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arXiv · 2105.12117

$L^2$-critical nonuniqueness for the 2D Navier-Stokes equations

Alexey Cheskidov·Xiaoyutao Luo
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Abstract

In this paper, we consider the 2D incompressible Navier-Stokes equations on the torus. It is well known that for any $L^2$ divergence-free initial data, there exists a global smooth solution that is unique in the class of $C_t L^2$ weak solutions. We show that such uniqueness would fail in the class $C_t L^p$ if $ p<2$. The non-unique solutions we constructed are almost $L^2$-critical in the sense that $(i)$ they are uniformly continuous in $L^p$ for every $p<2$; $(ii)$ the kinetic energy agrees with any given smooth positive profile except on a set of arbitrarily small measure in time.

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Alexey Cheskidov, Xiaoyutao Luo. 2021-05-25. $L^2$-critical nonuniqueness for the 2D Navier-Stokes equations. https://arxiv.org/abs/2105.12117

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