Stability Of Implicit Euler Method For The Navier-Stokes Equation With Time Delay

Navier-Stokes equation is used widely in fluid mechanics,In general,it is difficult to get exact solution of the Navier-Stokes equation,so the study on its numerical solu-tion is very necessary.The movement of fluid would change its state because of some external force containing genetic information,so there must be delay in Navier-Stokes equation,and delay often makes the system unstable. Generally, the Navier-Stokes equation with delay is more accurate than the Navier-Stokes equation without de-lay when describing fluid motion,then the research on numerical solution of delay Navier-Stokes equation has vitally great practical significance.Nowadays the theoret-ical researches in this area of scientists at home and abroad are still lacking.This article will consider the following two-dimensional delay Navier-Stokes equa-tions ut+?u·??u-v?u=f+g?u?t-???. By using implicit Euler scheme,we discrete its time discrete and abtain some stability results:1.We get sufficient conditions for L² long time stability of implicit Euler scheme in the space L.It means that under these condition,for any n>0,we get: |uⁿ|²?K₁(max 1?j?m|u^-j|²,|f|?), where |·| is the norm of the space L,K1?·,·?is a function that depends on the initial discrete value through its norm max 1?j?m|u^-j|² and also on |f|?.2.We get H¹ stability of implicit Euler scheme in finite time interval in the space H: ?un?2?K₄(max 0?j?m-1?u^-j?²,|f|?,?,n?t) ?n=1,…,N:=[T/?t], where ?·? is the norm of the space H,?=m?t is a delay,K₄?·,·,·,·?is a function that depends on the initial discrete value through its max0?j?m-1 ?u^-j?²,|f|_?,?, n?t.