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The Cauchy problem of the Navier–Stokes equations in Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^n$$\end{document} with the initial data a in the Besov space Bp,q-1+np(Rn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${B}^{-1+\frac{n}{p}}_{{p},{q}}(\mathbb {R}^n)$$\end{document} for n<p<∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n<p<\infty $$\end{document} and 1≤q≤∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1 \le q \le \infty $$\end{document} is considered. We construct the local solution in Lα,q(0,T;Br,10(Rn))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{\alpha ,q}(0,T;{B}^{0}_{{r},{1}}(\mathbb {R}^n))$$\end{document} for p≤r<∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p \le r< \infty $$\end{document} satisfying 2α+nr=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{2}{\alpha }+\frac{n}{r}=1$$\end{document} with the initial data a∈Bp,q-1+np(Rn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a \in {B}^{-1+\frac{n}{p}}_{{p},{q}}(\mathbb {R}^n)$$\end{document}, where Lα,q\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{\alpha ,q}$$\end{document} denotes the Lorentz space. Conversely, if the solution belongs to Lα,q(0,T;Lr(Rn))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{\alpha ,q}(0,T;L^{r}(\mathbb {R}^n))$$\end{document} with 2α+nr=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{2}{\alpha }+\frac{n}{r}=1$$\end{document}, then the initial data a necessarily belong to Br,q-1+nr(Rn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${B}^{-1+\frac{n}{r}}_{{r},{q}}(\mathbb {R}^n)$$\end{document}. It implies that the initial data in the Besov space Bp,q-1+np(Rn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B^{-1+\frac{n}{p}}_{p,q} (\mathbb {R}^n)$$\end{document} are a necessary and sufficient condition for the existence of solutions in the Serrin class.
Journal of Evolution Equations – Springer Journals
Published: Sep 1, 2021
Keywords: Navier–Stokes equations; Serrin class; Inhomogeneous Besov space; 35Q30; 76D05
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