Local and Global Well-Posedness for the Critical Schrodinger-Debye System

TL;DR

This paper proves local and global well-posedness for the critical Schrödinger-Debye system in 2D and 3D, extending previous results and showing solutions exist globally in the energy space, thus addressing singularity formation.

Contribution

It establishes new local well-posedness results in broader Sobolev spaces and proves global existence in the energy space for the critical case in 2D.

Findings

Local well-posedness in $H^s\times H^{\ell}$ for specified $s, \ell$

Global existence in the energy space $H^1\times L^2$ for the critical case in 2D

Negative answer to the singularity formation problem in the critical case

Abstract

We establish local well-posedness results for the Initial Value Problem associated to the Schr\"odinger-Debye system in dimensions $N=2, 3$ for data in $H^s\times H^{\ell}$, with $s$ and $\ell$ satisfying $\max \{0, s-1\} \le \ell \le \min\{2s, s+1\}$. In particular, these include the energy space $H^1\times L^2$. Our results improve the previous ones obtained in \cite{Bidegaray1}, \cite{Bidegaray2} and \cite{Corcho-Linares}. Moreover, in the critical case (N=2) and for initial data in $H^1\times L^2$, we prove that solutions exist for all times, thus providing a negative answer to the open problem mentioned in \cite{Fibich-Papanicolau} concerning the formation of singularities for these solutions.