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Statistical solutions of hyperbolic conservation laws I: Foundations

by U. Fjordholm and S. Lanthaler and S. Mishra

(Report number 2016-59)

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 $L^1$.

Keywords: Hyperbolic systems, statistical solutions, probability measures, Wasserstein Metric

@Techreport{FLM16_696,
  author = {U. Fjordholm and S. Lanthaler and S. Mishra},
  title = {Statistical solutions of hyperbolic conservation laws I: Foundations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-59},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-59.pdf },
  year = {2016}
}

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