包邮水波问题:数学分析与渐近:mathematical analysis and asymptotics

Preface Index of notations General notations Matrices and vectors Variables and standard operators Parameter depending quantities Functional spaces Functional spaces on Rd Functional spaces on a domain Ω C Rd+1 Chapter 1.The Water Waves Problem and Its Asymptotic Regimes 1.1.Mathematical formulation 1.1.1.Basic assumptions 1.1.2.The free surface Euler equations 1.1.3.The free surface Bernoulli equations 1.1.4.The Zakharov/Craig-Sulem formulation 1.2.Other formulations of the water waves problem 1.2.1.Lagrangian parametrizations of the free surface 1.2.1.1.Nalimov's formulation in dimension d=1 1.2.1.2.Wu's formulation 1.2.2.Other interface parametrizations and extension to two-fluids interfaces 1.2.3.Variational formulations 1.2.3.1.The geometric approach 1.2.3.2.Luke's variational formulation 1.2.4.Free surface Euler equations in Lagrangian formulation 1.3.The nondimensionalized equations 1.3.1.Dimensionless parameters 1.3.2.Linear wave theory 1.3.3.Nondimensionalization of the variables and unknowns 1.3.4.Nondimensionalization of the equations 1.4.Plane waves, waves packets, and modulation equations 1.5.Asymptotic regimes 1.6.Extension to moving bottoms 1.7.Extension to rough bottoms 1.7.1.Nonsmooth topographies 1.7.2.Rapidly varying topographies 1.8.Supplementary remarks 1.8.1.Discussion on the basic assumptions 1.8.2.Related frameworks Chapter 2.The Laplace Equation 2.1.The Laplace equation in the fluid domain 2.1.1.The equation 2.1.2.Functional setting and variational solutions 2.1.3.Existence and uniqueness of a variational solution 2.2.The transformed Laplace equation 2.2.1.Notations and new functional spaces 2.2.2.Choice of a diffeomorphism 2.2.3.Transformed equation 2.2.4.Variational solutions for data in H1/2 (Rd) 2.3.Regularity estimates 2.4.Strong solutions to the Laplace equation 2.5.Supplementary remarks 2.5.1.Choice of the diffeomorphism 2.5.2.Nonasymptotically flat bottom and surface parametrizations 2.5.3.Rough bottoms 2.5.4.Infinite depth 2.5.5.Nonhomogeneous Neumann conditions at the bottom 2.5.6.Analyticity Chapter 3.The Dirichlet-Neumann Operator 3.1.Definition and basic properties 3.1.1.Definition 3.1.2.Basic properties 3.2.Higher order estimates 3.3.Shape derivatives 3.4.Commutator estimates 3.5.The Dirichlet-Neumann operator and the vertically averaged velocity 3.6.Asymptotic expansions 3.6.1.Asymptotic expansion in shallow-water (μ＜＜1) 3.6.2.Asymptotic expansion for small amplitude waves (ε＜＜1) 3.7.Supplementary remarks 3.7.1.Nonasymptotically flat bottom and surface parametrizations 3.7.2.Rough bottoms 3.7.3.Infinite depth 3.7.4.Small amplitude expansions for nonflat bottoms 3.7.5.Self-adjointness 3.7.6.Invertibility 3.7.7.Symbolic analysis 3.7.8.The Neumann-Neumann, Dirichlet-Dirichlet, and Neumann-Dirichlet operators Chapter 4.Well-posedness of the Water Waves Equations 4.1.Linearization around the rest state and energy norm 4.2.Quasilinearization of the water waves equations 4.2.1.Notations and preliminary results 4.2.2.A linearization formula …… Chapter 5.Shallow Water Asymptotics: Systems. Part 1: Derivation Chapter 6.Shallow Water Asymptotics: Systems. Part 2: Justification Chapter 7.Shallow Water Asymptotics: Scalar Equations Chapter 8.Deep Water Models and Modulation Equations Chapter 9.Water Waves with Surface Tension Appendix A.More on the Dirichlet-Neumann Operator Appendix B.Product and Commutator Estimates Appendix C.Asymptotic Models: A Reader's Digest Bibliography Index