A kinetic theory of shock clustering in scalar conservation laws

We describe a kinetic theory for shock clustering in scalar conservation laws with random initial data. Our main discovery is that for a natural class of random data the shock clustering is described by a completely integrable Hamiltonian system. Thus, the problem is in a precise sense exactly solvable. Our results have implications in other areas: mathematical physics (limits of shell models of turbulence, and forced Burgers turbulence); probability theory (explicit computations of laws of excursions); and statistics (limit laws in the vicinity of maxima). Our work grew out of a study of Smoluchowski’s coagulation equation. This is a mean-field model of domain coarsening, first introduced to model coagulation in colloids. Quite remarkably, it also describes the clustering of shocks in Burgers equation for a class of random initial data [3, 7, 12]. This is a particular case of Burgers turbulence– the study of shock statistics in Burgers equation with random initial data or forcing. Our goal was to understand if this link between a mean-field model of coalescence and shock clustering was an isolated example, or part of a more general theory. It is in fact, a consequence of the theory outlined below.