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November  2018, 38(11): 5443-5460. doi: 10.3934/dcds.2018240

Periodic solutions for the N-vortex problem via a superposition principle

Björn Gebhard

Mathematisches Institut, Universität Gießen, Arndstr. 2, 35392 Gießen, Germany

Received  September 2017 Revised  April 2018 Published  August 2018

We examine the
$N$
-vortex problem on general domains
$Ω\subset\mathbb{R}^2$
concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the form
$Γ_k\dot{z}_k = J\nabla_{z_k}H(z_1,...,z_N),\ \ \ \ k = 1,...,N,$
where
$Γ_k∈\mathbb{R}\setminus\{0\}$
is the strength of the
$k$
th vortex at position
$z_k(t)∈Ω$
,
$J∈\mathbb{R}^{2× 2}$
is the standard symplectic matrix and
$H(z_1,...,z_N) = -\frac{1}{2π}\sum\limits_{\underset{k≠ j}{k,j = 1}}^NΓ_jΓ_k\log|z_k-z_j|-\sum\limits_{k,j = 1}^NΓ_jΓ_k g(z_k,z_j)$
with some regular and symmetric, but in general not explicitely known function
$g:Ω×Ω \to \mathbb{R}$
. The investigation relies on the idea to superpose a stationary solution of a system of less than
$N$
vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. We establish general conditions on both, the stationary solution and the configurations, under which multiple
$T$
-periodic solutions are shown to exist for every
$T>0$
small enough. The crucial condition holds in generic bounded domains and is explicitly verified for an example in the unit disc
$Ω = B_1(0)$
. In particular we therefore obtain various examples of periodic solutions in
$B_1(0)$
that are not rigidly rotating configurations.
Keywords: Vortex dynamics, singular first order Hamiltonian systems, periodic solutions, vortex clusters.
Mathematics Subject Classification: Primary: 37J45; Secondary: 37N10, 76B47.
Citation: Björn Gebhard. Periodic solutions for the N-vortex problem via a superposition principle. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5443-5460. doi: 10.3934/dcds.2018240
References:
[1]

H. Aref, Relative equilibria of point vortices and the fundamental theorem of algebra, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 2168-2184.  doi: 10.1098/rspa.2010.0580.  Google Scholar

[2]

H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda and D. L. Vainchtein, Vortex crystals, Adv. Appl. Mech., 39 (2003), 1-79.  doi: 10.1016/S0065-2156(02)39001-X.  Google Scholar

[3]

T. Bartsch, A generalization of the Weinstein-Moser theorems on periodic orbits of a Hamiltonian system near an equilibrium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 691-718.  doi: 10.1016/S0294-1449(97)80130-8.  Google Scholar

[4]

T. Bartsch and Q. Dai, Periodic solutions of the N-vortex Hamiltonian system in planar domains, J. Diff. Equ., 260 (2016), 2275-2295.  doi: 10.1016/j.jde.2015.10.002.  Google Scholar

[5]

T. Bartsch, Q. Dai and B. Gebhard, Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary, SIAM J. Appl. Math., 78 (2018), 977-995.  doi: 10.1137/16M1107085.  Google Scholar

[6]

T. Bartsch and B. Gebhard, Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type, Math. Ann., 369 (2017), 627-651.  doi: 10.1007/s00208-016-1505-z.  Google Scholar

[7]

T. Bartsch, A. M. Micheletti and A. Pistoia, The Morse property for functions of Kirchhoff-Routh path type, to appear: Discrete Contin. Dyn. Syst. Ser. S. Google Scholar

[8]

T. Bartsch and A. Pistoia, Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations, SIAM J. Appl. Math., 75 (2015), 726-744.  doi: 10.1137/140981253.  Google Scholar

[9]

T. Bartsch, A. Pistoia and T. Weth, N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-poisson and the Lane-Emden-Fowler equations, Comm. Math. Phys., 297 (2010), 653-686.  doi: 10.1007/s00220-010-1053-4.  Google Scholar

[10]

T. Bartsch and M. Sacchet, Periodic solutions with prescribed minimal period of vortex type problems in domains, Nonlinearity, 31 (2018), 2156-2172.  doi: 10.1088/1361-6544/aaaf2d.  Google Scholar

[11]

D. Cao, Z. Liu and J. Wei, Regularization of point vortices pairs for the Euler equation in dimension two, Arch. Rat. Mech. Anal., 212 (2014), 179-217.  doi: 10.1007/s00205-013-0692-y.  Google Scholar

[12]

O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik-Schnirelmann Category, American Mathematical Society, Providence Rhode Island, 2003. doi: 10.1090/surv/103.  Google Scholar

[13]

M. del Pino, M. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81.  doi: 10.1007/s00526-004-0314-5.  Google Scholar

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P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 227-257.  doi: 10.1016/j.anihpc.2004.12.001.  Google Scholar

[15]

M. Flucher, Variational Problems with Concentration, Birkhäuser, Basel Boston Berlin, 1999. doi: 10.1007/978-3-0348-8687-1.  Google Scholar

[16]

