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 [Formula: see text] disjoint disks with centers [Formula: see text] and radii [Formula: see text]. We assume that the initial velocities [Formula: see text] are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require, [Formula: see text], and we assume [Formula: see text] as [Formula: see text]. Let [Formula: see text] be the circulation of [Formula: see text] around the circle [Formula: see text]. We prove that the limit as [Formula: see text] retains information on the circulations as a time-independent coefficient. More precisely, we assume that: (1) [Formula: see text] has a uniform compact support and converges weakly in [Formula: see text], for some [Formula: see text], to [Formula: see text], (2) [Formula: see text] weak-∗ in [Formula: see text] for some bounded Radon measure μ, and (3) the radii [Formula: see text] are sufficiently small. Then the corresponding solutions [Formula: see text] 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 [Formula: see text], with initial data [Formula: see text], where the transporting velocity field is generated from ω, so that its curl is [Formula: see text]. As a byproduct, we obtain a new existence result for this modified Euler system.
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