Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp Hadamard local well-posedness, enhanced uniqueness and pointwise continuation criterion for the incompressible free boundary Euler equations

Mihaela Ifrim, Ben Pineau, Daniel Tataru, Mitchell A. Taylor

TL;DR

The paper develops a comprehensive Hadamard-style local well-posedness theory for the incompressible free boundary Euler equations with zero surface tension on a compact domain, working in low-regularity $H^s$ Sobolev spaces with $s> frac{d}{2}+1$. It introduces a nonlinear distance functional and a balanced elliptic framework to obtain stability, uniqueness, continuous dependence of the data-to-solution map, and refined energy estimates built from Alinhac-style good variables, all within an Eulerian setting that handles moving boundaries without flattening the domain. A sharp pointwise continuation criterion is established, asserting that solutions persist as long as the velocity is controlled in $L^1_t W^{1, abla}$ and the free surface remains in $L^1_t C^{1, rac{1}{2}}$, mirroring the Beale–Kato–Majda criterion in the boundaryless case. The analysis develops a robust suite of tools—balanced elliptic estimates, an extension of Littlewood–Paley theory to moving domains, and a constructive scheme for regular and rough solutions via frequency envelopes—providing a framework likely adaptable to broader free-boundary problems, including MHD in future work.

Abstract

We provide a complete local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$, which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains.

Sharp Hadamard local well-posedness, enhanced uniqueness and pointwise continuation criterion for the incompressible free boundary Euler equations

TL;DR

The paper develops a comprehensive Hadamard-style local well-posedness theory for the incompressible free boundary Euler equations with zero surface tension on a compact domain, working in low-regularity Sobolev spaces with . It introduces a nonlinear distance functional and a balanced elliptic framework to obtain stability, uniqueness, continuous dependence of the data-to-solution map, and refined energy estimates built from Alinhac-style good variables, all within an Eulerian setting that handles moving boundaries without flattening the domain. A sharp pointwise continuation criterion is established, asserting that solutions persist as long as the velocity is controlled in and the free surface remains in , mirroring the Beale–Kato–Majda criterion in the boundaryless case. The analysis develops a robust suite of tools—balanced elliptic estimates, an extension of Littlewood–Paley theory to moving domains, and a constructive scheme for regular and rough solutions via frequency envelopes—providing a framework likely adaptable to broader free-boundary problems, including MHD in future work.

Abstract

We provide a complete local well-posedness theory in based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in and the free surface is in , which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains.
Paper Structure (66 sections, 66 theorems, 758 equations, 1 figure)

This paper contains 66 sections, 66 theorems, 758 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Cauchy problem: scaling, Sobolev spaces and control parameters
  3. Historical comments
  4. An overview of the main results
  5. Uniqueness and stability
  6. Well-posedness
  7. Energy estimates
  8. Low regularity continuation criterion
  9. Outline of the paper
  10. The linearized equations
  11. Function spaces and the geometry of moving domains
  12. Stability estimates and uniqueness
  13. Elliptic theory
  14. Energy estimates
  15. Construction of regular solutions
  16. ...and 51 more sections

Key Result

Theorem 1.3

Let $\epsilon, T>0$ and let $\Omega_0$ be a domain with boundary $\Gamma_0$ of $C^{1,\frac{1}{2}}$ regularity. Then for every divergence free initial data $v_0\in W^{1,\infty}(\Omega_0)$, the free boundary Euler equations with the Taylor sign condition admit at most one solution $(v,\Gamma_t)$ with

Figures (1)

  • Figure 1: Inviscid fluid droplet in a gravitational field, where $\Omega_t$ is the fluid domain and $\Gamma_t$ is its moving boundary.

Theorems & Definitions (143)

  • Definition 1.1: State space
  • Theorem 1.3: Uniqueness
  • Theorem 1.4: Hadamard local well-posedness
  • Theorem 1.5: Energy estimates
  • Definition 1.6
  • Theorem 1.7: Continuation criterion
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 133 more