On Mixed Pressure-Velocity Regularity Criteria in Lorentz Spaces

In this paper the authors derive regular criteria in Lorentz spaces for Leray-Hopf weak solutions $v$ of the three-dimensional Navier-Stokes equations based on the formal equivalence relation $\pi\cong|v|^2$, where $\pi$ denotes the fluid pressure and $v$ denotes the fluid velocity. It is called the mixed pressure-velocity problem (the P-V problem for short). It is shown that if $\frac{\pi}{(\rme^{-|x|^2}+|v|)^{\theta}}\in L^p(0,T;L^{q,\infty}),$ where $0\leq\theta\leq1$ and $\frac{2}{p}+\frac{3}{q}=2-\theta$,then $v$ is regular on $(0,T]$. Note that, if $\Om$ is periodic,$\rme^{-|x|^2}$ may be replaced by a positive constant. This result improves a 2018 statement obtained by one of the authors.Furthermore, as an integral part of the contribution, the authors give an overview on the known results on the P-V problem, and also on two main techniques used by many authors to establish sufficient conditions for regularity of the so-called Ladyzhenskaya-Prodi-Serrin (L-P-S for short) type.