Scalar conservation laws with white noise initial data

The statistical description of the scalar conservation law with smooth convex Hamiltonian has been an object of interest when the initial profile is random. The special case of Burgers equation has in particular received extensive interest in the past and is now understood for various random initial conditions. I will present in this talk some recent results on the profile of the entropy solution at any time fixed t>0 for a general class of Hamiltonians H when the initial data is a Brownian white noise. This solves a conjecture of Menon and Srinivasan and proves the complete integrability of this system for this class of initial profiles. En route, the density function of the maximizer of W(z)−ϕ(z) is made explicit, where W is a two-sided Brownian motion and ϕ a C2 strictly convex function. Finally, I will elaborate on the structure of shocks of the entropy solution in the scenario above and extend the results of Avellaneda and Bertoin in Burgers turbulence.