X) . ε 1/p



Recalling Theorem 6.6, the following corollary is an immediate consequence of the Lipschitz estimates. Corollary 8.6 (Approximate differentiability of the flow). Let b be a vector field satisfying assumptions (A), (B) and (C) and denote by X its regular Lagrangian flow. Then X(t, ·) is approximately differentiable L d -a.e. in Rd , for every t ∈ [0, T ]. Now we are going to present some new compactness results. The strategy of the proof is quite elementary: thanks to the Lipschitz estimates, the flows will be equicontinuous on large sets and this, together with the equi-boundedness in L∞ , will

46

L. Ambrosio and G. Crippa

allow the use of Ascoli–Arzel`a theorem “on big sets”. From this, the compactness in L1loc easily follows. The following theorem gives a partial answer to a conjecture raised by Alberto Bressan in [34]: in fact, Bressan’s conjecture is the L1 version of the L p result that we are going to present. Presently this conjecture is still unsolved. Theorem 8.7 (Compactness of the flow). Let bn be a sequence of smooth vector fields. Denote by Xn their regular Lagrangian flows and let Ln be the compressibility constant of the flow Xn . Suppose that • |bn | are equi-bounded in L∞ ([0, T ] × Rd ), • |Dx bn | are equi-bounded in L1 (L p ) for some p > 1, • {Ln }n is a bounded sequence (in R). Then the sequence {Xn }n is relatively compact in L1loc ([0, T ] × Rn). Proof. Fix R > 0 and notice that, since the vector fields {bn }n are uniformly bounded in L∞ , the flows {Xn }n are uniformly bounded in L∞ ([0, T ] × BR (0)); denote by CR a common bound in L∞ for the flows. Applying Proposition 8.4 we obtain that the quantities A p (R, Xn ) are uniformly bounded with respect to n. Now, fix j ∈ N and apply Theorem 8.5 with ε = 1/ j to obtain, for every n ∈ N, a set Kn ⊂ BR (0) with L d (BR (0) \ Kn ) ≤ 1/ j such that Lip(X n (t, ·)|Kn )

is uniformly bounded w.r.t. n uniformly in t.

Since dtd Xn (t, x) = bn (t, X n (t, x)), thanks to the equi-boundedness in L∞ of the vector fields we also get that Lip(Xn |[0,T ]×Kn )

is uniformly bounded w.r.t. n.

Now, by applying some classical results on the extension of Lipschitz maps, we can extend every Xn |[0,T ]×Kn to a map X˜ n, j defined on [0, T ] × BR(0) in such a way that Lip(X˜ n, j |[0,T ]×BR (0) ) ≤ cd Lip(Xn |[0,T ]×Kn )

(42)

X˜ n, j L∞ ([0,T ]×BR (0)) ≤ Xn L∞ ([0,T ]×BR (0)) .

(43)

and Applying Ascoli–Arzel`a theorem and a standard diagonal argument we can find a subsequence (not relabeled) such that for each j the sequence {X˜ n, j }n converges uniformly (and hence strongly in L1 ([0, T ] × BR(0))) to a map X˜ ∞, j . Notice that 2 X˜ n, j − Xn L1 ([0,T ]×BR (0)) ≤ T L d (Bρ (0))CR . j Next, for any given ε > 0 select j such that 2 T L d (BR (0))CR ≤ ε /3, j

Flow of Weakly Differentiable Vector Fields

47

and then N > 0 such that X˜ i, j − X˜ k, j L1 ([0,T ]×BR (0)) ≤ ε /3

for all i, k > N.

Hence for every i, k > N we get ˜ i, j − X˜ k, j L1 ([0,T ]×B (0)) Xi − Xk L1 ([0,T ]×BR (0)) ≤ Xi − X˜ i, j L1 ([0,T ]×BR (0)) + X R + Xk − X˜ k, j L1 ([0,T ]×B (0)) ≤ ε . R

Hence {Xn }n is a Cauchy sequence in L1 ([0, T ] × BR (0)). A second diagonal argument yields a subsequence, not relabeled, which is a Cauchy sequence in L1 ([0, T ] × Bl (0)) for every l ∈ N. 

Remark 8.8 (Existence of the regular Lagrangian flow). As a corollary we obtain a new proof of the existence of a regular Lagrangian flow associated to a bounded vector field b satisfying assumptions (A), (B) and (C). Indeed, approximating b by convolution with a positive convolution kernel, we get a sequence which satisfies the assumptions of the previous theorem, hence in the limit we get a flow associated to b, thanks to the compactness in the strong topology. An analogous remark applies to Theorem 8.11. With similar techniques it is also possible to show directly compactness in the W 1,1 case, under the assumption that the maximal functions of the derivatives of the vector fields bn are uniformly bounded in L1 (L1 ). We start defining an integral quantity similar to the previous one, but without the supremum with respect to the radius r. For R > 0 and 0 < r < R/2 fixed we set a(r, R, X) =



 |X(t, x) − X(t, y)| + 1 dydx. log r Br (x)



sup

BR (0) 0≤t≤T



We first give a quantitative estimate for the quantity a(r, R, X), similar to the previous one for A p (R, X). Proposition 8.9. Let b be a vector field satisfying the assumptions (A) and (C) and such that Mλ Db ∈ L1 (L1 )

for every λ > 0.

Denote by X its regular Lagrangian flow and let L be the compressibility constant of the flow. Then we have  a(r, R, X) ≤ C R, L, MR˜ Dx bL1 (L1 ) , where R˜ = 3R/2 + 2Tb∞ .

48

L. Ambrosio and G. Crippa

Proof. We start as in the proof of Proposition 8.4, obtaining inequality (41) (but this time it is sufficient to set R˜ = 3R/2 + 2Tb∞ ). Integrating with respect to x over BR (0), we obtain

   |X(t, x) − X(t, y)| sup log a(r, R, X) = + 1 dydx r BR (0) 0≤t≤T Br (x) ≤ cR + cd + cd





 T

BR (0) 0  T 

BR (0) 0

MR˜ Db(t, X(t, x)) dtdx Br (x)

MR˜ Db(t, X(t, y)) dydtdx.

As in the previous computations, the first integral can be estimated with cd LMR˜ DbL1 ([0,T ];L1 (BR+T b

∞ (0)))

,

but this time we cannot bound the norm of the maximal function with the norm of the derivative. To estimate the last integral we compute 

cd = cd ≤ cd ≤ cd



 T  BR (0) 0

Br (x)

BR (0) 0

Br (0)

Br (0) 0

BR (0)

 T 

 

 T  T

Br (0) 0

MR˜ Db(t, X(t, y)) dydtdx MR˜ Db(t, X(t, x + z)) dzdtdx MR˜ Db(t, X(t, x + z)) dxdtdz



L

B3R/2+T b∞ (0)

MR˜ Db(t, w) dwdtdx

= cd LMR˜ DbL1 ([0,T ];L1 (B3R/2+T b

∞ (0)))

.

This concludes the proof of the estimate for a(r, R, X).



We recall the following classical criterion for strong compactness in L p , since it will be used in the proof of the compactness theorem. For the proof of the lemma we refer for example to [35]. Lemma 8.10 (Riesz–Fr´echet–Kolmogorov compactness criterion). Let F be a bounded subset of L p (RN ) for some 1 ≤ p < ∞. Suppose that lim  f (· − h) − f L p = 0

|h|→0

uniformly in f ∈ F .

p (RN ). Then F is relatively compact in Lloc

Theorem 8.11. Let bn be a sequence of smooth vector fields. Denote by Xn their regular Lagrangian flows and let Ln be the compressibility constant of the flow Xn . Suppose that

Flow of Weakly Differentiable Vector Fields

49

• |bn | are equi-bounded in L∞ ([0, T ] × Rd ), • Mλ Dx bn are equi-bounded in L1 (L1 ) for every λ > 0, • {Ln }n is a bounded sequence (in R). Then the sequence {Xn }n is relatively compact in L1loc ([0, T ] × Rd ). Proof. We apply Proposition 8.9 to obtain that, under the assumptions of the Theorem, the quantities a(r, R, X n ) are uniformly bounded with respect to n. Now observe that, for 0 ≤ z ≤ R¯ (with R¯ = 3R/2 + 2Tb∞ as in Proposition 8.9), thanks to the concavity of the logarithm we have  log Rr¯ + 1 z z. log + 1 ≥ R¯ r Since |X n (t, x) − Xn (t, y)| ≤ R¯ this implies that 



sup

BR (0) 0≤t≤T



≤ log

R¯ R¯ r

+1

Br (x)

|Xn (t, x) − Xn (t, y)| dydx

 C R, Ln , MR¯ Dx bn L1 (L1 ) ≤ g(r),

where the function g(r) does not depend on n and satisfies g(r) ↓ 0 for r ↓ 0. Changing the integration order this implies 

 Br (0) BR (0)

|X n (t, x) − Xn (t, x + z)| dxdz ≤ g(r),

uniformly with respect to t and n. Now notice the following elementary fact. There exists a dimensional constant αd > 0 with the following property: if A ⊂ B1 (0) is a measurable set with L d (B1 (0) \ A) ≤ αd , then A + A ⊃ B1/2 (0). Then fix αd as above and apply Chebyshev inequality for every n to obtain, for every 0 < r < R/2, a measurable set Kr,n ⊂ Br (0) with L d (Br (0) \ Kr,n) ≤ αd L d (Br (0)) and  BR (0)

|Xn (t, x + z) − Xn (t, x)| dx ≤

g(r) αd

for every z ∈ Kr,n .

For such a set Kr,n , thanks to the previous remark, we have that Kr,n + Kr,n ⊃ Br/2 (0). Now let v ∈ Br/2 (0) be arbitrary. For every n we can write v = z1,n + z2,n with z1,n , z2,n ∈ Kr,n . We can estimate the increment in the spatial directions as follows: 

=



BR/2 (0) BR/2 (0)

|X n (t, x + v) − Xn (t, x)| dx |X n (t, x + z1,n + z2,n ) − Xn (t, x)| dx

50

L. Ambrosio and G. Crippa

≤ ≤ ≤

 

BR/2 (0) BR (0)

|X n (t, x + z1,n + z2,n ) − Xn (t, x + z1,n )| + |Xn (t, x + z1,n ) − Xn (t, x)| dx

|X n (t, y + z2,n) − Xn (t, y)| dy +

 BR (0)

|Xn (t, x + z1,n ) − Xn (t, x)| dx

2g(r) . αd

Now notice that, by definition of regular Lagrangian flow, we have dXn (t, x) = bn (t, X n (t, x)). dt Then we can estimate the increment in the time direction in the following way   h   dX n  (t + s, x) ds |X n (t + h, x) − Xn (t, x)| ≤  dt 0 =

 h 0

|bn (t + s, Xn (t + s, x))| ds ≤ hbn ∞ .

Combining these two informations, for (t0 ,t1 ) ⊂⊂ [0, T ], R > 0, v ∈ Br/2 (0) and h > 0 sufficiently small we can estimate  t1 

≤

t0

BR/2 (0)

t0

BR/2 (0)

 t1 

≤T

2g(r) + αd

|X n (t + h, x + v) − Xn (t, x)| dxdt |X n (t + h, x + v) − Xn (t + h, x)| + |Xn (t + h, x) − Xn (t, x)| dxdt

 t1  t0

BR/2 (0)

hbn ∞ dxdt ≤ T

2g(r) + cd T Rd hbn . αd

The thesis follows applying the Riesz–Fr´echet–Kolmogorov compactness criterion, recalling that bn are uniformly bounded in L∞ . 

We conclude this section showing another result obtained in [48] with techniques which are very similar to the ones described so far. It is a result of quantitative stability for regular Lagrangian flows: it improves the stability result given in Theorem 5.10 in the sense that it gives a rate of convergence of the flows in terms of convergence of the vector fields. Theorem 8.12 (Quantitative stability). Let b and b˜ be vector fields satisfying assumptions (A), (B) and (C). Denote by X and X˜ the respective regular Lagrangian flows and let L and L˜ be the compressibility constants of the two flows. Then, for every time τ ∈ [0, T ], we have   −1  ˜ 1 ˜ τ , ·)L1 (B (0)) ≤ C log b − b X(τ , ·) − X( , L ([0,τ ]×BR (0))  r

Flow of Weakly Differentiable Vector Fields

51

˜ ∞ , L, L, ˜ where R = r + T b∞ and the constant C only depends on T , r, b∞ , b and Dx bL1 (L p ) . ˜ 1 Proof. Set δ := b − b L ([0,T ]×BR (0)) and consider the function  ˜ x)| |X(t, x) − X(t, g(t) := log + 1 dx. δ Br (0) 



Clearly g(0) = 0 and after some standard computations we get    ˜ x)    dX(t, x) d X(t,   |X(t, x) − X(t,  ˜ x)| + δ −1 dx g (t) ≤ −   dt dt Br (0)  ˜ X(t, ˜ x))| |b(t, X(t, x)) − b(t, = dx ˜ x)| + δ |X(t, x) − X(t, Br (0) ≤

1 δ +





Br (0)

˜ X(t, ˜ x)) − b(t, ˜ x))| dx |b(t, X(t,

˜ x))| |b(t, X(t, x)) − b(t, X(t, dx. ˜ x)| + δ |X(t, x) − X(t, Br (0)

(44)

˜ ∞ ) and we apply Lemma 8.3 to estimate the last We set R˜ = 2r + T (b∞ + b integral as follows: 

˜ x))| |b(t, X(t, x)) − b(t, X(t, dx ˜ |X(t, x) − X(t, x)| + δ Br (0) ≤ cd



Br (0)

˜ x)) dx. MR˜ Db(t, X(t, x)) + MR˜ Db(t, X(t,

˜ ∞ }, changing variInserting this estimate in (44), setting r˜ = r + T max{b∞ , b ables in the integrals and using Lemma 8.2 we get g (t) ≤ ≤ ≤

   L˜ ˜ y)| dy + L˜ + L |b(t, y) − b(t, MR˜ Db(t, y) dy δ Br+T b Br˜ (0) ˜ ∞ (0)

   L˜ ˜ y)| dy + cd r˜n−n/p L˜ + L M ˜ Db(t, ·) p |b(t, y) − b(t, R Lx δ Br+T b ˜ ∞ (0)    L˜ ˜ y)| dy + cd,pr˜n−n/p L˜ + L Db(t, ·) p . |b(t, y) − b(t, Lx δ Br+T b ˜ ∞ (0)

For any τ ∈ [0, T ], integrating the last inequality between 0 and τ we get

  ˜ τ , x)| |X(τ , x) − X( g(τ ) = log + 1 dx ≤ C1 , δ Br (0) ˜ ∞ , L, L, ˜ and Dx bL1 (L p ) . where the constant C1 depends on T , r, b∞ , b

(45)

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Next we fix a second parameter η > 0 to be chosen later. Using Chebyshev inequality we find a measurable set K ⊂ Br (0) such that L d (Br (0) \ K) ≤ η and 

˜ τ , x)| C1 |X(τ , x) − X( +1 ≤ for x ∈ K. log δ η Therefore we can estimate  Br (0)

˜ τ , x)| dx |X(τ , x) − X(

   ˜ τ , ·)L∞ (B (0)) + |X(τ , x) − X( ˜ τ , x)| dx ≤ η X(τ , ·)L∞ (Br (0)) + X( r K

≤ η C2 + cd rn δ (exp(C1 /η )) ≤ C3 (η + δ exp(C1 /η )) ,

(46)

˜ ∞ , L, L, ˜ and Dx bL1 (L p ) . with C1 , C2 and C3 which depend only on T , r, b∞ , b Without loss of generality we can assume δ < 1. Setting η = 2C1 | log δ |−1 = 2C1 (− log δ )−1 , we have exp(C1 /η ) = δ −1/2 . Thus we conclude   ˜ τ , x)| dx ≤ C3 2C1 | log δ |−1 + δ 1/2 ≤ C| log δ |−1 , |X(τ , x) − X( (47) Br (0)

˜ ∞ , L, L, ˜ and Dx bL1 (L p ) . This completes where C depends only on T , r, b∞ , b the proof. 

Remark 8.13 (Uniqueness of the regular Lagrangian flow). We observe that the previous theorem also gives a new proof of the uniqueness of the regular Lagrangian flow associated to a vector field b which satisfies assumptions (A), (B) and (C).

9 Bibliographical Notes and Open Problems Section 3. The material contained in this section is classical. Good references are [64], Chap. 8 of [14], [27] and [61]. For the proof of the area formula, see for instance [13, 67, 68]. The proof of the second local variant, under the stronger assumption 0T Rd |bt | d μt dt < ∞, is given in Proposition 8.1.8 of [14]. The same proof works under the weaker assumption (6). Section 4. Many ideas of this section, and in particular the idea of looking at measures in the space of continuous maps to characterize the flow and prove its stability, are borrowed from [6], dealing with BV vector fields. Later on, the arguments have been put in a more general form, independent of the specific class of vector fields under consideration, in [7]. Here we present the same version of [8]. The idea of a probabilistic representation is of course classical, and appears in many contexts (particularly for equations of diffusion type); to our knowledge the first reference in the context of conservation laws and fluid mechanics is [30], where a similar approach is proposed for building generalized geodesics in the space G of

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measure-preserving diffeomorphisms; this is related to Arnold’s interpretation of the incompressible Euler equation as a geodesic in G (see also [31, 32, 33]): in this case the compact (but neither metrizable, nor separable) space X [0,T ] , with X ⊂ Rd compact, has been considered. This approach is by now a familiar one also in optimal transport theory, where transport maps and transference plans can be thought in a natural way as measures in the space of minimizing geodesics [85], and in the so called irrigation problems, a nice variant of the optimal transport problem [23]. See also [20] for a similar approach within Mather’s theory. The lecture notes [94] (see also the Appendix of [78]) contain, among several other things, a comprehensive treatment of the topic of measures in the space of action-minimizing curves, including at the same time the optimal transport and the dynamical systems case (this unified treatment was inspired by [22]). Another related reference is [58]. The superposition principle is proved, under the weaker assumption 0T Rd |bt | p d μt dt < +∞ for some p > 1, in Theorem 8.2.1 of [14], see also [79] for the extension to the case p = 1 and to the nonhomogeneous continuity equation. Very closely related results, relative to the representation of a vector field as the superposition of “elementary” vector fields associated to curves, appear in [86, 20]. Section 5. The definition of renormalized solution and the strong convergence of commutators are entirely borrowed from [61]. See also [62] for the relevance of this concept in connection with the existence theory for Boltzmann equation. The proof of the comparison principle assuming only an L1 (L1loc ) bound (instead of an L1 (L∞ ) one, as in [61, 6]) on the divergence was suggested to us by G. Savar´e. See also [40] for a proof, using radial convolution kernels, of the renormalization property for vector fields satisfying Di b j + D j bi ∈ L1loc . No general existence result for Sobolev (or even BV ) vector fields seems to be known in the infinite-dimensional case: the only reference we are aware of is [24], dealing with vector-fields having an exponentially integrable derivative, extending previous results in [49, 50, 51]. Also the investigation of nonEuclidean geometries, e.g. Carnot groups and horizontal vector fields, could provide interesting results. Finally, notice that the theory has a natural invariance, namely if X is a flow relative to b, then X is a flow relative to b˜ whenever {b˜ = b} is L 1+d -negligible in (0, T ) × Rd . So a natural question is whether the uniqueness “in the selection sense” might be enforced by choosing a canonical representative b˜ in the equivalence class ˜ the ODE γ˙(t) = of b: in other words we may think that, for a suitable choice of b, ˜bt (γ (t)) has a unique absolutely continuous solution starting from x for L d -a.e. x. Concerning the case of BV vector fields, the main idea of the proof of Theorem 5.14, i.e. the adaptation of the convolution kernel to the local behaviour of the vector field, has been used at various level of generality in [25, 77, 45] (see also [41, 42] for related results independent of this technique), until the general result obtained in [6]. The optimal regularity condition on b ensuring the renormalization property, and therefore the validity of the comparison principle in Lb , is still not known. New results, both in the Sobolev and in the BV framework, are presented in [10, 73, 74].

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In [12] we investigate in particular the possibility to prove the renormalization property for nearly incompressible BVloc ∩ L∞ fields b: they are defined by the property that there exists a positive function ρ , with ln ρ ∈ L∞ , such that the space-time field (ρ , ρ b) is divergence free. As in the case of the Keyfitz-Kranzer system, the existence a function ρ with this property seems to be a natural replacement of the condition Dx · b ∈ L∞ (and is actually implied by it); as explained in [9], a proof of the renormalization property in this context would lead to a proof of a conjecture, due to Bressan, on the compactness of flows associated to a sequence of vector fields bounded in BVt,x . Section 6. The material of this section is entirely taken from [17]. See Chap. 3 of [68] for a deep study of approximate differentiability and calculus with the class of approximately differentiable maps. Section 7. Here we have presented the main result in [73]. Section 8. The first progress in the direction of proving approximate differentiability of the flow X(t, x) with respect to x has been achieved in [15]. Later on, these results have been substantially improved in [47], and for this reason we have chosen to present only the latter results and proofs in this section.

