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Localized Anisotropic Regularity Conditions for the Navier–Stokes Equations

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Abstract

We establish a sufficient regularity condition for local solutions of the Navier–Stokes equations. For a suitable weak solution (u, p) on a domain D we prove that if \(\partial _3 u\) belongs to the space \(L_t^{s_0}L_x^{r_0}(D)\) where \(2/s_0 + 3/r_0 \le 2 \) and \(9/4 \le r_0\le 5/2\), then the solution is Hölder continuous in D.

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Acknowledgements

I.K. was supported in part by the NSF Grant DMS-1615239, W.R. was supported in part by the NSF Grant DMS-1613831, while M.Z. was supported in part by the NSF Grant DMS-1109562.

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Authors and Affiliations

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

    Igor Kukavica & Mohammed Ziane

  2. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, USA

    Walter Rusin

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  1. Igor Kukavica
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  2. Walter Rusin
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  3. Mohammed Ziane
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Correspondence to Igor Kukavica.

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Communicated by Paul Newton.

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Kukavica, I., Rusin, W. & Ziane, M. Localized Anisotropic Regularity Conditions for the Navier–Stokes Equations. J Nonlinear Sci 27, 1725–1742 (2017). https://doi.org/10.1007/s00332-017-9382-5

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  • DOI: https://doi.org/10.1007/s00332-017-9382-5

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