The Cauchy problem for the elliptic-hyperbolic Davey-Stewartson system in Sobolev space
- Wang, Yuzhao
DOIAbstract
We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson systems [GRAPHICS] where Delta = partial derivative(2)(x1) + partial derivative(2)(x2), c(1), c(2) is an element of R, u is a complex valued function and phi 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) boolean AND 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. (C) 2010 Els...
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