Kinetic theory and Lax equations for shock clustering and Burgers turbulence

We study shock statistics in the scalar conservation law ∂tu+∂xf(u) = 0, x ∈ R, t > 0, with a convex flux f and random initial data. We show that a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps) is preserved by the entropy solution to the conservation law and we derive kinetic equations that describe the evolution of shock statistics. These kinetic equations are equivalent to a Lax pair. Moreover, they admit remarkable exact solutions for Burgers equation (f(u) = u/2) suggesting the complete integrability of Burgers turbulence. MSC classification: 60J75, 35R60, 35L67, 82C99