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.

}, issn = {2790-1939}, doi = {https://doi.org/10.4208/cmaa.2022-0022}, url = {http://global-sci.org/intro/article_detail/cmaa/21780.html} }