### 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 $\lambda\to +\infty$.

 [1] Asymptotic behaviour in planar vortex theory. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. (1990) 1: 285-291. [2] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Second edition. Applied Mathematical Sciences, 125. Springer, Cham, 2021. doi: 10.1007/978-3-030-74278-2 [3] Nonlinear desingularization in certain free-boundary problems. Comm. Math. Phys. (1980) 77: 149-172. [4] Variational problems on classes of rearrangements and multiple configurations for steady vortices. Ann. Inst. H. Poincaré Anal. Non Linéaire (1989) 6: 295-319. [5] Rearrangements of functions, saddle points and uncountable families of steady configurations for a vortex. Acta Math. (1989) 163: 291-309. [6] Regularization of point vortices for the Euler equation in dimension two. Arch. Ration. Mech. Anal. (2014) 212: 179-217. [7] Multiplicity of solutions for the plasma problem in two dimensions. Adv. Math. (2010) 225: 2741-2785. [8] Planar vortex patch problem in incompressible steady flow. Adv. Math. (2015) 270: 263-301. [9] Regularization of planar vortices for the incompressible flow. Acta Math. Sci. Ser. B (Engl. Ed.) (2018) 38: 1443-1467. [10] The Lazer-McKenna conjecture and a free boundary problem in two dimensions. J. Lond. Math. Soc. (2008) 78: 639-662. [11] Steady vortex flows with circulation past asymmetric obstacles. Comm. Partial Differential Equations (1987) 2: 1095-1115. [12] Two dimensional incompressible ideal flow around a small obstacle. Commun. Partial Diff. Equ. (2003) 28: 349-379. [13] Two dimensional incompressible ideal flow around a thin obstacle tending to a curve. Ann. Inst. H. Poincaré Anal. Non Linéaire (2009) 26: 1121-1148. [14] Vortex dynamics in a two dimensional domain with holes and the small obstacle limit. SIAM J. Math. Anal. (2007) 39: 422-436. [15] Desingulariation of vortices for the Euler equation. Arch. Rational Mech. Anal. (2010) 198: 869-925. [16] B. Turkington, On steady vortex flow in two dimensions. Ⅰ, Ⅱ, Comm. Partial Differential Equations, 8 (1983), 999–1030, 1031–1071. doi: 10.1080/03605308308820293 [17] Existence and asymptotic behavior in planar vortex theory. Math. Models Methods Appl. Sci. (1991) 1: 461-475.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142