The Water Waves Problem: Mathematical Analysis and Asymptotics

346 Pages·2013·3.074 MB·English

by  David Lannes

Mathematical
Surveys
and
Monographs
Volume 188
The Water Waves Problem
Mathematical Analysis and Asymptotics
David Lannes
American Mathematical Society
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
Michael A. Singer Michael I. Weinstein
2010 Mathematics Subject Classification. Primary 76B15, 35Q53, 35Q55, 35J05, 35J25.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-188
Library of Congress Cataloging-in-Publication Data
Lannes,David,1973–
Thewaterwavesproblem: mathematicalanalysisandasymptotics/DavidLannes.
pagescm. –(Mathematicalsurveysandmonographs;volume188)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-9470-5(alk.paper)
1.Waterwaves–Mathematicalmodels. 2.Hydrodynamics–Mathematicalmodels. I.Title.
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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(cid:3)1) 79
3.6.2. Asymptotic expansion for small amplitude waves (ε(cid:3)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. TheNeumann-Neumann,Dirichlet-Dirichlet,andNeumann-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(cid:4)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(cid:3)1) 122
5.1.1. Large amplitude models (μ(cid:3)1 and ε=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(cid:3)1 and ε=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(cid:3)1 and ε=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 h (I+μT ) 149
b b
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 V and the velocity at an arbitrary
θ,δ
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. The great lake equations 155
5.6.6. Bottom friction 156
Chapter 6. Shallow Water Asymptotics: Systems. Part 2: Justification 157
6.1. Mathematical analysis of some shallow water models 157
6.1.1. The Nonlinear Shallow Water equations 157
6.1.2. The Green-Naghdi equations 161
6.1.3. The Fully Symmetric Boussinesq systems 164
6.2. Full justification (convergence) of shallow water models 165
6.2.1. Full justification of the Nonlinear Shallow Water equations 165
6.2.2. Full justification of the Green-Naghdi equations 167
6.2.3. Full justification of the Fully Symmetric Boussinesq equations 168
6.2.4. (Almost) full justification of other shallow water systems 169
6.3. Supplementary remarks 172
6.3.1. Energy conservation 172
6.3.1.1. Nonlinear Shallow Water equations 172
6.3.1.2. Boussinesq systems 172
6.3.1.3. Green-Naghdi equations 173
6.3.2. Hamiltonian structure 174
Chapter 7. Shallow Water Asymptotics: Scalar Equations 177
7.1. The splitting into unidirectional waves in one dimension 178
7.1.1. The Korteweg-de Vries equation 178
7.1.2. Statement of the main result 179
7.1.3. BKW expansion 181
7.1.4. Consistency of the approximate solution and secular growth 182
7.1.5. Proof of Theorem 7.1 and Corollary 7.2 185
CONTENTS ix
7.1.6. An improvement 186
7.2. The splitting into unidirectional waves: The weakly transverse case 188
7.2.1. Statement of the main result 190
7.2.2. BKW expansion 191
7.2.3. Consistency of the approximate solution and secular growth 193
7.2.4. Proof of Theorem 7.16 196
7.3. A direct study of unidirectional waves in one dimension 196
7.3.1. The Camassa-Holm regime 197
7.3.1.1. Approximations based on the velocity 197
7.3.1.2. Equations on the surface elevation 199
7.3.1.3. Proof of Theorem 7.24 200
7.3.1.4. The Camassa-Holm and Degasperis-Procesi equations 202
7.3.2. The long-wave regime and the KdV and BBM equations 204
7.3.3. The fully nonlinear regime 205
7.4. Supplementary remarks 206
7.4.1. Historical remarks on the KDV equation 206
7.4.2. Large time well-posedness of (7.47) and (7.51) 208
7.4.3. The case of nonflat bottoms 211
7.4.3.1. Generalization of the KdV equation for nonflat bottoms 211
7.4.3.2. Generalization of the CH/DP equations for nonflat bottoms 211
7.4.4. Wave breaking 212
7.4.5. Full dispersion versions of the scalar shallow water approximations 213
7.4.5.1. One dimensional models 213
7.4.5.2. The weakly transverse case 214
Chapter 8. Deep Water Models and Modulation Equations 217
8.1. A deep water (or full-dispersion) model 218
8.1.1. Derivation 219
8.1.2. Consistency of the deep water (or full-dispersion) model 219
8.1.3. Almost full justification of the asymptotics 222
8.1.4. The case of infinite depth 223
8.2. Modulation equations in finite depth 223
8.2.1. Defining the ansatz 224
8.2.2. Small amplitude expansion of (8.11) 225
8.2.3. Determination of the ansatz 228
8.2.4. The “full-dispersion” Benney-Roskes model 231
8.2.5. The “standard” Benney-Roskes model 232
8.2.6. The Davey-Stewartson model (dimension d=2) 234
8.2.7. The nonlinear Schro¨dinger equation (dimension d=1) 237
8.3. Modulation equations in infinite depth 238
8.3.1. The ansatz 239
8.3.2. The nonlinear Schro¨dinger equation (dimension d=1 or 2) 239
8.4. Justification of the modulation equations 240
8.5. Supplementary remarks 241
8.5.1. Benjamin-Feir instability of periodic wave-trains 241
8.5.2. Full-dispersion Davey-Stewartson and Schro¨dinger equations 243
8.5.3. The nonlinear Schro¨dinger approximation with improved dispersion 244
8.5.4. Higher order approximation: The Dysthe equation 246
8.5.5. The NLS approximation in the neighborhood of |K|H =1.363 247
0