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

TL;DR

This paper demonstrates the sharp boundary of uniqueness for Navier-Stokes solutions in critical spaces, showing non-uniqueness in slightly larger spaces and classifying when uniqueness holds across various critical Besov spaces.

Contribution

It provides a complete classification of uniqueness versus non-uniqueness of mild solutions in critical Besov spaces for Navier-Stokes equations, revealing new non-unique solutions with non-trivial asymptotics.

Findings

Uniqueness holds in $C([0,T);L^n)$ but fails in larger critical spaces.

Constructs infinitely many global solutions from zero initial data.

First examples of non-dissipative unforced Navier-Stokes flow with critical regularity.

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 $n \geqslant 3$. 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.