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The low mach number limit of global solutions to the full compressible Navier-Stokes system in critical Besov spaces with large initial data

Sai Li

Abstract

We are concerned with global existence of regular solutions to full compressible Navier-Stokes equations and their asymptotic behavior when the Mach number is sufficiently small. We establish global existence in critical Besov spaces for arbitrary large initial date provided that the divergence-free component of initial velocity and the difference between initial temperature and density generate a global regular solution to incompressible Boussinesq systems. Moreover, we rigorously justify the convergence to the incompressible model as the Mach number tends to zero. The proof relies on a fine-grained analysis of the high-middle-low frequencies of density, velocity and temperature. Our result can be seen as an improvement on Danchin and He [Math. Ann., 366 (2016), no. 3-4, pp. 1365-1402], including the extension from small initial data to large initial data and new convergence results which hold at the level of critical regularity.

The low mach number limit of global solutions to the full compressible Navier-Stokes system in critical Besov spaces with large initial data

Abstract

We are concerned with global existence of regular solutions to full compressible Navier-Stokes equations and their asymptotic behavior when the Mach number is sufficiently small. We establish global existence in critical Besov spaces for arbitrary large initial date provided that the divergence-free component of initial velocity and the difference between initial temperature and density generate a global regular solution to incompressible Boussinesq systems. Moreover, we rigorously justify the convergence to the incompressible model as the Mach number tends to zero. The proof relies on a fine-grained analysis of the high-middle-low frequencies of density, velocity and temperature. Our result can be seen as an improvement on Danchin and He [Math. Ann., 366 (2016), no. 3-4, pp. 1365-1402], including the extension from small initial data to large initial data and new convergence results which hold at the level of critical regularity.
Paper Structure (8 sections, 13 theorems, 183 equations)

This paper contains 8 sections, 13 theorems, 183 equations.

Table of Contents

  1. Introduction and main result
  2. Preliminaries
  3. Function spaces
  4. Analytical tools in Besov spaces
  5. Linear estimates
  6. The energy functional $E^\varepsilon$ and quantity $M^{\varepsilon,\alpha}_{p,q}$
  7. A prior estimates
  8. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Let and with $\inf_{x\in\mathbb{R}^d} (1+\varepsilon a_0(x))>0.$ If systems equ5 supplement with initial data $\Theta_0=\frac{\vartheta_0-a_0}{\sqrt{2}}$ and $\mathbf{v}_0=\mathbb{P}{\boldsymbol u}_0$ admits a global regular solution $(\Theta,\mathbf{v})$ in the class then there exists constants $\varepsilon_0$ and $C$ depending on $p,q,d,\mu,\lambda,\kappa,a_0,{\boldsymbol u}_0,\vartheta_0,\Th

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • Lemma 3.2
  • ...and 7 more