Potentially Singular Behavior of the 3D Navier–Stokes Equations

Abstract

Whether the 3D incompressible Navier–Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompressible axisymmetric Navier–Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in a companion paper published in the same issue, see also Hou (Potential singularity of the 3D Euler equations in the interior domain. arXiv:2107.05870 [math.AP], 2021). We present numerical evidence that the 3D Navier–Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of \(10^7\). We have applied several blow-up criteria to study the potentially singular behavior of the Navier–Stokes equations. The Beale–Kato–Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier–Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya–Prodi–Serrin regularity criteria (Kiselev and Ladyzhenskaya in Izv Akad Nauk SSSR Ser Mat 21(5):655–690, 1957; Prodi in Ann Math Pura Appl 4(48):173–182, 1959; Serrin in Arch Ration Mech Anal 9:187–191, 1962) that are based on the growth rate of \(L_t^q L_x^p\) norm of the velocity with \(3/p + 2/q \le 1\). Our numerical results for the cases of \((p,q) = (4,8),\; (6,4),\; (9,3)\) and \((p,q)=(\infty ,2)\) provide strong evidence for the potentially singular behavior of the Navier–Stokes equations. The critical case of \((p,q)=(3,\infty )\) is more difficult to verify numerically due to the extremely slow growth rate in the \(L^3\) norm of the velocity field and the significant contribution from the far field where we have a relatively coarse grid. Our numerical study shows that while the global \(L^3\) norm of the velocity grows very slowly, the localized version of the \(L^3\) norm of the velocity experiences rapid dynamic growth relative to the localized \(L^3\) norm of the initial velocity. This provides further evidence for the potentially singular behavior of the Navier–Stokes equations.