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A New Criterion for Partial Regularity of Suitable Weak Solutions to the Navier-Stokes Equations

In: Advances in Mathematical Fluid Mechanics

Author

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  • JÖrg Wolf

    (Mathematical Institute, Humboldt University Berlin)

Abstract

In the present paper we study local properties of suitable weak solutions to the Navier-Stokes equation in a cylinder Q = Ω × (0, T). Using the local representation of the pressure we are able to define a positive constant ɛ⋆ such that for every parabolic subcylinder QR ⊂ Q the condition $$R^{-2}\int_{Q_R}|u|^3dxdt\leq\varepsilon_{\ast}$$ implies $${\bf U}\in L^{\infty}(Q_{R/2})$$ ). As one can easily check this condition is weaker then the well known Serrin's condition as well as the condition introduced by Farwig, Kozono and Sohr in a recent paper. Since our condition can be verified for suitable weak solutions to the Navier-Stokes system it improves the known results substantially.

Suggested Citation

  • JÖrg Wolf, 2009. "A New Criterion for Partial Regularity of Suitable Weak Solutions to the Navier-Stokes Equations," Springer Books, in: Rolf Rannacher & Adélia Sequeira (ed.), Advances in Mathematical Fluid Mechanics, pages 613-630, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-04068-9_34
    DOI: 10.1007/978-3-642-04068-9_34
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