Existence of a Periodic Attractor for the 3D Navier-Stokes Equations on T 3

Abstract

This paper presents a framework for solving the Navier-Stokes equations on the 3D torus T 3 . In this work, a review is made and a sequel follows with regard to the author’s work in 2021 in [1] where the Geometric Calculus method was used to obtain Equation (6) in that reference. Developing an algebraic and differential procedure to rewrite the Navier-Stokes equations in a form that renders solutions in a particular i -th direction of fluid flow, the expressions solved for give finite time singularities in the form of the LambertW function on the surface of the ball embedded in T 3 = [ 0 , 1 ] 3 . In this paper, it is argued that these singularities for a single u i direction can be shifted outwards by using fixed point methods via nth compositions of LambertW functions. In this paper, the solution to Equation (6) is provided, which proves that the triple product u x u y u z has both singular and no finite time blowup. We develop a geometric-analytic framework for the three-dimensional incompressible Navier-Stokes equations on the torus T 3 that yields a periodic global attractor and clarifies the role of singular structures arising in nonlinear velocity interactions. Building on a componentwise reformulation of the Navier-Stokes equations, the nonlinear transport term is resolved explicitly in physical space, revealing a rigid normal-form structure governed by compositions of the Lambert W function. Singular solutions initially appear on embedded spherical manifolds through branch-point behavior of W , but we show that these singularities are removable in finite space and can be shifted to infinity via fixed-point methods and iterated Lambert W compositions and increasing Tori approaching ℝ 3 in the infinite limit. A key mechanism is the synchronization of local Lambert W dynamics with global elliptic structure through a Weierstrass ℘ - ζ mapping. By exploiting homogeneity and quasi-periodicity, a drift-corrected Weierstrass zeta potential is constructed that is strictly periodic on T 3 . The resulting velocity field is bounded, oscillatory in time, and invariant under the Navier-Stokes evolution, with secular growth absorbed into the pressure gauge. Elliptic degeneracy controls the singular-regular transition and ensures that all accessible singularities are isolated and integrable. The analysis identifies Lambert W profiles as invariant manifolds of finite codimension and establishes the existence of a periodic attractor for the 3D Navier-Stokes equations on T 3 , providing a concrete mechanism by which nonlinear singular behavior is regularized globally.