The Energy Based Near Singularity for Fourier Spectral 3D Navier-Stokes Equations
For researchers studying the Navier-Stokes regularity problem, this work offers rigorous numerical diagnostics to detect potential finite-time singularities.
The paper establishes an energy-based conditional regularity framework for Fourier spectral discretizations of the 3D Navier-Stokes equations, proving exponential spatial convergence and algebraic temporal convergence, and provides an a posteriori criterion linking numerical blowup to loss of regularity.
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 in space, algebraic convergence in time, 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.