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水波问题：数学分析与渐近（影印版）

样章
作者：
David Lannes
定价：
135.00元
ISBN：
978-7-04-055634-6
版面字数：
563.000千字
开本：
16开
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
出版时间：
2021-03-08
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
偏微分方程
暂无
  • 前辅文
  • 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 ˙H 1/2(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
      • 4.2.3. The quasilinear system
    • 4.3. Main results
      • 4.3.1. Initial condition
      • 4.3.2. Statement of the theorems
      • 4.3.3. Asymptotic regimes
      • 4.3.4. Proof of Theorems 4.16 and 4.18
        • 4.3.4.1. The mollified quasilinear system
        • 4.3.4.2. Symmetrizer and energy
        • 4.3.4.3. Energy estimates
        • 4.3.4.4. Construction of a solution
        • 4.3.4.5. Uniqueness and stability
      • 4.3.5. The Rayleigh-Taylor criterion
        • 4.3.5.1. Reformulation of the equations
        • 4.3.5.2. Comments on the Rayleigh-Taylor criterion (4.56)
    • 4.4. Supplementary remarks
      • 4.4.1. Nonasymptotically flat bottom and surface parametrizations
      • 4.4.2. Rough bottoms
      • 4.4.3. Very deep water (μ ≫ 1) and infinite depth
      • 4.4.4. Global well-posedness
      • 4.4.5. Low regularity
  • Chapter 5. Shallow Water Asymptotics: Systems. Part 1: Derivation
    • 5.1. Derivation of shallow water models (μ ≪ 1)
      • 5.1.1. Large amplitude models (μ ≪ 1 and ε = O(1), β = O(1))
        • 5.1.1.1. The Nonlinear Shallow Water (NSW) equations
        • 5.1.1.2. The Green-Naghdi (GN) equations
      • 5.1.2. Medium amplitude models (μ ≪ 1 and ε = O(μ))
        • 5.1.2.1. Large amplitude topography variations: β = O(1)
        • 5.1.2.2. Medium amplitude topography variations: β = O(μ)
        • 5.1.2.3. Small amplitude topography variations: β = O(μ)
      • 5.1.3. Small amplitude models (μ ≪ 1 and ε = O(μ))
        • 5.1.3.1. Large amplitude topography variations: β = O(1)
        • 5.1.3.2. Small amplitude topography variations: β = O(μ)
    • 5.2. Improving the frequency dispersion of shallow water models
      • 5.2.1. Boussinesq equations with improved frequency dispersion
        • 5.2.1.1. A first family of Boussinesq-Peregrine systems with improved frequency dispersion
        • 5.2.1.2. A second family of Boussinesq-Peregrine systems with improved frequency dispersion
        • 5.2.1.3. Simplifications for the case of flat or almost flat bottoms
      • 5.2.2. Green-Naghdi equations with improved frequency dispersion
        • 5.2.2.1. A first family of Green-Naghdi equations with improved frequency dispersion
        • 5.2.2.2. A second family of Green-Naghdi equations with improved frequency dispersion
      • 5.2.3. The physical relevance of improving the frequency dispersion
    • 5.3. Improving the mathematical properties of shallow water models
    • 5.4. Moving bottoms
      • 5.4.1. The Nonlinear Shallow Water equations with moving bottom
      • 5.4.2. The Green-Naghdi equations with moving bottom
      • 5.4.3. A Boussinesq system with moving bottom.
