Inviscid limit for the derivative Ginzburg-Landau equation with small data in modulation and Sobolev spaces

Applying the frequency-uniform decomposition technique, we study the Cauchy problem for derivative Ginzburg-Landau equation u(t) = (nu + i)Delta u + del (vertical bar u vertical bar(2) u) + (lambda(2) . del u)vertical bar u vertical bar(2) + alpha vertical bar u vertical bar(2 delta)u, where delta is an element of N, lambda(1), lambda(2) are complex constant vectors, nu is an element of [0, 1], alpha is an element of C. For n >= 3, we show that it is uniformly global well posed for all v E [0,11 if initial data u0 in modulation space M-2,1(s) and Sobolev spaces Hs+n/2 (s > 3) and parallel to u(0)parallel to L-2 is small enough. Moreover. we show that its solution will converge to that of the derivative Schrodinger equation in C(0, T; L-2) if nu -> 0 and u(0) in M-2,1(s) or Hs+n/2 with s > 4. For n = 2, we obtain the local well-posedness results and inviscid limit with the Cauchy data in M-1,1(s) (s > 3) and parallel to u(0)parallel to L-1 <<1. (C) 2011 Elsevier Inc. All rights reserved.Mathematics, AppliedPhysics, MathematicalSCI(E)EI2ARTICLE2197-2223