New regularity criteria for NSE and SQG in critical spaces

In this paper, we investigate some priori estimates to provide the critical regularity criteria for incompressible Navier-Stokes equations on R³ and super critical surface quasi-geostrophic equations on R². Concerning the Navier-Stokes equation, we demonstrate that a Leray-Hopf solution u is regular if u LT^2{1-} Ḃ^-, (R³), or u in Lorentz space LT^p, r Ḃ^-1+2{p}, (R³), with 4 p r<. Additionally, an alternative regularity condition is expressed as u Lₓ^2{1-} Ḃ^-, (R³) +LT^̇^{-1, } (R³) ( (0, 1) ), contingent upon a smallness assumption on the norm LT^̇^-1,. For the SQG equation, we derive that a Leray-Hopf weak solution LT^{} Ċ^1-+ (R²) is smooth for any small enough. Similar to the case of Navier-Stokes equation, we derive regularity criterion in more refined spaces, i. e. Lorentz spaces LT^{, r}Ċ^1-+ (R²) and addition of two critical spaces Lₓ^{}Ċ^1-+ (R²) +LT^̇^{1- (R²) }, with smallness assumption on LT^̇^1- (R²).