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Statistical Solutions of Hyperbolic Conservation Laws: Foundations

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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 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. Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491, Trondheim, Norway

    U. S. Fjordholm

  2. Swiss Plasma Center, SB SPC-TH, PPB 313 (Bâtiment PPB), Station 13, CH-1015, Lausanne, Switzerland

    S. Lanthaler

  3. Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, Zürich, Switzerland

    S. Mishra

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  1. U. S. Fjordholm
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  2. S. Lanthaler
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  3. S. Mishra
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Correspondence to S. Mishra.

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Communicated by C. Dafermos

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Fjordholm, U.S., Lanthaler, S. & Mishra, S. Statistical Solutions of Hyperbolic Conservation Laws: Foundations. Arch Rational Mech Anal 226, 809–849 (2017). https://doi.org/10.1007/s00205-017-1145-9

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  • DOI: https://doi.org/10.1007/s00205-017-1145-9

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