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

Abstract

Applying the frequency-uniform decomposition technique, we study the Cauchy problem for derivative Ginzburg–Landau equation ut=(ν+i)Δu+λ→1⋅∇(|u|2u)+(λ→2⋅∇u)|u|2+α|u|2δu, where δ∈N, λ→1,λ→2 are complex constant vectors, ν∈[0,1], α∈C. For n≥3, we show that it is uniformly global well posed for all ν∈[0,1] if initial data u0 in modulation space M2,1s and Sobolev spaces Hs+n/2 (s>3) and ‖u0‖L2 is small enough. Moreover, we show that its solution will converge to that of the derivative Schrödinger equation in C(0,T;L2) if ν→0 and u0 in M2,1s 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 M1,1s (s>3) and ‖u0‖L1≪1.