Abstract:

In this paper, we are concerned with the time decay rates of smooth solutions to the Cauchy problem for the compressible Navier-Stokes equations. Under the assumptions that the initial data are close to the constant equilibrium state in $H^l(\mathbb{R}^3)$ with $l\geq3$ and belong to $\dot{H}^{-s}(\mathbb{R}^3)$ with $0 \le s < \frac52$, via decomposing the solutions into the low- and high-frequency parts, we establish the optimal convergence rates of all the derivatives of the solution by combining spectral analysis and the energy method.

Key words: Compressible flow, Optimal decay, Energy method