Large-time behavior of the spherically symmetric compressible Navier–Stokes equations with degenerate viscosity coefficients

This paper is concerned with the vacuum free boundary problem for the compressible spherically symmetric Navier–Stokes equations with an external force and degenerate viscosities in Rn(n≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{n}(n\ge 2)$$\end{document}. When the initial data are a small perturbation of the stationary profile and the viscosity coefficients are proportional to ρθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \rho ^{\theta } $$\end{document} with θ∈(0,2(γ-1))∩(0,γ2]n=2(0,γ2]n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in {\left\{ \begin{array}{ll} (0,2(\gamma -1))\cap (0,\frac{\gamma }{2}]&{}n=2\\ (0,\frac{\gamma }{2}]&{}n\ge 3 \end{array}\right. }$$\end{document}, a result on the global existence as well as sharper time decay rates of the weak solution is obtained which improves the one in Wei et al. (SIAM J Math Anal 40:869–904, 2008). The proof is based on some weighted energy estimates, and in our analysis, no smallness constraint is prescribed upon the derivatives of the initial data. It is also worth pointing out that our result covers the interesting case of the Saint-Venant shallow water model (i.e., γ=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =2$$\end{document} and θ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =1$$\end{document}).