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July  2015, 14(4): 1581-1601. doi: 10.3934/cpaa.2015.14.1581

The dynamics of vortex filaments with corners

Luis Vega 1,

1. 

Departamento de Matemáticas, UPV/EHU, Apdo 644, 48080 Bilbao, Spain

Received  October 2013 Revised  January 2014 Published  April 2015

This paper focuses on surveying some recent results obtained by the author together with V. Banica on the evolution of a vortex filament with one corner according to the so-called binormal flow. The case of a regular polygon studied in collaboration with F. de la Hoz is also considered.
Keywords: Vortex filaments, NLS..
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q35, 35Q4.
Citation: Luis Vega. The dynamics of vortex filaments with corners. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1581-1601. doi: 10.3934/cpaa.2015.14.1581
References:
[1]

V. Banica and L. Vega, On the stability of a singular vortex dynamics,, \emph{Comm. Math. Phys.}, (2009), 593.  doi: 10.1007/s00220-008-0682-3.  Google Scholar

[2]

V. Banica and L. Vega, Scattering for 1D cubic NLS and singular vortex dynamics,, \emph{J. Eur. Math. Soc.l}, 14 (2012), 209.  doi: 10.4171/JEMS/300.  Google Scholar

[3]

V. Banica and L. Vega, Stability of the self-similar dynamics of a vortex filament,, \emph{Arch. Ration. Mech. Anal.}, 210 (2013), 673.  doi: 10.1007/s00205-013-0660-6.  Google Scholar

[4]

V. Banica and L. Vega, The initial value problem for the binormal flow with rough data,, preprint, ().   Google Scholar

[5]

T. F. Buttke, A numerical study of superfluid turbulence in the Self Induction Approximation,, \emph{J. of Compt. Physics}, 76 (1988), 301.   Google Scholar

[6]

L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape,, \emph{Rend. Circ. Mat. Palermo}, 22 (1906).   Google Scholar

[7]

M. B. Erdogan and N. Tzirakis, Talbot effect for the cubic nonlinear Schrödinger equation on the torus,, preprint, ().  doi: 10.4310/MRL.2013.v20.n6.a7.  Google Scholar

[8]

S. Jaffard, The spectrum of singularities of Riemanns function,, \emph{Rev. Mat. Iberoamericana}, 12 (1996), 44.  doi: 10.4171/RMI/203.  Google Scholar

[9]

U. Frisch and G. Parisi, Fully developed turbulence and intermittency,, in \emph{Proc. Int. Sch. Phys. Enrico Fermi}, (1985).   Google Scholar

[10]

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov,, Cambridge University Press, (1995).   Google Scholar

[11]

L. Kapitanski and I. Rodnianski, Does a Quantum Particle Know the Time?, \emph{Emerging Applications of Number Theory, 109 (1999), 355.  doi: 10.1007/978-1-4612-1544-8_14.  Google Scholar

[12]

S. Gutiérrez and L. Vega, Self-similar solutions of the localized induction approximation: singularity formation,, \emph{Nonlinearity}, 17 (2004), 2091.  doi: 10.1088/0951-7715/17/6/006.  Google Scholar

[13]

S. Gutiérrez and L. Vega, On the stability of self-similar solutions of 1D cubic Schrödinger equations,, \emph{Math. Ann.}, 356 (2013), 259.  doi: 10.1007/s00208-012-0847-4.  Google Scholar

[14]

S. Gutiérrez, J. Rivas and L. Vega, Formation of singularities and self-similar vortex motion under the localized induction approximation,, \emph{Comm. Part. Diff. Eq.}, 28 (2003), 927.  doi: 10.1081/PDE-120021181.  Google Scholar

[15]

H. Hasimoto, A soliton in a vortex filament,, \emph{J. Fluid Mech.}, 51 (1972), 477.   Google Scholar

[16]

J. C. Hardin, The velocity field induced by a helical vortex filament,, \emph{Phys. Fluids}, 25 (1982), 1949.   Google Scholar

[17]

F. de la Hoz, Self-similar solutions for the 1-D Schrödinger map on the hyperbolic plane,, \emph{Math. Z.}, 257 (2007), 61.  doi: 10.1007/s00209-007-0115-6.  Google Scholar

