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

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