    December  2021, 29(6): 4229-4241. doi: 10.3934/era.2021081

Planar vortices in a bounded domain with a hole

 1 School of Mathematics and Statistics, Central China Normal University, Wuhan, China 2 Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Shusen Yan

Received  August 2021 Published  December 2021 Early access  October 2021

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
 $\begin{equation} \begin{cases} -\Delta \psi = \lambda(\psi-\frac{\kappa}{4\pi}\ln\lambda)_+^p,\quad &\text{in}\; \Omega,\\ \psi = \rho_\lambda,\quad &\text{on}\; \partial O_0,\\ \psi = 0,\quad &\text{on}\; \partial\Omega_0, \end{cases} \;\;\;\;\;\;\;\;(1)\end{equation}$
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. Electronic Research Archive, 2021, 29 (6) : 4229-4241. doi: 10.3934/era.2021081
References:
  A. Ambrosetti and J. Yang, Asymptotic behaviour in planar vortex theory, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 1 (1990), 285-291. Google Scholar  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.  Google Scholar  M. S. Berger and L. E. Fraenkel, Nonlinear desingularization in certain free-boundary problems, Comm. Math. Phys., 77 (1980), 149-172.  doi: 10.1007/BF01982715.  Google Scholar  G. R. Burton, Variational problems on classes of rearrangements and multiple configurations for steady vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 295-319.  doi: 10.1016/S0294-1449(16)30320-1.  Google Scholar  G. R. Burton, Rearrangements of functions, saddle points and uncountable families of steady configurations for a vortex, Acta Math., 163 (1989), 291-309.  doi: 10.1007/BF02392738.  Google Scholar  D. Cao, Z. Liu and J. Wei, Regularization of point vortices for the Euler equation in dimension two, Arch. Ration. Mech. Anal., 212 (2014), 179-217.  doi: 10.1007/s00205-013-0692-y.  Google Scholar  D. Cao, S. Peng and S. Yan, Multiplicity of solutions for the plasma problem in two dimensions, Adv. Math., 225 (2010), 2741-2785.  doi: 10.1016/j.aim.2010.05.012.  Google Scholar  D. Cao, S. Peng and S. Yan, Planar vortex patch problem in incompressible steady flow, Adv. Math., 270 (2015), 263-301.  doi: 10.1016/j.aim.2014.09.027.  Google Scholar  D. Cao, S. Peng and S. Yan, Regularization of planar vortices for the incompressible flow, Acta Math. Sci. Ser. B (Engl. Ed.), 38 (2018), 1443-1467.  doi: 10.1016/S0252-9602(18)30827-0.  Google Scholar  E. N. Dancer and S. Yan, The Lazer-McKenna conjecture and a free boundary problem in two dimensions, J. Lond. Math. Soc., 78 (2008), 639-662.  doi: 10.1112/jlms/jdn045.  Google Scholar  A. R. Elcrat and K. G. Miller, Steady vortex flows with circulation past asymmetric obstacles, Comm. Partial Differential Equations, 2 (1987), 1095-1115.  doi: 10.1080/03605308708820520.  Google Scholar  D. Iftimie, M. C. Lopes Filho and H. J. Nussenzveig Lopes, Two dimensional incompressible ideal flow around a small obstacle, Commun. Partial Diff. Equ., 28 (2003), 349-379.  doi: 10.1081/PDE-120019386.  Google Scholar  C. Lacave, Two dimensional incompressible ideal flow around a thin obstacle tending to a curve, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1121-1148.  doi: 10.1016/j.anihpc.2008.06.004.  Google Scholar  M. C. Lopes Filho, Vortex dynamics in a two dimensional domain with holes and the small obstacle limit, SIAM J. Math. Anal., 39 (2007), 422-436.  doi: 10.1137/050647967.  Google Scholar  D. Smets and J. Van Schaftingen, Desingulariation of vortices for the Euler equation, Arch. Rational Mech. Anal., 198 (2010), 869-925.  doi: 10.1007/s00205-010-0293-y.  Google Scholar  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.  Google Scholar  J. Yang, Existence and asymptotic behavior in planar vortex theory, Math. Models Methods Appl. Sci., 1 (1991), 461-475.  doi: 10.1142/S021820259100023X.  Google Scholar

