Stable Finite-Time Singularity Formation for 3D Navier--Stokes via 5D-Lifted Axisymmetric Reductions

TL;DR

This paper develops a computer-assisted method to demonstrate finite-time singularity formation in 3D Navier--Stokes equations using a 5D-lifted axisymmetric reduction and validated fixed-point analysis.

Contribution

It introduces a novel 5D-lifted analytic-profile approach combined with rigorous computer validation to establish singularity formation in the Navier--Stokes equations.

Findings

Constructed a stationary rescaled profile satisfying a nonlinear elliptic equation.

Validated the profile using interval arithmetic and Newton--Kantorovich method.

Reconstructed a nearly self-similar singular evolution in the 3D setting.

Abstract

We present a 5D-lifted analytic-profile program for finite-time singularity formation in the 3D incompressible Navier--Stokes equations on the periodic torus $\T^3$. The core of the construction is a stationary rescaled profile $\Ombar$ satisfying a nonlinear elliptic fixed-point equation in an analytically weighted Hilbert space $\Xspace$, together with a computer-assisted Newton--Kantorovich validation based on interval arithmetic. The profile is reconstructed into a nearly self-similar singular evolution and then transferred to $\T^3$ by periodic extension and exact Leray projection. The manuscript is organized in the style of a computer-assisted proof paper, with theorem statements, proof packages, and explicit validation constants for the residual, inverse stability, and Lipschitz bounds.