Collective Behavior: Macroscopic versus Kinetic Descriptions

Conservation laws with rough fluxes and kinetic formulation

Benoît Perthame

Université Pierre et Marie Curie


We present a theory of Scalar Conservation Laws with rough (stochastic) fluxes, meaning that the flux is written as $\dot W(t) {\rm div} A(u)$ with W(t) a continuous function. The difficulty is that, even with BV estimates in space, the equations does not make sense in distributional sense.

Our approach is based on the kinetic formulation of conservations laws. It uses a natural definition by integrating along characteristics as it is done in Hamilton-Jacobi equations. This approach allows us to define a unique 'stochastic entropy solution'.

This theory leads naturally to consider Kinetic Averaging Lemmas with random fluxes. For simple situations we will show the differences in the gain of regularity between the deterministic and stochastic cases.

This talk is based on works with P.-L. Lions and P. E. Souganidis