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The Navier-Stokes Equations in the Critical Lebesgue Space

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Abstract

We study regularity criteria for the d-dimensional incompressible Navier-Stokes equations. We prove in this paper that if \({u \in L_\infty^tL_{d}^x((0,T)\times \mathbb{R}^d)}\) is a Leray-Hopf weak solution, then u is smooth and unique in \({(0, T)\times \mathbb{R}^d}\) . This generalizes a result by Escauriaza, Seregin and Šverák [5]. Additionally, we show that if T = ∞ then u goes to zero as t goes to infinity.

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Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, 182 George Street, Box F, Providence, RI, 02912, USA

    Hongjie Dong

  2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. China

    Dapeng Du

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  1. Hongjie Dong
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  2. Dapeng Du
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Correspondence to Hongjie Dong.

Additional information

Communicated by P. Constantin

Hongjie Dong was partially supported by the National Science Foundation under agreement No. DMS-0111298 and DMS-0800129.

Dapeng Du was partially supported by China Postdoctor Science Fund CPSF 20070410683.

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Dong, H., Du, D. The Navier-Stokes Equations in the Critical Lebesgue Space. Commun. Math. Phys. 292, 811–827 (2009). https://doi.org/10.1007/s00220-009-0852-y

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