Non-uniqueness for the hyperdissipative Navier-Stokes equations with arbitrarily small subcritical data

Abstract

In this paper, we consider the hyperdissipative Navier-Stokes equations with fractional dissipation $(-Δ)^β$ with $β>1$. We prove that smooth solutions of the hyperdissipative Navier-Stokes equations are non-unique with arbitrarily small initial data in ${B}^{-β-α}_{\infty,1}(\mathbb{T}^d)$ for any $α>0$. Moreover, we show the existence of a solution with arbitrarily small initial data in ${B}^{-β-α}_{\infty,1}(\mathbb{T}^d)$ ($α>0$) that grows arbitrarily large in $\dot{B}^{-s}_{\infty,\infty}(\mathbb{T}^d)$ for all $s\in\mathbb{R}$ in arbitrarily small time. It is worth pointing out that ${B}^{-β-α}_{\infty,1}(\mathbb{T}^d)$ lies in the subcritical regime when $0<α<β-1$. To the best of our knowledge, this is the first non-uniqueness result of the Navier-Stokes equations with initial data at the subcritical regularity. To show the sharpness of the above results, we establish the local well-posedness of the hyperdissipative Navier-Stokes equations with initial data in $\dot{B}^{-β-α}_{\infty,\infty}(\mathbb{T}^d)$ with $α< 0$.