## 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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### References:

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