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# Communications in Mathematical Sciences

## Volume 18 (2020)

### Number 4

### Global strong solutions to compressible Navier–Stokes system with degenerate heat conductivity and density-depending viscosity

Pages: 973 – 985

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n4.a4

#### Authors

#### Abstract

We consider the compressible Navier–Stokes system where the viscosity depends on density and the heat conductivity is proportional to a positive power of the temperature under stress-free and thermally insulated boundary conditions. Under the same conditions on the initial data as those of the constant viscosity and heat conductivity case [Kazhikhov-Shelukhin. *J. Appl. Math. Mech.* 41, 1977], we obtain the existence and uniqueness of global strong solutions. Our result can be regarded as a natural generalization of Kazhikhov’s theory for the constant heat conductivity case to the degenerate and nonlinear case under stress-free and thermally insulated boundary conditions.

#### Keywords

compressible Navier–Stokes system, density-depending viscosity, degenerate heat conductivity, stress-free

#### 2010 Mathematics Subject Classification

60F10, 60J75, 62P10, 92C37

Received 14 March 2019

Accepted 13 January 2020

Published 28 July 2020