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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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Authors and Affiliations

  1. School of mathematics and statistics, Huaiyin Normal University, Huai’an, 223300, Jiangsu, China

    Zhengguang Guo

  2. Czech Technical University, Prague, Thakurova 7, 166 29, Prague 6, Czech Republic

    Petr Kucera & Zdenek Skalak

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  1. Zhengguang Guo
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  2. Petr Kucera
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  3. Zdenek Skalak
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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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