Numerical study on how advection delays and removes singularity formation in the Navier–Stokes equations

Abstract

We numerically study a distorted version of the Euler and Navier–Stokes equations, which are obtained by depleting the advection term systematically. It is known that in the inviscid case some solutions blow up in finite time when advection is totally discarded, Constantin (1986 Commun. Math. Phys. 104 311–26). Taking a pair of orthogonally offset vortex tubes and the Taylor–Green vortex as initial data, we show the following. (1) Blowup persists even with viscosity when advection is discarded, and (2) for small viscosity, the time of blowup increases logarithmically as we reinstate advection using a continuous parameter, which would be consistent with the regularity of the Navier–Stokes equations. A tiny mismatch in the coefficient of the advection term, as minute as parts per trillion, throws the system out of compactness and leads to blowup.