Hierarchies of N-Point Functions for Nonlinear Conservation Laws with Random Initial Data

TL;DR

This paper develops a hierarchy of equations for n-point functions in nonlinear conservation laws with random initial data, capturing shock interactions and rarefactions, and introduces methods to analyze their evolution.

Contribution

It introduces two approaches to derive and analyze hierarchies of n-point functions, with one method remaining valid through shock collisions.

Findings

Hierarchy of equations for n-point functions derived

Second approach remains valid through shock interactions

Provides a framework for understanding solution evolution with randomness

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.