M. Gelantalis and P. Sternberg, Rotating 2N-vortex solutions to the Gross-Pitaevskii equation on $ S^2$, J. Math. Phys., 53 (2012), 083701, 24pp. doi: 10.1063/1.4739748.  Google Scholar

[17]

R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech. Anal., 142 (1998), 99-125.  doi: 10.1007/s002050050085.  Google Scholar

[18]

K. Khanin, Quasi-periodic motions of vortex systems, Phys. D, 4 (1982), 261-269.  doi: 10.1016/0167-2789(82)90067-7.  Google Scholar

[19]

G. R. Kirchhoff, Vorlesungen Über Mathematische Physik, Teubner, Leipzig, 1876. Google Scholar

[20]

C. Kuhl, Symmetric equilibria for the N-vortex problem, J. Fixed Point Theory Appl., 17 (2015), 597-624.  doi: 10.1007/s11784-015-0242-3.  Google Scholar

[21]

C. Kuhl, Equilibria for the N-vortex-problem in a general bounded domain, J. Math. Anal. Appl., 433 (2016), 1531-1560.  doi: 10.1016/j.jmaa.2015.08.055.  Google Scholar

[22]

M. Kurzke, C. Melcher, R. Moser and D. Spirn, Ginzburg-Landau vortices driven by the Landau-Lifshitz-Gilbert equation, Arch. Rat. Mech. Anal., 199 (2011), 843-888.  doi: 10.1007/s00205-010-0356-0.  Google Scholar

[23]

C. C. Lin, On the motion of vortices in two dimensions i. Existence of the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 570-575.   Google Scholar

[24]

C. C. Lin, On the motion of vortices in two dimensions ii. Some further investigations on the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 575-577.   Google Scholar

[25]

C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, Springer, Berlin Heidelberg, 1984.  Google Scholar

[26]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Springer, New York, 1994. doi: 10.1007/978-1-4612-4284-0.  Google Scholar

[27]

J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747.  doi: 10.1002/cpa.3160290613.  Google Scholar

[28]

P. K. Newton, The N-Vortex Problem, Springer, New York, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[29]

G. E. Roberts, Stability of relative equilibria in the planar N-vortex problem, SIAM J. Appl. Dyn. Sys., 12 (2013), 1114-1134.  doi: 10.1137/130907434.  Google Scholar

[30]

E. J. Routh, Some applications of conjugate functions, Proc. London Math. Soc., 12 (1880), 73-89.  doi: 10.1112/plms/s1-12.1.73.  Google Scholar

[31]

P. G. Saffman, Vortex Dynamics, Cambridge University Press, 1992.  Google Scholar

[32]

R. Venkatraman, Periodic orbits of Gross-Pitaevskii in the disc with vortices following point vortex flow, Calc. Var. Partial Differential Equations, 56 (2017), Art. 64, 35 pp. doi: 10.1007/s00526-017-1168-y.  Google Scholar

[33]

A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Inventiones mathematicae, 20 (1973), 47-57.  doi: 10.1007/BF01405263.  Google Scholar

show all references

References:
[1]

H. Aref, Relative equilibria of point vortices and the fundamental theorem of algebra, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 2168-2184.  doi: 10.1098/rspa.2010.0580.  Google Scholar

[2]

H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda and D. L. Vainchtein, Vortex crystals, Adv. Appl. Mech., 39 (2003), 1-79.  doi: 10.1016/S0065-2156(02)39001-X.  Google Scholar

[3]

T. Bartsch, A generalization of the Weinstein-Moser theorems on periodic orbits of a Hamiltonian system near an equilibrium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 691-718.  doi: 10.1016/S0294-1449(97)80130-8.  Google Scholar

[4]

T. Bartsch and Q. Dai, Periodic solutions of the N-vortex Hamiltonian system in planar domains, J. Diff. Equ., 260 (2016), 2275-2295.  doi: 10.1016/j.jde.2015.10.002.  Google Scholar

[5]

T. Bartsch, Q. Dai and B. Gebhard, Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary, SIAM J. Appl. Math., 78 (2018), 977-995.  doi: 10.1137/16M1107085.  Google Scholar

[6]

T. Bartsch and B. Gebhard, Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type, Math. Ann., 369 (2017), 627-651.  doi: 10.1007/s00208-016-1505-z.  Google Scholar

[7]

T. Bartsch, A. M. Micheletti and A. Pistoia, The Morse property for functions of Kirchhoff-Routh path type, to appear: Discrete Contin. Dyn. Syst. Ser. S. Google Scholar

[8]

T. Bartsch and A. Pistoia, Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations, SIAM J. Appl. Math., 75 (2015), 726-744.  doi: 10.1137/140981253.  Google Scholar

[9]

T. Bartsch, A. Pistoia and T. Weth, N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-poisson and the Lane-Emden-Fowler equations, Comm. Math. Phys., 297 (2010), 653-686.  doi: 10.1007/s00220-010-1053-4.  Google Scholar

[10]

T. Bartsch and M. Sacchet, Periodic solutions with prescribed minimal period of vortex type problems in domains, Nonlinearity, 31 (2018), 2156-2172.  doi: 10.1088/1361-6544/aaaf2d.  Google Scholar

