New regularity criteria for Navier-Stokes and SQG equations in critical spaces

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 equations, we demonstrate that a Leray-Hopf solution $ u $ is regular if $ u\in L_T^{\frac{2}{1-\alpha}} \dot{B}^{-\alpha}_{\infty,\infty}(\mathbb{R}^3) $, or $ u $ in Lorentz space $ L_T^{p,r} \dot{B}^{-1+\frac{2}{p}}_{\infty,\infty}(\mathbb{R}^3) $, with $ 4\leq p\leq r<\infty $. Additionally, an alternative regularity condition is expressed as $ u\in L_{T}^{\frac{2}{1-\alpha}} \dot{B}^{-\alpha}_{\infty,\infty}(\mathbb{R}^3)+{L_T^\infty\dot{B}^{-1}_{\infty,\infty}}(\mathbb{R}^3) $($ \alpha\in(0,1) $), contingent upon a smallness assumption on the norm $ L_T^\infty\dot{B}^{-1}_{\infty,\infty} $. For the surface quasi-geostrophic equations, we derive that a Leray-Hopf weak solution $ \theta\in L_T^{\frac{\alpha}{\varepsilon}} \dot{C}^{1-\alpha+\epsilon}(\mathbb{R}^2) $ is smooth for any $ \varepsilon $ small enough. Similar to the case of Navier-Stokes equations, we derive regularity criteria 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) $.