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Global existence of a uniformly local energy solution for the incompressible fractional Navier–Stokes equations

Volume 160 / 2020

Jingyue Li Colloquium Mathematicum 160 (2020), 7-40 MSC: Primary 35Q30; Secondary 76D03. DOI: 10.4064/cm7997-10-2019 Published online: 29 November 2019

Abstract

We introduce the concept of local Leray solutions starting from locally square-integrable initial data to the fractional Navier–Stokes equations with $s\in [3/4,1)$. Furthermore, we prove their local-in-time existence when $s\in (3/4, 1)$. In particular, if a locally square-integrable initial datum vanishes at infinity, we show that the fractional Navier–Stokes equations admit a global-in-time local Leray solution when $s\in [5/6, 1)$. For such local Leray solutions starting from locally square-integrable initial data vanishing at infinity, a singularity only occurs in $B_R(0)$ for some $R$.

Authors

  • Jingyue LiThe Graduate School of
    China Academy of Engineering Physics
    Beijing 100088, P.R. China
    e-mail

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