Bilinear Strichartz estimates for the Schr{\"o}dinger map problem

TL;DR

This paper establishes bilinear Strichartz estimates for solutions to the Schr{"o}dinger map problem with small initial data in a critical space, aiding in proving local well-posedness.

Contribution

It introduces bilinear Strichartz estimates for the Schr{"o}dinger map problem using gauge techniques and an argument similar to prior work, advancing the analysis of well-posedness.

Findings

Bilinear estimates for small Schr{"o}dinger map solutions.

Use of gauge techniques from previous studies.

Foundation for proving local well-posedness.

Abstract

In this paper we prove bilinear Strichartz estimates for a solution to the Schr{\"o}dinger map problem whose size is small in the critical Strichartz space $| |\nabla|^{\frac{d - 2}{2}} \psi_{x} |_{L_{t,x}^{\frac{2(d + 2)}{d}}}$. These estimates will be useful in an upcoming paper in proving a local well - posedness result. Bilinear estimates make use of an argument similar to the argument found in Planchon and Vega (2009). We use the same gauges as in Bejenaru, Ionescu and Kenig (2007), Bejenaru et al. (2011), and Smith.