[11]

D. Cao, Z. Liu and J. Wei, Regularization of point vortices pairs for the Euler equation in dimension two, Arch. Rat. Mech. Anal., 212 (2014), 179-217.  doi: 10.1007/s00205-013-0692-y.  Google Scholar

[12]

O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik-Schnirelmann Category, American Mathematical Society, Providence Rhode Island, 2003. doi: 10.1090/surv/103.  Google Scholar

[13]

M. del Pino, M. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81.  doi: 10.1007/s00526-004-0314-5.  Google Scholar

[14]

P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 227-257.  doi: 10.1016/j.anihpc.2004.12.001.  Google Scholar

[15]

M. Flucher, Variational Problems with Concentration, Birkhäuser, Basel Boston Berlin, 1999. doi: 10.1007/978-3-0348-8687-1.  Google Scholar

[16]

M. Gelantalis and P. Sternberg, Rotating 2N-vortex solutions to the Gross-Pitaevskii equation on $ S^2$, J. Math. Phys., 53 (2012), 083701, 24pp. doi: 10.1063/1.4739748.  Google Scholar

[17]

R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech. Anal., 142 (1998), 99-125.  doi: 10.1007/s002050050085.  Google Scholar

[18]

K. Khanin, Quasi-periodic motions of vortex systems, Phys. D, 4 (1982), 261-269.  doi: 10.1016/0167-2789(82)90067-7.  Google Scholar

[19]

G. R. Kirchhoff, Vorlesungen Über Mathematische Physik, Teubner, Leipzig, 1876. Google Scholar

[20]

C. Kuhl, Symmetric equilibria for the N-vortex problem, J. Fixed Point Theory Appl., 17 (2015), 597-624.  doi: 10.1007/s11784-015-0242-3.  Google Scholar

[21]

C. Kuhl, Equilibria for the N-vortex-problem in a general bounded domain, J. Math. Anal. Appl., 433 (2016), 1531-1560.  doi: 10.1016/j.jmaa.2015.08.055.  Google Scholar

[22]

M. Kurzke, C. Melcher, R. Moser and D. Spirn, Ginzburg-Landau vortices driven by the Landau-Lifshitz-Gilbert equation, Arch. Rat. Mech. Anal., 199 (2011), 843-888.  doi: 10.1007/s00205-010-0356-0.  Google Scholar

[23]

C. C. Lin, On the motion of vortices in two dimensions i. Existence of the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 570-575.   Google Scholar

[24]

C. C. Lin, On the motion of vortices in two dimensions ii. Some further investigations on the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 575-577.   Google Scholar

[25]

C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, Springer, Berlin Heidelberg, 1984.  Google Scholar

[26]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Springer, New York, 1994. doi: 10.1007/978-1-4612-4284-0.  Google Scholar

[27]

J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747.  doi: 10.1002/cpa.3160290613.  Google Scholar

[28]

P. K. Newton, The N-Vortex Problem, Springer, New York, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[29]

G. E. Roberts, Stability of relative equilibria in the planar N-vortex problem, SIAM J. Appl. Dyn. Sys., 12 (2013), 1114-1134.  doi: 10.1137/130907434.  Google Scholar

[30]

E. J. Routh, Some applications of conjugate functions, Proc. London Math. Soc., 12 (1880), 73-89.  doi: 10.1112/plms/s1-12.1.73.  Google Scholar

[31]

P. G. Saffman, Vortex Dynamics, Cambridge University Press, 1992.  Google Scholar

[32]

R. Venkatraman, Periodic orbits of Gross-Pitaevskii in the disc with vortices following point vortex flow, Calc. Var. Partial Differential Equations, 56 (2017), Art. 64, 35 pp. doi: 10.1007/s00526-017-1168-y.  Google Scholar

[33]

A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Inventiones mathematicae, 20 (1973), 47-57.  doi: 10.1007/BF01405263.  Google Scholar

Figure 1.  This diagram illustrates the superposition idea. The $2$-vortex problem in the unit disc admits a stationary solution with $\Gamma^1 = -\Gamma^2$, cf. Example 1.2, say $\Gamma^1 = -2$ (blue star), $\Gamma^2 = 2$ (red star). As rigidly rotating configurations on $\mathbb{R}^2$ we take here for simplicity two identical vortices for $\Gamma^1$ and $\Gamma^2$, i.e. $\Gamma^1_1 = \Gamma^1_2 = -1$ rotate on the blue circle in clockwise direction and $\Gamma^2_1 = \Gamma^2_2 = 1$ rotate on the red circle in counterclockwise direction. The result on the right-hand side is a periodic solution of the $4$-vortex system in the disc with vorticities $\Gamma^1_1,\Gamma^1_2,\Gamma^2_1,\Gamma^2_2$, where each pair of vortices moves along a deformed circle in the same orientation as before. The shown trajectory is the actual numerically computed trajectory of the $4$-vortex problem. Suitable initial conditions can in this case be found due to symmetry considerations
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