Well-posedness for a system of quadratic derivative nonlinear Schr\"odinger equations in almost critical spaces

TL;DR

This paper establishes the existence of smooth solutions for a quadratic derivative nonlinear Schrödinger system in nearly optimal Sobolev spaces, advancing understanding of well-posedness in critical regimes.

Contribution

It determines an almost optimal Sobolev regularity for the system's well-posedness, filling a gap in previous research and excluding the critical scaling case.

Findings

Existence of smooth flow map in almost critical Sobolev spaces

Extension of well-posedness results to a broader regularity range

Clarification of the system's behavior near critical regularity

Abstract

In this paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schr\"odinger equations introduced by Colin and Colin (2004). We determine an almost optimal Sobolev regularity where the smooth flow map of the Cauchy problem exists, expect for the scaling critical case. This result covers a gap left open in papers of the first and second authors (2014, 2019).