Optimal Decay Rates of Higher-Order Derivatives of Solutions to the Compressible Navier–Stokes System

We investigate optimal decay rates of higher-order derivatives of solutions to the 3D compressible Navier–Stokes equations with large initial data, and the main purpose of this work is twofold: First, it is shown that if the initial data belong to H2∩Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2\cap L^p$$\end{document} with 1≤p≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p\le 2$$\end{document}, then the second-order spatial derivative of solution of the compressible Navier–Stokes equations converges to zero at the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-rate (1+t)-34(2p-1)-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+t)^{-\frac{3}{4}(\frac{2}{p}-1)-1}$$\end{document} for 1≤p≤65\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p\le \frac{6}{5}$$\end{document} and (1+t)-32(2p-1)-12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+t)^{-\frac{3}{2}(\frac{2}{p}-1)-\frac{1}{2}}$$\end{document} for 65<p≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{6}{5}< p\le 2$$\end{document}, which improves the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-rate (1+t)-34(2p-1)-12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+t)^{-\frac{3}{4}(\frac{2}{p}-1)-\frac{1}{2}}$$\end{document} in the previous related works. Second, if additionally the initial data satisfy some low-frequency assumption, the optimal lower decay rates of the first- and second-order spatial derivatives of solution are also obtained, which are totally new as compared to the results of the previous related works. Therefore, our decay rates are optimal in this sense.