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On the interior regularity criteria and the number of singular points to the Navier-Stokes equations

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Abstract

We establish some interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin’s criterion and Escauriza-Seregin-Šverák’s criterion. We also show that if a weak solution u satisfies

$\left\| {u( \cdot ,t)} \right\|_{L^p } \leqslant C( - t)^{(3 - p)/2p} $

for some 3 < p < ∞, then the number of singular points is finite.

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Correspondence to Wendong Wang.

Additional information

Wendong Wang is supported by the Fundamental Research Funds for the Central Universities.

Zhifei Zhang is partly supported by NSF of China under Grant 10990013 and 11071007.

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Wang, W., Zhang, Z. On the interior regularity criteria and the number of singular points to the Navier-Stokes equations. JAMA 123, 139–170 (2014). https://doi.org/10.1007/s11854-014-0016-7

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  • DOI: https://doi.org/10.1007/s11854-014-0016-7

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