A Contribution to the Theory of Regularity of a Weak Solution to the Navier–Stokes Equations via One Component of Velocity and Other Related Quantities

Abstract

We deal with a suitable weak solution $(\mathbf {v},p)$ to the Navier–Stokes equations in $\Omega \times (0,T)$, where $\Omega$ is a domain in ${\mathbb R}^3$, $T>0$ and $\mathbf {v}=(v_1,v_2,v_3)$. We show that the regularity of $(\mathbf {v},p)$ at a point $(\mathbf {x}_0, t_0)\in \Omega \times (0,T)$ is essentially determined by the Serrin-type integrability of the positive part of a certain linear combination of $v_1^2,\, v_2^2,\, v_3^2$ and p in a backward neighborhood of $(\mathbf {x}_0,t_0)$. An appropriate choice of the coefficients in the linear combination leads to the Serrin-type condition on the positive part of the Bernoulli pressure $\frac{1}{2}|\mathbf {v}|^2+p$ or the negative part of p (Theorem 1), or one component of $\mathbf {v}$ (Theorem 2), etc.