On the borderline regularity criterion in anisotropic Lebesgue spaces of the Navier-Stokes equations

In this paper, we are concerned with the critical mixed norm regularity of Leray-Hopf weak solutions of the Navier-Stokes equations in three dimensions and higher dimensions. It is shown that $u\in L^{\infty }(0,T;L^{\overrightarrow{q}}(R^n\left)\right)$ with $\sum _{i=1}^n\frac{1}{q_i}=1$ ensure that Leray-Hopf weak solutions are regular. A new ingredient is ε-regularity criterion derived by the De Giorgi iteration technique under this critical regularity in high spatial dimension.