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The Water Waves Problem Mathematical Analysis and Asymptotics Mathematical Surveys and Monographs Volume 188 The Water Waves Problem Mathematical Analysis and Asymptotics David Lannes American Mathematical Society http://dx.doi.org/10.1090/surv/188 The Water Waves Problem Mathematical Analysis and Asymptotics Mathematical Surveys and Monographs Volume 188 The Water Waves Problem Mathematical Analysis and Asymptotics David Lannes American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov MichaelA.Singer MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 76B15, 35Q53, 35Q55, 35J05, 35J25. For additional information and updates on this book, visit www.ams.org/bookpages/surv-188 Library of Congress Cataloging-in-Publication Data Lannes, David, 1973– The water waves problem : mathematical analysis and asymptotics / David Lannes. pages cm. – (Mathematical surveys and monographs ; volume 188) Includes bibliographical references and index. ISBN 978-0-8218-9470-5 (alk. paper) 1.Water waves–Mathematical models. 2. Hydrodynamics–Mathematical models. I. Title. TC172.L36 2013 551.46301515–dc23 2012046540 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 181716151413 Contents Preface xiii Index of notations xvii General notations xvii Matrices and vectors xvii Variables and standard operators xvii Parameter depending quantities xviii Functional spaces xix Functional spaces on Rd xix Functional spaces on a domain Ω ⊂ Rd+1 xix Chapter 1. The Water Waves Problem and Its Asymptotic Regimes 1 1.1. Mathematical formulation 1 1.1.1. Basic assumptions 1 1.1.2. The free surface Euler equations 2 1.1.3. The free surface Bernoulli equations 3 1.1.4. The Zakharov/Craig-Sulem formulation 4 1.2. Other formulations of the water waves problem 5 1.2.1. Lagrangian parametrizations of the free surface 5 1.2.1.1. Nalimov’s formulation in dimension d =1 6 1.2.1.2. Wu’s formulation 7 1.2.2. Other interface parametrizations and extension to two-fluids interfaces 8 1.2.3. Variational formulations 10 1.2.3.1. The geometric approach 11 1.2.3.2. Luke’s variational formulation 12 1.2.4. Free surface Euler equations in Lagrangian formulation 12 1.3. The nondimensionalized equations 13 1.3.1. Dimensionless parameters 13 1.3.2. Linear wave theory 14 1.3.3. Nondimensionalization of the variables and unknowns 16 1.3.4. Nondimensionalization of the equations 18 1.4. Plane waves, waves packets, and modulation equations 20 1.5. Asymptotic regimes 23 1.6. Extension to moving bottoms 25 1.7. Extension to rough bottoms 27 1.7.1. Nonsmooth topographies 27 1.7.2. Rapidly varying topographies 29 1.8. Supplementary remarks 30 1.8.1. Discussion on the basic assumptions 30 v vi CONTENTS 1.8.2. Related frameworks 33 Chapter 2. The Laplace Equation 37 2.1. The Laplace equation in the fluid domain 38 2.1.1. The equation 38 2.1.2. Functional setting and variational solutions 39 2.1.3. Existence and uniqueness of a variational solution 41 2.2. The transformed Laplace equation 42 2.2.1. Notations and new functional spaces 42 2.2.2. Choice of a diffeomorphism 43 2.2.3. Transformed equation 46 2.2.4. Variational solutions for data in H˙ 1/2(Rd)48 2.3. Regularity estimates 50 2.4. Strong solutions to the Laplace equation 55 2.5. Supplementary remarks 56 2.5.1. Choice of the diffeomorphism 56 2.5.2. Nonasymptotically flat bottom and surface parametrizations 56 2.5.3. Rough bottoms 57 2.5.4. Infinite depth 58 2.5.5. Nonhomogeneous Neumann conditions at the bottom 60 2.5.6. Analyticity 60 Chapter 3. The Dirichlet-Neumann Operator 61 3.1. Definition and basic properties 62 3.1.1. Definition 62 3.1.2. Basic properties 64 3.2. Higher order estimates 67 3.3. Shape derivatives 70 3.4. Commutator estimates 76 3.5. The Dirichlet-Neumann operator and the vertically averaged velocity 78 3.6. Asymptotic expansions 79 3.6.1. Asymptotic expansion in shallow-water (μ 1) 79 3.6.2. Asymptotic expansion for small amplitude waves (ε 1) 84 3.7. Supplementary remarks 85 3.7.1. Nonasymptotically