Scalar conservation laws with rough (stochastic) fluxes
Scalar conservation laws with rough (stochastic) fluxes
复制标题
具有粗糙（随机）通量的标量守恒定律
DOI：
10.1007/s40072-013-0021-3
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
P. Souganidis
中科院分区：
文献类型：
--
作者：
P. Lions;B. Perthame;P. Souganidis
我们发展了具有拟线性乘性粗路径依赖的标量守恒律的路径理论，其特例是具有拟线性随机依赖的随机守恒律。我们引入了路径随机熵解的概念，证明了它是适定的，即我们建立了路径L1\Docentclass[12pt]{Minimum}\usepackage{amsath}\usepackage{waysym}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setLong{\oddsidemarin}{-69pt}\begin{document}$$L^1$$\end{document}-contraction，形式的适定解，即建立了解的存在唯一性和连续依赖性。我们的方法是受到随机粘性解理论的启发，该理论是由两位作者提出并发展的，用于研究带有乘性噪声的一阶和二阶随机偏微分方程组。这一理论依赖于通过对随机特征的流动进行局部倒置而构造的特殊检验函数。对于守恒定律，这最好是在我们这里遵循的动力学公式的水平上实施。
We develop a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, a special case being stochastic conservation laws with quasilinear stochastic dependence. We introduce the notion of pathwise stochastic entropy solutions, which is closed with the local uniform limits of paths, and prove that it is well posed, i.e., we establish existence, uniqueness and continuous dependence, in the form of pathwise L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document}-contraction, as well as some explicit estimates. Our approach is motivated by the theory of stochastic viscosity solutions, which was introduced and developed by two of the authors, to study fully nonlinear first- and second-order stochastic pde with multiplicative noise. This theory relies on special test functions constructed by inverting locally the flow of the stochastic characteristics. For conservation laws this is best implemented at the level of the kinetic formulation which we follow here.