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Remark on Regularity Criterion Via Pressure in Anisotropic Lebesgue Spaces to the 3d Navier–Stokes Equations

Remark on Regularity Criterion Via Pressure in Anisotropic Lebesgue Spaces to the 3d... 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}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Remark on Regularity Criterion Via Pressure in Anisotropic Lebesgue Spaces to the 3d Navier–Stokes Equations

Liu, Qiao
Acta Applicandae Mathematicae , Volume 185 (1) – Jun 1, 2023
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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
DOI
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

References

  • On the regularity of the class of generalized Drazin-Riesz invertible operators
    Abad, O.; Zguitti, H.
  • Sufficient conditions for the regularity to the 3D Navier–Stokes equations
    Cao, C.
  • Two regularity criteria for the 3D MHD equations
    Cao, C.; Wu, J.
  • A new regularity class for 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}
    Beirãoda Veiga, H.
  • Regularity criteria involving the pressure for the weak solutions to the Navier–Stokes equations
    Berselli, L.; Galdi, G.
  • The analysis of the fractional-order Navier–Stokes equations by a novel approach
    Elsayed, E.M.; Shah, R.S.; Nonlaopon, K.
  • L3,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{3,\infty}$\end{document}-Solutions of the Navier–Stokes equations and backward uniqueness
    Escauriaza, L.; Seregin, G.
  • Regularity criterion for 3D Navier–Stokes equations in Besov spaces
    Fang, D.; Qian, C.
  • A new regularity criterion for the nematic liquid crystal flows
    Gala, S.; Liu, Q.; Ragusa, M.A.
  • A regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity field
    Guo, Z.; Gala, S.
  • A new regularity criterion for the 3D incompressible MHD equations in terms of one component of the gradient of the pressure
    Jia, X.; Zhou, Y.
  • Navier–Stokes equations with regularity in one direction
    Kukavica, I.; Ziane, M.
  • Sur le mouvement d’un liquide visqueux emplissant l’espace
    Leray, J.
  • A regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity
    Liu, Q.
  • On the 3d Navier–Stokes equations with regularity in pressure
    Liu, Q.; Dai, G.
  • Blowup criteria in terms of pressure for the 3D nonlinear dissipative system modeling electro-diffusion
    Liu, Q.; Zhao, J.
  • Un teorema di unicita per le equazioni di Navier–Stokes
    Prodi, G.
  • Some new regularity criteria for the Navier–Stokes equations containing gradient of the velocity
    Penel, P.; Pokorny, M.
  • Regularity of solutions of divergence form elliptic equations
    Ragusa, M.A.
  • On the regularity of the solutions to the Navier–Stokes equations via the gradient of one velocity component
    Skalak, Z.
  • An improved regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity field
    Zhang, Z.
  • Regularity criteria for the 3D MHD equations via one directional derivative of the pressure
    Zhang, Z.; Li, P.; Yu, G.
  • Some remarks on the Navier–Stokes equations with regularity in one direction
    Zhang, Z.; Yuan, W.; Zhou, Y.
  • Regularity criterion via two components of vorticity on weak solutions to the Navier–Stokes equations in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{3}$\end{document}
    Zhang, Z.; Chen, Q.
  • On regularity criteria in terms of pressure for the Navier-Stokes equations in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{3}$\end{document}
    Zhou, Y.
  • Regularity criteria in terms of pressure for the 3-D Navier–Stokes equations in a generic domain
    Zhou, Y.
  • On a regularity criterion in terms of the gradient of pressure for 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}
    Zhou, Y.