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- Springer Journals
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- Copyright © The Author(s), under exclusive licence to Springer Nature B.V. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
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- 0167-8019
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- 1572-9036
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- 10.1007/s10440-023-00573-7
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Abstract
In this remark, we consider regularity criterion for weak solutions to the 3d incompressible Navier–Stokes equations via pressure. It is proved that if the corressponding pressure P\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$P$\end{document} satisfies ∇h˜P∈Lβ(0,T;Lp(Rx1;Lq(Rx2;Lr(Rx3))))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\nabla _{\tilde{h}} P\in L^{\beta}(0,T;L^{p}(\mathbb{R}_{x_{1}};L^{q}( \mathbb{R}_{x_{2}};L^{r}(\mathbb{R}_{x_{3}}))))$\end{document} with 2β+1p+1q+1r=3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\frac{2}{\beta}+\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=3$\end{document}, 65<p,q,r≤3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\frac{6}{5}< p, q, r \leq 3$\end{document} and 2−(1p+1q+1r)≥0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$2-\big(\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\big) \geq 0$\end{document}, then the weak solution remains smooth on (0,T]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$(0,T]$\end{document}. Here, ∇h˜=(0,∂2,∂3)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\nabla _{\tilde{h}}=(0,\partial _{2},\partial _{3})$\end{document}.
Journal
Acta Applicandae Mathematicae – Springer Journals
Published: Jun 1, 2023
Keywords: Navier–Stokes equations; Weak solution; Regularity criterion; Pressure; Anisotropic Lebesgue spaces; 35B65; 35Q35