## Differential and Integral Equations

- Differential Integral Equations
- Volume 23, Number 3/4 (2010), 265-278.

### Low regularity well-posedness for some nonlinear Dirac equations in one space dimension

Sigmund Selberg and Achenef Tesfahun

#### Abstract

We prove that the Cauchy problem for a nonlinear Dirac equation with vector self-interaction (Thirring model) and for a nonlinear system of two Dirac equations coupled through a vector-vector interaction (Federbusch model) are locally well posed, in one space dimension, for initial data in Sobolev spaces of almost critical dimension; i.e., in $H^\varepsilon$, the critical space being $L^2$, and globally well posed for initial data in $H^{1/2+\varepsilon}$, for any $\varepsilon>0$. We also consider a nonlinear Dirac equation with quadratic nonlinearity which was studied earlier by S.~Machihara and N.~Bournaveas. We prove that the Cauchy problem for this equation is locally well posed for initial data in $H^{\varepsilon}$.

#### Article information

**Source**

Differential Integral Equations, Volume 23, Number 3/4 (2010), 265-278.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019318

**Mathematical Reviews number (MathSciNet)**

MR2588476

**Zentralblatt MATH identifier**

1240.35362

**Subjects**

Primary: 35Q40: PDEs in connection with quantum mechanics 35L70: Nonlinear second-order hyperbolic equations

#### Citation

Selberg, Sigmund; Tesfahun, Achenef. Low regularity well-posedness for some nonlinear Dirac equations in one space dimension. Differential Integral Equations 23 (2010), no. 3/4, 265--278. https://projecteuclid.org/euclid.die/1356019318