## Advances in Differential Equations

- Adv. Differential Equations
- Volume 16, Number 7/8 (2011), 643-666.

### Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension

#### Abstract

We prove global existence from $L^2$ initial data for a nonlinear Dirac equation known as the Thirring model [12]. Local existence in $H^s$ for $s>0$, and global existence for $s>\frac{1}{2}$, has recently been proven by Selberg and Tesfahun in [9] where they used $X^{s, b}$ spaces together with a type of null form estimate. In contrast, motivated by the recent work of Machihara, Nakanishi, and Tsugawa, [7] we first prove local existence in $L^2$ by using null coordinates, where the time of existence depends on the profile of the initial data. To extend this to a global existence result we need to rule out concentration of $L^2$ norm, or charge, at a point. This is done by decomposing the solution into an approximately linear component and a component with improved integrability. We then prove global existence for all $s\geqslant 0$.

#### Article information

**Source**

Adv. Differential Equations, Volume 16, Number 7/8 (2011), 643-666.

**Dates**

First available in Project Euclid: 17 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355703201

**Mathematical Reviews number (MathSciNet)**

MR2829499

**Zentralblatt MATH identifier**

1229.35225

**Subjects**

Primary: 35Q41: Time-dependent Schrödinger equations, Dirac equations 35A01: Existence problems: global existence, local existence, non-existence

#### Citation

Candy, Timothy. Global existence for an $L^2$ critical nonlinear Dirac equation in one dimension. Adv. Differential Equations 16 (2011), no. 7/8, 643--666. https://projecteuclid.org/euclid.ade/1355703201