New Regularity Criteria for Navier-Stokes and SQG Equations in Critical Spaces

Abstract

In this paper, we investigate some priori estimates to provide the critical regularity criteria for incompressible Navier-Stokes equations on ${\mathbb{R}}^3$ and super critical surface quasi-geostrophic equations on ${\mathbb{R}}^2$. Concerning the Navier-Stokes equation, we demonstrate that a Leray-Hopf solution $u$ is regular if $u\in L_T^{\frac{2}{1-\alpha }}{\dot{B}}_{\infty ,\infty }^{-\alpha }({\mathbb{R}}^3)$, or $u$ in Lorentz space $L_T^{p,r}{\dot{B}}_{\infty ,\infty }^{-1+\frac{2}{p}}({\mathbb{R}}^3)$, with $4\le p\le r<\infty$. Additionally, an alternative regularity condition is expressed as $u\in L_T^{\frac{2}{1-\alpha }}{\dot{B}}_{\infty ,\infty }^{-\alpha }({\mathbb{R}}^3)+L_T^{\infty }{\dot{B}}_{\infty ,\infty }^{-1}({\mathbb{R}}^3)$($\alpha \in (0,1)$), contingent upon a smallness assumption on the norm $L_T^{\infty }{\dot{B}}_{\infty ,\infty }^{-1}$. For the SQG equation, we derive that a Leray-Hopf weak solution $\theta \in L_T^{\frac{\alpha }{\epsilon }}{\dot{C}}^{1-\alpha +\epsilon }({\mathbb{R}}^2)$ is smooth for any $\epsilon$ small enough. Similar to the case of Navier-Stokes equation, we derive regularity criterion in more refined spaces, i.e. Lorentz spaces $L_T^{\frac{\alpha }{\epsilon },r}{\dot{C}}^{1-\alpha +\epsilon }({\mathbb{R}}^2)$ and addition of two critical spaces $L_T^{\frac{\alpha }{\epsilon }}{\dot{C}}^{1-\alpha +\epsilon }({\mathbb{R}}^2)+L_T^{\infty }{\dot{C}}^{1-\alpha }({\mathbb{R}}^2)$, with smallness assumption on $L_T^{\infty }{\dot{C}}^{1-\alpha }({\mathbb{R}}^2)$.