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A Note on Alberti’s Rank-One Theorem Camillo De Lellis Institut f¨ur Mathematik, Universit¨at Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland e-mail: [email protected] URL: http://www.math.unizh.ch/

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A Blow-Up Argument Leading to a Partial Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Fundamental Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Proof of Theorem 1.1 in the Planar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 63 65 66 68 74

1 Introduction The aim of these notes is to illustrate a proof of the following remarkable Theorem of Alberti (first proved in [1]). Here, when μ is a Radon measure on Ω ⊂ Rn , we denote by μ a its absolutely continuous part (with respect to the Lebesgue measure L n ), by μ s := μ − μ a its singular part, and by |μ | its total variation measure. Clearly, |μ |a = |μ a | and |μ |s = |μ s |. When μ = Du for some u ∈ BV (Ω, Rk ), we will write Ds u and Da u. If ν is a nonnegative measure, μ /ν will denote the Radon–Nykodim derivative of μ with respect to ν . Finally we recall the polar decomposition of Radon measures, namely the identity μ = |μμ | |μ |, which implies that the vetor Borel map μ /|μ | has modulus 1 μ -a.e. Theorem 1.1. Let u ∈ BV (Ω, Rk ) for some open set Ω ⊂ Rn . Then rank(Du/ |Du|(x)) = 1 for |Ds u|-a.e. x ∈ Ω. We start by discussing what can be inferred from the “standard theory” of BV functions without much effort. A first conclusion can be drawn from the BV Structure Theorem (see Sect. 3.6, Theorem 3.77, and Proposition 3.92 of [3]) for which we first need some terminology. Given an L1 function u we say that u is approximately continuous at x if there exists u(x) ˜ ∈ Rk such that limr r−n Br (x) |u(y) − u(x)| ˜ dy = 0. We denote by Su the set of points where u is not approximately continuous and we say that x ∈ Su is an approximate jump point if there exists ν (x) ∈ Sn−1

61

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and u± (x) ∈ Rk such that    1 + − lim n |u(y) − u (x)| dy + |u(y) − u (x)| dy = 0, r↓0 r B+ B− r (x) r (x) + − where B± r (x) = {y ∈ Br (x) : ±(y − x) · ν (x) > 0}. The triple (ν (x), u (x), u (x)) is + − unique up to a change of sign of ν (x) and a permutation of u (x) and u (x). The set of approximate jump points is denoted by Ju . Finally, we recall that an (n − 1)-dimensional rectifiable set R ⊂ Rn is a Borel set which can be covered H n−1 -almost all by a countable family of C1 (n − 1)dimensional surfaces. Here, H k denotes the k-dimensional Hausdorff measure.

Theorem 1.2 (Structure Theorem for BV functions). If Ω ⊂ Rn is open and u ∈ BV (Ω, Rk ), then Ju is a rectifiable (n − 1)-dimensional set, H n−1 (Su \ Ju ) = |Du|(Su \ Ju ) = 0 and Ds u can be decomposed as Dc u + D j u, where • |Dc u|(E) = 0 for every Borel set E with H n−1 (E) < ∞; • D j u = (u+ − u− ) ⊗ ν H n−1 Ju . Here and in what follows, given a measure μ and a Borel set E we denote by μ E the measure given by μ E(A) = μ (E ∩ A). Following [5], we call Dc u and D j u respectively Cantor part and Jump part of the measure Du. Thus, Theorem 1.2 implies the statement of Theorem 1.1 when we replace |Ds u| with |D j u|. A second fact that can be inferred from the “standard theory” of BV functions is the following dimensional reduction: Proposition 1.3. Theorem 1.1 holds if and only if it holds for Ω = B1 (0) ⊂ R2 and Rk = R2 . This proposition will be proved in Sect. 2. Thus, the key point of Theorem 1.1 is to show that M has rank one |Dc u|-a.e. when u is a BV planar map. A first heuristic idea of why this property indeed holds is given in Sect. 3. The key remark of that section is the following lemma, which has a quite simple proof. Lemma 1.4. Let Ω ⊂ R2 be connected and u ∈ BV (Ω, R2 ) be such that Du/|Du| is a constant matrix M of rank 2. Then, Du = cML 2 Ω for some c > 0. Building on this lemma and on a “blow-up” argument, we prove in Sect. 3 a particular case of Theorem 1.1. However, this simple strategy cannot prove Theorem 1.1 in its full generality (see Sect. 3, in particular Proposition 3.3). Alberti’s strategy relies on replacing Lemma 1.4 with Lemma 1.5 below. From now on a set C ⊂ R2 will be called a closed convex cone if there exist e ∈ S1 and 0 < a < 1 such that C = C(e, a) := {x : x · e ≥ a|x|}. Lemma 1.5. Let C1 and C2 be two closed convex cones such that C1 ∩C2 = (−C1 ) ∩ C2 = {0}. Let Ω ⊂ R2 be open and v1 , v2 ∈ BV (Ω) be two scalar functions such that Dvi /|Dvi |(x) ∈ Ci for |Dvi |-a.e. x. If μ ≥ 0 is a measure such that μ << |Dvi | for i = 1, 2, then μ << L 2 Ω.

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This lemma will be proved in Sect. 4. We want to stress here the analogies with Lemma 1.4. Set v = (v1 , v2 ). By the polar factorization, the main assumption of Lemma 1.5 could be restated as Dv/|Dv| belongs (|Dv|-almost everywhere) to a suitably small neighborhood of a constant matrix M of rank 2. Moreover the last sentence is equivalent to |Dv| << L 2 Ω. Thus, we can consider Lemma 1.4 as a rigidity result and Lemma 1.5 as its quantitative counterpart. Now consider u ∈ BV (Ω, R2 ) and the Borel set E := {x : rank (Du/|Du|(x)) = 2}. Standard arguments show that E can be decomposed in countably many Borel pieces Ei where Du/|Du| is very close to a single constant matrix Mi . Thus the relaxed assumption of Lemma 1.5 suggests that we could use a “decomposition” approach, in contrast with the “blow-up” argument which builds on the rigidity Lemma 1.4. More precisely, we will show in Sect. 5 that the decomposition in Borel pieces Ei s can be chosen so that • If we fix any i and set μ := |Du| Ei , then there are two BV scalar functions v1 and v2 such that v1 , v2 and μ satisfy the hypotheses of Lemma 1.5. Clearly, the decomposition stated above and Lemma 1.5 show that μ is absolutely continuous, i.e. they prove Theorem 1.1. The construction of the vi s is the second key idea of Alberti’s proof. The argument combines a simple geometric consideration on the level sets of the ui s together with a clever use of the coarea formula for BV scalar functions. Recently, Alberti, Csorniey and Preiss, (see [2]) have proposed a different proof of the Rank-One Theorem. This new proof uses as well the coarea formula, but it avoids Lemma 1.5, and relies instead on a general covering result for Lebesgue-null sets of the plane. Let us mention, in passing, that this last result has many other deep implications in real analysis and geometric measure theory; see [2].

2 Dimensional Reduction Proof of Proposition 1.3. Assume that Theorem 1.1 holds for maps u ∈ BV (B1 (0), R2 ) with B1 (0) ⊂ R2 . Clearly, by translating and rescaling, we immediately conclude the theorem when u ∈ BV (B, R2 ) for any two-dimensional ball B. The statement of Theorem 1.1 is trivially true if Ω ⊂ R or if k = 1 Moreover, any open set Ω ⊂ Rn can be written as countable union of balls. Hence it suffices to prove the theorem when Ω is a ball of Rn , n ≥ 2, and k ≥ 2. From n = 2 to n generic. Here we prove Theorem 1.1 for maps u ∈ BV (B, R2 ) whenever B is an n-dimensional ball. We argue by contradiction and let u ∈ BV (B, R2 ) be such that rank (Du/|Du|(x)) = 2 on some set E with |Ds u|(E) > 0. Set M = Du/|Du| and choose coordinates x1 , . . . , xn on B and u1 , u2 on R2 . Clearly, M has n(n − 1)/2 different minors, corresponding to the choice of coordinates xi , x j with 1 ≤ i < j ≤n: We denote them by M i j . If we set Ei j := {x : rank (M i j (x)) = 2}, then E = i j Ei j , and hence |Ds u|(Ei j ) > 0 for some i and j. Without loss

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of generality we assume i = 1 and j = 2. Consider the matrix valued measure (μ )l α = (∂xl uα )l α with l, α = 1, 2. Then, rank (μ /|μ |(x)) = 2 for |μ |-a.e. x ∈ E12 and |μ s |(E12 ) > 0. For any y ∈ Rn−2 we define By = {(x1 , x2 ) ∈ R2 : (x1 , x2 , y) ∈ B}. Clearly, By is either empty or it is an open two-dimensional ball. Moreover, we define v y : B y → R2

vy (x1 , x2 ) = u(x1 , x2 , y).

by

By the slicing theory of BV functions (see Theorem 3.103, Theorem 3.107, and Theorem 3.108 of [3]) we have: (a) vy ∈ BV (By , R2 ) for L n−2 -a.e. y ∈ Rn−2 ; (b) μ = Dvy ⊗ L n−2 and |μ | = |Dvy | ⊗ L n−2 . (Here, when α is a measure on Y and y → βy a weakly measurable map from Y into the space M (X) of Radon measures on X, the symbol βy ⊗ α denotes the measure γ on X × Y which satisfies  X×Y

ϕ (x, y) d γ (x, y) =

  Y

X

ϕ (x, y) d βy (x) d α (y)

for every ϕ ∈ Cc (X × Y ).) (b) implies two things. First of all, Dvy μ (x1 , x2 ) = (x1 , x2 , y) for L n−2 -a.e. y and |Dvy |-a.e. (x1 , x2 ). |Dvy | |μ |

(1)

Second, if for every y we set Ey := {(x1 , x2 ) : (x1 , x2 , y) ∈ E}, then  Rn−2

|Dvsy |(Ey )dL n−2 (y) = |μ s |(E) > 0.

(2)

Thus, from (a), (1) and (2), we conclude that there exists a y such that vy ∈ BV (By , R2 ), |Dvsy |(Ey ) > 0, and rank (Dvy /|Dvy |(x)) = 2 for |Dvy |-a.e. x ∈ Ey . Such vy contradicts our assumption that Theorem 1.1 holds for maps u ∈ BV (By , R2 ). From k = 2 to k generic. Fix any u ∈ BV (B, Rk ), with k ≥ 2 and B n-dimensional ball, and choose coordinates u1 , . . . , uk on Rk . For any pair of integers 1 ≤ i < j ≤ k, consider the map ui j := (ui , u j ) ∈ BV (B, R2 ). If M = Du/|Du| and Mi j is the corresponding 2 × n minor, then Dui j = Mi j |Du|. Thus, by the previous step, rank (Mi j (x)) ≤ 1 for |Ds ui j |-a.e. x, and hence for |Ds u|-a.e. x. Set Ei j :=



x : rank (Mi j (x)) ≤ 1



and

E :=



Ei j .

1≤i< j≤k

Then, |Ds u|(Rn \ E) = 0 and rank(M(x)) ≤ 1 for every x ∈ E. This concludes the proof. 

A Note on Alberti’s Rank-One Theorem

65

3 A Blow-Up Argument Leading to a Partial Result We start this section by proving Lemma 1.4. Proof of Lemma 1.4. We let M be the constant matrix Du/|Du| and μ = |Du|. By standard arguments, it suffices to prove the lemma when Ω is the unit ball B1 (0). Denote by u1 and u2 the two components of u. Then Dui = vi μ , where v1 , v2 ∈ R2 are two linearly independent vectors. Let {ϕε }ε >0 be a standard family of mollifiers supported in Bε (0) and consider the mollifications ui ∗ ϕε in B1−ε (0). Notice that D(ui ∗ ϕε ) = vi μ ∗ ϕε , and hence ui ∗ ϕε is constant on the direction orthogonal to vi . Therefore the density of the absolutely continuous measure μ ∗ ϕε is a function fε which is constant along two linearly independent directions. Thus, fε is constant. Letting ε ↓ 0 we complete the proof.  This simple remark leads to a partial answer to Theorem 1.1, given in Proposition 3.2. Definition 3.1. Let μ be a measure on Ω ⊂ R2 and for any x in the support of μ and any r ∈]0, dist (x, ∂ Ω)[ consider the measures μx,r on B1 (0) given by

μx,r (A) = μ (x + rA)/|μ |(Br (x))

for any Borel set A ⊂ B1 (0).

We say that a measure μ0 is tangent to μ at x if for some sequence rn ↓ 0 we have μx,rn ∗ μ0 . A nonnegative measure μ on Ω ⊂ R2 is said to have only trivial blow-ups at x, if every tangent measure to μ at x is of the form cL 2 B1 (0). For u ∈ BV (Ω, R2 ) we denote by T the set of points where |Ds u| has only trivial blow-ups. This definition of tangent measure is very similar to that introduced by Preiss in the fundamental paper [6]. We are now ready to state our Proposition 3.2. Let u ∈ BV (Ω, R2 ). Then rank (Du/|Du|(x)) = 1 for |Ds u|-a.e. x ∈ T . Proof. We argue by contradiction and assume that the proposition is false for some u. Denote by μ the measure |Ds u|. Then, by standard measure-theoretic arguments, it is possible to find a point x ∈ T and a sequence rn ↓ 0 such that the following properties hold: (i) μx,rn ∗ μ0 , and μ0 = cL n B1 (0); (ii) ||Du| − μ |(Br (x)) = o(μ (Br (x))); (iii) M = Du/|Du|(x) is a matrix of rank > 1 and lim r↓0

1 |Du|(Br (x))

 Br (x)



Du/|Du|(y) − M d|Du|(y) = 0.

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C. De Lellis

Let ur be the average of u on Br (x) and define the function ur ∈ BV (B1 (0), R2 ) as ur (y) =

rn−1 (u(x + ry) − ur ) . |Du|(Br (x))

It follows that Dur = [Du]x,r , and hence |Dur |(B1 (0)) = 1. Moreover, since the average of ur is 0, the Poincar´e inequality gives ur L1 ≤ C. Thus, we can assume that a subsequence, not relabeled, of {urn } converges to some u0 ∈ BV (B1 (0), R2 ) strongly in L1 . Now, from (ii) we get |Du|x,r − μx,r ∗ 0 and from (iii) we conclude [Du]x,r − M|Du|x,r ∗ 0. Therefore, by (i), Dur = [Du]x,r ∗ M μ0 . This implies Du0 = M μ0 , because urn converges to u0 . Applying Lemma 1.4 we conclude μ0 = cL 2 B1 (0), which contradicts (i). Unfortunately, we cannot hope to prove Theorem 1.1 by showing that singular parts of BV functions have necessarily nontrivial blow-ups. More precisely we have Proposition 3.3. There exist BV maps u such that |Ds u|(T ) > 0. Proof. The example 5.8(1) of [6] gives a nonnegative measure μ on a bounded interval I which is singular and such that μx,r ∗ 21 L 1 [−1, 1] for μ -a.e. x. Clearly, any primitive of μ is a bounded BV function which satisfies the requirements of the proposition. 

4 The Fundamental Lemma Before coming to the proof of the lemma, let us explain its basic ingredients. Assume for the moment that the vi s of the lemma are regular, and that μ = f L 2 ≤ C|∇vi |. Consider the map v = (v1 , v2 ). Since the gradients ∇vi belong everywhere to the cones Ci and C1 ∩ C2 = (−C1 ) ∩ C2 = {0}, a simple algebraic consideration shows that det ∇v controls, up to some constant depending on the Ci s, the product |∇v1 ||∇v2 |, and hence f 2 . Thus we can bound the L2 norm of f by the integral of det ∇v. A second key remark is that the geometric constraints on the Ci s imply that v is almost injective (more precisely, v would be injective if ∇vi ∈  Ci \ {0}). Thus, det ∇v can be computed using the area formula. This means that f 2 can be bound in terms, for instance, of the L∞ norm of v, but independently of ∇v. In the proof below we will extend such a priori estimate to the general case, using truncations and a suitable regularization procedure. Remark 4.1. In the rest of these notes we will use extensively the following elementary fact. Let C = C(e, a) = {x : x · e ≥ a|x|} be a closed convex cone, Ω ⊂ R2 an open set and v ∈ BV (Ω, R). Then, it follows easily from the polar decomposition of measures that Dv/|Dv|(x) ∈ C for |Dv|-a.e. x if and only if ∂e v ≥ a|Dv|. Proof of Lemma 1.5. We can assume without loss of generality that v1 , v2 ∈ L∞ . Indeed for every k ∈ N set vki = min(max(vi , −k), k) and Ek = {|v1 | < k}∩{|v2 | < k}. Then, by the locality of |Dv| (see Remark 3.93 of [3]):

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67

• vk1 , vk2 are bounded BV functions which satisfy the assumptions of the lemma;  • μ (Ω \ k Ek ) = 0 and μ Ek << |Dvi | Ek = |Dvki | Ek ≤ |Dvki |. Therefore, if the lemma holds for bounded BV functions, then we conclude that μ Ek << L 2 Ω, and hence that μ << L 2 Ω. In addition, since every open set Ω can be covered by a countable family of convex subsets, we will assume that Ω is convex. Finally, we can assume, without loss of generality, that μ ≤ N|Dvi | for some constant N. Indeed, for any N > 0 let EN be the set of points x where  the Radon–Nykodim derivatives μ /|Dvi |(x) ≤ N. Then μ (R2 \ N EN ) = 0 and μ EN ≤ N|Dvi |. Let any such vi s and Ω satisfy all these assumptions, and let C1 and C2 be the cones of the lemma. Recall that Ci = C(ei , ai ) for some 1 > ai > 0 and ei ∈ S1 . Given two vectors z1 , z2 ∈ R2 we measure the angle θ (z1 , z2 ) between z1 and z2 in counterclockwise direction. By possibly exchanging the indices we can assume θ (e1 , e2 ) < π . Then, the assumptions C1 ∩ C2 = (−C1 ) ∩ C2 = {0} translate into the existence of a constant δ 0 > 0 such that δ 0 ≤ θ ( f1 , f2 ) ≤ π − δ 0 for every pair ( f1 , f2 ) ∈ C1 × C2 . Therefore, for δ = sin δ 0 > 0, det( f1 , f2 ) = | f1 || f2 | sin θ ( f1 , f2 ) ≥ δ | f1 || f2 |

∀( f1 , f2 ) ∈ C1 × C2 .

(3)

By Remark 4.1, ∂ei vi ≥ ai |Dvi |. Set wi (x) = vi (x) + arctan(x · ei ) and w = (w1 , w2 ) and note that (a) ∂ei wi ≥ ai |Dwi |; (b) [∂ei wi ](Br (x)) > 0 for every ball Br (x) ⊂ Ω; −1 (c) μ ≤ N|Dvi | ≤ Na−1 i ∂ei vi ≤ Nai ∂ei wi . Let {ϕε } be a standard family of nonnegative mollifiers supported in Bε (0) and consider the mollifications w ∗ ϕε in the open sets Ωε := {x ∈ Ω : dist (x, ∂ Ω) > ε }. We claim that (a’) ∇(wi ∗ ϕε )(x) ∈ Ci for any i and any x ∈ Ωε ; (b’) w ∗ ϕε : Ωε → R2 is injective; (c’) μ ∗ ϕε ≤ Na−1 i ∂ei (wi ∗ ϕε ). From (a) we get ∂ei (wi ∗ ϕε ) ≥ ai |Dwi | ∗ ϕε ≥ ai |D(wi ∗ ϕε )|, which, by Remark 4.1 and the smoothness of wi ∗ ϕε , implies (a’). (c’) follows from μ ≤ Na−1 i ∂ei wi . We now come to (b’). Note that, by (b), ∂ei (wi ∗ ϕε ) > 0. So ∇wi ∗ ϕε (x) = 0 for every x ∈ Ωε , and hence belongs to Ci \ {0}. Let x = y ∈ Ωε , and set f := (x − y)/|x − y|. We claim that, for some i, | f · z| > 0 for all z ∈ Ci \ {0}.

(4)

Otherwise, there are z1 ∈ C1 and z2 ∈ C2 with |zi | = 1 and zi ⊥ f . Therefore, either z1 = z2 or z1 = −z2 , contradicting C1 ∩C2 = (−C1 ) ∩C2 = {0}. Next, write wi ∗ ϕε (y) − wi ∗ ϕε (x) =

 |y−x| 0

∇wi ∗ ϕε (x + σ f ) · f d σ .

(5)

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C. De Lellis

Recall that ∇wi ∗ ϕε (x + σ f ) ∈ Ci \ {0}. Moreover, since Ci \ {0} is connected, (4) implies that the integrand in (5) is either strictly positive, or strictly negative. In any case, wi ∗ ϕε (y) = wi ∗ ϕε (x), which gives (b’). We are now ready for the final step. (a’), (b’), (c’) and the area formula give w1 ∞ w2 ∞ ≥ w1 ∗ ϕε ∞ w2 ∗ ϕε ∞ ≥ L 2 (w ∗ ϕε (Ωε )) (b )

=



Ωε

(a )+(3)

≥

≥δ (c )

det(∇(w ∗ ϕε )(x)) dx

δ

 Ωε

 Ωε

|∇(w1 ∗ ϕε )(x)||∇(w2 ∗ ϕε )(x)| dx

[∂e1 (w1 ∗ ϕε )](x) [∂e2 (w2 ∗ ϕε )](x) dx

≥ δ N −2 a1 a2

 Ωε

(μ ∗ ϕε (x))2 dx.

Hence, μ ∗ ϕε 2L2 (Ω ) ≤ N 2 (a1 a2 δ )−1 w1 ∞ w∞ , which, letting ε ↓ 0, gives μ = ε

f L 2 for some f ∈ L2 (Ω).



