Asymptotic behavior of 2D incompressible ideal flow around small disks

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 ${\mathit{n}}_{\mathit{k}}$ disjoint disks with centers $\{{\mathit{z}}_{\mathit{i}}^{\mathit{k}}\}$ and radii ${\mathit{\epsilon }}_{\mathit{k}}$ . We assume that the initial velocities ${\mathit{u}}_0^{\mathit{k}}$ are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require, ${\mathit{n}}_{\mathit{k}}\to \infty$ , and we assume ${\mathit{\epsilon }}_{\mathit{k}}\to 0$ as $\mathit{k}\to \infty$ .

Let ${\mathit{\gamma }}_{\mathit{i}}^{\mathit{k}}$ be the circulation of ${\mathit{u}}_0^{\mathit{k}}$ around the circle $\{|\mathit{x}-{\mathit{z}}_{\mathit{i}}^{\mathit{k}}|={\mathit{\epsilon }}_{\mathit{k}}\}$ . We prove that the limit as $\mathit{k}\to \infty$ retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1) ${\mathit{\omega }}_0^{\mathit{k}}=curl{\mathit{u}}_0^{\mathit{k}}$ has a uniform compact support and converges weakly in ${\mathit{L}}^{{\mathit{p}}_0}$ , for some ${\mathit{p}}_0\mathrm{>}2$ , to ${\mathit{\omega }}_0\in {\mathit{L}}_{\mathit{c}}^{{\mathit{p}}_0}\mathrm{(}{\mathbb{R}}^2\mathrm{)}$ , (2) ${\sum }_{\mathit{i}=1}^{{\mathit{n}}_{\mathit{k}}}{\mathit{\gamma }}_{\mathit{i}}^{\mathit{k}}{\mathit{\delta }}_{{\mathit{z}}_{\mathit{i}}^{\mathit{k}}}\mathrm{\rightharpoonup }\mathit{\mu }$ weak-∗ in $\mathcal{BM}\mathrm{(}{\mathbb{R}}^2\mathrm{)}$ for some bounded Radon measure μ, and (3) the radii ${\mathit{\epsilon }}_{\mathit{k}}$ are sufficiently small. Then the corresponding solutions ${\mathit{u}}^{\mathit{k}}$ 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 $\mathit{\omega }=curl\mathit{u}$ , with initial data ${\mathit{\omega }}_0$ , where the transporting velocity field is generated from ω, so that its curl is $\mathit{\omega }+\mathit{\mu }$ . As a byproduct, we obtain a new existence result for this modified Euler system.