Concentration-diffusion effects in viscous incompressible flows

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 $t_2$, $t_3$, ...