Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On partial regularity for the Navier-Stokes equations

Pages: 717 - 728, Volume 21, Issue 3, July 2008      doi:10.3934/dcds.2008.21.717

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Igor Kukavica - Department of Mathematics, University of Southern California, Los Angeles, CA 90089, United States (email)

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 $\delta>0$.

Keywords:  Navier-Stokes equations, Navier-Stokes equation, partial regularity, singular set, Hausdorff dimension.
Mathematics Subject Classification:  Primary: 35Q30, 76D05, 35K55, 35K15.

Received: July 2007;      Revised: March 2008;      Available Online: April 2008.