修正的Navier-Stokes方程的再生性质和周期解

Reproductive Property and Periodic Solution of the Modified Navier-Stokes Equations

本文证明了由提出的修正的Navier-Stokes方程在小外力的条件下对t_1∈(0,T]存在唯一的初始速度分布使得相应的初边值问题的广义解具再生性质:(t_1)=(0)=.从而当外力还是时间t的周期函数时,是周期解.进而证明此周期解以指数方式吸引相应于同一外力但初值可任意的其它解.上述结论的证明基于对广义解v的导数v在空间L~∞(0,T;L^2(Ω))中估计.

A reproductive property of the modified Navier-Stokes equations proposed by O.A.Ladyzhenskaya is proved,that is,if the external force is small enough,then given t_1∈(0,T] there is a unique initial velocity and a corresponding weak solution that reproduces its initial value at t=t_1:(t_1)=(0)=.And if the external force is periodic in the time,then is also periodic.Moreover,it is proved that the periodic solution attracts exponently any solution with respect to same external force and arbitrary initial velocities.The proofs of these results are based on an estimate of ‖zv‖_(L~∞(0,T_1:L^2(Ω))) for any weak solution v.