Global well-posedness and optimal time-decay of 3D full compressible Navier-Stokes system

Abstract

In this paper, we investigate the global well-posedness and optimal time-decay of classical solutions for the 3-D full compressible Navier-Stokes system, which is given by the motion of the compressible viscous and heat-conductive gases. First of all, we study the global well-posedness of the Cauchy problem to the system when the initial data is small enough. Secondly, we show the optimal decay rates of the higher-order spatial derivatives of the ${\dot{H}}^{-s}$ $\left(0\le s<\frac{3}{2}\right)$ negative Sobolev norms. Finally, under the assumption that the initial data is bounded in $L^1$-norm, we establish the upper and lower bounds of the optimal decay rates for the classical solutions.