Archive for Rational Mechanics and Analysis

, Volume 226, Issue 2, pp 809–849 | Cite as

Statistical Solutions of Hyperbolic Conservation Laws: Foundations

  • U. S. Fjordholm
  • S. Lanthaler
  • S. Mishra


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 correlation measures. Each member of this family, termed a 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 entropy statistical solutions exist, are unique and are stable with respect to the 1-Wasserstein metric on probability measures on L 1.


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Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Swiss Plasma Center, SB SPC-THLausanneSwitzerland
  3. 3.Seminar for Applied MathematicsZürichSwitzerland

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