L²-critical nonuniqueness for the 2D Navier-Stokes equations

2021/05/25 by Cheskidov, Alexey, Luo, Xiaoyutao
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2105.12117

Abstract

In this paper, we consider the 2D incompressible Navier-Stokes equations on the torus. It is well known that for any L² divergence-free initial data, there exists a global smooth solution that is unique in the class of C_t L² 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 L²-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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