# Ill-posedness of the Thirring model below the critical regularity

## TL;DR

This paper investigates the ill-posedness of the one-dimensional Thirring model, a nonlinear Dirac equation, below the critical regularity, showing it fails to be well-posed in certain function spaces.

## Contribution

It establishes ill-posedness of the Thirring model in $L^p$ spaces for $1 \,< p < 2$ and in $H^s$ for $s < 0$ in the massless case, extending understanding of its regularity thresholds.

## Key findings

- Ill-posed in $L^p$ for $1 \,< p < 2$
- Ill-posed in $H^s$ for $s < 0$ in the massless case
- Global well-posedness in $L^2$ was previously known

## Abstract

We consider a nonlinear $L^2$-critical nonlinear Dirac equation in one space dimension known as the Thirring model. Global well-posedness in $L^2$ for this equation was proved by Candy. Here we prove that the equation is ill posed in $L^p$ for $1 \le p < 2$, and in the massless case also in $H^s$ with $s < 0$.