Global Smooth Solutions to the 3D Incompressible Navier-Stokes Equations: Weakly Regular Framework and Multi-Scenario Adaptation

This paper establishes a unified mathematical framework independent of strong regularity constraints on initial data and external forces, and rigorously proves the existence, uniqueness, and stability of global smooth solutions for the 3D incompressible Navier-Stokes equations. The framework covers three classes of initial data: $H^{s}$-bounded, purely $L^{2}$-bounded, and locally weakly singular, with external forces restricted only to $L^{2}([0, \infty) ; L^{2}(\mathbb{R}^{3}))$. The core innovation lies in the trinity framework of compactly supported mollifier regularization, uniform double limit energy estimates, and Galerkin iteration, which seamlessly adapts to both weakly regular practical scenarios and highly regular ideal scenarios without structural reconstruction. Key conclusions include: (1) Local weak singularities of initial data vanish instantaneously for $t>0$, and solutions are globally smooth in $C^{\infty}((0, \infty) ; H^{\infty}(\mathbb{R}^{3}))$; (2) High-regularity initial data and external forces yield solutions with arbitrary-order smoothness at $t=0$ and for all subsequent time, excluding finite-time blow-up; (3) Turbulent "apparent singularities" are interpreted as spatiotemporal high-frequency oscillations of smooth solutions, without relying on physical assumptions. This work fills the gap in weakly regular well-posedness theory and provides rigorous mathematical support for ideal scenario analysis.