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Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations. (English) Zbl 1126.35047

The authors consider a Leray-Hopf weak solution $\left(u,p\right)$ of the Navier-Stokes equations in ${ℝ}^{3}×\left(0,T\right)$. They prove that this solution is regular provided that the initial velocity ${u}_{0}$ belongs to ${L}^{2}\left({ℝ}^{3}\right)\cap {L}^{q}\left({ℝ}^{3}\right)$ for some $q>3$ and that the pressure satisfies

${\int }_{0}^{T}{\parallel p\left(t\right)\parallel }_{{\stackrel{˙}{B}}_{\infty ,\infty }^{0}}\phantom{\rule{0.166667em}{0ex}}dt<\infty ·$

The main tool in obtaining this result is an a-priori-estimate of the form

$\underset{0\le t\le T}{sup}{\parallel u\left(t\right)\parallel }_{{L}^{s}}\le C\left(\parallel {u}_{0}{{\parallel }_{{L}^{s}}+{\left(CT\right)}^{\frac{1}{s}}+e\right)}^{exp\left(C{\int }_{0}^{T}\parallel p\left(t\right){\parallel }_{{\stackrel{˙}{B}}_{\infty ,\infty }^{0}}\phantom{\rule{0.166667em}{0ex}}dt\right)},\phantom{\rule{1.em}{0ex}}3

which is derived with the help of the Paley-Littlewood decomposition.

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory 35B65 Smoothness and regularity of solutions of PDE