Georgia Institute of Technology College of Sciences

We consider the evolutionary first order nonlinear partial differential equations of the most general form \frac{\partial u}{\partial t} + H(x, u, d_x u)=0.By virtue of introducing a new type of solution semigroup, we establish the weak KAM theorem for such partial differential equations, i.e. the existence of weak KAM solutions or viscosity solutions. Indeed, by employing dynamical approach for characteristics, we develop the theory of associated global viscosity solutions in general. Moreover, the solution semigroup acting on any given continuous function will converge to a uniform limit as the time goes to infinity. As an application, we prove that such limit satisfies the the associated stationary first order partial differential equations: H(x, u, d_x u)=0.