
 
On Mixed PressureVelocity Regularity Criteria in Lorentz Spaces 
 
Citation： 
Hugo BEIR\~{A}O da,Jiaqi YANG.On Mixed PressureVelocity Regularity Criteria in Lorentz Spaces[J].Chinese Annals of Mathematics B,2021,42(1):1~16 
Page view： 202
Net amount： 149 
Authors： 
Hugo BEIR\~{A}O da; Jiaqi YANG 
Foundation： 
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 LerayHopf weak solutions $v$ of the threedimensional NavierStokes equations based on the formal equivalence relation $\pi\congv^2$, where $\pi$ denotes the fluid pressure and $v$ denotes the fluid velocity. It is called the mixed pressurevelocity problem (the PV 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 PV problem, and also on two main techniques used by many authors to establish sufficient conditions for regularity of the socalled LadyzhenskayaProdiSerrin (LPS for short) type. 
Keywords： 
NavierStokes equations, Pressure $\cong$ square velocity, Regularity criteria, Lorentz spaces 
Classification： 
35Q30, 76D03, 76D05 

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