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Global well-posedness and asymptotic behavior of large strong solutions to the 3D full compressible Navier-Stokes equations with temperature-dependent coefficients

Yachun Li, Peng Lu, Zhaoyang Shang

Abstract

It is well known that the global well-posedness of the Navier-Stokes equations with temperature-dependent coefficients is a challenging problem, especially in multi-dimensional space. In this paper, we study the 3D Navier-Stokes equations with temperature-dependent coefficients in the whole space. When the initial density and the initial temperature are linearly equivalent to some large constant states, we establish the first result on the global existence of large strong solution. Moreover, the optimal decay rates of the solution to its associated equilibrium are established when the initial data belong to $L^{p_0}(\mathbb{R}^3)$ for some $p_0\in[1,2]$.

Global well-posedness and asymptotic behavior of large strong solutions to the 3D full compressible Navier-Stokes equations with temperature-dependent coefficients

Abstract

It is well known that the global well-posedness of the Navier-Stokes equations with temperature-dependent coefficients is a challenging problem, especially in multi-dimensional space. In this paper, we study the 3D Navier-Stokes equations with temperature-dependent coefficients in the whole space. When the initial density and the initial temperature are linearly equivalent to some large constant states, we establish the first result on the global existence of large strong solution. Moreover, the optimal decay rates of the solution to its associated equilibrium are established when the initial data belong to for some .

Paper Structure

This paper contains 13 sections, 26 theorems, 364 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Global Existence of Large Strong Solution
  4. Proof of Proposition \ref{['p4.1']}
  5. Uniform zero-order estimates of the solution on $[0,T]$
  6. Uniform estimates of velocity and temperature on $[0,\sigma(T)]$
  7. Time-weighted uniform estimates of velocity and temperature on $[0,T]$
  8. Uniform first-order estimates of density on $[0,T]$
  9. Proof of Proposition \ref{['Lem:10']}
  10. Asymptotic behavior of the solution
  11. Decay rate of $\|(\rho-\bar{\rho}, u, \theta-\bar{\theta})\|_{L^2}$
  12. Decay rate of $\|(\nabla\rho, \nabla u, \nabla\theta)\|_{L^2}$
  13. Decay rate of $\|(\nabla^2\rho, \nabla^2u, \nabla^2\theta)\|_{L^2}$

Key Result

Theorem 1.1

$($Global existence of strong solutions$)$ Let the positive constants $\bar{\rho}, \bar{\theta}>1$, $2\leq\alpha\leq 10$, and $\alpha<\beta<\alpha+2$. Suppose that the initial data $(\rho_0, u_0, \theta_0)$ satisfy for some positive constant $C_0$ independent of $\bar{\rho}$ and $\bar{\theta}$. Then there exist positive constants $L$ and $\eta$ independent of $\bar{\rho}$ and $\bar{\theta}$, such

Theorems & Definitions (50)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Corollary 1
  • Remark 1.3
  • Theorem 1.2
  • Remark 1.4
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • ...and 40 more