移动超曲面上的随机标量守恒律
Stochastic Scalar Conservation Laws on Moving Hypersurfaces
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中文总结 AI 辅助
该研究针对由布朗运动驱动的移动超曲面上的随机标量守恒律，推导移动曲面伊藤公式、引入广义熵解，通过消失黏性法构造鞅熵解，适配Kruzhkov方法建立路径唯一性，结合Yamada-Watanabe定理得到问题适定性。
中文摘要 AI 辅助
我们建立了由布朗运动驱动的移动超曲面上的随机标量守恒律的适定性。为处理随机强迫与演化几何之间的相互作用，我们推导了移动曲面上的伊藤公式，并引入了结合相关随机相互作用项的广义熵解概念。通过消失黏性法构造了鞅熵解，该方法基于空间和时间上的一致$L^\text{∞}$界、空间梯度的$L^1$估计、时间上的$L^1$连续性估计以及合适的紧性论证。通过将Kruzhkov的变量加倍法适配到移动超曲面上建立了路径唯一性，得到$L^1$收缩性质。最后，结合Yamada-Watanabe定理，这些结果得出该问题的适定性。
英文摘要
We establish the well-posedness of stochastic scalar conservation laws on moving hypersurfaces driven by Brownian motion. To handle the interaction between stochastic forcing and evolving geometry, we derive an Itô formula on moving surfaces and introduce the notion of generalized entropy solutions incorporating the relevant stochastic interaction terms. A martingale entropy solution is constructed via the vanishing-viscosity method, based on a uniform $L^\infty$-bound in space and time, an $L^1$-estimate for the spatial gradient, an $L^1$-continuity estimate in time, and a suitable tightness argument. Pathwise uniqueness is established by adapting Kruzhkov's doubling-of-variables method to moving hypersurfaces, yielding an $L^1$-contraction property. Finally, together with the Yamada-Watanabe theorem, these results yield the well-posedness of the problem.