5 Proof of Theorem 1.1 in the Planar Case We will argue by contradiction, and hence in a different way with respect to what said in Sect. 1. However, this is only to make the presentation more transparent: The ideas presented in this section can be easily adapted to prove the general decomposition property claimed at the end of the introduction. So, let u = (u1 , u2 ) ∈ BV (B, R2 ) where B is a two-dimensional disk. Define   E := x : rank (Du/|Du|(x)) = 2 , (6) and assume that |Ds u|(E) > 0. Without loss of generality, we can assume u ∈ L∞ . Indeed, for every k truncate u1 and u2 by setting uki = min{max{ui , −k}, k}, and define   uk := (uk1 , uk2 ) and Ek := x : rank (Duk /|Duk |(x)) = 2 . Then, |Ds uk |(Ek ) → |Ds u|(E) as k ↑ ∞. Hence, from now on we assume that u ∈ BV ∩ L∞ . For each point x ∈ E, we set wi (x) := Dui /|Du|(x), which must be nonzero vectors. Thus, we can define ei (x) := wi (x)/|wi (x)|, which is paralell to Dui /|Dui |(x) and pointing in the same direction. Next, let • Fk be the set of pairs ( f1 , f2 ) ∈ S1 ×S1 which form an angle ≥1/k and ≤ π −1/k; • Fk := {x ∈ E : (e1 (x), e2 (x)) ∈ Fk }.

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69



Since E = k Fk , obviously |Ds u|(Fk ) > 0 for some k. Fix any such k and for any ( f1 , f2 ) ∈ Fk and any ε > 0 define  F( f1 , f2 , ε ) := x ∈ Fk : e1 (x) ∈ C( f1 , 1 − ε ), e2 (x) ∈ C( f2 , 1 − ε ) . We claim that there exist ( f1 , f2 ) ∈ Fk such that |Ds u|(F( f1 , f2 , ε )) > 0 for every ε > 0. Otherwise, by compactness of Fk , we can find N pairs ( f1j , f2j ) and N positive numbers ε j > 0 such that Fk ⊂

N

C( f1j , 1 − ε j ) × C( f2j , 1 − ε j )

j=1

and |Ds u|(F( f1j , f2j , ε j )) = 0. This would give |Ds u|(Fk ) ≤ ∑ j |Ds u|(F( f1j , f2j , ε j )) = 0. Therefore, fix ( f1 , f2 ) ∈ Fk such that |Ds u|(F( f1 , f2 , ε )) > 0 for every positive ε . Note that, since f1 and f2 are linearly independent, for ε sufficiently small the closed convex cones Ci = C( fi , 1 − ε ) satisfy C1 ∩C2 = (−C1 ) ∩C2 = {0}. We choose such an ε and we define   Dui (x) ∈ Ci for both is . (7) F  := x : |Dui | Theorem 1.1 is then implied by the following Proposition 5.1. Let C = C(e, a) be a closed convex cone, v ∈ BV ∩ L∞ (B, R) and   Dv G := x : (x) ∈ C . (8) |Dv| For any convex cone C = C(e, a ) with a < a there exists w ∈ BV ∩ L∞ (B, R) such that |Dv| G << |Dw|, and Dw (x) ∈ C |Dw|

for |Dw|-a.e. x.

(9)

Proof of Theorem 1.1. We recall that we argue by contradiction. The discussion above gives a bounded BV map u : B → R2 and two closed convex cones C1 and C2 such that • C1 ∩C2 = (−C1 ) ∩C2 = {0}; • If E and F  are defined as in (6) and (7), then |Ds u|(E ∩ F  ) > 0. Now, by definition of E, |Ds u| E << |Dui | for both i = 1, 2. Thus, if we set μ := |Ds u| (E ∩ F  ), then μ is a singular measure such that μ << |Dui | F  for both i = 1, 2. Next choose two larger closed convex cones C1 and C2 so that C1 ∩C2 = (−C1 ) ∩  C2 = {0}. Apply Proposition 5.1 to find v1 and v2 such that Dvi /|Dvi |(x) ∈ Ci for

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|Dvi |-a.e. x, and |Dui | F  << |Dvi |. Thus, we have μ << |Dvi | for both i = 1, 2. Applying Lemma 1.5 we conclude that μ is absolutely continuous, which is the desired contradiction.  Therefore, we are left with the task of proving Proposition 5.1. A special case of this proposition is when v is the indicator function of a set (which therefore is a Caccioppoli set). This case turns out to be an elementary geometric remark, but it is the key to prove the proposition in its full generality, via the coarea formula. Proof of Proposition 5.1 when v is the indicator function of a set A. Since v is a BV function, A is a Caccioppoli set. We denote by ∂ ∗ A its reduced boundary (see Sect. 3.5 of [3] for the definition) and by η the approximate exterior unit normal to ∂ ∗ A. Since Dv = η H 1 ∂ ∗ A, the set G is given by {x ∈ ∂ ∗ A : η (x) ∈ C}. Since ∂ ∗A ∞ is rectifiable (cp. with Theorem 3.59 of [3]), G can be decomposed as G0 ∪ i=1 Gi , where: • H 1 (G0 ) = 0 and for i ≥ 1 each Gi is the subset of a C1 curve γi ; • η |Gi coincides with the normal to the curve γi . Step 1. For each i we claim that there are Lipschitz open sets {Si, j } j∈N such that: the exterior normal to ∂ Si, j belongs H 1 -a.e. to C and {∂ Si, j } j is a covering of Gi . Recall that C = C(e, a ), and choose coordinates x1 , x2 in R2 in such a way that e = (0, 1). For any x ∈ Gi , the normal νi (x) belongs to C(e, a), and thus it is transversal to (1, 0). Since γi is C1 , this implies that we can choose an open ball Bx centered at x such that γi ∩ Bx is the graph {(x1 , f (x1 ))} of a C1 function f : I → R, where I is some bounded open interval of R. Moreover, by continuity of the normal νi , we can choose Bx so that νi (y) ∈ C for every y ∈ γi ∩ Bx . Fix any such y. Note that the angle θ between e and νi (y) is equal to the angle between (0, 1) and the tangent to γi at y. Since νi (y) ∈ C(e, a ), we conclude that

θ = arccos(νi (y) · e) ≤ arccos(1/a ). √  )) ≤ a2 − 1, and hence f is a Lipschitz function with Thus | f  | ≤ tan(arccos(1/a √ constant less than a2 − 1. It is an elementary well-known fact that f can be extended to a function f˜ : R → R with the same Lipschitz constant. If we define Sx := {(x1 , x2 ) : x2 < f˜(x1 )}, then Sx is a Lipschitz open set, the normal to Sx belongs everywhere to the cone C , and ∂ Sx covers Bx ∩ γi (Fig. 1). Since we can cover γi with a countable family of these balls Bx , the corresponding Sx form the desired countable covering {Si, j } j Step 2. Consider the sets {Si, j }i, j . Their boundaries have all finite lengths, which we denote by i, j and they cover H 1 -a.e. G. Let λi, j be a collection of positive numbers such that ∑i, j λi, j ≤ 1 and ∑i, j λi, j i, j ≤ 1. Let w be the function

∑ λi, j 1Si, j . i, j

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71

Bx same angle θ

γi

e

x

νi

Sx

Fig. 1 The set Sx and the ball Bx

First of all, w∞ ≤ ∑i, j λi, j ≤ 1. Second, w ∈ BV and |Dw| is the nonnegative measure ∑i, j λi, j H 1 ∂ Si, j . Thus, |Dv| G = H 1 G << |Dw|. Finally, wBV = wL1 + |Dw|(B) ≤ 2π + ∑ λi, j i, j ≤ 2π + 1. i, j

 Proof of Proposition 5.1. Fix v, C, G and C as in the statement, set c := v∞ and for every t ∈ [−c, c] consider the function vt := 1{v>t} . Then, it follows from the coarea formula (see Theorem 3.40 of [3]) that: (i) vt is a BV function for L 1 -a.e. t, i.e. {v > t} is a Caccioppoli set, and we denote by νt its exterior unit normal; (ii) νt (x) =Dv/|Dv|(x) for L 1 -a.e. t and H 1 -a.e. x ∈ ∂ ∗ {v > t}; c |Dvt | dL 1 (t). (iii) |Dv| = −c (Here, when α is a measure on Y and y → βy a weakly measurable map from Y  into the space M (X) of Radon measures on X, the symbol βy d α (y) denotes the measure γ on X which satisfies 

ϕ (x) d γ (x) =

  Y

X

ϕ (x) d βy (x) d α (y)

for every ϕ ∈ Cc (X).) Therefore, for L 1 -a.e. t, vt , C, G, and C satisfy the hypotheses of the proposition. We denote by wt the corresponding BV function given by the special case of this proposition, proved above. We will show below that wt can be selected in

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such a way that the map t → wt ∈ L∞ is weakly∗ measurable, i.e. that t → ϕ wt is measurable for every ϕ ∈ L1 (B). Having a map with this property, we choose λ ∈ L1 ([−c, c]) such that λ > 0 and  c −c

  λ (t) wt ∞ + wt BV dt < ∞. 

c Assuming this fact, we set w(x) := −c λ (t)wt (x) dL 1 (t). Then w is bounded, |Dw| c is a measure and |Dw| ≤ −c λ (t)|Dwt |dL 1 (t), which is a finite measure. Therefore w ∈ BV ∩ L∞ . Next, recall that C = C(e, a ) for some real a and some e ∈ S1 . By Remark 4.1, ∂e wt ≥ a |Dwt |. Thus ∂e wt is a nonnegative measure for L 1 -a.e. t. From this we conclude

∂e w =

 c −c

λ (t)∂e wt dL 1 (t) ≥ a

 c −c

λ (t)|Dwt |dL 1 (t) ≥ a |Dw|,

which, by Remark 4.1, gives Dw/|Dw|(x) ∈ C for |Dw|-a.e. x. Finally, |Dvt | G << |Dwt |, from which we get |Dv| G =

 c −c

≤ a

|Dvt | GdL 1 (t) <<

 c −c

 c −c

λ (t)|Dwt |dL 1 (t)

λ (t)∂e wt dL 1 (t) = a ∂e w ≤ a |Dw|.

Thus w satisfies the requirements of the proposition. Proof of the existence of a measurable selection t → wt . In order to show the existence of such a selection, we will use a general Measurable Selection Theorem due to Aumann (see Theorem III.2 in [4]). More precisely, consider the set S of functions z such that z ∈ BV ∩ L∞ and Dz/|Dz|(x) ∈ C for |Dz|-a.e. x. We endow S with the L∞ weak∗ topology. Next, set Ft := {z ∈ S : |Dvt | G << |Dz|} if vt ∈ BV , and Ft = 0/ otherwise. In order to apply Aumann’s Theorem we need that: • S and [−c, c] are both locally compact and separable; • The set F := {(t, u) : u ∈ Ft } ⊂ [−c, c] × S is a Borel set; • Ft =  0/ for L 1 -a.e. t. This last condition has been already shown. Moreover, [−c, c] is compact and separable. Thus it remains to show that S is locally compact and separable and that F is a Borel set. S is locally compact and separable. For every N ∈ N consider the set SN := S ∩ {z∞ ≤ N}. Since on bounded sets the L∞ weak∗ topology is metrizable, clearly SN is separable. Therefore, S is separable. We next show that SN is compact, which implies that S is locally compact. Indeed consider any sequence {zn } ⊂ SN . By weak∗ compactness we can assume that zn ∗ z for some z ∈ L∞ : Our task is to show that z ∈ S. Recall that a ∂e zn ≥ |Dzn |. Thus {∂e zn } is a sequence of nonnegative measures which converge distributionally to ∂e z. Therefore, these measures

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are uniformly bounded, i.e. zn BV is uniformly bounded. Thus Dzn ∗ Dz. Up to extraction of a subsequence we can assume that |Dzn | converges in the sense of measures to some ν . Then, |Dz| ≤ ν = w∗ lim |Dzn | ≤ a w∗ lim ∂e zn = a ∂e z. n

n

This implies that z ∈ S. F is a Borel set. Denote by M 2 the set of R2 -valued Radon measures on B and by M + the set of nonnegative Radon measures. Define T : M + × M 2 → R by T (ν , μ ) :=



ν (x) d|μ |(x). |μ |

Note that ν << |μ | if and only if T (ν , μ ) = ν (B). Thus,   F = (t, z) ∈ [−c, c] × S : T (|Dvt | G, Dz) = |Dvt |(B ∩ G) . Since the map t → |Dvt | can be chosen Borel-measurable, in order to prove that F is a Borel set it suffices to show that T is a Borel function. First of all, note that       ν ν /n (x) d|μ |(x) = sup n min 1, (x) d|μ |(x). T (ν , μ ) = sup min n, |μ | |μ | n∈N n∈N Therefore, it suffices to show that the map T˜ : M + × M 2 → R given by    α ˜ (x) d|μ |(x) T (α , μ ) = min 1, |μ | is Borel measurable. Note that T˜ (α , μ ) = inf{α (A) + |μ |(B \ A) : A ⊂ B is measurable}. Therefore,

T˜ (α , μ ) = =

 inf

f ∈Cc (B), 0< f <1

inf

f ∈Cc (B), 0< f <1



(1 − f ) d α + (1 − f ) d α +



 f d|μ | 

sup g∈Cc (B,R2 ),0≤|g|< f

 g · dμ .

Let F1 be a countable dense subset of { f ∈ Cc (B) : 0 < f < 1} and F2 a countable dense subset of Cc (B, R2 ). Then    T˜ (α , μ ) = inf sup (1 − f ) d α + g · d μ . (10) f ∈F1 g∈F , 0≤|g|< f 2

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Since for each ( f , g) ∈ F1 × F2 the map (α , μ ) →



(1 − f ) d α +



g · dμ

is weakly∗ continuous, (10) implies that T˜ is a Borel function.



Acknowledgements The author has been supported by a grant of the Swiss National Foundation. Moreover, he wishes to thank Giovanni Alberti and Matteo Focardi for reading the manuscript and suggesting many improvements.

References 1. A LBERTI , G. Rank-one properties for derivatives of functions with bounded variations. Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 239–274. 2. A LBERTI , G.; C SORNIEY, M.; P REISS , D. Structure of null-sets in the plane and applications European congress of mathematics, Stockholm, June 27–July 2, 2004. A. Laptev (ed.). European Mathematical Society, Z¨urich 2005. 3. A MBROSIO , G.; F USCO , N.; PALLARA , D. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, 2000. 4. C ASTAING , C.; VALADIER , M. Convex analysis and measurable multifunctions. Lecture Notes in Mathematics 580. Springer, Berlin, 1977. 5. D E G IORGI , E.; A MBROSIO , L. Un nuovo funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8) Mat. Appl. 82 (1988), 199–210. 6. P REISS , D. Geometry of measures in Rn : distribution, rectifiability, and densities. Ann. of Math. 125 (1987), 537–643.

Regularizing Effect of Nonlinearity in Multidimensional Scalar Conservation Laws Gianluca Crippa1 , Felix Otto2 , and Michael Westdickenberg3 1 Scuola

Normale Superiore, Piazza dei Cavalieri 7, I–56126 Pisa, Italy e-mail: [email protected] 2 Institut f¨ ur Angewandte Mathematik, Universit¨at Bonn, Wegelerstraße 10, D–53115 Bonn, Germany e-mail: [email protected] 3 School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160, U.S.A. e-mail: [email protected] URL: http://www.mathphys.iam.unibonn.de/∼otto/ URL: http://www.math.gatech.edu/∼mwest/ 1 2 3 4 5 6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entropy Solutions with BV-Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of Entropy Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinetic Formulation, Blow-Ups and Split States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of Split States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Special Split States: No Entropy Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Special Split States: ν Supported on a Hyperplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Special Split States: ν Supported on Half a Hyperplane . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Classification of General Split States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Proofs of the Regularity Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 79 84 87 91 98 98 101 103 105 106 112 127

1 Introduction In these notes we will deal with scalar multidimensional conservation laws, which are first order partial differential equations of the form

∂u + ∇ · f(u) = 0 ∀(t, x) ∈ R+ × Rn . ∂t

(1)

Here f : R −→ Rn is a smooth map, which will be called the flux function. The solution u : R+ × Rn −→ R can be viewed as a conserved quantity: indeed, if we integrate the equation over a bounded set Ω ⊂ Rn with smooth boundary and we apply the divergence theorem, we see that the total amount of u in Ω changes in time according to the flux of f(u) across the boundary ∂ Ω. This interpretation motivates the interest of this class of equations: many situations in nature are modeled on the

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general principle that physical quantities are neither created nor destroyed, and their variation in a domain is due to the flux across the boundary. The mathematical treatment of these equations, however, is challenging. As we will show in Sect. 2, there is a lack of existence of smooth solutions, even for onedimensional equations: starting from smooth initial data we can lose regularity in finite time, due to a blow-up of the space derivative. Hence we are forced to consider distributional solutions, but this implies a loss of uniqueness: we will indicate via some simple explicit examples how it is possible to associate to the same initial data several (in fact, infinitely many) weak solutions. To restore the uniqueness we are forced to add some condition to our notion of solution. This is encoded in the concept of entropy solution: we ask that some nonlinear functions of u are dissipated along the flow. This is inspired by the second principle of thermodynamics and is consistent with a widely used approximation procedure, the vanishing viscosity technique. A celebrated theorem by Kruˇzkov ensures existence, uniqueness and stability for this class of solutions of scalar conservation laws. A consequence of Kruˇzkov theory is the propagation of BV-regularity (see Sect. 3 for details): if we start with initial data with bounded variation, then the entropy solution has bounded variation for all times. We remark in passing that the BV framework is natural in the context of conservation laws, since entropy solutions in general develop discontinuities (which will be called shocks) in finite time, even starting with smooth initial data. If the initial data is just L∞ we cannot expect BV regularity of the solution (see the examples in Sect. 4). It turns out, however, that under quite general assumptions the equation has a regularizing effect. Clearly, this requires nondegeneracy of the flux f: remember that in the case of a linear flux function f the equation reduces to a transport equation with constant speed, hence no regularization will be possible. We will therefore require genuine nonlinearity of the flux function (see condition (15)). Under this assumption we will describe two different regularization results. For a flux function which is the higher dimensional analogue of Burgers’ flux we can obtain fractional regularity of the entropy solution; we will show that it belongs to a Besov space whose order depends on the space dimension. We will also show the sharpness of this result by constructing explicit examples. See Propositions 4.3 and 4.4. These results are original and we give a rather elementary proof of them in the Appendix. Our approach completely avoids the use of Fourier transform methods, which are otherwise typical in the context of velocity averaging lemmas, the usual tool in this framework. Even if BV regularity of the solution is not expected, we can prove that entropy solutions nevertheless have a similar structure as BV functions. We can identify a jump set on which the solution has strong traces; outside this set the solution has a weak form of continuity (see Theorem 4.5 and [11] for the precise statement). Our proof relies on the kinetic formulation of the conservation law (see Sect. 5), which encodes the entropy inequalities in form of a linear transport equation with measure-valued right-hand side. The structure theorem then follows from blow-up

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techniques and a complete characterization of the states that are obtained via blow-ups.

2 Background Material We are mainly interested in how the nonlinearity of f affects the regularity of entropy solutions of the scalar conservation law (1). It turns out that a nonlinear flux in general does not allow for smooth solutions. To see this, we first use the chain rule to rewrite (1) in the form ∂u + f  (u) · ∇u = 0. ∂t This identity implies that u is constant along the characteristic lines of (1), that is, along the trajectories of the ordinary differential equation   ˙ = f u(t, X(t)) ∀t ∈ R+ . X(t) In particular, the characteristics are straight lines. Assuming that the solution u of (1) attains initial data u(0, ·) = u0 , we obtain the implicit relation   (2) u(t, x) = u0 x − tf  (u(t, x)) ∀(t, x) ∈ R+ × Rn . If the flux function is nonlinear, however, and thus f  is nonconstant, then the characteristic lines typically intersect somewhere and so the identity (2) is no longer well-defined. This shows that smooth solutions of (1) in general cannot exist globally in time. Indeed, one can check that an upper bound for the lifespan of smooth solutions is given by T∞ = max(−κ , 0)−1, where

κ := ess inf f  (u0 (x)) · ∇u0 (x), n x∈R

(3)

see Theorem 6.1.1 in [10]. As an example, notice that in the one-dimensional case n = 1 with a convex flux f   0, formula (3) indicates global existence of smooth solutions, if the initial data is nondecreasing u0  0. One can easily check that indeed in this situation characteristic lines have a fan-like shape and therefore never cross. In general, however, the nonlinearity of the flux f forces us to consider weak solutions of (1) instead of smooth ones. In the following, we will be concerned with functions u ∈ L∞ (R+ × Rn ) satisfying the scalar conservation law (1) in distributional sense only. These functions are called weak solutions. The price we have to pay for this broader solution concept is a lack of uniqueness. Let us consider Burgers’ equation

∂ u 1 ∂ u2 + = 0 ∀(t, x) ∈ R+ × R ∂t 2 ∂x 

with initial data u0 (x) :=

−1 1

if x < 0 if x  0

(4)

 ∀x ∈ R.

(5)

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Initial value problems for one-dimensional conservation laws with piecewise constant initial data are called Riemann problems. One can check that ⎫ ⎧ ⎪ ⎬ ⎨ −1 if x/t < −1 ⎪ 1 if x/t > 1 ∀(t, x) ∈ R+ × R ur (t, x) := ⎪ ⎭ ⎩ x/t otherwise ⎪ is a weak solution of (4) and (5). The function ur is homogeneous of degree zero and locally Lipschitz continuous outside the origin. Notice that across the lines defined by |x/t| = 1 the x-derivative of ur is discontinuous. The solution ur is called a rarefaction wave. Another weak solution of (4) and (5) is us (t, x) := u0 (x)

∀(t, x) ∈ R+ × R.

This solution is called a shock solution because it is discontinuous along the timeaxis. More generally, if u is a solution of the scalar conservation law (1), then a shock is a discontinuity of u across a (sufficiently regular) hypersurface J ⊂ R+ × Rn that is timelike in the sense that in every point (t, x) ∈ J , the normal vector μ (t, x) ∈ Rn+1 is not parallel to the timeaxis. This implies that the normal vector can be written (up to some factor) in the form

μ = (−s, ν )T

with s ∈ R and ν ∈ Sn−1 .

