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Global Strong Well-posedness of the heat-conducting, compressible primitive Equations

Tarek Zöchling

Abstract

The full heat-conducting compressible primitive equations are considered, extending the compressible primitive-equation framework by coupling the temperature through the ideal gas law and the thermal energy balance in the presence of gravity. Global strong well-posedness is established for small perturbations of an equilibrium state, thereby providing a result beyond the isothermal regime. The proof relies on a structural representation of the density in terms of the temperature and on a new Lagrangian transformation.

Global Strong Well-posedness of the heat-conducting, compressible primitive Equations

Abstract

The full heat-conducting compressible primitive equations are considered, extending the compressible primitive-equation framework by coupling the temperature through the ideal gas law and the thermal energy balance in the presence of gravity. Global strong well-posedness is established for small perturbations of an equilibrium state, thereby providing a result beyond the isothermal regime. The proof relies on a structural representation of the density in terms of the temperature and on a new Lagrangian transformation.
Paper Structure (7 sections, 5 theorems, 133 equations)

This paper contains 7 sections, 5 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Analytical strategy
  3. Preliminaries
  4. Main Results
  5. Coordinate Transformation
  6. Linear Theory and nonlinear estimates
  7. Proof of the Main Theorem

Key Result

Theorem 3.1

Let $\tau>0$ and $\overline{\varrho}^\ast,\Theta^\ast>0$. Assume that the initial data $(\overline{\varrho}_0,v_0,\Theta_0)$ satisfy assu:data0. Then there exists such that for every $\varepsilon\in(0,\varepsilon_0)$ the reformulated system eq:full prim reformulated subject to the boundary conditions eq: bc admits a unique strong solution $(\overline{\varrho},v,\Theta)$ with With $\hat{B}(\Theta

Theorems & Definitions (9)

  • Theorem 3.1: Global, strong well-posedness of \ref{['eq:full prim']}
  • Remark 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Corollary 1
  • proof : Proof of \ref{['thm main']}