show all references

References:
  A. Ambrosetti and J. Yang, Asymptotic behaviour in planar vortex theory, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 1 (1990), 285-291. Google Scholar  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.  Google Scholar  M. S. Berger and L. E. Fraenkel, Nonlinear desingularization in certain free-boundary problems, Comm. Math. Phys., 77 (1980), 149-172.  doi: 10.1007/BF01982715.  Google Scholar  G. R. Burton, Variational problems on classes of rearrangements and multiple configurations for steady vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 295-319.  doi: 10.1016/S0294-1449(16)30320-1.  Google Scholar  G. R. Burton, Rearrangements of functions, saddle points and uncountable families of steady configurations for a vortex, Acta Math., 163 (1989), 291-309.  doi: 10.1007/BF02392738.  Google Scholar  D. Cao, Z. Liu and J. Wei, Regularization of point vortices for the Euler equation in dimension two, Arch. Ration. Mech. Anal., 212 (2014), 179-217.  doi: 10.1007/s00205-013-0692-y.  Google Scholar  D. Cao, S. Peng and S. Yan, Multiplicity of solutions for the plasma problem in two dimensions, Adv. Math., 225 (2010), 2741-2785.  doi: 10.1016/j.aim.2010.05.012.  Google Scholar  D. Cao, S. Peng and S. Yan, Planar vortex patch problem in incompressible steady flow, Adv. Math., 270 (2015), 263-301.  doi: 10.1016/j.aim.2014.09.027.  Google Scholar  D. Cao, S. Peng and S. Yan, Regularization of planar vortices for the incompressible flow, Acta Math. Sci. Ser. B (Engl. Ed.), 38 (2018), 1443-1467.  doi: 10.1016/S0252-9602(18)30827-0.  Google Scholar  E. N. Dancer and S. Yan, The Lazer-McKenna conjecture and a free boundary problem in two dimensions, J. Lond. Math. Soc., 78 (2008), 639-662.  doi: 10.1112/jlms/jdn045.  Google Scholar  A. R. Elcrat and K. G. Miller, Steady vortex flows with circulation past asymmetric obstacles, Comm. Partial Differential Equations, 2 (1987), 1095-1115.  doi: 10.1080/03605308708820520.  Google Scholar  D. Iftimie, M. C. Lopes Filho and H. J. Nussenzveig Lopes, Two dimensional incompressible ideal flow around a small obstacle, Commun. Partial Diff. Equ., 28 (2003), 349-379.  doi: 10.1081/PDE-120019386.  Google Scholar  C. Lacave, Two dimensional incompressible ideal flow around a thin obstacle tending to a curve, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1121-1148.  doi: 10.1016/j.anihpc.2008.06.004.  Google Scholar  M. C. Lopes Filho, Vortex dynamics in a two dimensional domain with holes and the small obstacle limit, SIAM J. Math. Anal., 39 (2007), 422-436.  doi: 10.1137/050647967.  Google Scholar  D. Smets and J. Van Schaftingen, Desingulariation of vortices for the Euler equation, Arch. Rational Mech. Anal., 198 (2010), 869-925.  doi: 10.1007/s00205-010-0293-y.  Google Scholar  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.  Google Scholar  J. Yang, Existence and asymptotic behavior in planar vortex theory, Math. Models Methods Appl. Sci., 1 (1991), 461-475.  doi: 10.1142/S021820259100023X.  Google Scholar
  Andrzej Nowakowski. Variational analysis of semilinear plate equation with free boundary conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3133-3154. doi: 10.3934/dcds.2015.35.3133  Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations & Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297  Xavier Fernández-Real, Xavier Ros-Oton. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6945-6959. doi: 10.3934/dcds.2019238  Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449  Aram L. Karakhanyan. Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 261-277. doi: 10.3934/dcds.2016.36.261  Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with cost depending on the free boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 605-615. doi: 10.3934/nhm.2012.7.605  Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003  Andrzej Nowakowski. Variational approach to stability of semilinear wave equation with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2603-2616. doi: 10.3934/dcdsb.2014.19.2603  Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122  Hiroshi Matsuzawa. A free boundary problem for the Fisher-KPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1821-1852. doi: 10.3934/cpaa.2018087  EL Hassene Osmani, Mounir Haddou, Naceurdine Bensalem. A new relaxation method for optimal control of semilinear elliptic variational inequalities obstacle problems. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021061  Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399  Chengchun Hao. Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2885-2931. doi: 10.3934/dcdsb.2015.20.2885  Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033  Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355  Marco Degiovanni, Michele Scaglia. A variational approach to semilinear elliptic equations with measure data. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1233-1248. doi: 10.3934/dcds.2011.31.1233  Yimei Li, Jiguang Bao. Semilinear elliptic system with boundary singularity. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2189-2212. doi: 10.3934/dcds.2020111  Xiu Ye, Shangyou Zhang. A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4131-4145. doi: 10.3934/dcdsb.2020277  Shuxing Chen, Gui-Qiang Chen, Zejun Wang, Dehua Wang. A multidimensional piston problem for the Euler equations for compressible flow. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 361-383. doi: 10.3934/dcds.2005.13.361  J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176

2020 Impact Factor: 1.833