    • 5.5. Reconstruction of the surface elevation from pressure measurements
      • 5.5.1. Hydrostatic reconstruction
      • 5.5.2. Nonhydrostatic, weakly nonlinear reconstruction
    • 5.6. Supplementary remarks
      • 5.6.1. Technical results
        • 5.6.1.1. Invertibility properties of hb(I + μTb)
        • 5.6.1.2. Invertibility properties of h(I + μT)
      • 5.6.2. Remarks on the “velocity” unknown used in asymptotic models
        • 5.6.2.1. Relationship between the averaged velocity V and the velocity at an arbitrary elevation
        • 5.6.2.2. Relationship between Vθ,δ and the velocity at an arbitrary elevation
        • 5.6.2.3. Recovery of the vertical velocity from ζ and V
      • 5.6.3. Formulation in (h, hV) variables of shallow water models
        • 5.6.3.1. The Nonlinear Shallow Water equations
        • 5.6.3.2. The Green-Naghdi equations
      • 5.6.4. Equations with dimensions
      • 5.6.5. The lake and great lake equations
        • 5.6.5.1. The lake equations
        • 5.6.5.2. The great lake equations
      • 5.6.6. Bottom friction
  • Chapter 6. Shallow Water Asymptotics: Systems. Part 2: Justification
    • 6.1. Mathematical analysis of some shallow water models
      • 6.1.1. The Nonlinear Shallow Water equations
      • 6.1.2. The Green-Naghdi equations
      • 6.1.3. The Fully Symmetric Boussinesq systems
    • 6.2. Full justification (convergence) of shallow water models
      • 6.2.1. Full justification of the Nonlinear Shallow Water equations
      • 6.2.2. Full justification of the Green-Naghdi equations
      • 6.2.3. Full justification of the Fully Symmetric Boussinesq equations
      • 6.2.4. (Almost) full justification of other shallow water systems
    • 6.3. Supplementary remarks
      • 6.3.1. Energy conservation
        • 6.3.1.1. Nonlinear Shallow Water equations
        • 6.3.1.2. Boussinesq systems
        • 6.3.1.3. Green-Naghdi equations
      • 6.3.2. Hamiltonian structure
  • Chapter 7. Shallow Water Asymptotics: Scalar Equations
    • 7.1. The splitting into unidirectional waves in one dimension
      • 7.1.1. The Korteweg-de Vries equation
      • 7.1.2. Statement of the main result
      • 7.1.3. BKW expansion
      • 7.1.4. Consistency of the approximate solution and secular growth
      • 7.1.5. Proof of Theorem 7.1 and Corollary 7.2
      • 7.1.6. An improvement
    • 7.2. The splitting into unidirectional waves: The weakly transverse case
      • 7.2.1. Statement of the main result
      • 7.2.2. BKW expansion
      • 7.2.3. Consistency of the approximate solution and secular growth
      • 7.2.4. Proof of Theorem 7.16
    • 7.3. A direct study of unidirectional waves in one dimension
      • 7.3.1. The Camassa-Holm regime
        • 7.3.1.1. Approximations based on the velocity
        • 7.3.1.2. Equations on the surface elevation
        • 7.3.1.3. Proof of Theorem 7.24
        • 7.3.1.4. The Camassa-Holm and Degasperis-Procesi equations
      • 7.3.2. The long-wave regime and the KdV and BBM equations
      • 7.3.3. The fully nonlinear regime
    • 7.4. Supplementary remarks
      • 7.4.1. Historical remarks on the KDV equation
      • 7.4.2. Large time well-posedness of (7.47) and (7.51)
      • 7.4.3. The case of nonflat bottoms
        • 7.4.3.1. Generalization of the KdV equation for nonflat bottoms
        • 7.4.3.2. Generalization of the CH/DP equations for nonflat bottoms
      • 7.4.4. Wave breaking
      • 7.4.5. Full dispersion versions of the scalar shallow water approximations
        • 7.4.5.1. One dimensional models
        • 7.4.5.2. The weakly transverse case
  • Chapter 8. Deep Water Models and Modulation Equations
    • 8.1. A deep water (or full-dispersion) model
      • 8.1.1. Derivation
      • 8.1.2. Consistency of the deep water (or full-dispersion) model
      • 8.1.3. Almost full justification of the asymptotics
      • 8.1.4. The case of infinite depth
    • 8.2. Modulation equations in finite depth
      • 8.2.1. Defining the ansatz
      • 8.2.2. Small amplitude expansion of (8.11)
