Springer INdAM Series

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1 Springer INdAM Series Volume 17 Editor-in-Chief G. Patrizio Series Editors C. Canuto G. Coletti G. Gentili A. Malchiodi P. Marcellini E. Mezzetti G. Moscariello T. Ruggeri

2 More information about this series at

3 Ferruccio Colombini Daniele Del Santo David Lannes Editors Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics 123

4 Editors Ferruccio Colombini Dipartimento di Matematica UniversitJa dipisa Pisa, Italy Daniele Del Santo Dipartimento di Matematica e Geoscienze UniversitJa ditrieste Trieste, Italy David Lannes Institut de Mathématiques de Bordeaux Université de Bordeaux - UMR5251 Talence, France ISSN X ISSN (electronic) Springer INdAM Series ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

5 ...,cosìcomeraia da l un, se si conosce, il cinque e l sei; Par. XV,

6 Preface The present volume is a collection of papers with a strong focus on the study of nonlinear hyperbolic systems, and it addresses general theoretical issues such as symmetrizability, singularities, low regularity and dispersive perturbations. The past decade has been marked by important breakthroughs in the study of physical phenomena where the theory of hyperbolic systems plays a central role. We refer in particular to results in nonlinear optics, shock theory, and fluid mechanics. All these topics are developed in the papers contained in this volume, which also includes two thorough surveys on important research areas. The first, by L. Saint- Raymond, is devoted to the derivation of fluid models as hydrodynamic limits of the mesoscopic Boltzmann equations, while the second, by K. Zumbrun, addresses recent developments in detonation theory. The papers in the volume have all been written by leading experts in their respective fields. They are expanded versions of talks given at the INDAM Workshop Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics, held in September 2015 at Istituto Nazionale di Alta Matematica, Dipartimento di Matematica G. Castelnovo, University of Rome La Sapienza. The workshop was organized in honor of Guy Métivier, on the occasion of his 65th birthday: it is our pleasure to dedicate to him this volume, to which a number of friends and collaborators promptly manifested their willingness to contribute. We would like to take this opportunity to thank all the contributors as well as the participants in the workshop. The Istituto Nazionale di Alta Matematica Francesco Severi and the GDR CNRS 2434 made it possible to hold the workshop through their financial support. We warmly thank them for their generosity. Pisa, Italy Trieste, Italy Talence, France Ferruccio Colombini Daniele Del Santo David Lannes vii

7 Contents Amplitude Equations for Weakly Nonlinear Surface Waves in Variational Problems... 1 Sylvie Benzoni-Gavage and Jean-François Coulombel Global Weak Solutions of PDEs for Compressible Media: A Compactness Criterion to Cover New Physical Situations Didier Bresch and Pierre-Emmanuel Jabin Second Microlocalization and Stabilization of Damped Wave Equations on Tori Nicolas Burq A Few Remarks on Hyperbolic Systems with Zygmund in Time Coefficients Francesco Fanelli Wellposedness of Linearized Taylor Equations in Magnetohydrodynamics Isabelle Gallagher and David Gérard-Varet Strictly Dissipative Nonuniqueness with Corners Guy Métivier and Jeffrey Rauch On the Solvability of Certain Degenerate Partial Differential Operators Alberto Parmeggiani Location and Weyl Formula for the Eigenvalues of Some Non Self-Adjoint Operators Vesselin Petkov A Microscopic Point of View on Singularities in Fluid Models Laure Saint-Raymond ix

8 x Contents Geometric Optics for Rayleigh Pulses in Nonlinear Elasticity Mark Williams Recent Results on Stability of Planar Detonations Kevin Zumbrun

9 Contributors Sylvie Benzoni-Gavage Institut Camille Jordan (CNRS UMR5208), Université Claude Bernard Lyon 1, Villeurbanne Cedex, France Didier Bresch LAMA CNRS UMR 5127, University of Savoie Mont-Blanc, Bat. Le Chablais, Campus scientifique, Chambéry, France Nicolas Burq Laboratoire de Mathématiques d Orsay, Université Paris-Sud, CNRS, Orsay Cedex, France Jean-François Coulombel Laboratoire de Mathématiques Jean Leray, CNRS, Université de Nantes (CNRS UMR6629), Nantes Cedex 3, France Francesco Fanelli Institut Camille Jordan CNRS UMR 5208, Université Claude Bernard Lyon 1, Villeurbanne Cedex, France Isabelle Gallagher Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Diderot, Sorbonne Paris Cité, Paris Cedex 13, France David Gérard-Varet Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Diderot, Sorbonne Paris Cité, Paris Cedex 13, France Pierre-Emmanuel Jabin CSCAMM and Department of Mathematics, University of Maryland, College Park, MD, USA Guy Métivier Institut de Mathématiques de Bordeaux, Université de Bordeaux, Talence Cedex, France Alberto Parmeggiani Department of Mathematics, University of Bologna, Bologna, Italy Vesselin Petkov Institut de Mathématiques de Bordeaux, Université de Bordeaux, Talence Cedex, France Jeffrey Rauch Department of Mathematics, University of Michigan, Ann Arbor, MI, USA xi

10 xii Contributors Laure Saint-Raymond Département de Mathématiques et Applications, Ecole Normale Supérieure & Université Pierre et Marie Curie, Paris 6, France Mark Williams Mathematics Department, University of North Carolina, Chapel Hill, NC, USA Kevin Zumbrun Department of Mathematics, Indiana University, Bloomington, IN, USA