Local well-posedness for the quadratic Schrodinger equation in two-dimensional compact manifolds with boundary

Abstract

We consider the quadractic NLS posed on a bidimensional compact Riemannian manifold $(M,g)$ with $\partial M\ne \varnothing$. Using bilinear and gradient bilinear Strichartz estimates for Schrödinger operators in two-dimensional compact manifolds proved by J. Jiang in \cite{JIANG} we deduce a new evolution bilinear estimates. Consequently, using Bourgain's spaces, we obtain a local well-posedness result for given data $u_0\in H^s(M)$ whenever $s>\frac{2}{3}$ in such manifolds.

25 pages