CRM: Centro De Giorgi

This is the old version of the CRM site. Please use the new site on the page crmdegiorgi.sns.it

logo sns
Equazioni alle Derivate Parziali nella Dinamica dei Fluidi

Analytic solutions of the 2D Navier-Stokes equations with concentrated vorticity

speaker: Marco Maria Luigi Sammartino (Università di Palermo)

abstract: In this talk we shall consider the 2D Navier-Stokes equations with initial data concentrated on a small set. A particularly significant instance of this configuration is when the vorticity \(\omega=O(\epsilon^{-1})\) is distributed close to a curve and decays away from exponentially fast.

In a recent paper the authors considered a similar problem for the analytic solutions of the Euler equations, proving that, for a short time and in the limit $\epsilon\rightarrow 0$, the solutions do not develop oscillations or concentrations. The authors proved also that the vorticity remains supported close to a curve $y=\phi(x,t)$ whose dynamics is ruled by the Birkhoff-Rott equation.

The aim of this talk is to take into account the viscosity effects, assuming the diffusive scaling \(\epsilon=O(\sqrt{\nu})\). We shall see that, for an almost flat layer, the vorticity follows, to the leading order, the Euler dynamics; and that the viscosity effects, confined to the correction equation, can be described by a controllable weakly nonlinear convection-diffusion equation. {\bf This is joint work with R.Caflisch and M.C.Lombardo. }


timetable:
Wed 7 Feb, 9:35 - 10:10, Aula Dini
<< Go back