July  2013, 33(7): 3011-3042. doi: 10.3934/dcds.2013.33.3011

Geometry of stationary solutions for a system of vortex filaments: A dynamical approach

1. 

Dipartimento di Matematica e Fisica "Ennio De Giorgi", Università del Salento, 73100, Lecce

2. 

Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, 73100 Lecce

Received  April 2012 Revised  October 2012 Published  January 2013

We give a detailed analytical description of the global dynamics of $N$ points interacting through the singular logarithmic potential and subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group $D_l$ of order $2l$. The main device in order to achieve our results is a technique very popular in Celestial Mechanics, usually referred to as McGehee transformation. After performing this change of coordinates that regularizes the total collision, we study the rest-points of the flow, the invariant manifolds and, with the help of a computer algebra system, we derive interesting information about the global dynamics for $l=2$. We observe that our problem is equivalent to studying the geometry of stationary configurations of nearly-parallel vortex filaments in three dimensions in the LIA approximation.
Citation: Francesco Paparella, Alessandro Portaluri. Geometry of stationary solutions for a system of vortex filaments: A dynamical approach. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3011-3042. doi: 10.3934/dcds.2013.33.3011
References:
[1]

Vivina Barutello, Davide L. Ferrario and Susanna Terracini, On the singularities of generalized solutions to $n$-body-type problems,, Int. Math. Res. Not. IMRN 2008, (2008).   Google Scholar

[2]

G. Bellettini, G. Fusco and G. F. Gronchi, Regularization of the two body problem via smoothing the potential,, Commun. Pure Appl. Anal., 2 (2003), 323.  doi: 10.3934/cpaa.2003.2.323.  Google Scholar

[3]

Anna Capietto, Francesca Dalbono and Alessandro Portaluri, A multiplicity result for a class of strongly indefinite asymptotically linear second order systems,, Nonlinear Anal., 72 (2010), 2874.  doi: 10.1016/j.na.2009.11.032.  Google Scholar

[4]

Castelli Roberto, "Moti Periodici di Filamenti Vorticosi Quasi-Paralleli,", Laurea Magistrale dissertation at University of Milano-Bicocca, (2004).   Google Scholar

[5]

R. Castelli, F. Paparella and A. Portaluri, Singular dynamics under a weak potential on a sphere,, To appear in NoDEA. , ().  doi: 10.1007/s00030-012-0182-1.  Google Scholar

[6]

Castelli Roberto and Terracini Susanna, On the regularization of the collision solutions of the one-center problem with weak forces,, Discrete Contin. Dyn. Syst., 31 (2011), 1197.  doi: 10.3934/dcds.2011.31.1197.  Google Scholar

[7]

F. Dalbono and A. Portaluri, Morse-Smale index theorems for elliptic boundary deformation problems,, Journal of Differential Equations, 253 (2012), 463.  doi: 10.1016/j.jde.2012.04.008.  Google Scholar

[8]

Ennio De Giorgi, Conjectures concerning some evolution problems,, Duke Math. J., 81 (1996), 255.  doi: 10.1215/S0012-7094-96-08114-4.  Google Scholar

[9]

R. L. Devaney, Triple collision in the planar isosceles three-body problem,, Invent. Math., 60 (1980), 249.  doi: 10.1007/BF01390017.  Google Scholar

[10]

R. L. Devaney, Singularities in classical mechanical systems,, in, 10 (1981), 1979.   Google Scholar

[11]

F. Diacu, Regularization of partial collisions in the N -body problem,, Differential Integral Equations, 5 (1992), 103.   Google Scholar

[12]

Davide L. Ferrario, Transitive decomposition of symmetry groups for the $n$-body problem,, Adv. in Math., 213 (2007), 763.  doi: 10.1016/j.aim.2007.01.009.  Google Scholar

[13]

Davide L. Ferrario and Alessandro Portaluri, On the dihedral $n$- body problem,, Nonlinearity, 21 (2008), 1307.  doi: 10.1088/0951-7715/21/6/009.  Google Scholar

[14]

Davide L. Ferrario and Alessandro Portaluri, Dynamics of the the dihedral four-body problem,, To appear in DCDS-S, ().   Google Scholar

[15]