The number s is called the shock speed because it determines how fast J is propagating in spatial direction ν . The fact that u is a weak solution of (1) implies that the Rankine–Hugoniot condition holds in every point (t, x) ∈ J : If u+ denotes the limit of u when approaching (t, x) from that side of the hyperplane the normal μ = (−s, ν )T is pointing to, and if u− is the limit from the opposite side (recall that u is discontinuous along J ), then 

 −s u+ − u− + f(u+ ) − f(u−) · ν = 0. The limits u+ and u− are called traces of u on the hypersurface J . In order to restore uniqueness, one typically imposes an entropy condition, which in the case of scalar conservation laws can be written in the form of a family of differential inequalities. That is, one only keeps those weak solutions that satisfy in distributional sense ∂ η (u) + ∇ · q(u)  0 (6) ∂t for all convex entropy–entropy flux pairs (η , q) defined by q = η  f  and η   0. Weak solutions satisfying the entropy condition are called entropy solutions. This entropy condition is inspired by the second law of thermodynamics. It is consistent with a widely used method for constructing weak solutions of conservation laws, called the vanishing viscosity method: One considers

∂ uε + ∇ · f(uε ) = ε Δuε ∂t

with ε > 0,

(7)

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which is easily shown to have unique smooth solutions, and then sends ε to zero. Notice that after multiplying (7) by η  (uε ) and using q = η  f  we get

∂ η (uε ) + ∇ · q(uε ) = ε Δη (uε ) − η  (uε )|∇uε |2 , ∂t which converges to (6) in the limit ε → 0, since η  (uε ) ≥ 0 by convexity of η . It turns out that the concept of entropy solutions restores exactly the right amount of rigidity to obtain well-posedness for the initial value problem: In his seminal paper [19], Kruˇzkov proved existence and uniqueness of entropy solutions of (1) for initial data u0 ∈ L∞ (R). They coincide with the solutions obtained by the vanishing viscosity method. Kruˇzkov’s proof is based on the observation that entropy solutions generate a contractive semigroup in L1 (Rn ): If u and v are entropy solutions of (1) with initial data u0 and v0 respectively, then u(t) − v(t) L1(Rn )  u0 − v0 L1 (Rn )

∀t  0.

(8)

As a consequence of this estimate one also obtains stability in the space BV(Rn ) of functions of bounded variation: If the data u0 ∈ BV(Rn ), then also u(t) ∈ BV(Rn ) for all t > 0. Recall that a function is of bounded variation if the distributional first derivative is a measure (we refer the reader to Sect. 3 for further information). Notice also that this stability holds trivially for linear fluxes. As a consequence of the well-known structure theorem, entropy solutions of scalar conservation laws with BV-regularity automatically have the structure we expect: The solution is (approximately) continuous outside a set J ⊂ R+ × Rn that locally looks like a Lipschitz continuous hypersurface and across which the solution is discontinuous. On this shock set, strong traces can be defined and the Rankine–Hugoniot condition is satisfied. Moreover, the entropy condition (6) reduces to the shock admissibility condition  

(9) −s η (u+ ) − η (u− ) + q(u+ ) − q(u−) · ν  0 for all points in J . This last statement follows from a generalization of the chain rule for functions of bounded variation (see Sect. 3). In that sense, BV(Rn ) is a natural space for entropy solutions of conservation laws. Recall, however, that Kruˇzkov’s result ensures well-posedness even for initial data u0 ∈ L∞ (Rn ). What can be said about the regularity of entropy solutions in this case? Clearly, if the flux is linear, then (1) is simply a linear transport equation with constant velocity, and therefore the solution u at any positive time cannot be more regular than the initial data. But for nonlinear fluxes more can be said: It turns out that while on the one hand the nonlinearity prevents global existence of smooth solutions of (1), on the other hand it also has a regularizing effect! The first result in that direction, due to Ole˘ınik [22], is that for one-dimensional scalar conservation laws with uniformly convex flux, entropy solutions satisfy the one-sided Lipschitz condition 1 ∂u ∀t > 0, (10) sup (t, x)  ct x∈R ∂ x

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where c := infu∈R f  (u). Notice that (10) only allows for decreasing jumps. Since u is bounded it follows that initial data in L∞ (R) is instantaneously regularized to BV(R) locally. The slightly more precise estimate  1 ∂  f u(t, x)  t x∈R ∂ x

sup

∀t > 0

was proved by Hoff [17]. We also refer to [27, 9, 5] for further generalizations. Ole˘ınik’s estimate (10) can be proved using the vanishing viscosity approximation. For simplicity we only consider the case of Burgers’ equation (4): Let uε be the unique smooth solution of the parabolic approximation

∂ 2 uε ∂ uε 1 ∂ u2ε + =ε ∂t 2 ∂x ∂ x2 and set vε :=

∂ uε ∂x .

(11)

Then vε satisfies the equation

∂ vε ∂ vε ∂ 2 vε + v2ε + uε =ε 2 . ∂t ∂x ∂x

(12)

Now we use the comparison principle for parabolic equations and the fact that the function V (t, x) := 1/t is a solution of (12) to obtain the estimate vε (t, x)  V (t, x) =

1 t

∀(t, x) ∈ R+ × R.

Let wε (t, x) := uε (t, x) − x/t. Then we have for all t > 0 and x0  x1 1 wε (t, x1 ) − wε (t, x0 ) = uε (t, x1 ) − uε (t, x0 ) − (x1 − x0)  0, t

(13)

so that wε (t, ·) is a decreasing function, and thus of bounded variation locally. The same is true for uε (t, ·). From (11) we now conclude that uε is of bounded variation both in space and time locally, which implies strong convergence. We obtain (13) for the limit u = limε →0 uε as well, and then (10) follows. A different approach to regularity results for entropy solutions of scalar conservation laws uses the kinetic formulation introduced by Lions, Perthame and Tadmor in [20]. To motivate this approach, we recall that smooth solutions u of the scalar conservation law (1) are constant along characteristics, which are straight lines. This implies that the level sets of u, that is, the sets Ev (t) := {x ∈ Rn : u(t, x)  v}

∀v ∈ R,

are moving with constant velocity given by f  (v). Notice that level sets are ordered: If v  v, ˆ then Ev ⊆ Evˆ . For nonlinear fluxes, level sets corresponding to different values of v are traveling with different speeds. Therefore the ordering usually breaks down at some time. This corresponds to the fact that smooth solutions of (1) typically do not exist globally. The ordering of the Ev can be restored by a projection

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step, which in fact is related to entropy being dissipated. We refer the reader to [6] for further details. This heuristics can be made rigorous and leads to the following result: A function u ∈ L∞ (R+ × Rn ) is an entropy solution of (1) if and only if the function defined by ⎧ ⎪ ⎨ +1 if 0 < v  u(t, x) χ (v, u(t, x)) := −1 if u(t, x)  v < 0 ⎪ ⎩ 0 otherwise for a.e. (v,t, x) ∈ R × R+ × Rn , satisfies the kinetic equation

∂ χ (v, u) ∂μ + f  (v) · ∇x χ (v, u) = ∂t ∂v

in D  (R × R+ × Rn),

(14)

n where μ ∈ M+ loc (R × R+ × R ) is some nonnegative measure. We refer the reader to [20] and to Sect. 5 below. Notice that μ is intrinsically defined by the entropy solution since it captures the entropy dissipation due to the nonsmoothness of u. A plethora of intriguing results on scalar conservation laws has been obtained from the kinetic formulation, in particular when combined with a technique called velocity averaging. This method was invented in the context of kinetic equations, and it played a central role in the global existence result for Boltzmann’s equation by DiPerna and Lions, see [13]. The key observation is that moments of solutions to kinetic equations enjoy more regularity than one might expect a priori. For scalar conservation laws this method implies compactness and regularity of entropy solutions. Consider the kinetic equation (14): For fixed v ∈ R this equation provides information about the derivative of χ (v, u) in the direction of (1, f  (v)) ∈ Rn+1 . On the in x. These other hand, the map v → χ (v, u(x)) is of bounded variation uniformly  observations can be combined to obtain regularity for the average R χ (v, u) dv = u. Here regularity means either boundedness in some Sobolev space or strong L1loc (Rn+1 )-precompactness for sequences of (approximate) entropy solutions. For these results an assumption on the nondegeneracy of the flux f is necessary: Notice that if the flux is linear, then the scalar conservation law (1) is simply a transport equation. That is, the initial data is uniformly transported into the direction f  (v). We can therefore not expect any regularizing effect: At any positive time, the solution is not more regular the initial data. A natural assumption on the nondegeneracy of the flux f : R −→ Rn is the following:

There is no open interval I ⊂ R such that f  (v) is contained in an (n − 1)-dimensional affine space for all v ∈ I. In the one-dimensional case this means that the flux f is not affine on any open set. A more formal restatement of the same assumption is that   ∀ξ ∈ Sn the set v ∈ R : ξ0 + f  (v) · ξ  = 0 contains no open intervals,

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where ξ = (ξ0 , ξ  ) ∈ R × Rn . In order to apply the velocity averaging argument we need a slightly stronger assumption, which takes the following form:   ∀ξ ∈ Sn L 1 v ∈ R : ξ0 + f  (v) · ξ  = 0 = 0. (15) That is, the set of values v ∈ R for which the characteristic directions f  (v) stay inside the hyperplane determined by the normal vector ξ ∈ Sn has zero Lebesgue measure. A sufficient condition for (15) to hold is that ∀v ∈ R

the vectors f  (v), . . . , f(n+1) (v) are linearly independent.

(16)

Indeed choose ξ = (ξ0 , ξ  ) ∈ Sn and let h(v) := ξ0 + f  (v) · ξ  for v ∈ R. Assume that there exists an open interval I ⊂ R with h(v) = 0 for all v ∈ I. Then  I ⊂ v ∈ R:  = v ∈ R:  = v ∈ R:  ⊂ v ∈ R:

h (v) = . . . = h(n) = 0



f  (v) · ξ  = . . . = f(n+1) (v) · ξ  = 0



 f(k) (v) ⊥ ξ  for all k ∈ {2, . . . , n + 1}

 f  (v), . . . , f(n+1) (v) are linearly dependent = ∅

because n vectors in Rn all contained in a hyperplane cannot form a basis. Assumption (16) is satisfied for the generalized Burgers’ flux   1 n+1 ∀v ∈ R, (17) f(v) := 12 v2 , . . . , n+1 v which we will study in more detail later on. If (15) holds, then any sequence of entropy solutions contains a subsequence that converges strongly in L1loc (Rn+1 ), see [20]. A more quantitative version of (15) is that   ∀ξ ∈ Sn L 1 v ∈ R : |ξ0 + f  (v) · ξ  |  δ  Cδ α (18) for all δ > 0 and some α ∈ (0, 1]. Assumption (18) yields Sobolev regularity for entropy solutions. The regularity one obtains depends on the nondegeneracy of f, that is, on α . We refer the reader to [16, 14, 3, 20, 4, 18, 26, 24] for further information about the velocity averaging argument. Its proof typically relies on Littlewood–Paley type decompositions, interpolation arguments and a spectral decomposition adapted to the “velocity direction”.

3 Entropy Solutions with BV-Regularity Let Ω ⊂ Rn+1 be an open subset. A function u ∈ L1 (Ω) is called of bounded variation if its distributional derivative Du is an Rn+1 -valued measure with finite total variation in Ω. We denote by BV(Ω) the space of functions of bounded variation,

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which is a Banach space with respect to the norm u BV(Ω) := u L1(Ω) + Du M(Ω). The local space BVloc (Ω) is then defined in the usual way. It turns out that u BV(Ω) ≈ u L1(Ω) + sup |h|−1 u(· + h) − u L1(Ω) h=0

(see Remark 3.25 in [2]), which together with the L1 -contraction estimate (8) and (1) implies that entropy solutions of (1) are BV-stable: If the initial data u0 ∈ L∞ ∩ BVloc (Rn ), then also the entropy solution u ∈ BVloc (R+ × Rn ). The importance of this observation for the theory of scalar conservation laws comes from the fact that BV-functions have a very particular structure. To explain this statement we need the following definitions. Definition 3.1 (Rectifiable sets).  Let Ω ⊂ Rn+1 be open. A subset J ⊂ Ω is n called H -rectifiable if J = E ∪ k∈N Ek , where H n (E) = 0 and each Ek is contained in an n-dimensional Lipschitz continuous submanifold of Ω. Here H n is the n-dimensional Hausdorff measure. Definition 3.2 (Orientation). Let Ω ⊂ Rn+1 be open and let J ⊂ Ω be an H n rectifiable set. An orientation of J is a Borel vector field μ : J −→ Sn with the property that for H n -a.e. x ∈ J , the vector μ (x) is normal to J . Definition 3.3 (Traces of u). Let Ω ⊂ Rn+1 be open and let J ⊂ Ω be an H n rectifiable set oriented by a normal vector field μ . We say that two Borel functions u± : J −→ R are the traces of u on J if for H n -a.e. y ∈ J    lim − + |u(x) − u+(y)| dx + − − |u(x) − u−(y)| dx = 0, r→0

Br (y)

Br (y)

n+1 : ± μ (y) · (x − y) > 0}. where B± r (y) := Br (y) ∩ {x ∈ R

The Structure Theorem (see Sect. 3.9 in [2]) states that the derivative of a BVfunction u can be decomposed into three parts: We have Du = (Du)a + (Du)c + (Du) j , where the first component (Du)a is absolutely continuous with respect to the Lebesgue measure L n+1 , and the second part (Du)c (called the Cantor part) is a singular measure that, however, is small in some sense. The third part (Du) j (called the jump part) can be written in the form (Du) j = (u+ − u−)μ H n J .

(19)

Here J ⊂ Ω is an H n -rectifiable set oriented by a unit normal vector field μ : J −→ Sn , and the functions u± : J −→ R are the traces of u on J . It is

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possible to generalize the classical chain rule to functions of bounded variation: If u is a BV-function and g : R −→ R is Lipschitz continuous, then also the composition g(u) is of bounded variation. Therefore its derivative Dg(u) can be decomposed into three terms as above, and in particular     Dg(u) j = g(u+ ) − g(u−) μ H n J , where the rectifiable set J and the functions μ and u± are the same as in (19). We refer the reader to Theorem 3.101 in [2]. A function u : R+ × Rn −→ R of bounded variation is then a weak solution of (1) if the following holds: • With the subscript a denoting the absolutely continuous parts, we have     ∂u ∂u + ∇ · f(u) = + f  (u) · (∇u)a = 0, ∂t ∂ t a a and a similar statement is true for the Cantor parts. • With the same notation as in (19), we have    −s u+ − u−) + f(u+ ) − f(u− ) · ν = 0

H n -a.e. in J ,

(20)

(21)

√ where (s, ν ) is defined by μ 1 + s2 = (−s, ν ) and ν ∈ Sn−1 . The number s is the shock speed, and (21) is called the Rankine–Hugoniot condition. Under the assumption of BV-regularity, also the entropy condition simplifies: Let (η , q) be any convex entropy/entropy flux pair with q = η  f  . • With the subscript a denoting the absolutely continuous parts, we have      ∂ η (u) ∂u   + ∇ · q(u) = η (u) + f (u) · (∇u)a = 0 ∂t ∂t a a because of (20), and a similar statement is true for the Cantor parts. • With the same notation as in (21), we have     −s η (u+ ) − η (u−) + q(u+ ) − q(u−) · ν  0 H n -a.e. in J . These considerations show that BV-regularity is a very desirable property for entropy solutions of scalar conservation laws. As we will discuss in Sect. 4, however, entropy solutions are typically not of bounded variation. Let us introduce one more definition that will be used there. Definition 3.4. Let Ω ⊂ Rn+1 be an open set and u ∈ L1loc (Ω). We say that the function u has vanishing mean oscillation (VMO) at a point y ∈ Ω if         lim − u(x) − − u(z) dz dx = 0. r→0 Br (y)

Br (y)

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We say that u is approximately continuous at y ∈ Ω if u has VMO there and 

lim −

r→0 Br (y)

u(z) dz = u(y).

All Lebesgue-measurable functions are approximately continuous L n+1 -a.e. For BV-functions this statement can be improved to approximate continuity H n -a.e. off the jump set J . We refer the reader to Sect. 3.9 in [2].

4 Structure of Entropy Solutions As explained in Sect. 2, unique entropy solutions do exist even for rough initial data that is not in BV(Rn ). What can be said about the structure of these solutions? Under appropriate assumptions on the nondegeneracy of the flux, entropy solutions do have some extra regularity (as follows from the velocity averaging arguments), but typically they are not of bounded variation. In this section we discuss in more detail the regularizing effects due to the interplay between the nonlinearity of the problem and the entropy condition. It turns out that the distinction between time and space variables is not essential. Therefore we consider a slightly more general situation: Definition 4.1. Let Ω ⊂ Rn be an open set, and let f : R −→ Rn be a smooth flux function. For any convex entropy η : R −→ R let the corresponding entropy flux qη : R −→ Rn be defined (up to a constant) by the condition qη (v) = η  (v)f  (v) ∀v ∈ R.

(22)

The function u : Ω −→ (0, 1) is called a generalized entropy solution if ∇ · f(u) = 0

in D  (Ω)

(23)

and if for all convex entropy/entropy flux pairs (η , qη ) ∇ · qη (u) ∈ Mloc (Ω).

(24)

Remark 4.2. Notice that in (24) we do not require that the entropy dissipation measure has a sign. Our definition therefore includes certain weak solutions of (23) that contain nonclassical (entropy violating) shocks. We first discuss the regularity of generalized entropy solutions. For definiteness, we consider only the higher-dimensional version of Burgers’ flux:   f(v) := v, 12 v2 , . . . , 1n vn ∀v ∈ R. (25)

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Proposition 4.3. Let Ω ⊂ Rn be open. There exists a constant C > 0 with the following property: Let u be a generalized entropy solution u corresponding to the generalized Burgers’ flux (25). Assume that there is no entropy dissipation in Ω: for all convex entropy/entropy flux pairs (η , qη ) defined by (22) we assume that ∇ · qη (u) = 0

in D  (Ω).

For any compact subset K ⊂ Ω and R := dist(K, Rn \ Ω) we then have |u(x) − u(y)|

sup (x,y)∈K×K

1

|x − y| n−1

 3 1  CR− n−1 1 + u L∞(K) .

The exponent 1/(n − 1) is optimal. For Burgers’ equation with n = 2 we recover the well-known fact that entropy solutions are locally Lipschitz continuous in open sets that do not meet the shock set. We postpone the proof of the first part of Proposition 4.3 to the Appendix, see page 114. To prove the optimality of the H¨older exponent 1/(n − 1) let Ω := (0, 1)n−1 × R. Then we construct a solution of (23) that only depends on x1 and xn and is constant along characteristics. Consider  xαn if xn > 0, u0 (xn ) := (26) 0 otherwise, for some number α > 0. Then the function u, implicitly defined by   u(x) = u0 xn − x1f n (u(x)) ∀x ∈ Ω, is a solution of (23) as follows from easy inspection. In view of (26), this gives  α xn − x1 u(x)n−1 if xn − x1 u(x)n−1  0, u(x) = (27) 0 otherwise. In particular, we have u(x)  0, thus u(x) = 0 if xn < 0. Indeed, for xn < 0 the first case in (27) never applies since x1 u(x)n−1  0. We rewrite (27) as u(x)1/α + x1 u(x)n−1 = xn Solving this identity for u(x) yields  xαn u(x) ≈ (xn /x1 )1/(n−1)

if xn  0.

(28)

if α > 1/(n − 1), otherwise,

for small xn > 0. The second case shows that rough data with α small is regularized to H¨older continuity with exponent 1/(n − 1). On the other hand, if α is large and thus u0 is smooth, then the solution u stays smooth. We now consider the case with entropy dissipation.

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Proposition 4.4. Let Ω ⊂ Rn be open and let em denote the mth unit basis vector of Rn . Then there exists a constant C > 0 with the following property: Let u be a compactly supported, generalized entropy solution corresponding to the generalized Burgers’ flux (25). For all m ∈ {1, . . . , n} we then have sup |h|− n+1 u(· + hem) − u L1(Ω) 1

h=0

  1 n+1 m n−m  C | spt u| u L1(Ω) μ M(R×Ω) ,

(29)

where μ is the kinetic entropy dissipation measure (introduced in Theorem 5.1 below), which is finite. The exponent 1/(n + 1) is optimal. 1/(n+1)

Notice that (29) actually implies that u is in the Besov space B1,∞ (Ω). For Burgers’ equation with n = 2 we obtain a differentiability of one third. The regularity 1/(n + 1) was obtained independently in [24] for entropy solutions of (23) (for which the entropy dissipation in (24) is a nonnegative measure). Their proof uses a new Fourier multiplier estimate combined with the bootstrap argument already employed in [20]. In contrast to this, we give a rather elementary proof of Proposition 4.4 in the Appendix, which completely avoids the use of Fourier methods, see page 117. The optimality of the result again follows from an explicit example: Let Ω := (0, 1)n−1 × R and consider nonincreasing sequences Δk ∈ 1 and ck ∈ n+1 to be specified later. Let a− k :=

k

∑ Δl

− and a+ k := ak − Δ k /2

l=1

for all k ∈ N. Then we define the function ∞

u(x) :=

∑ ck 1Ik (xn + sk x1)

∀x ∈ Ω,

(30)

k=1

− 1 n−1 where Ik := [a+ for all k ∈ N. We claim that (30) is a generalized k , ak ] and sk := n ck entropy solution of (23), and prove first that the Rankine–Hugoniot condition is satisfied along each discontinuity. Indeed, consider   Jk± := x ∈ Ω : xn + sk x1 = a± k

 with unit normal vectors νk := (−sk , 0, . . . , 0, 1)/ 1 + s2k . Then we have ∇ · f(u) =

 −sk f1 (ck ) + fn(ck )  n−1  H Jk+ − H n−1 Jk− = 0, k=1 1 + s2k ∞

∑

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using that f(0) = 0. To check assumption (24), consider for any convex entropy η : R −→ R the corresponding entropy flux qη defined by qη (u) :=

 u 0

f  (v)η  (v) dv

∀u ∈ R.

