Seminar

Ensembles of solutions to partial differential equations underly mathematical theories of turbulence and phase transitions. However, our understanding of such ensembles for even well-understood PDE is quite modest. This should be expected, since in rigorous terms, an "ensemble" is a probability measure supported on the space of solutions.

 

I will describe several results on the deterministic evolution of spatially random data by scalar conservation laws with a convex flux. The (surprising) punchline is that the evolution of the law of the solution is determined by a completely integrable system of kinetic equations that describes the clustering of shocks.