Asymptotic behavior of 2D incompressible ideal flow around small disks

Abstract

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 $k\to \infty$. 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.