Abstract:In this paper, we consider the hyperdissipative Navier-Stokes equations with fractional dissipation $(-\Delta)^{\beta}$ with $\beta>1$. We prove that smooth solutions of the hyperdissipative Navier-Stokes equations are non-unique with arbitrarily small initial data in ${B}^{-\beta-\alpha}_{\infty,1}(\mathbb{T}^d)$ 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 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 $\dot{B}^{-\beta-\alpha}_{\infty,\infty}(\mathbb{T}^d)$ with $\alpha< 0$.
From: Song Liu [view email]
[v1]
Thu, 28 May 2026 13:46:13 UTC (32 KB)
[v2]
Fri, 12 Jun 2026 02:26:26 UTC (34 KB)
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