Weak KAM theorem for the most general first-order Nonlinear partial differential equation

Dynamical Systems Working Seminar
Tuesday, March 26, 2013 - 16:30
1 hour (actually 50 minutes)
Skiles 006
Academy of Mathematics and Systems Science, Chinese Academy 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.