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

The dynamics of vortex filaments with corners

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.
Citation: Luis Vega. The dynamics of vortex filaments with corners. Communications on Pure and 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, Comm. Math. Phys., 286 (2009), 593-627. doi: 10.1007/s00220-008-0682-3.

[2]

V. Banica and L. Vega, Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc.l, 14 (2012), 209-253. doi: 10.4171/JEMS/300.

[3]

V. Banica and L. Vega, Stability of the self-similar dynamics of a vortex filament, Arch. Ration. Mech. Anal., 210 (2013), 673-712. doi: 10.1007/s00205-013-0660-6.

[4]

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

[5]

T. F. Buttke, A numerical study of superfluid turbulence in the Self Induction Approximation, J. of Compt. Physics, 76 (1988), 301-326.

[6]

L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo, 22 (1906), 117.

[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.

[8]

S. Jaffard, The spectrum of singularities of Riemanns function, Rev. Mat. Iberoamericana, 12 (1996), 44-460. doi: 10.4171/RMI/203.

[9]

U. Frisch and G. Parisi, Fully developed turbulence and intermittency, in Proc. Int. Sch. Phys. Enrico Fermi, North-Holland, Amsterdam, 1985.

[10]

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

[11]

L. Kapitanski and I. Rodnianski, Does a Quantum Particle Know the Time? Emerging Applications of Number Theory, IMA Vol. Math. Appl., 109 (1999), 355-371. doi: 10.1007/978-1-4612-1544-8_14.

[12]

S. Gutiérrez and L. Vega, Self-similar solutions of the localized induction approximation: singularity formation, Nonlinearity, 17 (2004), 2091-2136. doi: 10.1088/0951-7715/17/6/006.

[13]

S. Gutiérrez and L. Vega, On the stability of self-similar solutions of 1D cubic Schrödinger equations, Math. Ann., 356 (2013), 259-300. doi: 10.1007/s00208-012-0847-4.

[14]

S. Gutiérrez, J. Rivas and L. Vega, Formation of singularities and self-similar vortex motion under the localized induction approximation, Comm. Part. Diff. Eq., 28 (2003), 927-968. doi: 10.1081/PDE-120021181.

[15]

H. Hasimoto, A soliton in a vortex filament, J. Fluid Mech., 51 (1972), 477-485.

[16]

J. C. Hardin, The velocity field induced by a helical vortex filament, Phys. Fluids, 25 (1982), 1949-1952.

[17]

F. de la Hoz, Self-similar solutions for the 1-D Schrödinger map on the hyperbolic plane, Math. Z., 257 (2007), 61-80. doi: 10.1007/s00209-007-0115-6.

[18]

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

[19]

F. de la Hoz, C. García-Cervera and L. Vega, A numerical study of the self-similar solutions of the Schrödinger Map, SIAM J. Appl. Math., 70 (2009), 1047-1077. doi: 10.1137/080741720.

[20]

C. E. Kenig, G. Ponce, and L.Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633. doi: 10.1215/S0012-7094-01-10638-8.

[21]

M. Lakshmanan and M. Daniel, On the evolution of higher dimensional Heisenberg continuum spin systems, Physica A, 107 (1981), 533-552. doi: 10.1016/0378-4371(81)90186-2.

[22]

M. Lakshmanan, T. W. Ruijgrok and C. J. Thompson, On the the dynamics of a continuum spin system, Physica A, 84 (1976), 577-590.

[23]

T, Lipniacki, Quasi-static solutions for quantum vortex motion under the localized induction approximation, J. Fluid Mech., 477 (2003), 321-337. doi: 10.1017/S0022112002003282.

[24]

K. I. Oskolkov, A class of I. M. Vinogradov's series and its applications in harmonic analysis, in Progress in Approximation Theory (Tampa, FL, 1990), Springer Ser. Comput. Math. 19, Springer, New York, 1992, 353-402. doi: 10.1007/978-1-4612-2966-7_16.

