Global existence of small classical solutions to nonlinear Schrödinger equations

Abstract

We study the global Cauchy problem for nonlinear Schrödinger equations with cubic interactions of derivative type in space dimension $n⩾3$. The global existence of small classical solutions is proved in the case where every real part of the first derivatives of the interaction with respect to first derivatives of wavefunction is derived by a potential function of quadratic interaction. The proof depends on the energy estimate involving the quadratic potential and on the endpoint Strichartz estimates.