Then the entropy dissipation is given as ∇ · qη (u) =

 −sk qη ,1 (ck ) + qη ,n (ck )  n−1  H Jk+ − H n−1 Jk− k=1 1 + s2k ∞

∑

because qη (0) = 0. Notice that after an integration by parts  ck

  η  (v) − 1n cn−1 + vn−1 dv k 0  ck   n = − 1n η  (v) cn−1 k v − v dv,

−sk qη ,1 (ck ) + qη ,n (ck ) =

0

which implies that the total entropy dissipation satisfies ∞

∇ · qη (u) M(Ω)  C ∑ |ck |n+1 < ∞, k=1

for some constant C which only depends on n and the sup of η  on a compact set. This shows that u is indeed a generalized entropy solution of (23). On the other hand, we can estimate the finite difference h−α u(· + hen) − u L1(Ω)  h−α

∑

2h|ck | ∀h > 0,

(31)

Δ k /2h

where en is the nth standard basis vector of Rn and α > 0 is some number. Indeed, for all k with Δk /2  h we have (hen + Ik ) ∩ Ik+1 = ∅. Let

Δk := k−(1+ε )

and ck := k−(1+ε )/(n+1)

for all k ∈ N and some ε > 0. Then Δk ∈ 1 and ck ∈ n+1 as required and h −α

∑

2h|ck |  Ch−α +1/(n+1)+ε /(1+ε )

∀h > 0,

(32)

Δ k /2h

with C some constant independent of h. Assume now that α > 1/(n + 1). Then there exists ε > 0 small such that the exponent in (32) is negative. We conclude that the left-hand side of (31) blows up as h → 0. This shows that the maximal regularity we can hope for is α = 1/(n + 1) < 1. We conclude that typically a generalized entropy solution is not of bounded variation. One might wonder, however, whether a generalized entropy solution still has the same structure as a BV-function. This is indeed the case in the sense made precise in the following theorem.

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Theorem 4.5 (De Lellis, Otto, Westdickenberg). Let Ω ⊂ Rn be open and assume that the flux f : R −→ Rn satisfies the nondegeneracy condition   ∀ξ ∈ Sn−1 : L 1 {v ∈ R : f  (v) · ξ = 0} = 0. (33) Let u be a generalized entropy solution in the sense of Definition 4.1. Then there exists an H n−1 -rectifiable set J ⊂ Ω such that: • For all y ∈ J the function u has strong traces on J . • For all y ∈ J the function u has vanishing mean oscillation. Remark 4.6. To simplify the presentation, we will consider only the case of a classical entropy solution in the following, for which the entropy dissipation in (24) is a nonnegative measure. We refer the reader to [20] for the necessary modifications in the general case of measures which change sign. Theorem 4.5 shows that generalized entropy solutions have a BV-like fine structure without actually being of bounded variation. It is an open problem whether VMO can be improved to approximate continuity of u outside J (see however the result by De Lellis and Rivi`ere [12]). Another open problem is to prove that there is no entropy dissipation outside of J . For entropy solutions of bounded variation this follows from the BV-chain rule and (23). Our Theorem 4.5 only ensures that the entropy dissipation restricted to J has the correct structure: If ν : J −→ Sn−1 denotes a normal vector field along J , then we have for all convex entropy/entropy flux pairs (η , qη ) that     ∇ · qη (u) J = qη (u+ ) − qη (u− ) · ν H n−1 J . To prove that ∇·qη (u) = 0 outside J would probably require some analogue of the BV-chain rule for generalized entropy solutions. This is a hard problem. We refer the reader to [1] for some results in that direction. Notice that the fine structure result contains the existence of strong traces on codimension-one rectifiable subsets as a subproblem. This is relevant for understanding how entropy solutions of nondegenerate scalar conservation laws attain their initial or boundary data. The problem has first been studied by Vasseur [25] who uses techniques quite similar to ours. In particular, the idea of “blowing up” a neighborhood of a given point (see Definition 5.7) was introduced there. We also refer the reader to [7, 8, 23] for related results.

5 Kinetic Formulation, Blow-Ups and Split States In this section we provide some tools we will need later. We start by proving a variant of the kinetic formulation for scalar conservation laws, introduced by Lions, Perthame and Tadmor in their seminal paper [20]. In the following, we will systematically use the notation a = f  .

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Theorem 5.1 (Kinetic Formulation). Let Ω ⊂ Rn be an open set, and assume that u is an entropy solution. Let the function χ be defined by   1 if 0 < v  u(x) ∀(v, x) ∈ R × Ω. (34) χ (v, u(x)) := 0 otherwise Then there exists a nonnegative measure μ ∈ M+ loc (R × Ω) such that a(v) · ∇χ (v, u(x)) =

∂ ∂v μ

in D  (R × Ω).

(35)

Remark 5.2. A similar construction works also for generalized entropy solutions, for which we only assume that the entropy dissipation is a locally finite measure. Then the measure μ can change sign. We refer to reader to [11]. Proof. Consider the linear map Φ defined by Φ(η , ϕ ) :=

 Ω

qη (u(x)) · ∇ϕ (x) dx

∀(η , ϕ ) ∈ D(R) × D(Ω),

(36)

where qη is related to η by the compatibility condition qη (v) = η  (v)a(v) for all v ∈ R. Notice that the map Φ is indeed well-defined since qη is unique up to a constant and ϕ has compact support. We have

η linear

=⇒

Φ(η , ϕ ) = 0 ∀ϕ ∈ D(Ω),

since then qη (u) = α f(u) + β for some constants α , β , and ∇ · f(u) = 0 in D  (Ω). This implies that Φ depends on η only through η  . We also have

η convex

=⇒

Φ(η , ϕ )  0 ∀ϕ ∈ D(Ω), ϕ  0

because u is an entropy solution. (It suffices for η to be convex on the unit interval because u : Ω −→ (0, 1).) Recalling that a nonnegative distribution is in fact a measure, we can therefore find μ ∈ M+ loc (R × Ω) such that  Ω

qη (u(x)) · ∇ϕ (x) dx =



η  (v)ϕ (x) d μ (v, x)

R×Ω

(37)

for all (η , ϕ ) ∈ D(R) × D(Ω). Notice that by definition of χ and the compatibility condition for qη we have that (up to a constant) qη (u(x)) =

 R

χ (v, u(x))qη (v) dv

=

 R

χ (v, u(x))η  (v)a(v) dv.

Hence (37) turns into      χ (v, u(x))η (v)a(v) dv · ∇ϕ (x) dx = Ω

R

R×Ω

η  (v)ϕ (x) d μ (v, x),

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which can be rewritten as  R×Ω



 χ (v, u(x))∇ η  (v)ϕ (x) · a(v) dv dx =

R×Ω

∂ ∂v

 η  (v)ϕ (x) d μ (v, x).

Since linear combinations of products η  ϕ with η  ∈ D(R) and ϕ ∈ D(Ω) are dense in D(R × Ω) (up to a constant), we conclude that indeed  R×Ω

χ (v, u(x))a(v) · ∇ζ (v, x) dv dx =

 R×Ω

∂ ∂ v ζ (v, x) d μ (v, x)

for all ζ ∈ D(R × Ω). In general, the measure μ is only locally finite.

 

The kinetic formulation can be used to prove the following compactness result for bounded sequences of entropy solutions of (23), see [20]. Theorem 5.3. Consider a sequence of entropy solutions uk of (23) in some open set Ω ⊂ Rn . According to Theorem 5.1 there exists a sequence of nonnegative measures μk ∈ M+ loc (R × Ω) such that the pairs (uk , μk ) satisfy the kinetic equation (35). Assume that the flux f : R −→ Rn is sufficiently smooth and satisfies the nondegeneracy condition (33). Assume also that the measures μk are locally uniformly bounded. Then there exists a subsequence kl −→ ∞ such that ukl −→ u in L1loc (Rn ), ∗ μkl − μ

in Mloc (R × Rn)

as l → ∞, and the limit (u, μ ) satisfies the kinetic equation (35). In particular, u is a generalized entropy solution of the scalar conservation law (23). The first step in proving our Theorem 4.5 is to identify a candidate for the jump set J ⊂ Ω. We recall that in the case of an entropy solution u with BV-regularity, the jump set J is exactly the set where entropy dissipation takes place. By the kinetic formulation, this in turn is related to the support of the kinetic measure μ . Therefore the following definition is natural. Definition 5.4. Let u be an entropy solution of (23) in Ω ⊂ Rn open, and let μ be the corresponding entropy dissipation measure provided by the kinetic formulation. Let ν ∈ M+ loc (Ω) be the x-marginal of μ , defined as

ν (A) := μ (R × A) ∀Borel sets A ⊂ Ω. We denote by J the set of points with positive upper (n − 1)-density of ν :   ν (Br (y)) > 0 . J := y ∈ Ω : lim sup rn−1 r→0

(38)

The main task is then to prove that J is indeed an H n−1 -rectifiable set. For later reference, we first record the following observation.

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Lemma 5.5. Let u be an entropy solution of (23) in Ω ⊂ Rn open, and let μ be the entropy dissipation measure provided by the kinetic formulation. Let ν be the x-marginal of μ . Then there exists a constant C > 0 such that lim sup r→0

ν (Br (y)) C rn−1

∀y ∈ Ω.

(39)

Proof. Fix a test function ζ ∈ D(Rn ) with ζ  0 and ζ (x) = 1 for x ∈ B1 (0). Choose an entropy η ∈ D(R) such that η (v) = 12 v2 for v ∈ [0, 1], and let q be the corresponding entropy flux. Using the nonnegativity of μ , we have 

η  (v) d μ (v, x)    x−y  d μ (v, x) ∀Br (y) ⊂ Ω. η (v)ζ  r R×Ω

ν (Br (y)) =

R×Br (y)

Then the kinetic equation (35) yields        x−y  ν (Br (y))  η (v)a(v) χ v, u(x) dv · ∇ζ dx r Ω R    x−y dx = q(u(x)) · ∇ζ r Ω  q(u) L∞(Ω) ∇ζ L1 (Rn ) rn−1

∀Br (y) ⊂ Ω.  

The lemma follows.

There exist several criteria to check a set for rectifiability. We will use the following well-known result (see Theorem 15.19 in [21]). Theorem 5.6 (Rectifiability criterion). Let J ⊂ Rn be a set and assume that there n exists a measure ν ∈ M+ loc (R ) with the following properties: • For all y ∈ J we have

ν (Br (y)) > 0. (40) rn−1 • For ally ∈ J there exist an orthonormal  coordinate system x1 , . . . , xn and a cone Cy := x ∈ Rn : |x1 |  c|(x2 , . . . , xn )| with c > 0 such that   ν (y + Cy ) ∩ Br (y) lim = 0. (41) r→0 rn−1 lim inf r→0

Then J is an H

n−1 -rectifiable

set.

We use for ν the x-marginal of the entropy dissipation measure μ . As is suggested by the rectifiability criterion, we study blow-ups: That is, we look at the structure of entropy solutions after “zooming” into a point y ∈ Ω.

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Definition 5.7. Let u be an entropy solution of (23) in Ω ⊂ Rn open, and let μ be the corresponding entropy dissipation measure provided by the kinetic formulation. Let ν be the x-marginal of μ . For any y ∈ Ω and r > 0 let uy,r (x) := u(y + rx),

μ y,r (B × A) := r1−n μ (B × (y + rA)), ν y,r (A) := r1−n ν (y + rA)

(42)

for all x ∈ Rn and all Borel sets A ⊂ Ω and B ⊂ R. A sequence of rescaled quantities (uy,r , μ y,r , ν y,r ) for r → 0 will be called a blow-up sequence. Notice that the measures μ y,r and ν y,r are also characterized by     x−y y,r 1−n ζ (v, x) d μ (v, x) = r ζ v, d μ (v, x) r R×Rn R×Rn     x−y ϕ (x) d ν y,r (x) = r1−n ϕ d ν (x) n n r R R

(43) (44)

for all ζ ∈ D(R × Rn ) and ϕ ∈ D(Rn ). For simplicity, we will always assume that μ , ν are extended by zero to Rn \ Ω. The bound (39) translates into lim sup ν y,r (B1 (0)) < ∞ ∀y ∈ Ω.

(45)

r→0

We know introduce a class of special solutions of (35), for which the entropy dissipation measure μ has a tensor product form. Definition 5.8. A split state is a triple (u, h, ν ) consisting of • a function u ∈ L∞ (Rn ), • a left-continuous function h ∈ BV(R), n • a nonnegative measure ν ∈ M+ loc (R ) such that for every v ∈ R a(v) · ∇χ (v, u) = h(v)ν

in D  (Rn ).

(46)

The following proposition is a key step in the proof of our Theorem 4.5. Proposition 5.9 (Blow-ups are Split States). Let u be an entropy solution of (23) in some open set Ω ⊂ Rn . Then there exists a set E ⊂ Ω with H n−1 (E) = 0 and with the following property: for every point y ∈ Ω \ E there exists a left-continuous function hy ∈ BV(R) such that   uy,rk −→ u∞ in L1loc (Rn ) =⇒ (u∞ , hy , ν ∞ ) is a split state. ∗ ν y,rk − ν ∞ in Mloc (Rn ) For all y ∈ Ω there exists at least one rk → 0 for which (uy,rk , ν y,rk ) converge.

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Remark 5.10. We emphasize that the function hy only depends on the blow-up point y ∈ Ω \ E, not on the particular blow-up sequence rk → 0. On the contrary, the limits (u∞ , ν ∞ ) may depend on the sequence. Proof. The fact that there always exists a subsequence such that (uy,rk , ν y,rk ) converge as rk → 0 follows from Banach–Alaoglu Theorem and Theorem 5.3, once we have checked that the rescaled measures are locally uniformly bounded. For ν y,r this follows from Lemma 5.5, and ν y,r is the x-marginal of μ y,r . In particular, we have ∗ ν y,r − 0 in Mloc (Rn ) for any blow-up sequence around a point y ∈ J . Then also ∗ μ y,r − 0 in Mloc (R × Rn ); so we can choose hy := 0 in this case. The argument for y ∈ J is slightly more involved. Step 1. By disintegration of measures (see Theorem 2.28 of [2]) there exists a weakly* ν -measurable map H : Ω −→ M+ (R) such that  R×Ω

 

ζ (v, x) d μ (v, x) =

Ω R

ζ (v, x) dHx (v) d ν (x) ∀ζ ∈ D(R × Ω).

Select a countable family S ⊂ D(R) which is dense in D(R) with respect to the uniform topology. For every ψ ∈ S we define a map fψ : Ω −→ R by setting fψ (y) :=

 R

ψ (v) dHy (v) ∀y ∈ Ω.

(47)

Then fψ is ν -measurable and fψ ∈ L1loc (Ω, ν ). Let Leb(ψ ) be the set of Lebesgue points of fψ . By derivation of measures (see Corollary 2.23 in [2]), we have ν (Eψ ) = 0 for the complement Eψ := Ω \ Leb(ψ ). We even have ν (ES ) = 0 for  ES := ψ ∈S Eψ since S is countable. Now fix a point y ∈ Ω \ ES . For all r > 0 we define a linear map Fr by Fr (ζ ) :=

1 ν y,r (B)

 R×B

  ζ (v, x) dHy (v) d ν y,r (x) − d μ y,r (v, x)

for all ζ ∈ D(R × B) with B := B1 (0). Choose ϕ ∈ D(B) and ψ ∈ S . Using definitions (42)–(44), (47) and the decomposition μ = H ν , we obtain         x−y  1  |Fr (ψϕ )| =  ψ (v)ϕ dHy (v) − dHx (v) d ν (x) ν (Br (y)) R×Br (y) r 

 ϕ L∞(Rn ) −

Br (y)

| fψ (y) − fψ (x)| d ν (x) −→ 0 as r → 0,

because y is a Lebesgue point of fψ . Since linear combinations of products ψϕ with ψ ∈ D(R) and ϕ ∈ D(B) are dense in D(R × B), and since |Fr (ζ ) − Fr (ξ )|  2 ζ − ξ L∞(R×B)

∀ζ , ξ ∈ D(R × B),

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we conclude that Fr vanishes in D  (R × B) as r → 0. With (45) this gives 

lim

r→0

R×B

  ζ (v, x) dHy (v) d ν y,r (x) − d μ y,r (v, x) = 0

(48)

for all ζ ∈ D(R × B). Consider now any subsequence rk → 0 with ∗

ν y,rk − ν ∞

in Mloc (Rn ).

Extracting another subsequence if necessary we may then assume that also ∗ μ y,rk − μ∞

in Mloc (R × Rn).

Then (48) implies μ ∞ = Hy ν ∞ in R × B. Obviously, the same argument works for any ball BR (0) instead of B1 (0). Since the limit is uniquely determined, the sequence μ y,rk converges whenever ν y,rk does. If also uy,rk −→ u∞ in L1loc (Rn ), then the pair (u∞ , Hy ν ∞ ) satisfies equation (35). Notice that Hy does not depend on the blow-up sequence, but only on y ∈ Ω \ ES . Step 2. We use the following implication, which holds for any ν ∈ M+ loc (Ω) (see Theorem 2.56 in [2]): For any Borel set E ⊂ Ω and any t ∈ (0, ∞) 

 ν (Br (y)) t lim sup rn−1 r→0

∀y ∈ E

=⇒

ν  tH n−1 E.

(49)

We define E := J ∩ ES and write E = limm→∞ Em with increasing sets   ν (Br (y)) −m  2 Em := y ∈ E : lim sup . rn−1 r→0 Using (49), we obtain H n−1 (Em )  2m ν (Em ) = 0 for all m because ν (E) = 0, by Step 1. Therefore we also have H n−1 (E) = 0. For any y ∈ J \ E there exists at least one subsequence rk → 0 such that uy,rk −→ u∞

in L1loc (Rn ),

∗ ν y,rk − ν∞

in Mloc (Rn )

and ν ∞ = 0. Let ϕ ∈ D(Rn ) with hy (v) := −a(v) ·



Rn ϕ (x) d ν

 Rn

∞ (x)

= 1 and define

∇ϕ (x)χ (v, u∞ (x)) dx

∀v ∈ R.

(50)

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By Step 1, the pair (u∞ , Hy ν ∞ ) satisfies the kinetic equation (35). Thus     ψ (v) dHy (v) ϕ (x) d ν ∞ (x) Rn

=

R



=−

n R×R

R

ψ (v)a(v) · ∇ϕ (x)χ (v, u∞ (x)) dx dv

ψ (v) hy (v) dv ∀ψ ∈ D(R).

By choice of ϕ , we get hy = ∂∂v Hy in D  (R). Notice that for any x ∈ Rn , the map v → χ (v, u∞ (x)) is a BV-function, see definition (34). Its total variation is bounded uniformly in x. Moreover, we have by dominated convergence

χ (v − ε , u∞ ) −→ χ (v, u∞ ) in L1loc (Rn ) as ε → 0+. Therefore the function hy defined by (50) is left-continuous and in BV(R), and the kinetic equation holds pointwise in v ∈ R as desired.  

6 Classification of Split States In this section we first give a complete classification of the simplest possible split states: those for which ν is either vanishing or supported on a hyperplane. These results are then used to study general split states.

6.1 Special Split States: No Entropy Dissipation If there is no entropy dissipation, then the solution is continuous. Lemma 6.1. Let (u, h, ν ) be a split state with hν = 0 in an open set Ω ∈ Rn . Then u is continuous and constant along characteristic lines in Ω:  for all y in the connected component of  ∀x ∈ Ω u(y) = u(x)  (51) x + Ra(u(x)) ∩ Ω that contains x. Proof. For any function g, let Leb(g) denote the set of Lebesgue points of g. Step 1. We first prove that for every ε > 0 and u0 ∈ [0, 1] there exists a δ > 0 such that, for any R > 0 and y ∈ Leb(u) with BR (y) ⊂ Ω         u u0 − ε a.e. in Bδ R (y) . (52) u(y) u0 =⇒   Let ε > 0 and u0 ∈ [0, 1] be given, and assume without loss of generality that y = 0 and R = 1. By nondegeneracy of a we can find n real values u 0 > v1 > v2 > . . . > vn  u 0 − ε

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such that Ra(v1 ) + . . . + Ra(vn ) = Rn . Since 0 ∈ Leb(u) and u(0)  u0 > v1 , by definition of χ (see (34)) we obtain that 0 ∈ Leb(χ (v1 , u)) and χ (v1 , u(0)) = 1. The kinetic equation (46) with v = v1 then implies ∀x ∈ Ra(v1 ) ∩ B1 (0) x ∈ Leb(χ (v1 , u)) and χ (v1 , u(x)) = 1. By monotonicity of v → χ (v, u), then also ∀x ∈ Ra(v1 ) ∩ B1 (0) x ∈ Leb(χ (v2 , u)) and χ (v2 , u(x)) = 1 since v2  v1 . We apply the kinetic equation (46) with v = v2 and find that   ∀x ∈ (Ra(v1 ) ∩ B1 (0)) + Ra(v2) ∩ B1 (0) x ∈ Leb(χ (v2 , u)) and χ (v2 , u(x)) = 1. A simple geometric consideration shows that there exists a δ2 > 0 such that     (Ra(v1 ) ∩ B1 (0)) + Ra(v2) ∩ B1 (0) ⊃ Ra(v1 ) + Ra(v2) ∩ Bδ2 (0). Since by assumption Ra(v1 ) + . . . + Ra(vn ) = Rn , by iterating this argument we obtain the existence of a δ = δn > 0 such that ∀x ∈ Bδ (0) x ∈ Leb(χ (vn , u)) and χ (vn , u(x)) = 1, which implies u  vn  u 0 − ε

a.e. in Bδ (0).

