  We consider the mean curvature flow of entire Lagrangian graphs with
Lipschitz continuous initial data. Assuming only a certain bound on the
Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that
the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime
solution which is smooth for all positive time and satisfies uniform estimates
away from time $t=0$. In particular, under the mean curvature flow the graph
immediately becomes smooth and the solution exists for all time such that the
second fundamental form decays uniformly to 0 on the graph as $t\to \infty$.
Our assumption on the Lipschitz norm is equivalent to the assumption that the
underlying Lagrangian potential $u$ is uniformly convex with its Hessian
bounded in $L^\infty$. We apply this result to prove a Bernstein type theorem
for translating solitons, namely that if such an entire Lagrangian graph is a
smooth translating soliton, then it must be a flat plane. We also prove
convergence of the evolving graphs under additional conditions.
