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arXiv 1605.05960math.AP
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双曲守恒律的统计解I：基础

Statistical solutions of hyperbolic conservation laws I: Foundations

Ulrik Skre Fjordholm, Samuel Lanthaler, Siddhartha Mishra

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摘要：

我们试图将双曲守恒律方程组的统计解定义为$p$ -可积函数上的时间参数化概率测度。为此，我们证明了$L^p$空间上的概率测度与\textit{correlation measures}无限族之间的等价性。这个家族的每个成员，称为\textit{correlation marginal}，是一个有限维张量积域上的Young测度，并提供了有关基础可积函数的多点相关性的信息。证明了$L^p$空间上的任何概率测度都是由等价相关测度的某些矩（相关函数）唯一确定的.我们利用这种等价性来定义多维守恒律的统计解的一组无限的方程，每个演变的相关边际的时刻。这些演化方程可以解释为增广熵测度值的解决方案，与所有可能的多点相关函数的演变的额外信息。我们的统计解决方案的概念，可以容纳不确定的初始数据，以及可能的非原子的解决方案，甚至原子的初始数据。对于多维标量守恒律，我们施加额外的熵条件，并证明了所得到的\textit{entropy statistical solutions}存在，是唯一的，是稳定的概率测度的$1$ -Wasserstein度量$L^1$。

英文摘要：

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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