具有分数次耗散的Navier-Stokes方程弱解的正则准则

Regularity criteria of weak solutions for Navier-Stokes equations of fractional dissipation

文章考虑在三维情形时,具有分数次耗散项-(-Δ)αu速度场的Navier-Stokes方程解的正则性;证明了:当0<α≤5/4,如果速度场的其中任意2个分量的梯度,例如▽u1,▽u2∈Lp(0,T;Lq(R3))=LtpLqx且2α/p+3/q≤2α时,或者当1/2<α≤5/4,如果速度场的其中2个分量属于Lp(0,T;Lq(R3))=LtpLqx,且2α/p+3/q≤2α-1,Navier-Stokes方程的弱解在(0,T]上是正则的。

This paper is concerned with the regularity criteria of weak solutions for the 3D Navier- Stokes equations with fractional dissipative term --（--△）au velocity vector. It is proved that the weak solutions of Navier-Stokes equation is regular on （0, T], if there exist two solution components, for example, u1 and u2 satisfying the condition either 0 〈 a ≤ 5/4, V ul ,↓△u1,↓△u2∈Lp（0,T;Lq（R3））=LipLxq, 2a/p＋3/q≤2a or 1/2〈a≤5/4,u2 ∈ LP（0,T;Lq（R3）） = Lip Lxq, 2a/p＋3/q≤ 2a- 1.