The Cauchy problem for the elliptic–hyperbolic Davey–Stewartson system in Sobolev space

Abstract

We study the initial value problem for the elliptic–hyperbolic Davey–Stewartson systems (0.1) { i ∂ t u + Δ u = c 1 | u | 2 u + c 2 u ∂ x 1 φ , ( t , x ) ∈ R 3 , ( ∂ x 1 2 − ∂ x 2 2 ) φ = ∂ x 1 | u | 2 , u ( 0 , x ) = ϕ ( x ) , where Δ = ∂ x 1 2 + ∂ x 2 2 , c 1 , c 2 ∈ R , u is a complex valued function and φ is a real valued function. Our purpose is to prove the local existence and uniqueness of the solution for (0.1) in the Sobolev space H 3 / 2 + ( R 2 ) with small mass. Our methods rely heavily on Hayashi and Hirata (1996) [11], but we improve partial results of it, which got global existence of small solutions to (0.1) in weighted Sobolev space H 3 , 0 ∩ H 0 , 3 . Our main new tools are Kenig–Ponce–Vega type commutator estimate in Kenig, Ponce and Vega (1993) [16] and its variant form.