Affiliations: [a] Univ. Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, France. E-mail: [email protected] | [b] Instituto de Matemática, Universidade Federal do Rio de Janeiro, Cidade Universitária – Ilha do Fundão, Caixa Postal 68530, 21941-909 Rio de Janeiro, RJ, Brazil. E-mails: [email protected], [email protected]
Abstract: In this article, we study the limit of a family of solutions to the incompressible 2D Euler equations in the exterior of a family of nk disjoint disks with centers {zik} and radii εk. We assume that the initial velocities u0k are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require, nk→∞, and we assume εk→0 as k→∞. Let γik be the circulation of u0k around the circle {|x−zik|=εk}. We prove that the limit as k→∞ retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1) ω0k=curlu0k has a uniform compact support and converges weakly in Lp0, for some p0>2, to ω0∈Lcp0(R2), (2) ∑i=1nkγikδzik⇀μ weak-∗ in BM(R2) for some bounded Radon measure μ, and (3) the radii εk are sufficiently small. Then the corresponding solutions uk converge strongly to a weak solution u of a modified Euler system in the full plane. This modified Euler system is given, in vorticity formulation, by an active scalar transport equation for the quantity ω=curlu, with initial data ω0, where the transporting velocity field is generated from ω, so that its curl is ω+μ. As a byproduct, we obtain a new existence result for this modified Euler system.
Keywords: Euler equations, homogenization in perforated domain, shrinking obstacles and porous medium
