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 relationπ≌|v|^(2),whereπdenotes th...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 relationπ≌|v|^(2),whereπdenotes 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(π/(e-^|(x)|^(2)+|v|^(θ)∈L^(p)(0,T;L^(q,∞)),where 0≤θ≤1 and 2/p+3/q=2-θ,then v is regular on(0,T].Note that,ifΩ,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.展开更多
基金supported by FCT(Portugal)under the project UIDB/MAT/04561/2020the Fundamental Research Funds for the Central Universities under grant G2019KY05114。
文摘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 relationπ≌|v|^(2),whereπdenotes 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(π/(e-^|(x)|^(2)+|v|^(θ)∈L^(p)(0,T;L^(q,∞)),where 0≤θ≤1 and 2/p+3/q=2-θ,then v is regular on(0,T].Note that,ifΩ,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.
