Regularity Of Weak Solutions For 3d Incompressible Navier-Stokes Equations

In this paper)we consider Cauchy problem for the incompressible Navier-Stokes equationsin which u=(u1(x,t),u2(x,t),u3(x,t))is the unknown velocity field in R忌3,u0 is the initial velocity field with▽·u0=0,p(x,t)is a scalar pressure,νis the kinematic viscosity coefficient,we will assume thatν=1.Here we use the classical notion:In 1934,Leray proved that existing a globall weak solution on the condition that u0∈L2(R3)and▽·u0=0.It is called the Leray-Hopf weak solution which satisfies the energy inequality.We say if‖u(t)‖H1 is continuous,then u(t)is regular.In this paper,we will give some new criterions as follow,which ensure that the weak solutions of the Navier-Stockes equations is regular in Lebesgue space LtsLxr= Ls(0,T；Lr(R3)).(1)If▽u3∈LtsLxp;with 2/s+3/p≤2,3/2≤p≤∞,and w3∈LtαLxγ,2/α+3/γ≤2, 3/2≤γ≤∞,then u is regular.(2)If (?)3u3∈LtsLsr,with 2/s+3/r≤1/4,12≤r≤∞,then u is regular.(3)If (?)3u3∈Lts1Lxr1,and ui∈Lts2Lxr2,with 2/s1+3/r1≤1,3≤r1≤∞,2/s2+3/r2≤1, 3≤r2≤9,i=1,2,then u is regular.(4)If (?)3u3∈Lts1Lxr1,and ui∈Lts2Lxrs,with 2/s1+3/r1≤2,3/2≤r1≤∞,2/s2+3/r2≤1, 3≤r2≤9,i=1,2,then u is regular.(5)If u is Leray-Hopf weak solution,we have where constant C depends on‖u0‖L2 and‖u0‖H1.(6)If u3∈LtsLxr,with 2/s+3/r≤7/11,33/7≤r≤∞,and(u14-u24)((?)1u1-(?)2u2)=0,then u is regular.