On Mixed Pressure-Velocity Regularity Criteria to the Navier-Stokes Equations in Lorentz Spaces

In this paper the authors derive regular criteria in Lorentz spaces for LerayHopf weak solutions v of the three-dimensional Navier-Stokes equations based on the formal equivalence relation pi congruent to |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 pi/(e(-|x|2) +| v|)(theta) is an element of L-p (0, T; L-q,L-infinity), where 0 <= theta <= 1 and 2/p + 3/q = 2 - theta, then v is regular on (0, T]. Note that, if Omega is periodic, e(-|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.