flat bottom and surface parametrizations 85 3.7.2. Rough bottoms 85 3.7.3. Infinite depth 86 3.7.4. Small amplitude expansions for nonflat bottoms 89 3.7.5. Self-adjointness 89 3.7.6. Invertibility 89 3.7.7. Symbolic analysis 89 3.7.8. The Neumann-Neumann, Dirichlet-Dirichlet, and Neumann-Dirichlet operators 89 Chapter 4. Well-posedness of the Water Waves Equations 91 4.1. Linearization around the rest state and energy norm 92 4.2. Quasilinearization of the water waves equations 93 4.2.1. Notations and preliminary results 93 4.2.2. A linearization formula 93 CONTENTS vii 4.2.3. The quasilinear system 97 4.3. Main results 101 4.3.1. Initial condition 101 4.3.2. Statement of the theorems 102 4.3.3. Asymptotic regimes 103 4.3.4. Proof of Theorems 4.16 and 4.18 104 4.3.4.1. The mollified quasilinear system 104 4.3.4.2. Symmetrizer and energy 105 4.3.4.3. Energy estimates 106 4.3.4.4. Construction of a solution 109 4.3.4.5. Uniqueness and stability 112 4.3.5. The Rayleigh-Taylor criterion 112 4.3.5.1. Reformulation of the equations 113 4.3.5.2. Comments on the Rayleigh-Taylor criterion (4.56) 113 4.4. Supplementary remarks 115 4.4.1. Nonasymptotically flat bottom and surface parametrizations 115 4.4.2. Rough bottoms 117 4.4.3. Very deep water (μ 1) and infinite depth 118 4.4.4. Global well-posedness 119 4.4.5. Low regularity 120 Chapter 5. Shallow Water Asymptotics: Systems. Part 1: Derivation 121 5.1. Derivation of shallow water models (μ 1) 122 5.1.1. Large amplitude models (μ 1andε = O(1), β = O(1)) 123 5.1.1.1. The Nonlinear Shallow Water (NSW) equations 123 5.1.1.2. The Green-Naghdi (GN) equations√ 125 5.1.2. Medium amplitude models (μ 1andε = O( μ)) 128 5.1.2.1. Large amplitude topography variations: β = O(1)√ 128 5.1.2.2. Medium amplitude topography variations: β = O( μ) 128 5.1.2.3. Small amplitude topography variations: β = O(μ) 128 5.1.3. Small amplitude models (μ 1andε = O(μ)) 129 5.1.3.1. Large amplitude topography variations: β = O(1) 129 5.1.3.2. Small amplitude topography variations: β = O(μ) 129 5.2. Improving the frequency dispersion of shallow water models 131 5.2.1. Boussinesq equations with improved frequency dispersion 132 5.2.1.1. A first family of Boussinesq-Peregrine systems with improved frequency dispersion 132 5.2.1.2. A second family of Boussinesq-Peregrine systems with improved frequency dispersion 134 5.2.1.3. Simplifications for the case of flat or almost flat bottoms 137 5.2.2. Green-Naghdi equations with improved frequency dispersion 139 5.2.2.1. A first family of Green-Naghdi equations with improved frequency dispersion 139 5.2.2.2. A second family of Green-Naghdi equations with improved frequency dispersion 140 5.2.3. The physical relevance of improving the frequency dispersion 140 5.3. Improving the mathematical properties of shallow water models 141 5.4. Moving bottoms 143 5.4.1. The Nonlinear Shallow Water equations with moving bottom 144 viii CONTENTS 5.4.2. The Green-Naghdi equations with moving bottom 145 5.4.3. A Boussinesq system with moving bottom. 145 5.5. Reconstruction of the surface elevation from pressure measurements 146 5.5.1. Hydrostatic reconstruction 147 5.5.2. Nonhydrostatic, weakly nonlinear reconstruction 148 5.6. Supplementary remarks 149 5.6.1. Technical results 149 5.6.1.1. Invertibility properties of hb(I + μTb) 149 5.6.1.2. Invertibility properties of h(I + μT ) 151 5.6.2. Remarks on the “velocity” unknown used in asymptotic models 151 5.6.2.1. Relationship between the averaged velocity V and the velocity at an arbitrary elevation 151 5.6.2.2. Relationship between Vθ,δ and the velocity at an arbitrary elevation 153 5.6.2.3. Recovery of the vertical velocity from ζ and V 153 5.6.3. Formulation in (h, hV ) variables of shallow water models 153 5.6.3.1. The Nonlinear Shallow Water equations 153 5.6.3.2. The Green-Naghdi equations 154 5.6.4. Equations with dimensions 154 5.6.5. The lake and great lake equations 154 5.6.5.1. The lake equations 154 5.6.5.2.
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