On mixed pressure-velocity regularity criteria to the Navier-Stokes equations in Lorentz spaces

In this paper we 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 $π\cong|v|^2$, where $π$ denotes the fluid pressure and $v$ the fluid velocity. It is called the mixed pressure-velocity problem (the P-V problem). It is shown that if $\fπ{(e^{-|x|^2}+|v|)^θ}\in L^p(0,T;L^{q,\infty})\,,$ where $0\leqθ\leq1$ and $\f2p+\f3q=2-θ$, then $v$ is regular on $(0,T]$. Note that, if $\Om$ is periodic, we may replace $\,e^{-|x|^2} \,$ by a positive constant. This result improves a 2018 statement obtained by one of the authors. Furthermore, as an integral part of our contribution, we 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) type.

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