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.

Keywords

• euler equations
• asymptotic behavior of vortex patches
• scattering theory