Methods and Applications of Analysis

Volume 28 (2021)

Number 2

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part I

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

Local classical solutions to the full compressible Navier–Stokes system with temperature-dependent heat conductivity

Pages: 105 – 152

DOI: https://dx.doi.org/10.4310/MAA.2021.v28.n2.a2

Authors

Yue Cao (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)

Yachun Li (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China; and SHL-MAC, Shanghai Jiao Tong University, Shanghai, China)

Shengguo Zhu (§School of Mathematical Sciences, CMA-Shanghai and SHL-MAC, Shanghai Jiao Tong University, Shanghai, China; and Mathematical Institute, University of Oxford, United Kingdom)

Abstract

In this paper, we study the full compressible Navier–Stokes equations in a bounded domain $\Omega \subset \mathbb{R}^3$, where the heat conductivity depends on the temperature $\theta$ in a power law ($\theta^b$ for some constant $b \gt 0$) of Chapman–Enskog. We first prove the existence of the unique strong solution with non-negative mass density and arbitrarily large data, and then lift the regularities to get a classical one. The corresponding proof is nontrivial due to the appearance of the vacuum and the strong nonlinearity of the temperature-dependent heat conductivity. We introduce a new variable $\theta^{b+1}$ to reformulate and simplify the system, and require that the measure of the initial vacuum domain is sufficiently small, for example, the initial vacuum only appears in some one-dimensional curves or two-dimensional surfaces.

Keywords

full compressible Navier–Stokes system, three dimensions, classical solutions, vacuum, temperature-dependent heat conductivity

2010 Mathematics Subject Classification

35A01, 35A09, 35B45, 35B65, 35Q35

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Received 28 August 2020

Accepted 8 February 2021

Published 10 June 2022