The sharp time decay rate of the three-dimensional compressible Navier-Stokes-Fourier system

In this paper, we are concerned with the sharp time decay rate of the solution U(t, x) for the compressible Navier-Stokes-Fourier system for x∈R³ to the constant steady state bar U=(barρ>0, 0, barþeta>0) when the initial data are small smooth perturbations near bar UThis is achieved by proving that U(t, x) shares the same decay rate with widetilde Ut, x), the solution to the corresponding linearized problem for the evolution of the initial perturbation.Under a mild non-degenerate condition on the initial perturbations, we show that the L²-norm of U-widetilde U decays at least at the rate of (1+t)^-f54 which is faster than the rate (1+t)^-f34for widetilde U Our method combines the linear sharp time decay rate (for both upper and lower bounds) obtained from the spectral analysis and the carefully designed energy method.