Bilinear Strichartz estimates for Schr\"odinger operators in 2 dimensional compact manifolds with boundary and cubic NLS

TL;DR

This paper develops bilinear Strichartz estimates for Schr"odinger operators on 2D compact manifolds with boundary, enabling local well-posedness results for cubic NLS in low regularity spaces.

Contribution

It introduces new bilinear and gradient bilinear Strichartz estimates for Schr"odinger operators on 2D manifolds with boundary, advancing understanding of nonlinear Schr"odinger equations.

Findings

Establishment of bilinear Strichartz estimates for Schr"odinger operators.

Proof of local well-posedness of cubic NLS in $H^s$ for $s > 2/3$.

Extension of analysis techniques to manifolds with boundary.

Abstract

In this paper, we establish bilinear and gradient bilinear Strichartz estimates for Schr\"odinger operators in 2 dimensional compact manifolds with boundary. Using these estimates, we can infer the local well-posedness of cubic nonlinear Schr\"odinger equation in $H^s$ for every $s>{2/3}$ on such manifolds.