AI 中文总结
研究依赖历史的熵公式相关问题，通过建立特定结构置于路径依赖熵拉格朗日构造中，利用端点校准等确定相关量，在不同正则性下制定扩散分配，组合要素得到泛函及相关性质，还可特殊化为Cahn-Hilliard方程。
AI 中文摘要
依赖历史的熵公式需要通道终端变化的精确含义以及对弱扩散场仍然有效的空间功率分配。本文建立了这两种结构，并将它们置于单个路径依赖熵拉格朗日构造中。端点校准和上循环可加性唯一地确定定向支出增量，而分解功率端口的通道扩展产生标量耗散和势加权通量的显式终端微分。扩散分配在索伯列夫、有限能量分布和有界变差/散度测度正则性下制定。一个自然对偶空间反例证明化学势加权残差并不意味着守恒，一个独立乘数恢复未加权平衡。这些要素在同步乘积空间上的一个增强热扩散PDEL泛函中组合。其允许的方向平稳性产生能量和热共轭、物种平衡、通量闭合、熵产生以及两个终端功率路由通道。相同构造允许有限能量弱形式并专门化为Cahn-Hilliard方程。
英文摘要
History-dependent entropic variational formulations require a calibrated terminal differential for accumulated power and a spatial power split that remains meaningful for weak diffusion fields. Endpoint calibration and cocycle additivity determine a unique oriented spending increment, while independent local selector fields give its distributional channel form. The diffusion identity for a potential-weighted flux is established at $H(\Div)$ regularity and extended to the finite-energy class $H^1\times L^2$ through a distributional balance component. For regular diffusion models, the weighted species channel yields the local balance whenever persistent zero-potential states are dynamically isolated in the admissible state class. An independent multiplier extends the same balance to the natural weak space. These ingredients are assembled in one synchronized thermo-diffusion functional whose directional stationarity yields energetic and thermal conjugacy, species balance, flux closure, the entropy equation, and both terminal routing rules. A Cahn--Hilliard specialization verifies the construction at finite-energy regularity.
Comments21 pages
