Existence and Smoothness of Three-Dimensional Navier-Stokes Solutions via Hodge Theory and Weighted Sobolev Decay

We prove the existence and smoothness of solutions to the three-dimensional Navier-Stokes equations. Specifically, we establish that for viscosity greater than zero and dimension equal to three, given any smooth, divergence-free vector field with spatial decay satisfying appropriate decay conditions with decay parameter greater than five, and taking the forcing term identically zero, there exist smooth pressure and velocity functions on three-dimensional Euclidean space times the nonnegative time interval satisfying the Navier-Stokes equations with smooth solutions and bounded energy for all nonnegative time. Our approach reformulates the problem geometrically using differential forms and Hodge theory on Riemannian manifolds. We demonstrate that the Navier-Stokes equations are equivalent to geometric consistency conditions on a velocity field section of a bundle over spacetime, governed by the Hodge-Laplace operator. The existence of smooth, globally defined solutions follows from the Hodge decomposition theorem, elliptic regularity theory for the Laplace-de Rham operator, and weighted Sobolev transport estimates along Lagrangian trajectories. A key contribution is establishing that spatial decay of initial data implies temporal integrability of the velocity gradient through geometric necessity: the vanishing energy flux at spatial infinity, acting as a boundary condition, prevents gradient accumulation. This closes the gap in the Beale-Kato-Majda conditional regularity criterion, demonstrating that decay structure of initial data governs global regularity via geometric constraints rather than dynamical evolution mechanisms. The method provides a pathway to global smoothness that complements existing approaches based on smallness or critical Sobolev regularity.