Regularity of Navier--Stokes flows with bounds for the velocity gradient along streamlines and an effective pressure: Navier--Stokes regularity via moderated pressure

Chuong Van Tran, xinwei yu

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Regularity criteria for solutions of the three-dimensional Navier--Stokes equations are derived in this paper. Let $$\Omega(t,q) \assign \left\{x:|u(x,t)| > C(t,q)\norm{u}_{L^{3q-6}(\mathbb{R}^3)}\right\} \cap\left\{x:\widehat{u}\cdot\nabla|u|\neq0\right\},$$ $$\tilde\Omega(t,q) \assign \left\{x:|u(x,t)| \le C(t,q)\norm{u}_{L^{3q-6}(\mathbb{R}^3)}\right\} \cap\left\{x:\widehat{u}\cdot\nabla|u|\neq0\right\},$$ where $q\ge3$ and $$C(t,q) \assign \left(\frac{\norm{u}_{L^4(\mathbb{R}^3)}^2\norm{|u|^{(q-2)/2}\,\nabla|u|}_{L^2(\mathbb{R}^3)}}{cq\norm{u_0}_{L^2(\mathbb{R}^3)} \norm{p+\mathcal{P}}_{L^2(\tilde\Omega)}\norm{|u|^{(q-2)/2}\, \widehat{u}\cdot\nabla|u|}_{L^2(\tilde\Omega)}}\right)^{2/(q-2)}.$$ Here $u_0=u(x,0)$, $\mathcal{P}(x,|u|,t)$ is a pressure moderator of relatively broad form, $\widehat{u}\cdot\nabla|u|$ is the gradient of $|u|$ along streamlines, and $c=(2/\pi)^{2/3}/\sqrt3$. If $$\norm{\widehat{u}\cdot\nabla|u|}_{L^{3/2}(\Omega)}\le\frac{2}{c^2q^2}\, \frac{\norm{u}_{L^{3q}(\mathbb{R}^3)}^2}{\norm{p+\mathcal{P}}_{L^{3q/2}(\Omega)}}$$ or $$\norm{p+\mathcal{P}}_{L^{3/2}(\Omega)}\le\frac{1}{c^2q^2}\, \frac{\norm{u}_{L^{3q}(\mathbb{R}^3)}^2}{\norm{p+\mathcal{P}}_{L^{3q/2}(\Omega)}}$$ or $$\norm{p+\mathcal{P}}_{L^{(q+2)/2}(\Omega)} \le \frac{1}{cq}\frac{\norm{u}_{L^{3q} (\mathbb{R}^3)}^{q/2}}{\norm{u}_{L^{q+2}(\Omega)}^{(q-2)/2}},$$ then $\norm{u}_{L^q(\mathbb{R}^3)}$ decreases and no singularities can develop.
Original languageEnglish
Title of host publicationLondon Mathematical Society Lecture Note series 452
Subtitle of host publicationPartial Differential Equations in Fluid Mechanics
EditorsCharles Fefferman, James Robinson, Jose Rodrigo
Place of PublicationCambridge
PublisherCambridge University Press
Pages252--267
Number of pages16
Volume452
ISBN (Print)978-1-108-46096-5
DOIs
Publication statusPublished - 2018

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