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August  2008, 21(3): 717-728. doi: 10.3934/dcds.2008.21.717

On partial regularity for the Navier-Stokes equations

Igor Kukavica 1,

1. 

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

Received  July 2007 Revised  March 2008 Published  April 2008

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 $\delta>0$.
Keywords: Navier-Stokes equations, Navier-Stokes equation, singular set, Hausdorff dimension., partial regularity.
Mathematics Subject Classification: Primary: 35Q30, 76D05, 35K55, 35K1.
Citation: Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717
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