Lax Equations and Kinetic Theory for Shock Clustering and Burgers Turbulence

Much of our current understanding of statistical theories of turbulence relies on vastly simplified caricatures. One such caricature is Burgers turbulence. This is the study of the statistics of shocks in Burgers equation with random initial data or forcing. This model also arises in statistics, combinatorics, and models of coagulation and surface growth. It is of wide interest as a benchmark, even if it describes phenomena that are not entirely turbulent.

I will describe a kinetic theory for shock clustering that applies to all scalar conservation laws with convex flux and a basic class of random initial data. A remarkable feature of the kinetic theory is that it is presented as a Lax pair, admits remarkable exact solutions, and seems to have deep connections with completely integrable systems.

The talk relies heavily on joint work with Bob Pego and Ravi Srinivasan.

Govind Menon, Brown University

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