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

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.

arXiv commentv2: minor corrections

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• Non Uniqueness of power-law flows
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• Weak solutions obtained by the vortex method for the 2D Euler equations are Lagrangian and conserve the energy