The Inviscid Limit of the Complex Ginzburg–Landau Equation

Naturally the question of inviscid limit arises. Does the solution u of the CGL equation (1.1) tend to (in an appropriate space norm) the solution v of the NLS equation (1.2) as the parameters a and b tend to 0? What is the convergence rate? The answers are not immediate especially when the initial data for these equations are not smooth. Because of its importance in both mathematical theory and physical applications, the inviscid limit has been extensively investigated for many partial differential equations such as Burgers' equation [3], the quasigeostrophic equation [17] and most notably the Navier Stokes equations. For smooth initial data and in absence of boundary, the inviscid limit of the Navier Stokes equations is the corresponding Euler equations and the rate is the optimal O(&) where & is the viscosity coefficient ([2, 4, 13]). But the situation changes if the initial data is not that smooth. It's shown in article no. DE973347