Asymptotic behavior of 2D incompressible ideal flow around small disks

In this article, we study the homogenization limit of a family of solutions to the incompressible 2D Euler equations in the exterior of a family of $n_k$ disjoint disks with centers ${z^k_i}$ and radii $varepsilon_k$. We assume that the initial velocities $u_0^k$ are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require, $n_k to infty$, and we assume $varepsilon_k to 0$ as $kto infty$. Let $gamma^k_i$ be the circulation of $u_0^k$ around the circle ${|x-z^k_i|=varepsilon_k}$. We prove that the homogenization limit retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1) $omega_0^k = mbox{ curl }u_0^k$ has a uniform compact support and converges weakly in $L^{p_0}$, for some $p_0>2$, to $omega_0 in L^{p_0}_{c}(mathbb{R}^2)$, (2) $sum_{i=1}^{n_k} gamma^k_i delta_{z^k_i} rightharpoonup mu$ weak-$ast$ in $mathcal{BM}(mathbb{R}^2)$ for some bounded Radon measure $mu$, and (3) the radii $varepsilon_k$ are sufficiently small. Then the corresponding solutions $u^k$ 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 $omega=mbox{ curl } u$, with initial data $omega_0$, where the transporting velocity field is generated from $omega$ so that its curl is $omega + mu$. As a byproduct, we obtain a new existence result for this modified Euler system.