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• W3101686413 endingPage "849" @default.
• W3101686413 startingPage "809" @default.
• W3101686413 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 $L^1$." @default.
• W3101686413 created "2020-11-23" @default.
• W3101686413 creator A5029463202 @default.
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• W3101686413 date "2017-07-14" @default.
• W3101686413 modified "2023-10-16" @default.
• W3101686413 title "Statistical Solutions of Hyperbolic Conservation Laws: Foundations" @default.
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• W3101686413 doi "https://doi.org/10.1007/s00205-017-1145-9" @default.
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