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 |
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Title of host publication | London Mathematical Society Lecture Note series 452 |
Subtitle of host publication | Partial Differential Equations in Fluid Mechanics |
Editors | Charles Fefferman, James Robinson, Jose Rodrigo |
Place of Publication | Cambridge |
Publisher | Cambridge University Press |
Pages | 252--267 |
Number of pages | 16 |
Volume | 452 |
ISBN (Print) | 978-1-108-46096-5 |
DOIs | |
Publication status | Published - 2018 |