The Finite-Dimensional Geometry of Turbulence: A Global Attractor Approach to 3D Navier-Stokes Regularity

The Navier-Stokes Existence and Smoothness problem, a central challenge in mathematics and fluid dynamics, is reformulated here as a geometric question concerning the asymptotic behavior of infinite-dimensional dynamical systems. The traditional approach focuses on short-time analytical estimates, which are hampered by the complex, chaotic nature of turbulence. We propose a solution based on the long-term dynamics by analyzing the properties of the Global Attractor (A3D). For any dissipative system, including the 3D Navier-Stokes equations (NSE), all long-term trajectories must converge onto this compact invariant set. The core theorem asserts that if the Global Attractor possesses a finite fractal dimension, dimF (A3D) = N < ∞, then A3D is necessarily contained entirely within the space of smooth functions, H1(Ω), thus structurally precluding the formation of finite-time singularities (blow-up). This finite dimension, which mathematically quantifies the effective degrees of freedom in turbulence, provides a rigorous topological constraint. The proof path centers on establishing an unconditional bound on the sum of the Lyapunov exponents, thereby confirming that viscous dissipation is strong enough to limit the asymptotic complexity N, reducing the original infinite-dimensional PDE to a manageable finite-dimensional system of ODEs.