Notice that the value of δ depends only on a, u0 and ε . The opposite inequality in (52) can be proved in a similar fashion, so our claim follows. Step 2. We now prove that for every ε > 0 there exists a δ > 0 such that, for any R > 0 and y ∈ Leb(u) with BR (y) ⊂ Ω |u − u(y)|  ε

a.e. in Bδ R (y).

(53)

This fact follows from Step 1 by a standard compactness argument. Indeed, let ε > 0 be fixed and select finitely many numbers {uk }k such that [0, 1] ⊂



[uk , uk + ε /2].

(54)

k

For each k apply Step 1 with uk and ε /2 instead of u0 and ε , and let δk > 0 be the corresponding radius. Define the minimum δ := mink δk > 0. Now fix some

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point y ∈ Leb(u) ∩ Ω. By (54), there exists k such that u(y) ∈ [uk , uk + ε /2], and in particular we have u(y)  uk . By our choice of δk we get u  uk − ε /2 a.e. in Bδk R (y) ⊃ Bδ R (y). On the other hand, we have uk  u(y) − ε /2, so we finally get u  u(y) − ε

a.e. in Bδ R (y).

The reverse inequality is proved in a similar way. We conclude that there is a locally uniform modulus of continuity in every Lebesgue point of u. Since the Lebesgue points are dense, u admits a continuous representative in Ω.  Step 3. To prove (51), fix x ∈ Ω and let y be in the connected component of x + Ra(u(x)) ∩ Ω that contains x. Since Ω is open, there exists a neighborhood U  y with U ⊂ Ω. Consider now the set C− := Ω ∩



x + Ra(v).

v
For any sequence yk → y with yk ∈ C− ∩ U, let vk be defined by yk ∈ x + Ra(vk ) for all k. We may assume that the lines connecting x and yk are all contained in Ω and that vk → u(x). From (46) we obtain χ (vk , u(yk )) = χ (vk , u(x)) = 1, which is equivalent to u(yk ) > vk . The continuity of u in y then yields u(y) = lim u(yk )  lim vk = u(x). k→∞

k→∞

Similarly, we consider the set C+ := Ω ∩



x + Ra(v).

u(x)v

For any yk → y with yk ∈ C+ ∩ U, let vk be defined by yk ∈ x + Ra(vk ) for all k. We may again assume that the segments connecting x and yk are all contained in Ω and that vk → u(x). From (46) we obtain χ (vk , u(yk )) = χ (vk , u(x)) = 0, which is equivalent to u(yk )  vk . The continuity of u in y then yields u(y) = lim u(yk )  lim vk = u(x). k→∞

k→∞

This proves that indeed u is constant along the characteristic lines in Ω.

 

Proposition 6.2 (Liouville Theorem). Let (u, h, ν ) be a split state such that hν = 0 in all of Rn . Then the function u is constant.

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Proof. From Lemma 6.1 with Ω = Rn we already know that u is continuous. Moreover, from Step 2 of the previous proof we find that for any ε > 0 and any point y ∈ Rn , there exists δ > 0 such that for all R > 0 |u − u(y)|  ε

in Bδ R (y).

Sending R → ∞ we obtain the result since ε was arbitrary.

 

6.2 Special Split States: ν Supported on a Hyperplane These split states are typically obtained from blow-ups at shock points. Lemma 6.3. Let (u, h, ν ) be a split state with h = 0 and ν = H n−1 J , where the support J ⊂ {η · x = 0} is relatively open, for some unit vector η . Then the conclusion of Lemma 6.1 holds with Ω = Rn \ J¯ . Moreover, there exist strong traces u+ and u− that are constant along J , and   h(v) = a(v) · η χ (v, u+ ) − χ (v, u−) ∀v ∈ R a.e. in J . (55) The traces u± and η (up to orientation) are completely determined by h. Proof. The main step consists in proving the existence of strong traces. Step 1. Without loss of generality, we may assume that η = (1, 0, . . . , 0)T . We denote by ˆ the projection onto the last (n − 1) components. Because of (46), the function χ (v, u) is freely transported in the set {η · x > 0} = {x1 > 0}. That is, for any v ∈ R with a1 (v) = 0 and for a.e. (x1 , x) ˆ ∈ R+ × Rn−1      aˆ (v) χ v, u(x1 , x) ˆ = χ v, u 1, xˆ + (1 − x1) . (56) a1 (v) We define    aˆ (v) χ (v, x) ˆ := χ v, u 1, xˆ + ∀xˆ ∈ Rn−1 . a1 (v) +

(57)

By (56) and the Lebesgue lemma, we then obtain that for all R > 0 

ess lim x1 →0+

Bˆ R (0)

    χ v, u(x1 , x) ˆ − χ + (v, x) ˆ  d xˆ

     χ v, u 1, xˆ + (1 − x1) aˆ (v) = ess lim x1 →0+ Bˆ R (0)  a1 (v)     aˆ (v)  d xˆ = 0. − χ v, u 1, xˆ +  a1 (v) 

(58)

This shows that (57) is the upper trace of χ (v, u) in L1loc (Rn−1 ) on the hyperplane {x1 = 0}. Since χ (v, u) takes values in {0, 1} only, we also have that χ + (v, ·) ∈ {0, 1}

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a.e. Notice also that the set of v ∈ R such that a1 (v) = 0 is a null set since a is nondegenerate. Thus (58) holds for a.e. v ∈ R. Let 

ˆ := u+ (x)

R

χ + (v, x) ˆ dv for a.e. xˆ ∈ Rn−1 .

(59)

By (58) and dominated convergence, we then obtain that for all R > 0 

ess lim x1 →0+

Bˆ R (0)

  u(x1 , x) ˆ − u+(x) ˆ  d xˆ = 0.

(60)

That is, the function (59) is the upper trace of u in L1loc (Rn−1 ) on the hyperplane {x1 = 0}. Consider now the set    G := v ∈ R : {xˆ ∈ Rn−1 : u+ (x) ˆ = v} > 0 and notice that G is at most countable. Then (60) implies that ess lim u(x1 , x) ˆ = u+ (x) ˆ for a.e. xˆ ∈ Rn−1 , x1 →0+

and thus for all v ∈ R \ G ess lim χ (v, u(x1 , x)) ˆ = χ (v, u+ (x)) ˆ x1 →0+

for a.e. xˆ ∈ Rn−1 .

By dominated convergence, we obtain for any R > 0 that  

ess lim x1 →0+

R Bˆ R (0)

  χ (v, u(x1 , x)) ˆ − χ (v, u+(x)) ˆ  d xˆ dv = 0.

This shows that the upper trace of χ (v, u) is in fact given by χ (v, u+ ) a.e. The same reasoning can be applied to prove the existence of a lower trace u− in L1loc (Rn−1 ) on the hyperplane, with analogous properties. Step 2. Consider now an even test function ϕ1 ∈ D(R) with 0  ϕ1  1 and ϕ1 (0) = 1, and let ε > 0. Testing the kinetic equation (46) against

ϕ1 (x1 /ε ) ϕˆ (x) ˆ ψ (v) with ϕˆ ∈ D(Rn−1 ) and ψ ∈ D(R), we obtain in the limit ε → 0 that 

R

ψ (v)a(v) · η =



R



 Rn−1

 χ (v, u+ (x)) ˆ − χ (v, u−(x)) ˆ ϕˆ (x) ˆ d xˆ dv

ψ (v)h(v) dv



ˆ Ω

ϕˆ (x) ˆ d x, ˆ

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ˆ := {xˆ ∈ Rn−1 : (0, x) where Ω ˆ ∈ Ω}. Since ϕˆ and ψ were arbitrary,  ˆ and is constant in Ω + − χ (v, u ) − χ (v, u ) ˆ vanishes outside Ω for a.e. v ∈ R. This proves (55), which implies {u+ , u− } = {inf spt h, sup spt h}. Recall that h = 0, by assumption. Finally, there exist v1 , . . . , vn ∈ spt h such that Ra(v1 ) + . . . + Ra(vn ) = Rn because a is nondegenerate. Therefore η is determined up to orientation by the n conditions   h(vk ) = a(vk ) · η χ (vk , u+ ) − χ (vk , u− ) , each of which determines a hyperplane in Rn . This proves the lemma.

 

Proposition 6.4. Let (u, h, ν ) be a split state with h = 0 and ν = H n−1 J , where the support J = {η · x = 0} for some unit vector η . Then the conclusion of Lemma 6.3 holds. Moreover, u is constant on either side of J . Proof. From Lemma 6.3 we already know that there exist strong traces u+ and u− which are constant along the hyperplane. Now fix some point y ∈ J and consider a sequence yk → y with yk ∈ {η · x > 0} and u(yk ) −→ u+ (y). By Lemma 6.1, the function u is continuous and constant along characteristics in Rn \ J . Therefore for any z ∈ y + Ra(u+(y)) with η · z > 0 there exists a sequence zk → z such that zk ∈ yk + Ra(u(yk )) for all k. Then u(z) = lim u(zk ) = lim u(yk ) = u+ (y). k→∞

k→∞

This shows that u = u+ in the upper halfspace {η ·x > 0} since y ∈ J was arbitrary. The same argument applies to the lower halfspace.  

6.3 Special Split States: ν Supported on Half a Hyperplane The following result will be used for second blow-ups. Lemma 6.5. Let (u, h, ν ) be a split state with h = 0 and ν = H

n−1 J ,

J = {η · x = 0, ω · x > 0} for some pair of orthonormal vectors η ⊥ ω . Then the conclusion of Lemma 6.3 holds. Moreover, the vector ω is fixed in the sense that there exists a cone C, which is not a halfspace and depends only on h, such that ω ∈ C. Proof. We explicitly construct the cone C. Step 1. By Lemma 6.3, the function h has the form h(v) = 1(u− ,u+ ] (v) a(v) · η

∀v ∈ R,

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where the constant traces u+ and u− and the unit normal vector η (up to orientation) are completely determined by h. Let I ⊂ (u− , u+ ) be an interval such that a(v)· η = 0 for all v ∈ I. Then, for every v < v¯ in I we have ¯ a(v) · ω a(v) ¯ ·ω  . (61) ¯ a(v) · η a(v) ¯ ·η ¯ We argue by contradiction. From (46) we deduce that for any v ∈ I a(v) · ∇χ (v, u) = H a(v) · η and therefore

n−1

J

⎧ a(v) ⎪ ⎪ , ⎨ 1 in J + R+ a(v) · η χ (v, u) = a(v) ⎪ ⎪ ⎩ 0 in J − R+ . a(v) · η

Recall that u is continuous and constant along characteristics outside J , by Lemma 6.1. Assume now that we can find v < v¯ such that (61) does not hold. By the mean value theorem there also exists v ∈ (v¯ , v) ¯ such that ¯ a(v) · ω a(v) ¯ ·ω a(v) · ω > > . (62) ¯ a(v) · η a(v) · η a(v) ¯ ·η ¯ Fix any point x such that the line L := x + Ra(v)/(a(v) · η ) does not intersect J . Then χ (v, u) is constant along L . Thanks to (62) there exist points     a(v) a(v) ¯ x ∈ L ∩ J + R+ and x¯ ∈ L ∩ J − R+ . ¯ a(v) · η a(v) ¯ ·η ¯ ¯ But using the monotonicity of v → χ (v, u) we obtain the contradiction 1 = χ (v, u(x)) = χ (v, u(x)) = χ (v, u(x) ¯ = χ (v, ¯ u(x)) ¯ = 0. ¯ ¯ ¯   We conclude that the map v → a(v) · ω /(a(v) · η ) is increasing and          a a(v) ·ω η (v) − a (v) · η a(v) · d a(v) · ω 0 =  2 dv a(v) · η a(v) · η   for all v ∈ I. Notice that the set v ∈ (u− , u+ ) : a(v) · η = 0 is open and dense in the interval (u− , u+ ) since a is continuous and nondegenerate. Therefore      a(v) · η a (v) − a (v) · η a(v) · ω  0 ∀v ∈ (u− , u+ ).

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Step 2. We now define the cone     C∗ := R+ a(v) · η a (v) − a (v) · η a(v) : v ∈ (u− , u+ ) , and clearly C∗ ⊂ {η · x = 0}. On the other hand, C∗ cannot be contained in a proper subspace of {η · x = 0}. That is, there cannot exist a ξ ⊥ η with      (63) a(v) · η a (v) − a (v) · η a(v) · ξ = 0 ∀v ∈ (u− , u+ ), since this would allow us to rewrite (63) in the form   d a(v) · ξ = 0 ∀v ∈ I, dv a(v) · η

(64)

where I ⊂ (u− , u+ ) is an open interval with a(v) · η = 0 for all v ∈ I. From (64) we could find a constant c such that a(v) · (ω − cη ) = 0 for all v ∈ I, in contradiction to the nondegeneracy of a. This proves that the cone C∗ must be genuinely (n − 1)dimensional. Let C be the dual cone to C∗ relative to the hyperplane {η · x = 0}. Then C is not a halfspace and ω ∈ C.  

6.4 Classification of General Split States Finally, we completely classify general split states. Proposition 6.6. Let (u, h, ν ) be a split state with hν = 0. Then there exist constants L, g > 0 and an orthonormal coordinate system x1 , . . . , xn (all depending on h only) with the following property: There exist • A constant e ∈ R and • A Lipschitz continuous function w : Rn−2 −→ R with Lip(w) ≤ L such that ν = gH n−1 J for some set J of the form     J = x1 = e or J = x1 = e, xn  w(x2 , . . . , xn−1 ) . Proof. Since the proof is rather technical, we only give a sketch of it and refer the reader to [11] for further details. We proceed in four steps. Step 1. We define the set J ⊂ Rn as above as   ν (Br (x)) > 0 . J := x ∈ Rn : lim sup rn−1 r→0 We can then show that J is contained in two Lipschitz graphs. In particular, J is a codimension-one rectifiable set and ν takes the form ν = gH n−1 J , where g is some Borel function that is strictly positive in the set J .

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Step 2. We then perform a second blow-up around a generic point x ∈ J . By rectifiability of J , we obtain a special split state of the form described in Lemma 6.3, which implies that both the function value g(x) and the unit normal vector η (x) to J (up to orientation) are independent of x, thus constant along J . They are completely determined by the function h. This allows to conclude that J is in fact contained in a single Lipschitz graph. Step 3. Since the unit normal vector η is constant along J , we can show that up to H n−1 -negligible sets J is contained in at most countably many distinct hyperplanes Πk . Each component J ∩ Πk is given as the intersection of 2n Lipschitz supergraphs that are completely determined by h. In particular, each J ∩ Πk is a set of locally finite perimeter. Step 4. We perform a second blow-up at a generic point x ∈ ∂ (J ∩ Πk ). By rectifiability of the boundary, we obtain a split state of the form described in Lemma 6.5, which implies that each J ∩ Πk is contained in a single Lipschitz supergraph with respect to a universal coordinate system x1 , . . . , xn . This forces J to be contained in only one hyperplane. The result follows.  

7 Proof of the Main Theorem In Sect. 6, we gave a complete classification of split states. In order to conclude the proof of Theorem 4.5, however, we need an extra piece of information that is provided by Proposition 7.1 below. Recall that     ν Br (x) J = x ∈ Ω : lim sup >0 , (65) rn−1 r→0 where ν is the x-marginal of the entropy dissipation measure μ . From Proposition 5.9 we know that for a generic point y ∈ J there exists hy ∈ BV(R) leftn continuous, such that the limits (u∞ , ν ∞ ) ∈ L∞ (Rd ) × M+ loc (R ) of any converging blow-up sequence form a split state: for every v ∈ R we have a(v) · ∇χ (v, u∞(x)) = hy (v)ν ∞

in D  (Rn ).

According to Proposition 6.6 we then have ν ∞ = gy H n−1 J ∞ for a constant gy > 0, and in an appropriate coordinate system J ∞ has the form     J ∞ = x1 = e∞ (66) or J ∞ = x1 = e∞ , xn  w∞ (x2 , . . . , xn−1 ) for a Lipschitz continuous function w∞ with Lip(w∞ )  Ly . All quantities with subscript y only depend on the point y ∈ J , while the superscript ∞ indicates a dependence on the particular converging blow-up sequence.

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Proposition 7.1. Let (u∞ , hy , ν ∞ ) be a split state obtained from a converging blowup sequence at a point y ∈ J . In the notation above, we have 0 ∈ J ∞ . Proof. The result will follow from the behavior of a discriminating functional along blow-up sequences. Let x1 , . . . , xn and Ly , gy > 0 be the coordinate system and constants discussed above. They all depend only on the blow-up point y ∈ J . Assuming from now on that y is fixed, we do not write the subscript y anymore to simplify notation. We define the wedge   W := xn  L|(x2 , . . . , xn−1 )| and the (n − 1)-dimensional cone C := W ∩ {x1 = 0}. Since the limit measure ν ∞ of any converging blow-up sequence is supported in a set J ∞ as in (66) with Lip(w∞ )  L, we obtain the following implication: x∈J∞

=⇒

x + C ⊂ J ∞.

(67)

0 and ϕ  (r) < 0 for r ∈ [0, 1), and Now fix a function ϕ ∈ D([0, ∞)) with ϕ (r) >  ϕ (r) = 0 for r ∈ [1, ∞). Then we define b := g C ϕ (|x|) H n−1 (x) and F (ν ) :=

1 b

 W

ϕ (|x|) d ν (x)

∀ν ∈ M+ (Rn ).

We divide the proof of the proposition into three steps. Step 1. The functional F has the following properties: For any limit ν ∞ = gH n−1 J ∞ from a converging blow-up sequence 1) 2) 3) 4)

F (ν ∞ ) ∈ [0, 1]. F (ν ∞ ) = 1 if and only if 0 ∈ J ∞ . There exists R > 0 such that F (ν ∞ ) = 0 implies ν ∞ (BR (0)) = 0. For the rescaled measure (ν ∞ )0,s defined as in (42) we have    d  F (ν ∞ )0,s  0,  ds s=1 with equality if and only if F (ν ∞ ) ∈ {0, 1}.

Indeed, (1) is obvious from the definitions and the shape of ν ∞ . If now 0 ∈ J ∞ , then C ⊂ J ∞ because of (67), and thus F (ν ∞ ) = 1. On the other hand, if F (ν ∞ ) = 1, then J ∞ ⊂ {x · η = 0} since ϕ is strictly decreasing, and even (W ∩ J ∞ ) ∩ B1 (0) = C ∩ B1 (0), up to H n−1 -negligible sets. Since J ∞ is closed, we obtain that 0 ∈ J ∞ . This proves (2). Assume now that F (ν ∞ ) = 0. Then (W ∩ J ∞ ) ∩ B1 (0) = ∅, up to H n−1 -negligible sets, and then ν ∞ (BR (0)) = 0 for some constant R > 0 that does

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not depend on the particular measure ν ∞ . This is statement (3). Finally, by definition (42) of rescaled measures we have  1    ϕ (|x|) d (ν ∞ )0,s (x) F (ν ∞ )0,s = b W    |x| 1 ϕ = n−1 d ν ∞ (x) bs s W    |x| g = n−1 ϕ dH n−1 (x). ∞ bs s W ∩J We differentiate with respect to s and pass to polar coordinates. This gives      ∞ 0,s  d  g (n − 1)ϕ (|x|) + ϕ (|x|)|x| dH n−1 (x) = − F ( ν )  ds s=1 b W ∩J ∞   g 1 d  n−1 (68) =− r ϕ (r) ω (r) dr, b 0 dr where   (W ∩ J ∞ ) ∩ ∂ Br (0) ω (r) = . rn−2 By some technical argument, for which we refer the reader to [11], one can show that the map r → ω (r) is monotone increasing. Integrating by parts in (68), we then obtain the first part of (4). Notice that (68) vanishes if and only if ω (r) is constant for a.e. r ∈ [0, 1], which means that   either H n−2 (W ∩ J ∞ ) ∩ ∂ Br (0) > 0 for a.e. r ∈ [0, 1]   or H n−2 (W ∩ J ∞ ) ∩ ∂ Br (0) = 0 for a.e. r ∈ [0, 1]. H

n−2

Since J ∞ is closed, this in turn is equivalent to either 0 ∈ J ∞

or (W ∩ J ∞ ) ∩ B1 (0) = ∅,

and then (4) follows by definition of F and (2). Step 2. We now consider the behavior of the functional F under rescaling. We define f (r) := F (ν y,r ) for r > 0, where the measure ν y,r is given in (42). If rk → 0 ∗ is a sequence such that ν y,rk − ν ∞ in M+ (Rn ), then lim f (rk ) = F (ν ∞ ),    d lim rk f  (rk ) =  F (ν ∞ )0,s . k→∞ ds s=1 k→∞

(69) (70)

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Indeed, notice that for the interior W˚ and closure W¯ of the wedge W  

˚ W

ϕ (|x|) d ν ∞ (x)  lim inf k→∞

∞

W¯



ϕ (|x|) d ν (x)  lim sup

W



k→∞

W

ϕ (|x|) d ν y,rk (x) = lim inf f (rk ), k→∞

ϕ (|x|) d ν

y,rk

(x) = lim sup f (rk ) k→∞

(see Example 1.63 in [2]). This implies (69) because the limit measure ν ∞ does not concentrate mass on the boundary ∂ W of W since  ∂W

ϕ (|x|) d ν ∞ (x)  g

 ∂ W ∩{x1 =e∞ }

ϕ (|x|) dH n−1 (x) = 0,

see (66). Notice also that f (sr) = F ((ν y,r )0,s ) for all r, s > 0. We compute    d   r f (r) =  F (ν y,r )0,s ds s=1     1 d  ϕ (|x|) d (ν y,r )0,s (x) = b ds s=1 W     |x| 1 1 d  ϕ = d ν y,r (x) b ds s=1 sn−1 W s    1 (n − 1)ϕ (|x|) + ϕ (|x|)|x| d ν y,r (x). =− b W

(71)

Repeating the arguments above for (71), we obtain (70). Step 3. We now prove that for every δ > 0 there exist ε > 0 and R > 0 such that for every r < R the following implication holds: f (r) ∈ [δ , 1 − δ ]

=⇒

r f  (r)  ε .

