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
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