# 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 language English London Mathematical Society Lecture Note series 452 Partial Differential Equations in Fluid Mechanics Charles Fefferman, James Robinson, Jose Rodrigo Cambridge Cambridge University Press 252--267 16 452 978-1-108-46096-5 https://doi.org/10.1017/9781108610575 Published - 2018

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