Global solution to a cubic nonlinear Dirac equation in 1+1 dimensions

TL;DR

This paper proves the global existence and stability of solutions to a class of cubic nonlinear Dirac equations in 1+1 dimensions, including models like the Thirring and Gross-Neveu models, under small initial charge conditions.

Contribution

It introduces a novel approach using Bony functionals and Lyapunov functionals to establish global existence and stability for these nonlinear Dirac equations.

Findings

Global solutions exist for small initial charge

Solutions are stable in L^2 norm

Weak solutions are globally available in L^2

Abstract

This paper studies a class of nonlinear Dirac equations with cubic terms in $R^{1+1}$, which include the equations for the massive Thirring model and the massive Gross-Neveu model. Under the assumptions that the initial data has small charge, the global existence of the solution in $H^1$ are proved. The proof is given by introducing some Bony functional to get the uniform estimates on the nonlinear terms and the uniform bounds on the local smooth solution, which enable us to extend the local solution globally in time. Then $L^2$-stability estimates for these solutions are also established by a Lyapunov functional and the global existence of weak solution in $L^2$ is obtained.