Statistical solutions of hyperbolic conservation laws I: Foundations
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2016-12
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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 \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 \(L^1\).
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2016-59
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Seminar for Applied Mathematics, ETH Zurich
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03851 - Mishra, Siddhartha / Mishra, Siddhartha
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