Homogenization of a level-set equation in periodic and random media

The G-equation is a nonlinear, level-set PDE that is used in models of turbulent combustion. Level sets of the solution represent a flame surface which moves with normal velocity that is the sum of the laminar flame velocity and the fluid velocity. I will present recent work on the large-time asymptotics of the solutions when the fluid velocity is given by a prescribed, nearly incompressible vector field which is either periodic or random and stationary. The main analytical challenge comes from the fact that the Hamiltonian in the equation is not coercive. This includes joint work with Pierre Cardaliaguet, Takis Souganidis, and Alexei Novikov.