| Abstract: |
| We deal with a suitable weak solution $(v,p)$ to the Navier--Stokes equations, where $v=(v_1,v_2,v_3)$ is the velocity and $p$ is the pressure. We show that the regularity of $(v,p)$ at
a space-time point $(x_0, t_0)$ 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 $(x_0,t_0)$. An appropriate choice of the coefficients in the linear combination leads to the Serrin--type condition on on the positive part of the Bernoulli pressure
$\frac{1}{2}\, |v|^2+p$, or the negative part of $p$, or one component of $v$, etc. |
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