On partial regularity for the Navier-Stokes equations

Igor Kukavica

doi:10.3934/dcds.2008.21.717

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2008, Volume 21, Issue 3: 717-728. Doi: 10.3934/dcds.2008.21.717

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On partial regularity for the Navier-Stokes equations

Igor Kukavica^1,

1.
Department of Mathematics, University of Southern California, Los Angeles, CA 90089

Received: June 30, 2007

Revised: February 29, 2008

Published: August 2008

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Abstract

We consider the partial regularity of suitable weak solutions of the Navier-Stokes equations in a domain $D$. We prove that the parabolic Hausdorff dimension of space-time singularities in $D$ is less than or equal to 1 provided the force $f$ satisfies $f\in L^{2}(D)$. Our argument simplifies the proof of a classical result of Caffarelli, Kohn, and Nirenberg, who proved the partial regularity under the assumption $f\in L^{5/2+\delta}$ where $\delta>0$.

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Mathematics Subject Classification: Primary: 35Q30, 76D05, 35K55, 35K15.

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