  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.
