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 $(-\hat{I}”{)}^2$ with ${\hat{I}}^2>1$. We prove that smooth solutions of the hyperdissipative Navier-Stokes equations are non-unique with arbitrarily small initial data in $B_{\infty ,1}^{-{\hat{I}}^2-\hat{I}\pm }({\mathbb{T}}^d)$ for any $\hat{I}\pm >0$. Moreover, we show the existence of a solution with arbitrarily small initial data in $B_{\infty ,1}^{-{\hat{I}}^2-\hat{I}\pm }({\mathbb{T}}^d)$ ($\hat{I}\pm >0$) that grows arbitrarily large in ${\dot{B}}_{\infty ,\infty }^{-s}({\mathbb{T}}^d)$ for all $s\in \mathbb{R}$ in arbitrarily small time. It is worth pointing out that $B_{\infty ,1}^{-{\hat{I}}^2-\hat{I}\pm }({\mathbb{T}}^d)$ lies in the subcritical regime when $0<\hat{I}\pm <{\hat{I}}^2-1$. 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 ${\dot{B}}_{\infty ,\infty }^{-{\hat{I}}^2-\hat{I}\pm }({\mathbb{T}}^d)$ with $\hat{I}\pm <0$.