Abstract

We study elementary excitations for describing the late stage of decaying Navier-Stokes flows. For 2D Navier-Stokes flows it is well-known that the late stage of the time evolution is represented by the so-called Burgers vortices. Here we will seek an analogue of such elementary excitations (i.e. building blocks) for 3D Navier-Stokes flows.   We begin by clarifying the two kinds of critical scale-invariance of the Navier-Stokes equations. We obtain the leading-order solution of the dynamically-scaled Navier-Stokes equations explicitly and characterise its spatial structure.   We will raise and answer the following questions about solutions to the dynamically-scaled Navier-Stokes equations: Can we find functions which map the linearised solutions to the nonlinear ones ? If so, in which dependent variables can we do that ? Also addressed is the implications on the non-integrability of the Navier-Stokes equations.   (Part of this work is joint with R. Vanon.)