Global well-posedness and optimal time-decay of 3D full compressible Navier-Stokes system
Wenwen Huo, Chao Zhang
TL;DR
The paper establishes global well-posedness for the 3D full compressible Navier–Stokes system under small initial data in $H^{N}$, and derives sharp, optimal time-decay rates for the solution and its derivatives using negative Sobolev norms and Fourier-based techniques. It introduces a scaled reformulation $(n,\boldsymbol{\omega},\vartheta)$, develops robust energy estimates to control high-order norms, and analyzes negative Sobolev norms to obtain decay in $\dot{H}^{-s}$ with $s\in[0,\tfrac{3}{2})$. Moreover, it sharpens decay results for initial data in $H^{N}\cap L^{1}$ by combining low/high frequency decomposition with Green’s function and Duhamel arguments, yielding matching upper and lower bounds of the form $(t+1)^{-{\tfrac{3}{4}}-{\tfrac{k}{2}}}$ for derivatives. The results advance understanding of the long-time behavior of compressible flows and provide precise decay rates for both the solution and its spatial derivatives, with implications for stability and asymptotic analysis of viscous, heat-conductive gases.
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 $\dot{H}^{-s}$ $\left(0\leq s<\frac{3}{2}\right)$ 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.