On the continuity of the solutions to the Navier–Stokes equations with initial data in critical Besov spaces

被引:0
|
作者
Reinhard Farwig
Yoshikazu Giga
Pen-Yuan Hsu
机构
[1] Technische Universität Darmstadt,Fachbereich Mathematik
[2] University of Tokyo,Graduate School of Mathematical Sciences
关键词
Nonstationary Navier–Stokes system; Initial values; Weighted Serrin condition; Limiting type of Besov space; Continuity of solutions; Stability of solutions; 35Q30; 76D05;
D O I
暂无
中图分类号
学科分类号
摘要
It is well known that there exists a unique local-in-time strong solution u of the initial boundary value problem for the Navier–Stokes system in a three-dimensional smooth bounded domain when the initial velocity u0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0$$\end{document} belongs to critical Besov spaces. A typical space is B=Bq,s-1+3/q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B=B^{-1+3/q}_{q,s}$$\end{document} with 3<q<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3<q<\infty $$\end{document}, 2<s<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2<s<\infty $$\end{document} satisfying 2/s+3/q≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2/s+3/q \le 1$$\end{document} or B=B˚q,∞-1+3/q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B=\mathring{B}^{-1+3/q}_{q,\infty }$$\end{document}. In this paper, we show that the solution u is continuous in time up to initial time with values in B. Moreover, the solution map u0↦u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0\mapsto u$$\end{document} is locally Lipschitz from B to C[0,T];B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\left( [0,T];B\right) $$\end{document}. This implies that in the range 3<q<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3<q<\infty $$\end{document}, 2<s≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2<s\le \infty $$\end{document} with 3/q+2/s≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3/q +2/s \le 1$$\end{document} the problem is well posed which is in strong contrast to norm inflation phenomena in the space B∞,s-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^{-1}_{\infty ,s}$$\end{document}, 1≤s<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le s <\infty $$\end{document} proved in the last years for the whole space case.
引用
收藏
页码:1495 / 1511
页数:16
相关论文
共 50 条