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The Lagrangian approach to the compressible primitive equations

Matthias Hieber, Yoshiki Iida, Arnab Roy, Tarek Zöchling

Abstract

This article develops the hydrostatic Lagrangian approach to the compressible primitive equations. A fundamental aspect in the analysis is the investigation of the compressible hydrostatic Lamé and Stokes operators. Local strong well-posedness for large data and global strong well-posedness for small data are established under various assumptions on the pressure law, both in the presence and absence of gravity.

The Lagrangian approach to the compressible primitive equations

Abstract

This article develops the hydrostatic Lagrangian approach to the compressible primitive equations. A fundamental aspect in the analysis is the investigation of the compressible hydrostatic Lamé and Stokes operators. Local strong well-posedness for large data and global strong well-posedness for small data are established under various assumptions on the pressure law, both in the presence and absence of gravity.

Paper Structure

This paper contains 8 sections, 15 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Results
  3. Hydrostatic Lagrangian coordinates
  4. Linear Theory: The Hydrostatic Lamé Operator
  5. Local-wellposedness
  6. Global in time existence
  7. The compressible hydrostatic Stokes operator
  8. Estimates for the nonlinear terms

Key Result

Theorem 2.1

Let $T>0$ and assume that $(\xi_0,v_0)$ satisfy assumption (A). Then there exists $0 < a =a(v_0) \leq T$ such that

Theorems & Definitions (32)

  • Remark 1
  • Theorem 2.1: local strong well-posedness
  • Theorem 2.2: global strong well-posedness for small data
  • Theorem 2.3: local strong and global strong well-posedness for small data
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • proof
  • ...and 22 more