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:
[1]

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

[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.  Google Scholar

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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

[4]

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

[5]

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

[6]

D. CaoZ. 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

[7]

D. CaoS. 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

[8]

D. CaoS. 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

[9]

D. CaoS. 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

[10]

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

[11]

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

[12]

D. IftimieM. 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

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[17]

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:
[1]

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

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

D. CaoZ. 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

[7]

D. CaoS. 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

[8]

D. CaoS. 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

[9]

D. CaoS. 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

[10]

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

[11]

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

[12]

D. IftimieM. 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

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[17]

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

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