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Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions

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Abstract

We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with (−1)-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main technical tools are local-in-space regularity estimates near the initial time, which are of independent interest.

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Notes

  1. Such decomposition is well-known. One can for example first localize u 0 using a smooth cutoff function, and then use Bogovskii’s lemma to deal with the divergence-free condition. See for example [1, 9].

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Acknowledgements

We thank Gregory Seregin for valuable discussions. We also thank the anonymous referees for constructive comments which helped to improve the paper.

This work was supported in part by grant DMS 1101428 from the National Science Foundation.

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  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Hao Jia & Vladimír Šverák

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  1. Hao Jia
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  2. Vladimír Šverák
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Correspondence to Vladimír Šverák.

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Jia, H., Šverák, V. Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions. Invent. math. 196, 233–265 (2014). https://doi.org/10.1007/s00222-013-0468-x

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