Optimal Time-decay Estimates for the Compressible Navier–Stokes Equations in the Critical Lp Framework

The global existence issue for the isentropic compressible Navier–Stokes equations in the critical regularity framework was addressed in Danchin (Invent Math 141(3):579–614, 2000) more than 15 years ago. However, whether (optimal) time-decay rates could be shown in critical spaces has remained an open question. Here we give a positive answer to that issue not only in the L2 critical framework of Danchin (Invent Math 141(3):579–614, 2000) but also in the general Lp critical framework of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010), Chen et al. (Commun Pure Appl Math 63(9):1173–1224, 2010), Haspot (Arch Ration Mech Anal 202(2):427–460, 2011): we show that under a mild additional decay assumption that is satisfied if, for example, the low frequencies of the initial data are in Lp/2(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^{p/2}(\mathbb{R}^{d})}$$\end{document}, the Lp norm (the slightly stronger B˙p,10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot B^0_{p,1}}$$\end{document} norm in fact) of the critical global solutions decays like t-d(1p-14)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${t^{-d(\frac 1p-\frac14)}}$$\end{document} for t→+∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${t\to+\infty,}$$\end{document} exactly as firstly observed by Matsumura and Nishida in (Proc Jpn Acad Ser A 55:337–342, 1979) in the case p = 2 and d = 3, for solutions with high Sobolev regularity.