Statistical solutions of hyperbolic conservation laws I: Foundations

TL;DR

This paper develops a rigorous framework for statistical solutions of hyperbolic conservation laws using probability measures and correlation measures, establishing existence, uniqueness, and stability results for entropy solutions in multiple dimensions.

Contribution

It introduces a novel equivalence between probability measures on $L^p$ spaces and correlation measures, enabling a new formulation of statistical solutions for conservation laws.

Findings

Established the equivalence between probability measures and correlation measures.

Defined statistical solutions via infinite moment equations for multi-dimensional laws.

Proved existence, uniqueness, and stability of entropy statistical solutions.

Abstract

We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on $p$-integrable functions. To do so, we prove the equivalence between probability measures on $L^p$ spaces and infinite families of \textit{correlation measures}. Each member of this family, termed a \textit{correlation marginal}, is a Young measure on a finite-dimensional tensor product domain and provides information about multi-point correlations of the underlying integrable functions. We also prove that any probability measure on a $L^p$ space is uniquely determined by certain moments (correlation functions) of the equivalent correlation measure. We utilize this equivalence to define statistical solutions of multi-dimensional conservation laws in terms of an infinite set of equations, each evolving a moment of the correlation marginal. These evolution equations can be interpreted as augmenting entropy measure-valued solutions, with additional information about the evolution of all possible multi-point correlation functions. Our concept of statistical solutions can accommodate uncertain initial data as well as possibly non-atomic solutions even for atomic initial data. For multi-dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting \textit{entropy statistical solutions} exist, are unique and are stable with respect to the $1$-Wasserstein metric on probability measures on $L^1$.