Roberto Giambò, Paolo Piccione and Alessandro Portaluri, Computation of the Maslov index and the spectral flow via partial signatures,, C. R. Math. Acad. Sci. Paris, 338 (2004), 397.  doi: 10.1016/j.crma.2004.01.004.  Google Scholar

[16]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (1977).   Google Scholar

[17]

Rupert Klein, Andrew J. Majda and Kumaran Damodaran, Simplified equations for the interaction of nearly parallel vortex filaments,, J. Fluid Mech., 288 (1995), 201.  doi: 10.1017/S0022112095001121.  Google Scholar

[18]

R. McGehee, Triple collision in the collinear three-body problem,, Invent. Math., 27 (1974), 191.   Google Scholar

[19]

Paul Newton, "The $N$-Vortex Problem. Analytical Techniques,", Applied Mathematical Sciences, 145 (2001).  doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[20]

F. Paparella and A. Portaluri, Dynamics of (4 + 1)-Dihedrally symmetric nearly parallel vortex filaments,, Acta. Appl. Math., 122 (2012), 349.  doi: 10.1007/s10440-012-9748-5.  Google Scholar

[21]

H. Pollard and D. G. Saari, Singularities of the n-body problem. I,, Arch. Rational Mech. Anal., 30 (1968), 263.   Google Scholar

[22]

H. Pollard and D. G. Saari, Singularities of the n-body problem. II. In Inequalities, II,, (Proc. Second Sympos., (1970), 255.   Google Scholar

[23]

Alessandro Portaluri, Maslov index for Hamiltonian systems,, Electron. J. Differential Equations, 2008 ().   Google Scholar

[24]

D. G. Saari, Singularities and collisions of Newtonian gravitational systems,, Arch. Rational Mech. Anal., 49 (): 311.   Google Scholar

[25]

H. J. Sperling, On the real singularities of the $N$ -body problem,, J. Reine Angew. Math., 245 (1970), 15.   Google Scholar

[26]

K. F. Sundman, Nouvelles recherches sur le probleme des trois corps,, Acta Soc. Sci. Fenn., 35 (1909).   Google Scholar

[27]

Cristina Stoica and Andreea Font, Global dynamics in the singular logarithmic potential,, J. Phys. A, 36 (2003), 7693.  doi: 10.1088/0305-4470/36/28/302.  Google Scholar

[28]

A. Wintner, "The Analytical Foundations of Celestial Mechanics,", Princeton Mathematical Series, 5 (1941).   Google Scholar

show all references

References:
[1]

Vivina Barutello, Davide L. Ferrario and Susanna Terracini, On the singularities of generalized solutions to $n$-body-type problems,, Int. Math. Res. Not. IMRN 2008, (2008).   Google Scholar

[2]

G. Bellettini, G. Fusco and G. F. Gronchi, Regularization of the two body problem via smoothing the potential,, Commun. Pure Appl. Anal., 2 (2003), 323.  doi: 10.3934/cpaa.2003.2.323.  Google Scholar

[3]

Anna Capietto, Francesca Dalbono and Alessandro Portaluri, A multiplicity result for a class of strongly indefinite asymptotically linear second order systems,, Nonlinear Anal., 72 (2010), 2874.  doi: 10.1016/j.na.2009.11.032.  Google Scholar

[4]

Castelli Roberto, "Moti Periodici di Filamenti Vorticosi Quasi-Paralleli,", Laurea Magistrale dissertation at University of Milano-Bicocca, (2004).   Google Scholar

[5]

R. Castelli, F. Paparella and A. Portaluri, Singular dynamics under a weak potential on a sphere,, To appear in NoDEA. , ().  doi: 10.1007/s00030-012-0182-1.  Google Scholar

[6]

Castelli Roberto and Terracini Susanna, On the regularization of the collision solutions of the one-center problem with weak forces,, Discrete Contin. Dyn. Syst., 31 (2011), 1197.  doi: 10.3934/dcds.2011.31.1197.  Google Scholar

[7]

F. Dalbono and A. Portaluri, Morse-Smale index theorems for elliptic boundary deformation problems,, Journal of Differential Equations, 253 (2012), 463.  doi: 10.1016/j.jde.2012.04.008.  Google Scholar

