Type: Article

Global existence for two dimensional quadratic derivative nonlinear Schrödinger equations

Nakao Hayashi; Pavel I. Naumkin;

Abstract:

We study the global in time existence of small classical solutions to the nonlinear Schrodinger equation with quadratic interactions of derivative type in two space dimensions {i partial derivative(t)u + 1/2 Delta u = N(u) , t >= 1, x is an element of R-2, (0.1) u(1, x) = u(0)(x), x is an element of R-2, where N(u) = Sigma(j,k=1,2) lambda(jk)(partial derivative(j)u)((partial derivative(k)u) over bar) lambda(jk) is an element of C. We prove that if the initial data u(0) is an element of H-10 boolean AND H-0,H-10 satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore we prove the existence of the usual scattering states in homogeneous Sobolev space of order one. The proof depends on the energy type estimates, and smoothing property by Doi.