Sharp non-uniqueness for the Navier-Stokes equations in scaling critical spaces

Abstract

It is known that uniqueness of mild solutions to the incompressible Navier-Stokes equations holds in the critical class $C([0,T);L^n({\mathbb{R}}^n))$ for $n⩾3$. In this paper, we prove that this result is sharp in the sense that uniqueness fails if $L^n({\mathbb{R}}^n)$ is replaced by some scaling critical spaces that are even slightly larger. We achieve this through a complete classification for every pair $(p,q)$ of whether uniqueness of mild solutions in the critical Besov class $C([0,T);{\dot{B}}_{p,q}^{n/p-1}({\mathbb{R}}^n))$ holds or not. Our non-uniqueness mechanism produces infinitely many global solutions emanating even from zero initial state, whose large-time asymptotics are governed by non-trivial stationary flow. To the best of our knowledge, such non-unique solutions provide the first examples of non-dissipative unforced Navier-Stokes flow with critical regularity.

The author corrected some harmless errors in the paper and added some comments on the very recent results that is closely related to this paper