      • 8.2.3. Determination of the ansatz
      • 8.2.4. The “full-dispersion” Benney-Roskes model
      • 8.2.5. The “standard” Benney-Roskes model
      • 8.2.6. The Davey-Stewartson model (dimension d = 2)
      • 8.2.7. The nonlinear Schr¨odinger equation (dimension d = 1)
    • 8.3. Modulation equations in infinite depth
      • 8.3.1. The ansatz
      • 8.3.2. The nonlinear Schr¨odinger equation (dimension d = 1 or 2)
    • 8.4. Justification of the modulation equations
    • 8.5. Supplementary remarks
      • 8.5.1. Benjamin-Feir instability of periodic wave-trains
      • 8.5.2. Full-dispersion Davey-Stewartson and Schr¨odinger equations
      • 8.5.3. The nonlinear Schr¨odinger approximation with improved dispersion
      • 8.5.4. Higher order approximation: The Dysthe equation
      • 8.5.5. The NLS approximation in the neighborhood of |K|H0 = 1.363
      • 8.5.6. Modulation equations for capillary gravity waves
  • Chapter 9. Water Waves with Surface Tension
    • 9.1. Well-posedness of the water waves equations with surface tension
      • 9.1.1. The equations
      • 9.1.2. Physical relevance
      • 9.1.3. Linearization around the rest state and energy norm
      • 9.1.4. A linearization formula
      • 9.1.5. The quasilinear system
      • 9.1.6. Initial condition
      • 9.1.7. Well-posedness of the water waves equations with surface tension
    • 9.2. Shallow water models (systems) with surface tension
      • 9.2.1. Large amplitude models
      • 9.2.2. Small amplitude models
    • 9.3. Asymptotic models: Scalar equations
      • 9.3.1. Capillary effects and the KdV approximation
      • 9.3.2. The Kawahara approximation
      • 9.3.3. Capillary effects and the KP approximation
      • 9.3.4. The weakly transverse Kawahara approximation
    • 9.4. Asymptotic models: Deep and infinite water
    • 9.5. Modulation equations
  • Appendix A. More on the Dirichlet-Neumann Operator
    • A.1. Shape analyticity of the Dirichlet-Neumann operator
      • A.1.1. Shape analyticity of the velocity potential
      • A.1.2. Shape analyticity of the Dirichlet-Neumann operator
        • A.1.2.1. The case of finite depth
        • A.1.2.2. The case of infinite depth
    • A.2. Self-Adjointness of the Dirichlet-Neumann operator
    • A.3. Invertibility of the Dirichlet-Neumann operator
    • A.4. Remarks on the symbolic analysis of the Dirichlet-Neumann operator
    • A.5. Related operators
      • A.5.1. The Laplace equation with nonhomogeneous Neumann condition at the bottom
      • A.5.2. The Neumann-Neumann, Dirichlet-Dirichlet, and Neumann-Dirichlet operators
      • A.5.3. A generalized shape derivative formula
      • A.5.4. Asymptotic expansion of the Neumann-Neumann operator
      • A.5.5. Asymptotic expansion of the averaged velocity
  • Appendix B. Product and Commutator Estimates
    • B.1. Product estimates
      • B.1.1. Product estimates for functions defined on Rd
      • B.1.2. Product estimates for functions defined on the flat strip S
    • B.2. Commutator estimates
      • B.2.1. Commutator estimates for functions defined in Rd
      • B.2.2. Commutator estimates for functions defined on S
    • B.3. Product and commutator with Ck functions
    • B.4. Product and commutator in uniformly local Sobolev spaces
      • B.4.1. Uniformly local Sobolev spaces
      • B.4.2. Product estimates
      • B.4.3. Commutator estimates
  • Appendix C. Asymptotic Models: A Reader's Digest
    • C.1. What is a fully justified asymptotic model?
    • C.2. Shallow water models
      • C.2.1. Low precision models
      • C.2.2. High precision models
      • C.2.3. Approximation by scalar equations
    • C.3. Deep water and infinite depth models
    • C.4. Modulation equations
      • C.4.1. Modulation equations in finite depth
      • C.4.2. Modulation equations in infinite depth
    • C.5. Influence of surface tension
      • C.5.1. On shallow water models
      • C.5.2. On deep water models and modulation equations
  • Bibliography
  • Index

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