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

Abstract

In this paper the authors derive regular criteria in Lorentz spaces for Leray-Hopf weak solutions υ of the three-dimensional Navier-Stokes equations based on the formal equivalence relation π ≅ ∣υ∣², where π denotes the fluid pressure and υ denotes the fluid velocity. It is called the mixed pressure-velocity problem (the P-V problem for short). It is shown that if ${\pi \over {{{({e^{ - {{\left| x \right|}^2}}} + \left| v \right|)}^\theta }}} \in {L^p}\left( {0,T;{L^{q,\infty }}} \right)$, where 0 ≤ θ ≤ 1 and ${2 \over p} + {3 \over q} = 2 - \theta $, then {itυ} is regular on (0, {itT}]. Note that, if Ω is periodic, ${{e^{ - {{\left| x \right|}^2}}}}$ 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.

Keywords

Navier-Stokes equations / Pressure ≌ square velocity / Regularity criteria / Lorentz spaces

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