Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local well-posedness of strong solutions to the non-isentropic compressible primitive equations with vertical diffusion

Rupert Klein, Jinkai Li, Xin Liu, Edriss S. Titi

Abstract

Due to the absence of dynamical equation in the vertical momentum component of the primitive equations (PEs) of atmospheric dynamics, the vertical component of the velocity can be recovered only from the information on the other physical quantities, while utilizing the hydrostatic balance. This causes one spatial derivative loss while leads to a stronger nonlinearity, comparing to the classic compressible Navier-Stokes equations. As a result, the mathematical analysis on the compressible primitive equations is mathematically more challenging than that on the compressible Navier-Stokes equations. In this paper, we consider the initial-boundary value problem to the non-isentropic compressible primitive equations with only vertical diffusion for the temperature, but without gravity. Local existence and uniqueness as well as the continuous dependence on the initial data of strong solutions are established for any suitably regular initial data. The initial velocity and pressure are assumed here to be of one order derivative higher regularity than that of the initial density.

Local well-posedness of strong solutions to the non-isentropic compressible primitive equations with vertical diffusion

Abstract

Due to the absence of dynamical equation in the vertical momentum component of the primitive equations (PEs) of atmospheric dynamics, the vertical component of the velocity can be recovered only from the information on the other physical quantities, while utilizing the hydrostatic balance. This causes one spatial derivative loss while leads to a stronger nonlinearity, comparing to the classic compressible Navier-Stokes equations. As a result, the mathematical analysis on the compressible primitive equations is mathematically more challenging than that on the compressible Navier-Stokes equations. In this paper, we consider the initial-boundary value problem to the non-isentropic compressible primitive equations with only vertical diffusion for the temperature, but without gravity. Local existence and uniqueness as well as the continuous dependence on the initial data of strong solutions are established for any suitably regular initial data. The initial velocity and pressure are assumed here to be of one order derivative higher regularity than that of the initial density.
Paper Structure (13 sections, 16 theorems, 240 equations)

This paper contains 13 sections, 16 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. The $\epsilon$-regularized system
  3. Some calculus inequalities
  4. The regularized system and local well-posedness
  5. $\epsilon$-independent conditional energy estimates
  6. $\epsilon$-dependent conditional energy estimates
  7. $\epsilon$-independent a priori estimates
  8. Proof of the main result
  9. Appendix A: proof of \ref{['CAL']} and \ref{['COME']}
  10. Appendix B: local existence of the regularized system
  11. Some calculations on $N_i$, $i=1,2,3$
  12. Estimates on the linear system
  13. Solvability of the nonlinear system

Key Result

Theorem 1.1

Given $v_{0}\in H^3(\mathcal{O})$, $\sigma_{0} \in H^2(\mathcal{O})$, and $p_{0}\in H^3(\mathbb T^2),$ such that for two positive numbers $\underline\sigma$ and $\underline p$. Then, there is a positive time $\mathcal{T}_0$, depending only on $\gamma$, $\nu$, $\mu$, $\lambda$, $\underline\sigma$, $\underline p$, and $\|v_{0}\|_{H^3}^2+\|\sigma_{0}\|_{H^2}^2+\|p_0\|_{H^3}^2$, such that system (EQv

Theorems & Definitions (32)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 22 more