(72)

We argue by contradiction: Assume that there exist δ > 0 and rk → 0 with f (rk ) ∈ [δ , 1 − δ ] and rk f  (rk ) < 1/k for all k. Up to a subsequence, we may suppose ∗ that ν y,rk − ν ∞ in Mloc (Rn ). From Step 2, we obtain    d  F (ν ∞ ) ∈ [δ , 1 − δ ] and F (ν ∞ )0,s = 0,  ds s=1 but this obviously contradicts Statement (4) of Step 1. This proves (72). Now fix any δ > 0 and find ε > 0 and R > 0 satisfying (72). Assume that for some r0 < R we have f (r0 ) ∈ [δ , 1 − δ ]. Because of (72), we then have f (r) ≤ f (r0 ) − ε log(r0 /r)

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for all r with f ([r, r0 ]) ⊂ [δ , 1 − δ ]. Therefore we can find a number 0 < r1 < r0 such that f (r1 ) < δ . Applying (72) again we then conclude that f (r) < δ for all r < r1 . In summary, we have shown the following implication: lim inf f (r) < 1 − δ r→0

=⇒

lim sup f (r)  δ . r→0

Since δ > 0 was arbitrary, this finally proves that either

lim f (r) = 0

r→0

or

lim f (r) = 1.

r→0

In particular, we have the same limit for all blow-up sequences rk → 0. Using the Statements (2) and (3) of Step 1, we obtain the following alternative: either 0∈J∞ or ν ∞ (BR (0)) = 0

∀ν ∞ ∀ν ∞ .

(73)

We want to rule out the second possibility. To achieve this, notice that if ν ∞ is the limit measure of some converging blow-up sequence rk → 0, then the rescaled measure (ν ∞ )0,s is the limit of the blow-up sequence srk → 0, for any s > 0. If now the second possibility in (73) holds, then  ∞ 0,s  ν ∞ (BsR (0)) (ν ) (BR (0)) = = 0 ∀s > 0. sn−1 This implies ν ∞ = 0, in contradiction to our choice of y ∈ J .

 

Proof (of Theorem 4.5). We divide the proof into three steps. Step 1. We apply Theorem 5.6 to prove that the set J defined in (65) is codimension-one rectifiable: From Propositions 6.6 and 7.1 we know that for H n−1 -a.e. y ∈ J there exist constants Ly , gy > 0 and an orthonormal coordinate ∗ system x1 , . . . , xn such that, for any sequence rk → 0 with ν y,rk − ν ∞ in Mloc (Rn ), we have ν ∞ = gy H n−1 J ∞ with 0 ∈ J ∞ ⊂ {x1 = 0}. Now we argue as follows: We define the (n − 1)-dimensional cone   Cy := x1 = 0, xn  Ly |(x2 , . . . , xn−1 )| ⊂ J ∞ . Then we can estimate lim inf k→∞

ν (Brk (y)) = lim inf ν y,rk (B1 (0))  ν ∞ (B1 (0))  gy H k→∞ rkn−1 ∗

n−1

  Cy ∩ B1 (0) > 0,

by weak* convergence of ν y,rk − ν ∞ and  openness of B1 (0)  (see Example 1.63 in [2]). On the other hand, defining Cy := |x1 |  |(x2 , . . . , xn )| we have       ν (y + Cy ) ∩ Brk (y) lim sup  lim sup ν y,rk Cy ∩ B¯ 1 (0)  ν ∞ Cy ∩ B¯ 1 (0) = 0, n−1 rk k→∞ k→∞

Regularizing Effect of Nonlinearity in Multi-d Scalar Conservation Laws

111

by compactness of Cy ∩ B¯ 1 (0) (see again Example 1.63 in [2]). This gives (41) and (40). Since rk → 0 and y ∈ J were arbitrary, rectifiability follows. Step 2. Fix a point y ∈ J . Then all rescaled measures ν y,r converge to the zero measure as r → 0, and the only split states (u∞ , hy , ν ∞ ) that can be obtained from the blow-up are those satisfying ν ∞ = 0. By Proposition 6.2, the function u∞ must be constant. We claim that y ∈ J is a point of vanishing mean oscillation. Indeed, let rk → 0 be a sequence such that uy,rk −→ u∞

in L1loc (Rn )

(74)

for some constant u∞ . Substituting x := y + rk z, we can write 



B1 (0)

|uy,rk (z) − u∞ | dz = ωn −

Brk (y)

|u(x) − u∞| dx,

(75)

with ωn the measure of the unit ball in Rn . Then          ∞  ∞ u(x) − −  dx  − u − − − u(z) dz |u(x) − u | dx +    Brk (y)

Brk (y)

Brk (y)



 2−

Brk (y)

Brk (y)

  u(z) dz

|u(x) − u∞| dx −→ 0 as k → ∞,

by (75), (74) and the triangle inequality. This proves the claim. Step 3. We know from Step 1 that J is codimension-one rectifiable. Therefore  we can decompose J = k Jk up to a H n−1 -null set, where each Jk is contained in a Lipschitz graph. This implies in particular that H n−1 Jk is a locally finite measure. From Besicovitch derivation theorem we obtain

ν = gk H n−1 Jk + νs ,

(76)

where νs and gk H n−1 Jk are mutually singular nonnegative measures, and the density gk can be computed for H n−1 -a.e. y ∈ Jk as the limit gk (y) = lim

r→0

ν (Br (y)) H n−1 (Jk ∩ Br (y))

(77)

(see Theorem 2.22 in [2]). Now notice that by definition of J ⊃ Jk lim sup r→0

ν (Br (y)) > 0 for all y ∈ Jk . rn−1

On the other hand, the rectifiability of Jk implies H n−1 (Jk ∩ Br (y)) =1 r→0 rn−1 lim

for H n−1 -a.e. y ∈ Jk

(78)

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(see Theorem 2.83(i) in [2]). Since the limit in (77) exists we conclude that the lim sup in (78) can in fact be replaced by a lim and therefore gk (y) > 0 for H n−1 -a.e. y ∈ Jk . Neglecting the singular part νs in (76) we obtain

ν  gH n−1 J

with g(y) ∈ (0, ∞) for H n−1 -a.e. y ∈ J .

(79)

Fix a generic point y ∈ J . According to Propositions 6.6 and 7.1, there exist a unit vector ηy and a constant gy > 0 such that for any limit measure ν ∞ from a converging blow-up sequence we have the representation

ν ∞ = gy H n−1 J ∞

with J ∞ ⊂ {ηy · x = 0}.

(80)

On the other hand, by (79) and rectifiability of J , we also have

ν ∞  g(y)H n−1 {η (y) · x = 0},

(81)

where η (y) is a unit vector normal to J in y. We conclude that g(y) = gy and η (y) = ±ηy . Moreover, we obtain J ∞ = {η (y) · x = 0}, thereby improving (80). Now Proposition 6.4 yields constants u+ (y) and u− (y) such that  u+ (y) in {η (y) · x > 0} ∞ u = u− (y) in {η (y) · x < 0} for all limit functions u∞ from converging blow-up sequences. We claim that u+ (y) and u− (y) are the strong traces of u in y ∈ J . Indeed, since uy,r −→ u∞

in L1loc (Rn ) as r → 0

by uniqueness of the limit, we can use the substitution x := y + rz to get 

−+



Br (y)

|u(x) − u+(y)| dx + − −

=

1 ωn

Br (y)



B1 (0)

|u(x) − u−(y)| dx

|uy,r (z) − u∞ (z)| dz −→ 0 as r → 0.

This concludes the proof of the main theorem.

 

8 Proofs of the Regularity Theorems In preparation for proving the regularity results stated in Sect. 4, we first collect some facts about matrices of Vandermonde type: Let c1 < . . . < cn be given numbers and consider for any (v, ε ) ∈ R × (0, ∞) the matrix

Regularizing Effect of Nonlinearity in Multi-d Scalar Conservation Laws

113

  V(v, ε ) := a(v + c1ε ) · · · a(v + cnε ) ⎞ ⎛ 1 ··· 1 ⎜ v + c1 ε · · · v + cn ε ⎟ ⎟ ⎜ =⎜ ⎟. .. .. .. ⎠ ⎝ . . . (v + c1ε )n−1 · · · (v + cnε )n−1 Then V−1 (v, ε ) exists and can be computed explicitly. It takes the form 1−n V−1 kl (v, ε ) = ε

pkl (v, ε ) ∏ (ci − ck )

∀k, l ∈ {1, . . . , n},

(82)

i=k

where the pkl (v, ε ) are suitable polynomials in the arguments (v + ci ε ). Sharp estimates for the inverse matrix are known: For all (v, ε ) ∈ R × (0, ∞) 1 + |v + ciε | , |ci − ck | i=k

V−1 (v, ε )  ε 1−n max ∏ k

(83)

with · the maximum absolute row sum norm (see Theorem 3.1 in [15]). The invertibility of V(v, ε ) implies that the vectors a(v + ciε ) span Rn . Another basis is given by the derivatives of the flux a: For any v ∈ R let   W(v) := a(v) a (v) · · · a(n−1) (v) ⎞ ⎛ 1 0 ⎟ ⎜ v 1 ⎟ ⎜ =⎜ . ⎟. . . . . . ⎠ ⎝ . . . vn−1 (n − 1)vn−2 · · · (n − 1)! Since a is a polynomial, Taylor expansion gives the identity a(w) =

n−1

∑ k!1 (w − v)k a(k) (v)

∀(w, v) ∈ R × R.

(84)

k=0

In particular, we can express the matrix V(v, ε ) in terms of W(v). We have ⎞⎛ ⎛ ⎞ 1 0 1 1 ··· 1 ⎟ ⎜ c1 c2 · · · cn ⎟ ⎜ ε ⎟⎜ ⎜ V(v, ε ) = W(v) ⎜ .. . . .. ⎟ .. ⎟ ⎝ .. . ⎠ ⎝ . . ⎠ . . 1 n−1 n−1 n−1 0 c1 c2 · · · cn−1 n (n−1)! ε for all (v, ε ) ∈ R × (0, ∞). Since the last factor is just the invertible matrix V(0, 1), for any k ∈ {0, . . . , n − 1} we can find numbers βlk such that

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a(k) (v) = ε −k ∑ βlk a(v + cl ε )

∀(v, ε ) ∈ R × (0, ∞).

(85)

l=1

Yet another basis of Rn can be obtained by rescaling v in a(v). Let      U(v, ε ) := a v(1 + c1ε ) · · · a v(1 + cnε ) ⎞ ⎛ 1 ··· 1 ⎜ v(1 + c1ε ) · · · v(1 + cnε ) ⎟ ⎟ ⎜ .. .. =⎜ ⎟ .. . ⎠ ⎝ . .  n−1  n−1 v(1 + c1ε ) · · · v(1 + cnε ) for (v, ε ) ∈ R × (0, ∞). Then U(v, ε ) is invertible if v = 0, and ⎛

1

0

0

vn−1

⎜ v U(v, ε ) = ⎜ .. ⎝ .

⎞⎛ ⎟⎜ ⎟⎜ ⎠⎝

··· ··· .. .

1 1 + c1 ε .. .

1 1 + cn ε .. .

⎞ ⎟ ⎟, ⎠

(1 + c1ε )n−1 · · · (1 + cnε )n−1

where the last factor is just the invertible matrix V(1, ε ). Let em denote the mth vector of the standard basis of Rn and recall the representation (82). For any m ∈ {1, . . . , n} we can then find polynomials γlm (ε ) such that n   em = ε 1−n v1−m ∑ γlm (ε )a v(1 + cl ε )

∀(v, ε ) ∈ R × (0, ∞), v = 0.

(86)

l=1

Notice that this formula holds for all v ∈ R if m = 1. Proof (of Proposition 4.3). Repeating the proof of Theorem 5.1 we obtain a(v) · ∇χ (v, u(x)) = 0

in D  (R × Ω),

(87)

because there is no entropy dissipation in the open set Ω ⊂ Rn , by assumption. The function χ is again defined by (34). Then Lemma 6.1 implies that u is continuous in Ω and constant along characteristics. The proof of Proposition 4.3 is just a more quantitative version of the one of Lemma 6.1. 1/(n−1)

(Ω) is equivalent to the following Step 1. We first argue that u ∈ Cloc statement: For all compact subsets K ⊂ Ω there exists CK < ∞ such that ∀ε > 0

∀x ∈ K

∀y ∈ Bε n−1 (x) ∩ K

|u(y) − u(x)|  CK ε .

(88)

The inf over all admissible constants CK coincides with the C1/(n−1)-norm of u over K. One direction is trivial: Fix some K ⊂ Ω compact and choose CK := u C1/(n−1)(K) < ∞.

Regularizing Effect of Nonlinearity in Multi-d Scalar Conservation Laws

115

Pick ε > 0 and x ∈ K arbitrary. Then 1

|u(x) − u(y)|  CK |x − y| n−1  CK ε ,

∀y ∈ Bε n−1 (x) ∩ K

which is (88). For the converse direction, we argue indirectly. Assume that there exists a compact subset K ⊂ Ω such that u C1/(n−1) (K) = ∞. That is, there exist sequences of numbers xk , yk ∈ K such that 1

|u(xk ) − u(yk )| > k|xk − yk | n−1

∀k ∈ N.

Defining εk := |xk − yk |1/(n−1), we obtain that ∀k ∈ N

∃εk , xk

∃yk ∈ Bε n−1 (xk ) ∩ K k

|u(xk ) − u(yk )| > kεk .

Therefore (88) does not hold in this case, which proves the claim. Step 2. Fix a compact subset K ⊂ Ω and let R := dist(K, Rn \ Ω) > 0. For a given point x ∈ K and numbers 0 > c1 > . . . > cn > −1 we define vk := u(x) + ck ε

with k ∈ {1, . . . , n} and ε > 0.

(89)

The vectors a(vk ) form a basis for Rn , therefore the parallelotope   Tˆ (u(x), ε ) :=

n

∑ λi a(u(x) + ciε ) : λi ∈ (−1, 1)

(90)

i=1

is genuinely n-dimensional. The rescaled parallelotope  −1 T (u(x), ε ) := Tˆ (u(x), ε ) R ∑i a(u(x) + ci ε )

(91)

is contained in BR (x). We claim that u(y)  u(x) − ε for all y ∈ x + T (u(x), ε ). Indeed, for any such y there exist numbers λi ∈ (−1, 1) such that n

y = x + α ∑ λi a(vi ). i=1

Since u(x) > v1 we first obtain that χ (v1 , u(x)) = 1, by definition (34). The kinetic equation (87) with v = v1 then implies that

χ (v1 , u(y1 )) = 1 with y1 := x + αλ1a(v1 ). By monotonicity of v → χ (v, u), then also χ (v2 , u(y1 )) = 1 since v2 < v1 . We apply the kinetic equation (87) with v = v2 and find that 2

χ (v2 , u(y2 )) = 1 with y2 := x + α ∑ λi a(vi ). i=1

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Iterating this argument, we obtain n

χ (vn , u(y)) = 1 with y = x + α ∑ λi a(vi ), i=1

which implies u(y) > vn > u(x) − ε . Similar reasoning gives an upper bound. To finish the proof, we must now estimate the radius of the biggest ball contained in the parallelogram T (u(x), ε ) in terms of ε , uniformly in u(x). Step 3. Let K ⊂ Ω be the compact set of the previous step and let R > 0 and ck be the numbers introduced there. For all x ∈ K and ε > 0 let   V(u(x), ε ) := a(u(x) + c1ε ) · · · a(u(x) + cnε ) . To simplify notation a bit, we occasionally do not write the argument x ∈ K. Let cof V(u, ε ) be the cofactor matrix of V(u, ε ), which is defined by (cof V)i j (u, ε ) := (−1)i+ j det Vˆ i j (u, ε )

with i, j ∈ {1, . . . , n}.

Here Vˆ i j (u, ε ) is obtained from V(u, ε ) by eliminating the ith row and the jth column. Let (cof V) j (u, ε ) be the jth column of cof V(u, ε ). By expansion by minors, the scalar product a(u + ci ε ) · (cof V) j (u, ε ) is just the determinant of the matrix that is obtained from V(u, ε ) by replacing its jth column by the vector a(u + ci ε ). For any j we therefore have  detV(u, ε ) if i = j, (92) a(u + ciε ) · (cof V) j (u, ε ) = 0 otherwise, which shows that (cof V) j (u(x), ε ) is orthogonal to the hyperplane H j (x, ε ) := ∑ Ra(u(x) + ciε ) i= j

∀x ∈ K.

The maximal ball contained between the hyperplanes a(u + c j ε ) ± H j (·, ε ) therefore has radius given by the scalar product         1 a(u + c j ε ) · (cof V) j (u, ε )  =  det V(u, ε )  = ,    |(cof V) j (u, ε )| |(cof V) j (u, ε )|  |V−1 (u, ε )| j where V−1 j (u, ε ) is the jth row of the inverse matrix of V(u, ε ). We used again identity (92). It follows that the maximal ball contained in the parallelotope defined in (90) has radius given by min j

1 |V−1 j (u, ε )|

=

1  C V−1 (u, ε ) −1 , max j |V−1 j (u, ε )|

with · some matrix norm and C a constant depending only on n. Recall that on a finite-dimensional vector space all norms are equivalent. Consider now the

Regularizing Effect of Nonlinearity in Multi-d Scalar Conservation Laws

117

parallelotope T (u(x), ε ) defined in (91). Then the radius r(x, ε ) of the maximal ball contained in T (u(x), ε ) is bounded below by −1  r(x, ε )  CR V(u(x), ε )−1 −1 ∑i a(u(x) + ci ε )  DRε n−1 for all x ∈ K and ε ∈ (0, 1), where for some constant Cˆ > 0 we defined  −2(n−1) . D := Cˆ −(n−1) 1 + u L∞(K) We used (83) and the bound |a(v)|  C(1 + |v|)n−1, which holds for all v ∈ R, with C some constant. We therefore conclude that ∀ε ∈ (0, 1) ∀x ∈ K

∀y ∈ BDRε n−1 (x) ∩ K

u(y) − u(x)  ε .

An upper bound can be proved in the same way, and for simplicity we assume that we obtain the same constants. Let CK := (DR)1/(n−1) and εˆ := CK ε . Recalling the equivalence established in Step 1, we find the estimate |u(x) − u(y)|

sup (x,y)∈K×K |x−y|CKn−1

|x − y|

1 n−1

 2 1 ˆ − n−1 1 + u L∞(K) .  CR

(93)

On the other hand, we can use the triangle inequality to get |u(x) − u(y)|

sup (x,y)∈K×K |x−y|CKn−1

|x − y|

1 n−1

 2 1 ˆ − n−1  2CR u L∞(K) 1 + u L∞(K) .

Combining (93) and (94) gives the result. The proposition is proved.

(94)

 

Proof (of Proposition 4.4). We will actually prove a slightly more precise version of the proposition, without the assumption of compact support. By Theorem 5.1 and Remark 5.2 any generalized entropy solution u satisfies a(v) · ∇χ (v, u(x)) =

∂ ∂ v μ (v, x)

in D  (R × Ω),

where χ is defined by (34) and μ is a locally finite measure (vanishing outside the support of u). Given some ϕ ∈ D(Ω) we define for all (v, x) ∈ R × Ω

χˆ (v, x) := ϕ (x)χ (v, u(x)),

(95)

μˆ (v, x) := ϕ (x)μ (v, x),   rˆ(v, x) := a(v) · ∇ϕ (x) χ (v, u(x)).

(96) (97)

To simplify notation, we treat measures as if they were functions, and we assume that χˆ , μˆ and rˆ are extended by zero to R × Rn . Notice that these terms are all

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integrable in R × Rn. They satisfy the kinetic equation a(v) · ∇χˆ (v, x) =

in D  (R × Rn ).

∂ ˆ ∂ v μ (v, x) + rˆ(v, x)

(98)

For all functions g : R × Rn −→ R we define the operator y g(v, x) := g(v, x + y) − g(v, x) ∀(v, x) ∈ R × Rn, y ∈ Rn . ˆ be defined by (95)–(97). For some D > 0 let Lemma 8.1. Let (χˆ , μˆ , g)   A := D

R×Rn

|ˆr| dv dx

  R×Rn

  Rk := D

k+2 R×Rn

∂   χˆ  dv dx ∂v

∂   χˆ  dv dx ∂v

−1   R×Rn

−(k+1)  

|μˆ | dv dx

|μˆ | dv dx

R×Rn

−1

,

−1

for k ∈ {0, . . . , n − 1}. Then there exist constants Ck > 0 such that 1

sup |h|− k+2

 R×Rn

|h|Rk

   Ck (1 + A)

   (k) χˆ (v, x) dv dx ha (v)

R×Rn

∂   χˆ  dv dx ∂v

 k+1  



k+2

R×Rn

|μˆ | dv dx

1 k+2

.

