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 u t = ( ν + i ) Δ u + λ → 1 ⋅ ∇ ( | u | 2 u ) + ( λ → 2 ⋅ ∇ u ) | u | 2 + α | u | 2 δ u , 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 u 0 in modulation space M 2 , 1 s and Sobolev spaces H s + n / 2 ( s > 3 ) and ‖ u 0 ‖ 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 ν → 0 and u 0 in M 2 , 1 s or H s + 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 ‖ u 0 ‖ L 1 ≪ 1 .