The fractal dimension of the singular set for solutions of the Navier–Stokes system

doi:10.1088/0951-7715/22/12/005

The fractal dimension of the singular set for solutions of the Navier–Stokes system

Igor Kukavica 2009 Nonlinearity 22 2889-2900 doi: 10.1088/0951-7715/22/12/005

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Igor Kukavica
Department of Mathematics, University of Southern California, Los Angeles, CA 90089
E-mail: kukavica@usc.edu

Recommended by K Ohkitani

Abstract. We consider suitable weak solutions of the Navier–Stokes system in a bounded space-time domain D. We prove that the parabolic fractal dimension of the singular set is less than or equal to 135/82. We also introduce the concept of the parabolic fractal measure ${\mathcal F}_{\mathcal P}^{\alpha}$ and prove that the fractal measure ${\mathcal F}_{p}^{135/82}$ of the singular set is zero. For the Leray–Hopf weak solutions, we prove ${\mathcal F}^{1/2}(\Sigma_T)=0$ , where Σ_T denotes the set of singular times on [0, T] and ${\mathcal F}^{1/2}$ stands for the 1/2-dimensional fractal measure.

Mathematics Subject Classification: 35Q30, 76D05, 35K55

Print publication: Issue 12 (December 2009)
Received 16 April 2009, in final form 29 September 2009
Published 30 October 2009

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The fractal dimension of the singular set for solutions of the Navier–Stokes system

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