Decay of the Compressible Navier-Stokes Equations with Hyperbolic Heat Conduction

Decay of the Compressible Navier-Stokes Equations with Hyperbolic Heat Conduction

Year:    2023

Author:    Zhigang Wu, Wenyue Zhou, Yujie Li

Communications in Mathematical Analysis and Applications, Vol. 2 (2023), Iss. 2 : pp. 115–141

Abstract

The global solution to the Cauchy problem of the compressible Navier-Stokes equations with hyperbolic heat conduction in dimension three is constructed when the initial data in $H^3$ norm is small. By using several elaborate energy functionals together with the interpolation trick, we simultaneously obtain the optimal $L^2$-decay estimate of the solution and its derivatives when the initial data is bounded in negative Sobolev (Besov) space or $L^1(\mathbb{R}^3).$ Specially speaking, the fluid density, the fluid velocity and the fluid temperature in $L^2$-norm have the same decay rate as the Navier-Stokes-Fourier equations, while the flux $q$ has faster $L^2$-decay rate as $(1+t)^{−2}.$ Our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis for a 8×8 Green matrix of the system. To the best of our knowledge, it is the first result on the large time behavior of this system.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cmaa.2022-0022

Communications in Mathematical Analysis and Applications, Vol. 2 (2023), Iss. 2 : pp. 115–141

Published online:    2023-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    27

Keywords:    Decay rate Navier-Stokes equations hyperbolic heat conduction energy method.

Author Details

Zhigang Wu

Wenyue Zhou

Yujie Li

  1. Space-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction

    Liu, Mengqian

    Wu, Zhigang

    Journal of Mathematical Physics, Vol. 64 (2023), Iss. 10

    https://doi.org/10.1063/5.0146449 [Citations: 1]