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 R-3 and super critical surface quasi-geostrophic equations on R-2. Concerning the Navier-Stokes equations, we demonstrate that a Leray-Hopf solution u is regular if u is an element of L-T(2/1-alpha) B-infinity,infinity(-alpha)(R-3), or u in Lorentz space L-T(p),(r) B-infinity,infinity(-1+2/p)(R-3), with 4 <= p <= r < infinity. Additionally, an alternative regularity condition is expressed as u is an element of L-T(1-alpha) B-infinity,infinity(-alpha)(R-3)+ L-T(infinity) B-infinity infinity(-1)(R-3)(alpha is an element of (0; 1)), contingent upon a smallness assumption on the norm L-T(infinity) B-infinity infinity(-1). For the surface quasi-geostrophic equations, we derive that a Leray-Hopf weak solution theta is an element of L-T (alpha/epsilon) C1-alpha+epsilon(R-2) is smooth for any epsilon small enough. Similar to the case of Navier-Stokes equations, we derive regularity criteria in more refined spaces, i.e. Lorentz spaces L-T (alpha/epsilon,) (r) C1-alpha+epsilon(R-2) and addition of two critical spaces L-T (alpha/epsilon) C1-alpha+epsilon(R-2)+(LTC1-alpha)-C-infinity(R-2), with smallness assumption on (LTC1-alpha)-C-infinity(R-2).