Burgers turbulence (1-D inviscid Burgers equation with random initial data) is a fundamental non-equilibrium model of stochastic coalescence. In this work we demonstrate that at the level of the 2-point correlation function, the entropy solution to Burgers equation yields a closed, completely integrable system. We show that the statistical evolution is given by a Lax pair. Finally, we demonstrate that this equation has an equivalent kinetic description with a rich family of self-similar solutions, and in particular admits an explicit solution derived by Groeneboom in nonparametric statistics. Finally, the closure property and complete integrability are shown to hold in the general case of 1-D scalar conservation laws with strictly convex flux.
"Closure and complete integrability in Burgers turbulence"
Applied Mathematics Theses and Dissertations.
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