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A geometrical approach to the problem of Navier--Stokes regularity

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Author(s)

Chuong Van Tran, Xinwei Yu

School/Research organisations

Abstract

In this article we present a new kind of regularity criterion for the global well-posedness problem of the three-dimensional Navier--Stokes equations in the whole space. It is shown that if for every fixed $t\in (0,T)$, the region $\Omega:=\{(x,t)\mid |u(x,t)|>C(q)\norm{u}_{L^{3 q-6}}\}$, for some $C(q)$ appropriately defined, shrinks fast enough as $q\nearrow \infty$, then the solution remains regular beyond $T$. The novelty of this new criterion is that it involves the shape of the magnitude of the velocity. We argue that reasonable flows satisfy our criterion, and singularity in Navier--Stokes flows is highly unlikely.
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Original languageEnglish
Number of pages15
JournalNonlinearity
StatePublished - 2018

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