Concentration-diffusion effects in viscous incompressible flows

TL;DR

This paper constructs smooth solutions to the Navier-Stokes equations demonstrating repeated concentration and diffusion phenomena in velocity fields over time.

Contribution

It provides explicit examples of solutions exhibiting multiple concentration-diffusion cycles in viscous incompressible flows.

Findings

Velocity concentrates before $t_1$, then diffuses afterward.

Phenomena recur near subsequent times $t_2$, $t_3$, ...

Solutions are smooth and explicitly constructed.

Abstract

Given a finite sequence of times $0<t_1<...<t_N$, we construct an example of a smooth solution of the free nonstationnary Navier--Stokes equations in $\R^d$, $d=2,3$, such that: (i) The velocity field $u(x,t)$ is spatially poorly localized at the beginning of the evolution but tends to concentrate until, as the time $t$ approaches $t_1$, it becomes well-localized. (ii) Then $u$ spreads out again after $t_1$, and such concentration-diffusion phenomena are later reproduced near the instants $t_2$, $t_3$, ...