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Paralinearization of free boundary problems in fluid dynamics

Thomas Alazard

Abstract

A classical topic in the mathematical theory of hydrodynamics is to study the evolution of the free surface separating air from an incompressible perfect fluid. The goal of this survey is to examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. These equations are different in nature, dispersive or parabolic, but we will see that they can be studied using related tools. In particular, we will discuss a paradifferential approach to these problems.

Paralinearization of free boundary problems in fluid dynamics

Abstract

A classical topic in the mathematical theory of hydrodynamics is to study the evolution of the free surface separating air from an incompressible perfect fluid. The goal of this survey is to examine this problem for two important sets of equations: the water wave equations and the Hele-Shaw equations, including the Muskat problem. These equations are different in nature, dispersive or parabolic, but we will see that they can be studied using related tools. In particular, we will discuss a paradifferential approach to these problems.
Paper Structure (23 sections, 20 theorems, 127 equations, 1 figure)

This paper contains 23 sections, 20 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The equations
  3. The Dirichlet-to-Neumann operator
  4. The Lipschitz threshold of regularity
  5. Elliptic estimates
  6. The shape derivative
  7. Paralinearization of the Dirichlet-to-Neumann operator
  8. Pseudo- and paradifferential operators
  9. Calderón's analysis of the Dirichlet-to-Neumann operator
  10. Paraproduct
  11. Paralinearization formula
  12. A toy quadratic model.
  13. The good unknown of Alinhac
  14. Elliptic factorization
  15. Applications
  16. ...and 8 more sections

Key Result

Theorem 1.1

Consider an irrotational, incompressible velocity field $u=\nabla_{x,y}\phi$ satisfying $\partial_tu+u\cdot\nabla_{x,y}u=-\nabla_{x,y}(P+gy)$ and $P\arrowvert_{y=\eta}=0$. Then $\eta$ and $\psi$ are conjugated:

Figures (1)

  • Figure :

Theorems & Definitions (28)

  • Theorem 1.1: Zakharov Zakharov1968
  • Remark 1.2
  • Proposition 2.1: Noether's theorem implies Rellich inequality
  • proof
  • Proposition 2.2: Alazard-Nguyen
  • Remark 2.3: A trace inequality
  • Remark 2.4: An optimal inequality
  • proof
  • Proposition 2.5: from ABZ3
  • Proposition 2.6: Lannes' shape derivative formula
  • ...and 18 more