I develop a comprehensive theoretical framework for dynamic spatial treatment effect boundaries using continuous functional definitions grounded in Navier-Stokes partial differential equations. Rather than discrete treatment effect estimators, the framework characterizes treatment intensity as a continuous function $\tau(\mathbf{x}, t)$ over space-time, enabling rigorous analysis of propagation dynamics, boundary evolution, and cumulative exposure patterns. Building on exact self-similar solutions expressible through Kummer confluent hypergeometric and modified Bessel functions, I establish that treatment effects follow scaling laws $\tau(d, t) = t^{-\alpha} f(d/t^\beta)$ where exponents characterize diffusion mechanisms. Empirical validation using 42 million TROPOMI satellite observations of NO$_2$ pollution from U.S. coal-fired power plants demonstrates strong exponential spatial decay ($\kappa_s = 0.004$ per km, $R^2 = 0.35$) with detectable boundaries at 572 km. Monte Carlo simulations confirm superior performance over discrete parametric methods in boundary detection and false positive avoidance (94\% vs 27\% correct rejection). Regional heterogeneity analysis validates diagnostic capability: positive decay parameters within 100 km confirm coal plant dominance; negative parameters beyond 100 km correctly signal when urban sources dominate. The continuous functional perspective unifies spatial econometrics with mathematical physics, providing theoretically grounded methods for boundary detection, exposure quantification, and policy evaluation across environmental economics, banking, and healthcare applications.

翻译：本文基于纳维-斯托克斯偏微分方程，通过连续泛函定义建立了一套用于动态空间处理效应边界的完整理论框架。该框架摒弃离散的处理效应估计量，将处理强度表征为时空上的连续函数 $\tau(\mathbf{x}, t)$，从而能够严格分析传播动力学、边界演化及累积暴露模式。基于可通过库默合流超几何函数与修正贝塞尔函数表达的精确实自相似解，本文论证了处理效应遵循标度律 $\tau(d, t) = t^{-\alpha} f(d/t^\beta)$，其中指数表征扩散机制。利用TROPOMI卫星对美国燃煤电厂NO$_2$污染的4200万次观测数据进行实证验证，结果显示强烈的指数型空间衰减（$\kappa_s = 0.004$ 每公里，$R^2 = 0.35$），可检测边界达572公里。蒙特卡洛模拟证实，在边界检测与误报规避方面，本方法优于离散参数化方法（正确拒绝率94%对27%）。区域异质性分析验证了其诊断能力：100公里内的正衰减参数确认了燃煤电厂的主导影响；100公里外的负衰减参数则准确指示了城市污染源占主导的情况。这一连续泛函视角将空间计量经济学与数学物理相统一，为环境经济学、金融与医疗健康等领域的边界检测、暴露量化及政策评估提供了理论扎实的方法体系。