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∈ LT2/1-α Ḃ-α∞,∞(R³), or u in Lorentz space LTp,r Ḃ-1+2/p∞,∞(R³), with 4≤ p≤ r<∞. Additionally, an alternative regularity condition is expressed as u∈ LT2/1-α Ḃ-α∞,∞(R³)+LT^̇⁻¹∞,∞(R³)(α∈(0,1)), contingent upon a smallness assumption on the norm LT^̇⁻¹∞,∞. For the SQG equation, we derive that a Leray-Hopf weak solution θ∈ LTα/ε Ċ1-α+ε(R²) is smooth for any ε small enough. Similar to the case of Navier-Stokes equation, we derive regularity criterion in more refined spaces, i.e. Lorentz spaces LTα/ε,rĊ1-α+ε(R²) and addition of two critical spaces LTα/εĊ1-α+ε(R²)+LT^̇1-α(R²), with smallness assumption on LT^̇1-α(R²).