A note on local interior regularity of a suitable       weak solution to the Navier--Stokes problem

Jiří Neustupa

doi:10.3934/dcdss.2013.6.1391

Article Contents

2013, Volume 6, Issue 5: 1391-1400. Doi: 10.3934/dcdss.2013.6.1391

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A note on local interior regularity of a suitable weak solution to the Navier--Stokes problem

Jiří Neustupa^1,

1.
Czech Academy of Sciences, Mathematical Institute, Žitná 25, 115 67 Prague 1

Received: December 2011

Revised: January 2012

Published: October 2013

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Abstract

We formulate a criterion which guarantees a local regularity of a suitable weak solution $v$ to the Navier--Stokes equations (in the sense of L. Caffarelli, R. Kohn and L. Nirenberg [3]). The criterion shows that if $(x_0,t_0)$ is a singular point of solution $v$ then the $L^3$--norm of $v$ concentrates in an amount greater than or equal to some $\epsilon>0$ in an arbitrarily small neighbourhood of $x_0$ at all times $t$ in some left neighbourhood of $t_0$. As a partial result, we prove that a localized solution satisfies the strong energy inequality.

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Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 76D05.

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