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Planar vortices in a bounded domain with a hole. (English) Zbl 1478.49004

Summary: 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 \[ \begin{cases} -\Delta \psi = \lambda(\psi-\frac{\kappa}{4\pi}\ln\lambda)_+^p &\text{in } \Omega,\\ \psi = \rho_\lambda, &\text{on }\partial O_0\\ \psi = 0, &\text{on }\partial\Omega_0 \end{cases} \tag{1} \] 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 \(\lambda\to+\infty\).

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49S05 Variational principles of physics
76B47 Vortex flows for incompressible inviscid fluids
35J61 Semilinear elliptic equations
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