Scalar Conservation Laws with Random Initial Data

Abstract

In this talk, instead of considering the pathwise property of the entropy solution of a scalar conservation law, we are interested in the (probability) law of the solution. In 2010, Menon and Srinivasan conjectured that a certain class (spectrally negative) of Markov processes (in \(x\)) is preserved by the entropy solution and proposed the evolution (in \(t\)) of the generator of the solution. Bertoin and Sinai characterized Burgers’ equation with Brownian initial data; Rezakhanlou and Kaspar verified this conjecture for bounded spectrally negative Lévy processes for general scalar conservation laws. We will discuss their proofs and show how probability theory intertwines with nonlinear PDEs.

See slides for more details.