Rendiconti di Matematica e delle Sue Applicazioni (Jan 1997)
Scattering theory: a possible approach to the homogenization problem for the Euler equations
Abstract
We are interested in the analysis of the asymptotic behavior of a vortex patch that evolves according to the two dimensional Euler equation for incompressible fluids. More precisely, we consider the problem of the existence of initial data, such that, as t → ∞, the vorticity ω(x, t) weakly converges, in the sense of measures, to a stationary solution, say ω_∞(|x|), of the Euler equations: in other words, we want to study if or not the vorticity is “homogenized”. In this paper we show that a characterization of homogenization can be given in terms of a scattering problem for the Euler equations. Moreover, via an iterative approach to the Euler problem, we show that the solutions of the equations of the first non trivial order homogenize.