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

TL;DR

This paper establishes local well-posedness for the quadratic nonlinear Schrödinger equation on two-dimensional compact manifolds with boundary, utilizing bilinear estimates and Bourgain spaces to extend previous results.

Contribution

It introduces new evolution bilinear estimates for Schrödinger operators on manifolds and proves local well-posedness for initial data in H^s with s>2/3.

Findings

Local well-posedness for quadratic NLS in 2D manifolds with boundary

New bilinear estimates for Schrödinger operators on manifolds

Well-posedness for initial data in H^s, s>2/3

Abstract

We consider the quadractic NLS posed on a bidimensional compact Riemannian manifold $(M, g)$ with $ \partial M \neq \emptyset$. Using bilinear and gradient bilinear Strichartz estimates for Schr\"odinger 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.