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The local regularity conditions for the Navier–Stokes equations via one directional derivative of the velocity

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Abstract

We study the local regularity of solutions to the Navier–Stokes equations. We show for a suitable weak solution (u, p) on an open space-time domain D that if \( {\partial}_3u\in {L}_t^p{L}_x^q(D) \), where 2/p + 3/q = 2 and q ∈ (27/16, 5/2), then the solution is regular in D.

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Correspondence to Zdenek Skalak.

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Zhengguang Guo was partially supported by NSFC under grant No. 11301394.

Petr Kucera and Zdenek Skalak were supported by the European Regional Development Fund, project No. CZ.02.1.01/0.0/0.0/16_019/0000778.

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Guo, Z., Kucera, P. & Skalak, Z. The local regularity conditions for the Navier–Stokes equations via one directional derivative of the velocity. Lith Math J 62, 333–348 (2022). https://doi.org/10.1007/s10986-022-09573-w

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  • DOI: https://doi.org/10.1007/s10986-022-09573-w

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