# American Institute of Mathematical Sciences

December  2021, 29(6): 4229-4241. doi: 10.3934/era.2021081

## Planar vortices in a bounded domain with a hole

 1 School of Mathematics and Statistics, Central China Normal University, Wuhan, China 2 Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Shusen Yan

Received  August 2021 Published  December 2021 Early access  October 2021

In this paper, we consider the inviscid, incompressible planar flows in a bounded domain with a hole and construct stationary classical solutions with single vortex core, which is closed to the hole. This is carried out by constructing solutions to the following semilinear elliptic problem
 $$$\begin{cases} -\Delta \psi = \lambda(\psi-\frac{\kappa}{4\pi}\ln\lambda)_+^p,\quad &\text{in}\; \Omega,\\ \psi = \rho_\lambda,\quad &\text{on}\; \partial O_0,\\ \psi = 0,\quad &\text{on}\; \partial\Omega_0, \end{cases} \;\;\;\;\;\;\;\;(1)$$$
where
 $p>1$
,
 $\kappa$
is a positive constant,
 $\rho_\lambda$
is a constant, depending on
 $\lambda$
,
 $\Omega = \Omega_0\setminus \bar{O}_0$
and
 $\Omega_0$
,
 $O_0$
are two planar bounded simply-connected domains. We show that under the assumption
 $(\ln\lambda)^\sigma\leq\rho_\lambda\leq (\ln\lambda)^{1-\sigma}$
for some
 $\sigma>0$
small, (1) has a solution
 $\psi_\lambda$
, whose vorticity set
 $\{y\in \Omega:\, \psi(y)-\kappa+\rho_\lambda\eta(y)>0\}$
shrinks to the boundary of the hole as
 $\lambda\to +\infty$
.
Citation: Shusen Yan, Weilin Yu. Planar vortices in a bounded domain with a hole. Electronic Research Archive, 2021, 29 (6) : 4229-4241. doi: 10.3934/era.2021081
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