Helicity and regularity of weak solutions to 3D Navier–Stokes equations View Full Text


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Article Info

DATE

2021-07-31

AUTHORS

Myong-Hwan Ri

ABSTRACT

We show that a Leray–Hopf weak solution to the three-dimensional Navier–Stokes the initial value problem is regular in (0, T] if ‖∇u0+‖2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \nabla u_0^+\Vert _2$$\end{document} (or ‖∇u0-‖2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \nabla u_0^-\Vert _2$$\end{document}) for initial value 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} and max{dHdt,0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max \{\frac{d{\mathcal H}}{dt},0\}$$\end{document} (or max{-dHdt,0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max \{-\frac{d{\mathcal H}}{dt},0\}$$\end{document}) are suitably small depending on the initial kinetic energy and viscosity, where u0+=∫0∞dEλu0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0^+=\int _0^{\infty } dE_\lambda u_0$$\end{document}, u0-=∫-∞0dEλu0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0^-=\int _{-\infty }^0 dE_\lambda u_0$$\end{document}, {Eλ}λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{E_\lambda \}_{\lambda \in {\mathbb R}}$$\end{document} is the spectral resolution of the curl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{curl}$$\end{document} operator and H≡∫R3u·curludx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal H}\equiv \int _{\mathbb R^3}u\cdot \mathrm{curl} u\,dx$$\end{document} is the helicity of the fluid flow. The results suggest that the helicity change rate rather than the magnitude of the helicity itself affects regularity of the viscous incompressible flows. More precisely, an initially regular viscous incompressible flow with suitably small positive or negative maximal helical component does not lose its regularity as long as the total helical behavior of the flow with respect to time is not decreasing, or even weakened at a moderate rate in accordance with the initial kinetic energy and viscosity. More... »

PAGES

435-445

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s11565-021-00370-w

DOI

http://dx.doi.org/10.1007/s11565-021-00370-w

DIMENSIONS

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