On the pressure regularity criterion of the 3D Navier–Stokes equations

In the study of regularity criteria for the weak solutions of the 3D Navier–Stokes equations, an improved regularity criterion is obtained. More precisely, it is proved that if the pressure satisfies the critical growth condition π ( x , t ) ∈ L 2 2 + r ( 0 , T ; B ̇ ∞ , ∞ r ( R 3 ) ) for  − 1 ≤ r ≤ 1 , then the weak solution u ( x , t ) is regular on ( 0 , T ] . The finding is mainly based on the innovative function decomposition methods together with Besov space techniques.