Global Existence and Asymptotic Behavior of Large Strong Solutions to the 3D Full Compressible Navier-Stokes Equations with Density-dependent Viscosities

TL;DR

This paper proves the global existence and describes the long-term behavior of large strong solutions to the 3D compressible Navier-Stokes equations with density-dependent viscosities, without restrictions on initial velocity or temperature.

Contribution

It introduces a novel approach for establishing global large solutions with density-dependent viscosities, extending previous results to large initial data and analyzing asymptotic decay.

Findings

Global-in-time strong solutions exist for large initial data.

Solutions converge to equilibrium with explicit decay rates.

First application of Fourier splitting method to Navier-Stokes with variable viscosity.

Abstract

The purpose of this work is to investigate the Cauchy problem of global-in-time existence of large strong solutions to the Navier-Stokes equations for compressible viscous and heat conducting fluids. A class of density-dependent viscosity is considered. By introducing the modified effective viscous flux and using the bootstrap argument, we establish the global existence of large strong solution when the initial density is linearly equivalent to a large constant state. It is worthy of mentioning that, different from the work of Matsumura and Nishida (J. Math. Kyoto Univ., 1980) with small initial perturbation and the work of Huang and Li (Arch. Ration. Mech. Anal., 2018) with small energy but possibly large oscillations, our global large strong solution is uniform-in-time in $H^2$ Sobolev space and the uniform-in-time bounds of both density and temperature are obtained without any restrictions on the size of initial velocity and initial temperature. In addition, when the initial data belongs to $L^{p_0}\cap H^2$ with $p_0\in[1,2]$, we establish the convergence of the solution to its associated equilibrium with an explicit decay rate whether the initial data close to or far away from the equilibrium in the whole space. As a result, we give a specific large strong solution in Sobolev space satisfying the global existence assumptions proposed by Villani (Mem. Amer. Math. Soc., 2009), He, Huang, and Wang (Arch. Ration. Mech. Anal., 2019), Zhang and Zi (Ann. Inst. H. Poincare Anal. Non Lineaire, 2020) in studying the asymptotic behavior of solution, and extend the second above result to the nonisentropic case. This paper considers for the first time the application of Fourier splitting method to Navier-Stokes equations with variable viscosity.