TY - JOUR
T1 - Decay of the Compressible Navier-Stokes Equations with Hyperbolic Heat Conduction
AU - Wu , Zhigang
AU - Zhou , Wenyue
AU - Li , Yujie
JO - Communications in Mathematical Analysis and Applications
VL - 2
SP - 115
EP - 141
PY - 2023
DA - 2023/06
SN - 2
DO - http://doi.org/10.4208/cmaa.2022-0022
UR - https://global-sci.org/intro/article_detail/cmaa/21780.html
KW - Decay rate, Navier-Stokes equations, hyperbolic heat conduction, energy method.
AB - <p style="text-align: justify;">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.</p>