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Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations
 
T. Alazard École Normale Supérieure, Paris, France
N. Burq Université Paris-Sud, Orsay, France
C. Zuily Université Paris-Sud, Orsay, France
Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations
eBook ISBN:  978-1-4704-4921-6
Product Code:  MEMO/256/1229.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations
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Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations
T. Alazard École Normale Supérieure, Paris, France
N. Burq Université Paris-Sud, Orsay, France
C. Zuily Université Paris-Sud, Orsay, France
eBook ISBN:  978-1-4704-4921-6
Product Code:  MEMO/256/1229.E
List Price: $78.00
MAA Member Price: $70.20
AMS Member Price: $46.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2562018; 108 pp
    MSC: Primary 35; 76

    This memoir is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves, the authors consider solutions such that the curvature of the initial free surface does not belong to \(L^2\).

    The proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and Hölder estimates. The authors first prove tame estimates in Sobolev spaces depending linearly on Hölder norms and then use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these Hölder norms.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Strichartz estimates
    • 3. Cauchy problem
    • A. Paradifferential calculus
    • B. Tame estimates for the Dirichlet-Neumann operator
    • C. Estimates for the Taylor coefficient
    • D. Sobolev estimates
    • E. Proof of a technical result
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2562018; 108 pp
MSC: Primary 35; 76

This memoir is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves, the authors consider solutions such that the curvature of the initial free surface does not belong to \(L^2\).

The proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and Hölder estimates. The authors first prove tame estimates in Sobolev spaces depending linearly on Hölder norms and then use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these Hölder norms.

  • Chapters
  • 1. Introduction
  • 2. Strichartz estimates
  • 3. Cauchy problem
  • A. Paradifferential calculus
  • B. Tame estimates for the Dirichlet-Neumann operator
  • C. Estimates for the Taylor coefficient
  • D. Sobolev estimates
  • E. Proof of a technical result
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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