On regularity criteria in terms of pressure for the Navier-Stokes equations in ℝ³

Abstract

In this paper we establish a Serrin-type regularity criterion on the gradient of pressure for the weak solutions to the Navier-Stokes equations in R 3 \mathbb {R}^3 . It is proved that if the gradient of pressure belongs to L α , γ L^{\alpha ,\gamma } with 2 / α + 3 / γ ≤ 3 2/\alpha +3/\gamma \leq 3 , 1 ≤ γ ≤ ∞ 1\leq \gamma \leq \infty , then the weak solution is actually regular. Moreover, we give a much simpler proof of the regularity criterion on the pressure, which was showed recently by Berselli and Galdi (Proc. Amer. Math. Soc. 130 (2002), no. 12, 3585–3595).