Hierarchies of N-point functions for nonlinear conservation laws with random initial data

Abstract

Nonlinear conservation laws subject to random initial conditions pose fundamental problems in the evolution and interactions of shocks and rarefactions. Using a discrete set of values for the solution, we derive a hierarchy of equations in terms of the states in two different methods. This hierarchy involves the n-point function, the probability that the solution takes on various values at different positions, in terms of the (n+1)-point function. In the first approach, these equations can be closed but the resulting solutions do not persist through shock interactions. In our second approach, the n-point function is expressed in terms of the (n+1)-point functions, and remains valid through collisions of shocks.