On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $\mathbb{R}^{N}$

Abstract.

In this paper we establish a Serrin’s type regularity criterion on the gradient of pressure for weak solutions to the Navier–Stokes equations in $\mathbb{R}^{N} ,N = 3,4.$ It is proved that if the gradient of pressure belongs to L^{α, γ} with $2/\alpha + N/\gamma \leq 3,\;N/3 \leq \gamma \leq \infty ,$ then the weak solution actually is regular and unique.