AI 中文总结
该研究针对乘性噪声驱动的一阶标量守恒律障碍问题，适配障碍替换策略到动力学公式，消除障碍-噪声相容性条件，建立适定性并证明$L^1$空间的Freidlin-Wentzell大偏差原理，为抛物型反射SPDE的大偏差理论提供双曲型对应结果。
AI 中文摘要
本文研究由乘性噪声驱动的一阶标量守恒律的障碍问题。通过将障碍替换策略适配到动力学公式，我们允许反射测度为一般的Radon测度，并消除了常规的障碍-噪声相容性条件。我们建立了连续障碍下动力学解的存在性，且在更强的空间正则性下进一步推导了$L^1$收缩性与唯一性。此外，我们在$L^1(0,T;L^1(\boldsymbol{\top}^N))$中证明了Freidlin-Wentzell大偏差原理。与现有基于障碍的控制一致惩罚的大偏差论证不同，我们直接处理反射骨架方程，全程保留可能奇异的Radon反射测度，同时粘性近似提供了紧致性所需的空间$H^1$正则性，之后过渡到无粘极限。我们的结果为Matoussi、Sabbagh和Zhang（2021）发展的抛物型反射SPDE的大偏差理论提供了双曲型对应结果。
英文摘要
This paper studies the obstacle problem for first-order scalar conservation laws driven by multiplicative noise. By adapting a barrier-substitution strategy to the kinetic formulation, we permit the reflection measure to be a general Radon measure and eliminate the usual obstacle-noise compatibility condition. We establish the existence of kinetic solutions for continuous obstacles and further derive $L^1$-contraction and uniqueness under stronger spatial regularity. Moreover, we prove a Freidlin--Wentzell large deviation principle in $L^1(0,T;L^1(\mathbb{T}^N))$. Unlike existing large-deviation arguments based on control-uniform penalization of the obstacle, we work directly with the reflected skeleton equation and retain the possibly singular Radon reflection measure throughout, while a viscous approximation provides the spatial $H^1$-regularity required for compactness before passing to the inviscid limit. Our results provide a hyperbolic counterpart to the large deviation theory for parabolic reflected SPDEs developed by Matoussi, Sabbagh, and Zhang (2021).
Comments61 pages
