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Asymptotic behavior of 2D incompressible ideal flow around small disks

By Christophe Lacave, Milton C. Lopes Filho and Helena J. Nussenzveig Lopes


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

Topics: [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP]
Publisher: HAL CCSD
Year: 2015
OAI identifier: oai:HAL:hal-01218374v1
Provided by: Hal-Diderot
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