I develop a continuous functional framework for spatial treatment effects grounded in Navier-Stokes partial differential equations. Rather than discrete treatment parameters, the framework characterizes treatment intensity as continuous functions $\tau(\mathbf{x}, t)$ over space-time, enabling rigorous analysis of boundary evolution, spatial gradients, and cumulative exposure. Empirical validation using 32,520 U.S. ZIP codes demonstrates exponential spatial decay for healthcare access ($\kappa = 0.002837$ per km, $R^2 = 0.0129$) with detectable boundaries at 37.1 km. The framework successfully diagnoses when scope conditions hold: positive decay parameters validate diffusion assumptions near hospitals, while negative parameters correctly signal urban confounding effects. Heterogeneity analysis reveals 2-13 $\times$ stronger distance effects for elderly populations and substantial education gradients. Model selection strongly favors logarithmic decay over exponential ($\Delta \text{AIC} > 10,000$), representing a middle ground between exponential and power-law decay. Applications span environmental economics, banking, and healthcare policy. The continuous functional framework provides predictive capability ($d^*(t) = \xi^* \sqrt{t}$), parameter sensitivity ($\partial d^*/\partial \nu$), and diagnostic tests unavailable in traditional difference-in-differences approaches.

翻译：本文基于纳维-斯托克斯偏微分方程，建立了空间处理效应的连续泛函框架。该框架将处理强度表征为时空连续函数 $\tau(\mathbf{x}, t)$，而非离散处理参数，从而能够严格分析边界演化、空间梯度与累积暴露效应。通过对美国32,520个邮政编码区的实证验证，发现医疗可及性存在指数型空间衰减（$\kappa = 0.002837$ 每公里，$R^2 = 0.0129$），可检测边界位于37.1公里处。该框架能有效诊断适用范围条件：正衰减参数验证了医院附近的扩散假设，而负参数则准确指示了城市混杂效应。异质性分析显示老年人群的距离效应强度高出2-13倍，且存在显著的教育梯度。模型选择强烈支持对数衰减而非指数衰减（$\Delta \text{AIC} > 10,000$），这代表了指数衰减与幂律衰减之间的折中方案。本框架可应用于环境经济学、银行业与医疗政策领域。连续泛函框架提供了传统双重差分法所不具备的预测能力（$d^*(t) = \xi^* \sqrt{t}$）、参数敏感性（$\partial d^*/\partial \nu$）及诊断检验功能。