Abstract :
Whether the 3D incompressible Euler and Navier–Stokes equations can develop
a finite-time singularity from smooth initial data with finite energy
has been one of the most long-standing open questions. We review some recent
theoretical and computational studies which show that there is a subtle
dynamic depletion of nonlinear vortex stretching due to local geometric regularity
of vortex filaments. We also investigate the dynamic stability of the 3D
Navier–Stokes equations and the stabilizing effect of convection. A unique
feature of our approach is the interplay between computation and analysis.
Guided by our local non-blow-up theory, we have performed large-scale computations
of the 3D Euler equations using a novel pseudo-spectral method
on some of the most promising blow-up candidates. Our results show that
there is tremendous dynamic depletion of vortex stretching. Moreover, we
observe that the support of maximum vorticity becomes severely flattened
as the maximum vorticity increases and the direction of the vortex filaments
near the support of maximum vorticity is very regular. Our numerical observations
in turn provide valuable insight, which leads to further theoretical
breakthrough. Finally, we present a new class of solutions for the 3D Euler
and Navier–Stokes equations, which exhibit very interesting dynamic growth
properties. By exploiting the special nonlinear structure of the equations, we
prove nonlinear stability and the global regularity of this class of solutions.
