一类修正Navier-Stokes方程解衰减速率的上下界估计

Upper and Lower Bounds of Decay Rates for a Solution of a Modified Navier-Stokes Equations

Navier-Stokes方程描述了具有小速度梯度的不可压缩粘性流体运动规律,在流体动力学研究中有着重要的应用。1966年,Ladyzhenskaya O.A.放弃了速度梯度很小的限制,提出了几种描述不可压缩粘性流体运动规律的修正Navier-Stokes方程。为估计整个三维空间上一类修正Navier-Stokes方程解衰减速率的上下界,使用改进的Fourier分解方法得到当初值u0∈Lp(1≤p<2)时,解的L2模衰减速率上界为(t+1)-3/2(1/p-1/2);对某些初值u0∈Lp(1≤p<97)时,解的L2模衰减速率下界为34(t+1)-3/4。

The Navier-Stokes equations have many important application in fluid dynamic,which describe motion characteristics of viscous incompressible fluids for small gradients of the velocities.In 1966,Ladyzhenskaya O.A.suggested several variants of modified Navier-Stokes equations to a determinate description of the nonstationary flows of viscous incompressible fluids for large gradients of the velocities.For estimating upper and lower bounds of decay rates for a modified Navier-Stokes equations in the whole three-dimensional space,by improving the Fourier splitting methods,the paper proves that upper bounds of decay rates of L2 norm to the solution are （t＋1）-3/2（1/p-1/2） for initial value u0 ∈Lp（1 ≤ p 2） and lower bounds of ones are34（t ＋ 1）？for some initial value 0u ∈Lp（1 ≤ p 97）.-3/4