The sharp time decay rate of the isentropic Navier-Stokes system in <inline-formula><tex-math id="M1">$ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $</tex-math></inline-formula>

Abstract

<p style='text-indent:20px;'>We investigate the sharp time decay rates of the solution <inline-formula><tex-math id="M2">$ U $</tex-math></inline-formula> for the compressible Navier-Stokes system (1.1) in <inline-formula><tex-math id="M3">$ {\mathop{\mathbb R\kern 0pt}\nolimits}^3 $</tex-math></inline-formula> to the constant equilibrium <inline-formula><tex-math id="M4">$ (\bar\rho>0, 0) $</tex-math></inline-formula> when the initial data is a small smooth perturbation of <inline-formula><tex-math id="M5">$ (\bar\rho,0) $</tex-math></inline-formula>. Let <inline-formula><tex-math id="M6">$ \widetilde U $</tex-math></inline-formula> be the solution to the corresponding linearized equations with the same initial data. Under a mild non-degenerate condition on initial perturbations, we show that <inline-formula><tex-math id="M7">$ \|U-\widetilde U\|_{L^2} $</tex-math></inline-formula> decays at least at the rate of <inline-formula><tex-math id="M8">$ (1+t)^{-\frac54} $</tex-math></inline-formula>, which is faster than the rate <inline-formula><tex-math id="M9">$ (1+t)^{-\frac34} $</tex-math></inline-formula> for the <inline-formula><tex-math id="M10">$ \widetilde U $</tex-math></inline-formula> to its equilibrium <inline-formula><tex-math id="M11">$ (\bar\rho ,0) $</tex-math></inline-formula>. Our method is based on a combination of the linear sharp decay rate obtained from the spectral analysis and the energy estimates.</p>