On regularity criteria for the Navier-Stokes equations based on one directional derivative of the velocity or one diagonal entry of the velocity gradient

It is proved that if the solution of the Navier-Stokes system satisfies partial derivative(3)u is an element of L-p(0, T; L-q(R-3)), 2/p + 3/q = 22/13 + 3/13q, 3 < q < 4, or partial derivative(3)u(3) is an element of L-beta(0, T; L-alpha(R-3)), 2/beta + 3/alpha = 3(root 65 alpha(2) - 78 alpha + 49 + 7 - alpha)/16 alpha, 3+root 17/4 <= alpha <= infinity, then the solution is smooth on (0, T]. These two improve many previous results.

Automated summary

It is proven that the solution to the Navier-Stokes system is smooth under specific conditions regarding the Sobolev spaces L-p and L-q, which improves upon previous research findings.