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Global Well-posedness of Compressible Viscous Surface Waves without Surface Tension

Ting Sun, Yanjin Wang

Abstract

We consider the free boundary problem for a layer of compressible viscous barotropic fluid lying above a fixed rigid bottom and below the atmosphere of positive constant pressure. The fluid dynamics is governed by the compressible Navier--Stokes equations with gravity, and the effect of surface tension is neglected on the upper free boundary. We prove the global well-posedness of the reformulated problem in flattening coordinates near the equilibrium in both two and three dimensions without any low frequency assumption of the initial data. The key ingredients here are the new control of the {\it Eulerian spatial derivatives} of the solution, which benefits a crucial nonlinear cancellation of the highest order spatial regularity of the free boundary, and the time weighted energy estimates.

Global Well-posedness of Compressible Viscous Surface Waves without Surface Tension

Abstract

We consider the free boundary problem for a layer of compressible viscous barotropic fluid lying above a fixed rigid bottom and below the atmosphere of positive constant pressure. The fluid dynamics is governed by the compressible Navier--Stokes equations with gravity, and the effect of surface tension is neglected on the upper free boundary. We prove the global well-posedness of the reformulated problem in flattening coordinates near the equilibrium in both two and three dimensions without any low frequency assumption of the initial data. The key ingredients here are the new control of the {\it Eulerian spatial derivatives} of the solution, which benefits a crucial nonlinear cancellation of the highest order spatial regularity of the free boundary, and the time weighted energy estimates.

Paper Structure

This paper contains 29 sections, 34 theorems, 280 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem.
  3. Related results.
  4. Main results
  5. Reformulation in flattening coordinates.
  6. Statement of the results
  7. Strategy of the proof
  8. Notation
  9. Preliminaries
  10. Perturbed linear form
  11. Interpolation estimates
  12. Nonlinear estimates
  13. Tangential energy evolution
  14. Basic $L^2$ energy estimate
  15. Energy evolution in perturbed linear form
  16. ...and 14 more sections

Key Result

Theorem 2.1

Let $d=2$ or $3$ and $N\geq 4$. Assume that the initial data $q_0, u_{0}\in H^{4N}(\Omega)$ and $\eta_{0}\in H^{4N+1/2}(\Sigma)$ are given such that the necessary $2N$ compatibility conditions for the local well-posedness of neweq are satisfied. There exists an $\varepsilon_{0}>0$ so that if $\mathc

Theorems & Definitions (71)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 61 more