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Regularity of Navier--Stokes flows with bounds for the pressure Tran, Chuong Van


This talk is concerned with global regularity of solutions of the Navier--Stokes equations. Let $\Omega(t) \assign \{x:|u(x,t)| > c\norm{u}_{L^r}\}$, for some $r\ge3$ and constant $c$ independent of $t$, with measure $|\Omega|$. It is shown that if $\int_\Omega|p+\mathcal{P}|^{3/2}\mathd x$ becomes sufficiently small as $|\Omega|$ decreases, then $\norm{u}_{L^{(r+6)/3}}$ decays and regularity is secured. Here $p$ is the physical pressure and $\mathcal{P}$ is a pressure moderator of relatively broad forms. The implications of the results are discussed and regularity criteria in terms of bounds for $|p+\mathcal{P}|$ within $\Omega$ are deduced. This is joint work with Xinwei Yu.

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