[18]

F. de la Hoz and L. Vega, Vortex Filament Equation for a Regular Polygon,, prepint, ().   Google Scholar

[19]

F. de la Hoz, C. García-Cervera and L. Vega, A numerical study of the self-similar solutions of the Schrödinger Map,, \emph{SIAM J. Appl. Math.}, 70 (2009), 1047.  doi: 10.1137/080741720.  Google Scholar

[20]

C. E. Kenig, G. Ponce, and L.Vega, On the ill-posedness of some canonical dispersive equations,, \emph{Duke Math. J.}, 106 (2001), 617.  doi: 10.1215/S0012-7094-01-10638-8.  Google Scholar

[21]

M. Lakshmanan and M. Daniel, On the evolution of higher dimensional Heisenberg continuum spin systems,, \emph{Physica A}, 107 (1981), 533.  doi: 10.1016/0378-4371(81)90186-2.  Google Scholar

[22]

M. Lakshmanan, T. W. Ruijgrok and C. J. Thompson, On the the dynamics of a continuum spin system,, \emph{Physica A}, 84 (1976), 577.   Google Scholar

[23]

T, Lipniacki, Quasi-static solutions for quantum vortex motion under the localized induction approximation,, \emph{J. Fluid Mech.}, (2003), 321.  doi: 10.1017/S0022112002003282.  Google Scholar

[24]

K. I. Oskolkov, A class of I. M. Vinogradov's series and its applications in harmonic analysis,, in \emph{Progress in Approximation Theory (Tampa, (1990), 353.  doi: 10.1007/978-1-4612-2966-7_16.  Google Scholar

[25]

C. S. Peskin and D. M. McQueen, Mechanical equilibrium determines the fractal fiber architecture of aortic heart valve leaflets,, \emph{Am. J. Physiol. 266 (Heart Circ. Physiol. 35)}, (1994).   Google Scholar

[26]

R. L. Ricca, The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics,, \emph{Fluid Dynam. Res.}, (1996), 245.  doi: 10.1016/0169-5983(96)82495-6.  Google Scholar

[27]

R. L. Ricca, Rediscovery of Da Rios equations,, \emph{Nature}, 352 (1991), 561.   Google Scholar

[28]

P. G. Saffman, Vortex dynamics,, in \emph{Cambridge Monographs on Mechanics and Applied Mathematics}, (1992).   Google Scholar

show all references

References:
[1]

V. Banica and L. Vega, On the stability of a singular vortex dynamics,, \emph{Comm. Math. Phys.}, (2009), 593.  doi: 10.1007/s00220-008-0682-3.  Google Scholar

[2]

V. Banica and L. Vega, Scattering for 1D cubic NLS and singular vortex dynamics,, \emph{J. Eur. Math. Soc.l}, 14 (2012), 209.  doi: 10.4171/JEMS/300.  Google Scholar

[3]

V. Banica and L. Vega, Stability of the self-similar dynamics of a vortex filament,, \emph{Arch. Ration. Mech. Anal.}, 210 (2013), 673.  doi: 10.1007/s00205-013-0660-6.  Google Scholar

[4]

V. Banica and L. Vega, The initial value problem for the binormal flow with rough data,, preprint, ().   Google Scholar

[5]

T. F. Buttke, A numerical study of superfluid turbulence in the Self Induction Approximation,, \emph{J. of Compt. Physics}, 76 (1988), 301.   Google Scholar

[6]

L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape,, \emph{Rend. Circ. Mat. Palermo}, 22 (1906).   Google Scholar

[7]

M. B. Erdogan and N. Tzirakis, Talbot effect for the cubic nonlinear Schrödinger equation on the torus,, preprint, ().  doi: 10.4310/MRL.2013.v20.n6.a7.  Google Scholar

[8]

S. Jaffard, The spectrum of singularities of Riemanns function,, \emph{Rev. Mat. Iberoamericana}, 12 (1996), 44.  doi: 10.4171/RMI/203.  Google Scholar

[9]

