Concentration diffusion Effects in Viscous Incompressible Flows

Concentration-diffusion Effects in Viscous Incompressible Flows LORENZO BRANDOLESE ABSTRACT. Given a finite sequence of times 0 < t1 < < tN , we construct an example of a smooth solution of the free nonstationnary Navier-Stokes equations in Rd, 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 t1, it becomes well-localized. (ii) Then u spreads out again after t1, and such concentration-diffusion phenomena are later reproduced near the instants t2, t3, . . . . 1. INTRODUCTION One of the most important questions in mathematical Fluid Mechanics, which is still far from being understood, is to know whether a finite energy, and initially smooth, nonstationnary Navier-Stokes flow will always remain regular during its evolution, or can become turbulent in finite time. As a first step toward the understanding of possible blow-up mechanisms, it is interesting to exhibit examples of smooth and decaying initial data such that, even if the corresponding solutions remain regular for all time, “something strange” happens around a given point ?x0; t0? in space-time. This is the goal of the present paper. Our main result is the construction of a class of (smooth) solutions to the incompressible Navier-Stokes equations such that, in the absence of any external forces, the motion of the fluid particles tends to be more concentrated around x0, as the time t approaches t0

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