(99)

For simplicity of notation, we do not write the accentˆ in the following. Proof (of Lemma 8.1). The main difficulty is to prove inequality (99) for k = 0. This will be done in the Steps 2 and 3 below. In the first step we show how the case k  1 can be reduced to the case k = 0. Step 1. Choose numbers c1 < . . . < cn and consider the vectors a(v + cl ε ) for (v, ε ) ∈ R × (0, ∞). As noted before, they form a basis of Rn . In particular, for each index k ∈ {1, . . . , n − 1} the derivative a(k) (v) can be expanded in terms of a(v+ cl ε ), see formula (85). We decompose   ha(k) (v) χ (v, x) = χ v, x + hε −k β1k a(v + c1ε ) − χ (v, x) + ...  + χ v, x + hε

−k



n

∑



βlk a(v + cl ε )

l=1

− χ v, x + hε =

n

∑ hε −k βlk a(v+cl ε ) χ

l=1



−k

n−1

∑



βlk a(v + cl ε )

l=1

v, x + hε

−k

l−1

∑

j=1



β jk a(v + c j ε )

.

Regularizing Effect of Nonlinearity in Multi-d Scalar Conservation Laws

119

Integrating with respect to x we can use the triangle inequality and the invariance of the L1 (Rn )-norm under translations to simplify terms. Then  Rn

|ha(k) (v) χ (v, x)| dx 

n

∑



n l=1 R

 

  dx.

hε −k βlk a(v+cl ε ) χ (v, x)

(100)

In each term of the right-hand side we need to adjust the v-argument in order to be able to use (99) for k = 0. Using again the invariance of the L1 (Rn )-norm under translations, we find that for all functions g : R × Rn −→ R          dx   −k k g v, x + hε −k β k a(v + cl ε )  + |g(v, x)| dx g(v, x) l hε β a(v+cl ε ) Rn

Rn

l

=2



Rn

|g(v, x)| dx.

(101)

Applying this inequality with g(v, x) := χ (v, x) − χ (v + cl ε , x) we get  Rn

 

hε −k βlk a(v+cl ε )

  χ (v, x) dx  2

+

|χ (v, x) − χ (v + cl ε , x)| dx

R  n

Rn

   −k k hε β a(v+cl ε ) χ (v + cl ε , x) dx. l

The map v → χ (v, x) has bounded variation uniformly in x. Moreover χ has compact x-support. We integrate with respect to v and obtain  R×Rn

 

 2|cl |ε

  dv dx

hε −k βlk a(v+cl ε ) χ (v, x)

 

   ∂ χ  dv dx + ∂v

R×Rn

 

  dv dx.

hε −k βlk a(v) χ (v, x)

(102)

Assume now that |h|  Rk . We make the ansatz 1

ε := αk |h| k+2

 

∂   χ  dv dx ∂v

−

1 k+2

 

 |μ | dv dx

1 k+2

,

for some αk > 0 that will be chosen later (see page 124). Here we only assume that αk is large enough such that αk−k maxl |βlk |  1. Then   − k   k   k+2 k+2 ∂  2  χ  dv dx |hε −k βlk | = |h| k+2 αk−k max |βlk | | μ | dv dx ∂v 

 

l

|χ | dv dx

2  

∂   χ  dv dx ∂v

−1  

|μ | dv dx

−1

= R0 .

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Recalling (100) we find a constant Bk = Bk (cl , βlk ) such that sup |h|− k+2 1

|h|Rk



 Bk αk

 R×Rn

 

−k + αk 2

|ha(k) (v) χ (v, x)| dv dx

∂   χ  dv dx ∂v  

 k+1   k+2

∂   χ  dv dx ∂v

ˆ − 12 sup |h|

 R×Rn

ˆ |h|R 0



 |μ | dv dx

k 2(k+2)

 

1 k+2

|μ | dv dx

−

k 2(k+2)

   ha(v) χ (v, x) dv dx .

  ˆ

(103)

The last term can be estimated by (99) with k = 0. For all k  1 we get 1

sup |h|− k+2

|h|Rk

 R×Rn

|ha(k) (v) χ (v, x)| dv dx

 k+1    1    k+2 k+2 ∂  −k  χ  dv dx | μ | dv dx .  Bk αk + C0 (1 + A)αk 2 ∂v with  Step 2. Consider now the case k = 0. Select a test function ρ ∈ D(R) −1 R ρ (v) dv = 1 and ρ  0. We define the family of mollifiers ρε (v) := ε for all (v, ε ) ∈ R × (0, ∞). For any w ∈ R we then have the estimate        ρε (v − w) ha(w) χ (w, x) dv dx |ha(w) χ (w, x)| dx =   Rn

Rn





ρ (v/ε )

R

   ρε (v − w)ha(w) χ (w, x) − χ (v, x)  dv dx R×Rn       + (104) ρε (v − w) ha(w) χ (v, x) dv dx.  Rn R

As in (101) we can get rid of the operator ha(w) in the first term on the right-hand side by using the triangle inequality and the invariance of the L1 (Rn )-norm under translations. We integrate (104) with respect to w. Since the function v → χ (v, x) has bounded variation uniformly in x we obtain

Regularizing Effect of Nonlinearity in Multi-d Scalar Conservation Laws

121

 

ρε (v − w) |χ (w, x) − χ (v, x)| dv dw     ρε (z) |χ (w, x) − χ (w + z, x)| dw dz R R    ∂  χ (v, x) dv ρε (z)|z| dz  ∂v

R R

R

(105)

R

for all x ∈ Rn . There exists a constant M1 > 0 such that  R

ρε (z)|z| dz = M1 ε

∀ε > 0.

We arrive at the following estimate: For all (h, ε ) ∈ R × (0, ∞)  R×Rn

 

  ∂ χ  dv dx ∂v       ρε (v − w) ha(w) χ (v, x) dv dw dx. +   R×Rn R

|ha(w) χ (w, x)| dw dx  2M1 ε

(106) Step 3. To estimate the second term on the right-hand side of (106) we define R := ∂∂v μ + r. Without loss of generality we may assume that h > 0. Using (84) and (98), we obtain for all w ∈ R that in D  (R × Rn)     n−1

R v, x + ∑ sl a(l) (w) l=0

n−1

= a(v) · ∇χ v, x + ∑ sl a(l) (w) l=0

=

∑ k!1 (v − w)k a(k)(w) · ∇χ

k=0

=



n−1

n−1

∑

k=0

 k ∂ 1 k! (v − w) ∂ sk χ



n−1

v, x + ∑ sl a(l) (w) l=0

n−1



v, x + ∑ sl a (w) . (l)

l=0

We average in s over the rectangle H := [0, h0 ]×· · ·×[0, hn−1] with suitable numbers hi > 0 to be specified later. By Gauss–Green theorem we obtain    n−1

∑ k!1 (v − w)k h−1 k − hk a(k) (w) χ H

k=0





n−1

v, x + ∑ sl a(l) (w) ds



= − R v, x + ∑ sl a(l) (w) ds. H

l=0

l=k

(107)

122

G. Crippa et al.

Our goal is to single out one term in (107) that does not depend on s anymore. To achieve this, we fix k = 0 and express χ on the left-hand side as   n−1

χ v, x + ∑ sl a(l) (w)

= χ (v, x)

l=1

  + χ v, x + s1 a (w) − χ (v, x) + ... 



n−1





n−2

+ χ v, x + ∑ sl a (w) − χ v, x + ∑ sl a (w) (l)

l=1

l=1



n−1

k−1

(l)



= χ (v, x) + ∑ sk a(k) (w) χ v, x + ∑ sl a(l) (w) . k=1

Recollecting terms, we can now write  n−1

h−1 0 h0 a(w) χ (v, x) = − ∑

k=1

l=1

v, x + ∑ sl a(l) (w) ds

k −1 1 k! (v − w) hk − hk a(k) (w) χ H

l=k









+ − R v, x + ∑ sl a(l) (w) ds. H



(l)

l=1



n−1

k−1

v, x + ∑ sl a (w) ds

+ h−1 0 − h0 a(w) sk a(k) (w) χ H









(108)

l=0

We first integrate (108) in v against the mollifier ρε (v− w) and then take the L1 (Rn )norm with respect to x. Using the triangle inequality we find      −1  ρε (v − w) h0 a(w) χ (v, x) dv dx h0  n R R    

n−1

∑ h−1 k

k=1

R

ρε (v − w)

+ 2h−1 0

k 1 k! |v − w|

 hk  0

Rn

Rn

|hk a(k) (w) χ (v, x)| dx 

|sk a(k) (w) χ (v, x)| dx dsk dv

     ρε (v − w)R(v, x) dv dx. +  Rn R 

(109)

We used the invariance of the L1 (Rn )-norm under translations to get rid of the operator h0 a(w) on the right-hand side. The same argument gives  Rn

|hk a(k) (w) χ (v, x)| dx  2

 Rn

|χ (v, x) − χ (w, x)| dx +

 Rn

|hk a(k) (w) χ (w, x)| dx

Regularizing Effect of Nonlinearity in Multi-d Scalar Conservation Laws

123

and the analogous estimate with hk replaced by sk . We use this inequality in (109) and integrate with respect to w in R. Then     ρε (v − w)|v − w|k |hk a(k) (w) χ (v, x)| dx dv dw R R Rn     ρε (v − w)|v − w|k |χ (v, x) − χ (w, x)| dv dw dx 2 Rn R×R    + ρε (v − w)|v − w|k |hk a(k) (w) χ (w, x)| dx dv dw. R×R

Rn

For the first term on the right-hand side a similar reasoning as for (105) applies. The second term is a convolution in w, which can be estimated with Young’s inequality. Therefore we obtain the following bound     k ρε (v − w)|v − w| |hk a(k) (w) χ (v, x)| dx dv dw R R Rn    ∂   χ  dv dx 2 ρε (z)|z|k+1 dz ∂v R    ρε (z)|z|k dz |hk a(k) (w) χ (w, x)| dw dx. + R

R×Rn

Since ρ ∈ D(R), there exist constants M j > 0 such that  R

ρε (z)|z| j dz = M j ε j

∀ε > 0

for all j  0. For the corresponding term in (109) with sk instead of hk , we can argue in a similar way. Notice that in this case the |v − w|k do not appear and we obtain different powers in ε . For the last term in (109) we find            ρε (v − w)R(v, x) dv dx dw  C ε −1 |μ | dv dx + |r| dv dx ,   R Rn

R

with C > 0 some constant. Collecting all terms we arrive at h−1 0



|h0 a(w) χ (w, x)| dw dx R×Rn     n−1   −1 k+1 −1 1 −1  ∂ χ  dv dx  h0 2M1 ε + ∑ k! hk 2Mk+1 ε + 2h0 2M1 ε ∂v n−1 

+∑

k=1

k=1

1 −1 k! hk

Mk ε k + h−1 0 2



 sup sk ∈[0,hk ]

R×Rn

    + C ε −1 |μ | dv dx + |r| dv dx .

|sk a(k) (w) χ (w, x)| dw dx (110)

124

G. Crippa et al.

For any k ∈ {1, . . . , n − 1} we choose hk in such a way that we obtain the correct homogeneities. With hk := h0 ε k the inequality (110) simplifies to h−1 0

 R×Rn



|h0 a(w) χ (w, x)| dw dx

 C h−1 0 ε n−1

+∑

      ∂ χ  dv dx + ε −1 | μ | dv dx + |r| dv dx ∂v

 

h−1 0

k=1





sup sk ∈[0,hk ]

R×Rn

|sk a(k) (w) χ (w, x)| dw dx ,

(111)

with C > 0 some constant. Assume now that |h0 |  R0 . We make the ansatz 1

ε := h02

 

∂   χ  dv dx ∂v

− 1   2

1 |μ | dv dx

2

,

(112)

which implies the inequalities |hk | = |h0 ε k |  Rk and 

|r| dv dx  Aε −1



|μ | dv dx,

1/2

with A defined above. Multiplying (111) by h0 , we get the estimate − 21

h0

 R×Rn



 B0

|h0 a(w) χ (w, x)| dw dx

1  1     2 2 ∂    |μ | dv dx (1 + A) ∂ v χ dv dx n−1

+∑

1 −1 h0 2 hkk+2

k=1

1 − k+2

ˆ sup |h|

 R×Rn

ˆ |h|R k

    ˆ (k) χ (v, x) dv dx , ha (v)

(113)

with B0 = B0 (ρ ) some constant. Since hk = h0 ε k and by (112), we have 1 −1 h0 2 hkk+2

=

 

∂   χ  dv dx ∂v

−

k 2(k+2)

 

 |μ | dv dx

k 2(k+2)

.

The right-hand side of (113) can now be estimated using (103). For each k  1 we choose αk > 0 large enough such that −k/2

B0 Bk αk Summing up we obtain



1 . 2(n − 1)

Regularizing Effect of Nonlinearity in Multi-d Scalar Conservation Laws

sup |h0 |− 2

1

|h0 |R0

 R×Rn



|h0 a(w) χ (w, x)| dw dx

n−1

 B0 (1 + A) + ∑ Bk αk

  

k=1

− 12

ˆ + 12 sup |h| ˆ |h|R 0

125



∂   χ  dv dx ∂v

 1   2

1 |μ | dv dx

2

|ha(w) χ (w, x)| dw dx. ˆ

R×Rn

 

The last term can then be absorbed into the left-hand side.

We can now conclude the proof of Proposition 4.4. Recall from (86), that for given numbers c1 < . . . < cn the standard basis vector e1 can be expanded in terms of the a(v(1 + cl ε )) for all (v, ε ) ∈ R × (0, ∞). For em with m  2 the expansion (86) contains negative powers of v. Notice, however, that ⎛ ⎞ 0 0 ⎟    ⎜  ⎜1 0 ⎟ a v(1 + cl ε ) = ⎜ . ⎟ a v(1 + cl ε ) . ⎝ .. .. ⎠ n−1 0

0

for all (v, ε ) ∈ R × (0, ∞) and l ∈ {1, . . . , n}. By decreasing the dimension by one, we can then use (86) again to find an expansion of e2 in terms of the derivatives a (v(1 + cl ε )) for all (v, ε ) ∈ R × (0, ∞). This argument can be iterated. There exist polynomials δlm (ε ) such that em = ε m−n

n−m+1

∑

  δlm (ε )a(m−1) v(1 + cl ε ) ∀(v, ε ) ∈ R × (0, ∞)

l=1

  for all m ∈ {1, . . . , n}. If m = n, then a(n−1) v(1 + c1ε ) = (n − 1)!en. Now hem χ (v, x)    = χ v, x + hε m−nδ1m (ε )a(m−1) v(1 + c1ε ) − χ (v, x) + ...  + χ v, x + hε m−n

n−m+1

∑

   δlm (ε )a(m−1) v(1 + cl ε )

l=1

 − χ v, x + hε

n−m m−n

∑

δlm (ε )a(m−1)

  v(1 + cl ε )



l=1

=

n−m+1

∑

l=1

hε m−n δ m (ε )a(m−1) (v(1+cl ε )) l 

χ v, x + hε

l−1

m−n

∑

j=1

δ jm (ε )a(m−1)

   v(1 + c j ε ) .

(114)

126

G. Crippa et al.

We integrate with respect to x and use the triangle inequality and the invariance of the L1 (Rn )-norm under translations to simplify terms. Then we integrate with respect to v. We obtain the estimate 

|hem χ (v, x)| dv dx R×Rn  n−m+1  

∑

R×Rn

l=1

   hε m−n δ m (ε )a(m−1) (v(1+cl ε )) χ (v, x) dv dx.

(115)

l

For each term on the right-hand side we need to adjust the v-argument in order to be able to use (99) with k = m − 1. Proceeding as before, we get      hε m−n δ m (ε )a(m−1) (v(1+cl ε )) χ (v, x) dv dx R×Rn

l



2

   χ (v, x) − χ v(1 + cl ε ), x  dv dx

R×Rn



+

R×Rn

    hε m−n δ m (ε )a(m−1) (v(1+cl ε )) χ v(1 + cl ε ), x  dv dx.

(116)

l

For the first term on the right-hand side recall (95) and (34). Then  R×Rn

=

   χ (v, x) − χ v(1 + cl ε ), x  dv dx



Rn

 |ϕ (x)| 1 R

u(x) 1+cl ε ,u(x)

  (v) dv dx =

cl ε 1 + cl ε



|χ | dv dx.

(117)

Without loss of generality let us assume that |cl |  12 . We require that ε  1, so the right-hand side of (117) is finite. We now make the ansatz 1

ε := |h| n+1

 

|χ | dv dx

− m+1   n+1

∂   χ  dv dx ∂v



m n+1

 

 |μ | dv dx

1 n+1

,

which implies the bound |h| 

 

|χ | dv dx

m+1  

∂   χ  dv dx ∂v

−m  

|μ | dv dx

−1

=: L.

Then there exists a constant C > 0 such that |hε m−n δlm (ε )|  Cm+1 L. We want to apply Lemma 8.1 to estimate the last term in (116). We have     1  sup |h|− n+1 hε m−n δ m (ε )a(m−1) (v(1+cl ε )) χ v(1 + cl ε ), x  dv dx R×Rn

|h|L

1 m−n − n+1 m+1

 CL

l

ˆ − m+1 sup |h| 1

ˆ |h|R m−1

 R×Rn

   (m−1) χˆ (v, x) dv dx, ha (v)

Regularizing Effect of Nonlinearity in Multi-d Scalar Conservation Laws

127

where we defined Rm−1 as in Lemma 8.1 with 

D := C

|χ | dv dx.

Collecting all terms we find a constant C > 0 such that 1 − n+1

sup |h|

 R×Rn

|h|L

|hem χ (v, x)| dv dx

 m      n−m    1 n+1 n+1 n+1 ∂     C(1 + A) |χ | dv dx |μ | dv dx , ∂ v χ dv dx

with A given by Lemma 8.1. For large h we use the triangle inequality and the invariance of the L1 (R × Rn)-norm under translations to get 

sup |h|− n+1 1

R×Rn

|h|L

1 − n+1

 2L =2

 

 

|hem χ (v, x)| dv dx 

|χ | dv dx

|χ | dv dx

 n−m   n+1

∂   χ  dv dx ∂v



m n+1

 

 |μ | dv dx

1 n+1

.

We conclude that there exists a universal constant C > 0 such that 1

sup |h|− n+1

|h|=0



R×Rn

|hem χ (v, x)| dv dx

 m      n−m    1 n+1 n+1 n+1 ∂     C(1 + A) |χ | dv dx |μ | dv dx ∂ v χ dv dx

for all m ∈ {1, . . . , n}. The proposition now follows easily: If u (and thus μ ) has compact support in Ω, then we can choose the cut-off function ϕ that we used in (95)–(97) equal to one on spt u. Then A vanishes (see the definition in Lemma 8.1) and the terms simplify a bit. Since χ has total v-variation equal to two in spt u, we obtain the inequality (29). The proposition is proved.  

References 1. L. Ambrosio, C. De Lellis, and J. Maly. On the chain rule for the divergence of BV like vector fields: Applications, partial results, open problems. In Perspectives in Nonlinear Partial Differential Equations: in honor of Haim Brezis. Birkh¨auser, 2006. 2. L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000.

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Index

μ

E, 62

approximate continuity, 61, 87 approximate differentiability, 36 approximate jump points, 62 area formula, 8 Besov space, 89 blow-up, 95 bounded variation, 84 Bressan’s compactness conjecture, 46 Burgers’ equation, 79 BV Structure Theorem, 61, 62 chain rule, 86 characteristics, 79 Chebyshev inequality, 41 cofactor matrix, 115 commutators, 23 compactness, 83 compatibility condition, 92 compressibility constant of the flow, 42 continuity equation, 5 convergence in measure, 33 convex entropy, 80

Gˆateaux derivative in measure, 33 genuine nonlinearity, 78 global existence, 81 H¨older continuity, 88 jump set, 87, 93 kinetic formulation, 82, 92 L1 -contraction, 85 Lagrangian flows, 16 lifespan, 79 Liouville theorem, 100 Lusin theorem for approximately differentiable maps, 36 maximal function, 40 narrow convergence, 13 nondegeneracy, 83 nonlinearity, 79 ordinary differential equations, 3 Prokhorov compactness theorem, 13

entropy flux, 80 entropy solution, 81

Rademacher theorem, 8 Rankine–Hugoniot condition, 80 rarefaction wave, 80 rectifiability criterion, 94 rectifiable set, 85 regularizing effect, 81 renormalized solution, 19 Riemann problem, 79 Riesz–Fr´echet–Kolmogorov compactness criterion, 48

flux function, 77 Fr´echet derivative in measure, 33

Su , 61 scalar conservation law, 77, 80

Da u, Dc u, D j u, Ds u, 62, 99 Depauw’s example, 20 directional differentiability in measure, 33 discontinuity, 89 disintegration of measures, 96 distributional solutions, 78

129

130 self-similar solution, 80 shock, 80 shock speed, 80 Sobolev space, 83 split state, 95, 98, 101, 103, 105 square root example, 4 stability, 81 structure theorem, 85, 91 superposition solution, 11 tangent measure, 65 tightness, 13

Index trace, 85 transport equation, 5 uniqueness, 81 vandermonde matrix, 112 vanishing mean oscillation, 86 vanishing viscosity method, 80 velocity averaging, 84 weak solution, 79

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