On Mixed Pressure-Velocity Regularity Criteria in Lorentz Spaces


Hugo BEIR\~{A}O da,Jiaqi YANG.On Mixed Pressure-Velocity Regularity Criteria in Lorentz Spaces[J].Chinese Annals of Mathematics B,2021,42(1):1~16
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Hugo BEIR\~{A}O da; Jiaqi YANG


This work was supported by FCT (Portugal) under the project: UIDB/MAT/04561/2020 and the Fundamental Research Funds for the Central Universities under grant: G2019KY05114.
Abstract: 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.


Navier-Stokes equations, Pressure $\cong$ square velocity, Regularity criteria, Lorentz spaces


35Q30, 76D03, 76D05
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