标题： 标量模型问题的统计守恒定律：层次演化方程 显示英文标题

标题： Statistical conservation laws for scalar model problems: Hierarchical evolution equations

摘要： 不可压缩纳维-斯托克斯方程解的概率密度函数（PDFs）可以通过一组线性方程来表示。 本文以随机初始数据的标量守恒定律为模型问题，发展了新的PDF层次演化方程。 开发了两种框架，包括多点PDFs和单点高阶导数PDFs。 这些层次结构捕捉统计相关性并指导闭合策略。 显示英文摘要

摘要： The probability density functions (PDFs) for the solution of the incompressible Navier-Stokes equation can be represented by a hierarchy of linear equations. This article develops new hierarchical evolution equations for PDFs of a scalar conservation law with random initial data as a model problem. Two frameworks are developed, including multi-point PDFs and single-point higher-order derivative PDFs. These hierarchies capture statistical correlations and guide closure strategies.