[8]

Ennio De Giorgi, Conjectures concerning some evolution problems,, Duke Math. J., 81 (1996), 255.  doi: 10.1215/S0012-7094-96-08114-4.  Google Scholar

[9]

R. L. Devaney, Triple collision in the planar isosceles three-body problem,, Invent. Math., 60 (1980), 249.  doi: 10.1007/BF01390017.  Google Scholar

[10]

R. L. Devaney, Singularities in classical mechanical systems,, in, 10 (1981), 1979.   Google Scholar

[11]

F. Diacu, Regularization of partial collisions in the N -body problem,, Differential Integral Equations, 5 (1992), 103.   Google Scholar

[12]

Davide L. Ferrario, Transitive decomposition of symmetry groups for the $n$-body problem,, Adv. in Math., 213 (2007), 763.  doi: 10.1016/j.aim.2007.01.009.  Google Scholar

[13]

Davide L. Ferrario and Alessandro Portaluri, On the dihedral $n$- body problem,, Nonlinearity, 21 (2008), 1307.  doi: 10.1088/0951-7715/21/6/009.  Google Scholar

[14]

Davide L. Ferrario and Alessandro Portaluri, Dynamics of the the dihedral four-body problem,, To appear in DCDS-S, ().   Google Scholar

[15]

Roberto Giambò, Paolo Piccione and Alessandro Portaluri, Computation of the Maslov index and the spectral flow via partial signatures,, C. R. Math. Acad. Sci. Paris, 338 (2004), 397.  doi: 10.1016/j.crma.2004.01.004.  Google Scholar

[16]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (1977).   Google Scholar

[17]

Rupert Klein, Andrew J. Majda and Kumaran Damodaran, Simplified equations for the interaction of nearly parallel vortex filaments,, J. Fluid Mech., 288 (1995), 201.  doi: 10.1017/S0022112095001121.  Google Scholar

[18]

R. McGehee, Triple collision in the collinear three-body problem,, Invent. Math., 27 (1974), 191.   Google Scholar

[19]

Paul Newton, "The $N$-Vortex Problem. Analytical Techniques,", Applied Mathematical Sciences, 145 (2001).  doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[20]

F. Paparella and A. Portaluri, Dynamics of (4 + 1)-Dihedrally symmetric nearly parallel vortex filaments,, Acta. Appl. Math., 122 (2012), 349.  doi: 10.1007/s10440-012-9748-5.  Google Scholar

[21]

H. Pollard and D. G. Saari, Singularities of the n-body problem. I,, Arch. Rational Mech. Anal., 30 (1968), 263.   Google Scholar

[22]

H. Pollard and D. G. Saari, Singularities of the n-body problem. II. In Inequalities, II,, (Proc. Second Sympos., (1970), 255.   Google Scholar

[23]

Alessandro Portaluri, Maslov index for Hamiltonian systems,, Electron. J. Differential Equations, 2008 ().   Google Scholar

[24]

D. G. Saari, Singularities and collisions of Newtonian gravitational systems,, Arch. Rational Mech. Anal., 49 (): 311.   Google Scholar

[25]

H. J. Sperling, On the real singularities of the $N$ -body problem,, J. Reine Angew. Math., 245 (1970), 15.   Google Scholar

[26]

K. F. Sundman, Nouvelles recherches sur le probleme des trois corps,, Acta Soc. Sci. Fenn., 35 (1909).   Google Scholar

[27]

Cristina Stoica and Andreea Font, Global dynamics in the singular logarithmic potential,, J. Phys. A, 36 (2003), 7693.  doi: 10.1088/0305-4470/36/28/302.  Google Scholar

[28]

A. Wintner, "The Analytical Foundations of Celestial Mechanics,", Princeton Mathematical Series, 5 (1941).   Google Scholar

[1]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

[2]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[3]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[4]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[5]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225

[6]

Simone Calogero, Juan Calvo, Óscar Sánchez, Juan Soler. Dispersive behavior in galactic dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 1-16. doi: 10.3934/dcdsb.2010.14.1

[7]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[8]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[9]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637

[10]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

[11]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53

[12]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

[13]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[14]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206

[15]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[16]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

[17]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109

[18]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

[19]

Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198

[20]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]