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 $n \geqslant 3$. 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.

Figures (1)

• Figure 1: Let $X_{p,q}$ be either $\dot{B}_{p,q}^{n/p-1}(\mathbb{R}^n)$ or $\dot{F}_{p,q}^{n/p-1}(\mathbb{R}^n)$. The blue region indicates the range of $(1/q,1/p)$ for which mild solutions to \ref{['eq:NS-intro']} are unique in $C([0,T);X_{p,q})$ and small stationary solutions in $X_{p,q}$ are unique, whereas the red region indicates the range where uniqueness fails for both non-stationary and stationary solutions. Note that the continuous embedding $X_{p_1,q_1}\hookrightarrow X_{p_2,q_2}$ for $1\leqslant p_1\leqslant p_2\leqslant \infty$ and $1\leqslant q_1\leqslant q_2\leqslant \infty$ implies that the underlying function space becomes larger as a point in the figure moves leftward or downward.

Theorems & Definitions (23)

• Proposition 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 2.1
• Lemma 2.2: Fuj-AIPHC
• Proposition 3.1
• Remark 3.2
• Lemma 3.3
• proof
• proof : Proof of Proposition \ref{['prop:F-Besov']}