Planar vortices in a bounded domain with a hole

Abstract

<p style='text-indent:20px;'>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</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1111"> \begin{document}$ \begin{equation} \begin{cases} -\Delta \psi = \lambda(\psi-\frac{\kappa}{4\pi}\ln\lambda)_+^p,\quad &amp;\text{in}\; \Omega,\\ \psi = \rho_\lambda,\quad &amp;\text{on}\; \partial O_0,\\ \psi = 0,\quad &amp;\text{on}\; \partial\Omega_0, \end{cases} \;\;\;\;\;\;\;\;(1)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \kappa $\end{document}</tex-math></inline-formula> is a positive constant, <inline-formula><tex-math id="M3">\begin{document}$ \rho_\lambda $\end{document}</tex-math></inline-formula> is a constant, depending on <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \Omega = \Omega_0\setminus \bar{O}_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \Omega_0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ O_0 $\end{document}</tex-math></inline-formula> are two planar bounded simply-connected domains. We show that under the assumption <inline-formula><tex-math id="M8">\begin{document}$ (\ln\lambda)^\sigma\leq\rho_\lambda\leq (\ln\lambda)^{1-\sigma} $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M9">\begin{document}$ \sigma&gt;0 $\end{document}</tex-math></inline-formula> small, (1) has a solution <inline-formula><tex-math id="M10">\begin{document}$ \psi_\lambda $\end{document}</tex-math></inline-formula>, whose vorticity set <inline-formula><tex-math id="M11">\begin{document}$ \{y\in \Omega:\, \psi(y)-\kappa+\rho_\lambda\eta(y)&gt;0\} $\end{document}</tex-math></inline-formula> shrinks to the boundary of the hole as <inline-formula><tex-math id="M12">\begin{document}$ \lambda\to +\infty $\end{document}</tex-math></inline-formula>.</p>