### Electronic Research Archive

2021, Issue 6: 4229-4241. doi: 10.3934/era.2021081
Special Issues

# Planar vortices in a bounded domain with a hole

• Received: 01 August 2021 Published: 08 October 2021
• Primary: 58F15, 58F17; Secondary: 53C35

• 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[J]. Electronic Research Archive, 2021, 29(6): 4229-4241. doi: 10.3934/era.2021081

### Related Papers:

• 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 $\lambda\to +\infty$.

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