Non-uniqueness of generalized Navier-Stokes equations in subcritical spaces

In this paper, we consider the generalized Navier-Stokes equations with fritional dissipation $(-\Delta)^{\beta}$ with $\beta>\frac{1}{2}$. When $\beta\in(1,2)$, We prove that smooth solutions of the generalized Navier-Stokes equations are non-unique with arbitrarily small initial data in $\dot{B}^{-\beta-\alpha}_{\infty,1}(\mathbb{T}^d)$ for any $\alpha>0$. It is worth pointing out that the space $\dot{B}^{-\beta-\alpha}_{\infty,1}(\mathbb{T}^d)$ is subcritical for $0<\alpha<\beta-1$. To the best of our knowledge, this is the first non-uniqueness result of Navier-Stokes equations with initial data at the critical regularity. To show the sharpness of the above results, for $\beta>\frac{1}{2}$, we establish the local well-poseness of the generalized Navier-Stokes equations with small initial data in $\dot{B}^{-\beta-\alpha}_{\infty,\infty}(\mathbb{T}^d)$ with $\alpha<0$ and $\alpha\leq\beta-1$.