U. Frisch and G. Parisi, Fully developed turbulence and intermittency,, in \emph{Proc. Int. Sch. Phys. Enrico Fermi}, (1985).   Google Scholar

[10]

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov,, Cambridge University Press, (1995).   Google Scholar

[11]

L. Kapitanski and I. Rodnianski, Does a Quantum Particle Know the Time?, \emph{Emerging Applications of Number Theory, 109 (1999), 355.  doi: 10.1007/978-1-4612-1544-8_14.  Google Scholar

[12]

S. Gutiérrez and L. Vega, Self-similar solutions of the localized induction approximation: singularity formation,, \emph{Nonlinearity}, 17 (2004), 2091.  doi: 10.1088/0951-7715/17/6/006.  Google Scholar

[13]

S. Gutiérrez and L. Vega, On the stability of self-similar solutions of 1D cubic Schrödinger equations,, \emph{Math. Ann.}, 356 (2013), 259.  doi: 10.1007/s00208-012-0847-4.  Google Scholar

[14]

S. Gutiérrez, J. Rivas and L. Vega, Formation of singularities and self-similar vortex motion under the localized induction approximation,, \emph{Comm. Part. Diff. Eq.}, 28 (2003), 927.  doi: 10.1081/PDE-120021181.  Google Scholar

[15]

H. Hasimoto, A soliton in a vortex filament,, \emph{J. Fluid Mech.}, 51 (1972), 477.   Google Scholar

[16]

J. C. Hardin, The velocity field induced by a helical vortex filament,, \emph{Phys. Fluids}, 25 (1982), 1949.   Google Scholar

[17]

F. de la Hoz, Self-similar solutions for the 1-D Schrödinger map on the hyperbolic plane,, \emph{Math. Z.}, 257 (2007), 61.  doi: 10.1007/s00209-007-0115-6.  Google Scholar

[18]

F. de la Hoz and L. Vega, Vortex Filament Equation for a Regular Polygon,, prepint, ().   Google Scholar

[19]

F. de la Hoz, C. García-Cervera and L. Vega, A numerical study of the self-similar solutions of the Schrödinger Map,, \emph{SIAM J. Appl. Math.}, 70 (2009), 1047.  doi: 10.1137/080741720.  Google Scholar

[20]

C. E. Kenig, G. Ponce, and L.Vega, On the ill-posedness of some canonical dispersive equations,, \emph{Duke Math. J.}, 106 (2001), 617.  doi: 10.1215/S0012-7094-01-10638-8.  Google Scholar

[21]

M. Lakshmanan and M. Daniel, On the evolution of higher dimensional Heisenberg continuum spin systems,, \emph{Physica A}, 107 (1981), 533.  doi: 10.1016/0378-4371(81)90186-2.  Google Scholar

[22]

M. Lakshmanan, T. W. Ruijgrok and C. J. Thompson, On the the dynamics of a continuum spin system,, \emph{Physica A}, 84 (1976), 577.   Google Scholar

[23]

T, Lipniacki, Quasi-static solutions for quantum vortex motion under the localized induction approximation,, \emph{J. Fluid Mech.}, (2003), 321.  doi: 10.1017/S0022112002003282.  Google Scholar

[24]

K. I. Oskolkov, A class of I. M. Vinogradov's series and its applications in harmonic analysis,, in \emph{Progress in Approximation Theory (Tampa, (1990), 353.  doi: 10.1007/978-1-4612-2966-7_16.  Google Scholar

[25]

C. S. Peskin and D. M. McQueen, Mechanical equilibrium determines the fractal fiber architecture of aortic heart valve leaflets,, \emph{Am. J. Physiol. 266 (Heart Circ. Physiol. 35)}, (1994).   Google Scholar

[26]

R. L. Ricca, The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics,, \emph{Fluid Dynam. Res.}, (1996), 245.  doi: 10.1016/0169-5983(96)82495-6.  Google Scholar

[27]

R. L. Ricca, Rediscovery of Da Rios equations,, \emph{Nature}, 352 (1991), 561.   Google Scholar

[28]

P. G. Saffman, Vortex dynamics,, in \emph{Cambridge Monographs on Mechanics and Applied Mathematics}, (1992).   Google Scholar

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