The Energy Based Near Singularity for Fourier Spectral 3D Navier-Stokes Equations

TL;DR

This paper develops an energy-based framework and diagnostics for detecting potential finite-time singularities in 3D incompressible Navier-Stokes equations using Fourier spectral methods.

Contribution

It introduces an analytical energy-based regularity framework and diagnostics linking numerical blowup to loss of regularity in spectral simulations.

Findings

Proves exponential and algebraic convergence of the spectral method.

Establishes resolution conditions for accurate simulations.

Develops diagnostics for potential finite-time singularities.

Abstract

We investigate the three-dimensional incompressible Navier-Stokes equations. The equations are discretized with Fourier spectral method and a fourth-order Runge-Kutta scheme in time. The spectral accuracy, resolution conditions, and an energy based conditional regularity framework are established analytically. Then we prove exponential convergence, algebraic convergence, and an a posteriori criterion that links numerical blowup to loss of regularity. This work develops a suite of diagnostics for detecting potential finite time singular behavior.