[25]

C. S. Peskin and D. M. McQueen, Mechanical equilibrium determines the fractal fiber architecture of aortic heart valve leaflets, Am. J. Physiol. 266 (Heart Circ. Physiol. 35), (1994), H319-H328.

[26]

R. L. Ricca, The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics, Fluid Dynam. Res., 18 (1996), 245-268. doi: 10.1016/0169-5983(96)82495-6.

[27]

R. L. Ricca, Rediscovery of Da Rios equations, Nature, 352 (1991), 561-562.

[28]

P. G. Saffman, Vortex dynamics, in Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992.

show all references

References:
[1]

V. Banica and L. Vega, On the stability of a singular vortex dynamics, Comm. Math. Phys., 286 (2009), 593-627. doi: 10.1007/s00220-008-0682-3.

[2]

V. Banica and L. Vega, Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc.l, 14 (2012), 209-253. doi: 10.4171/JEMS/300.

[3]

V. Banica and L. Vega, Stability of the self-similar dynamics of a vortex filament, Arch. Ration. Mech. Anal., 210 (2013), 673-712. doi: 10.1007/s00205-013-0660-6.

[4]

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

[5]

T. F. Buttke, A numerical study of superfluid turbulence in the Self Induction Approximation, J. of Compt. Physics, 76 (1988), 301-326.

[6]

L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo, 22 (1906), 117.

[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.

[8]

S. Jaffard, The spectrum of singularities of Riemanns function, Rev. Mat. Iberoamericana, 12 (1996), 44-460. doi: 10.4171/RMI/203.

[9]

U. Frisch and G. Parisi, Fully developed turbulence and intermittency, in Proc. Int. Sch. Phys. Enrico Fermi, North-Holland, Amsterdam, 1985.

[10]

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

[11]

L. Kapitanski and I. Rodnianski, Does a Quantum Particle Know the Time? Emerging Applications of Number Theory, IMA Vol. Math. Appl., 109 (1999), 355-371. doi: 10.1007/978-1-4612-1544-8_14.

[12]

S. Gutiérrez and L. Vega, Self-similar solutions of the localized induction approximation: singularity formation, Nonlinearity, 17 (2004), 2091-2136. doi: 10.1088/0951-7715/17/6/006.

[13]

S. Gutiérrez and L. Vega, On the stability of self-similar solutions of 1D cubic Schrödinger equations, Math. Ann., 356 (2013), 259-300. doi: 10.1007/s00208-012-0847-4.

[14]

S. Gutiérrez, J. Rivas and L. Vega, Formation of singularities and self-similar vortex motion under the localized induction approximation, Comm. Part. Diff. Eq., 28 (2003), 927-968. doi: 10.1081/PDE-120021181.

[15]

H. Hasimoto, A soliton in a vortex filament, J. Fluid Mech., 51 (1972), 477-485.

[16]

J. C. Hardin, The velocity field induced by a helical vortex filament, Phys. Fluids, 25 (1982), 1949-1952.

[17]

F. de la Hoz, Self-similar solutions for the 1-D Schrödinger map on the hyperbolic plane, Math. Z., 257 (2007), 61-80. doi: 10.1007/s00209-007-0115-6.

[18]

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

[19]

F. de la Hoz, C. García-Cervera and L. Vega, A numerical study of the self-similar solutions of the Schrödinger Map, SIAM J. Appl. Math., 70 (2009), 1047-1077. doi: 10.1137/080741720.

[20]

C. E. Kenig, G. Ponce, and L.Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633. doi: 10.1215/S0012-7094-01-10638-8.

[21]

M. Lakshmanan and M. Daniel, On the evolution of higher dimensional Heisenberg continuum spin systems, Physica A, 107 (1981), 533-552. doi: 10.1016/0378-4371(81)90186-2.

[22]

M. Lakshmanan, T. W. Ruijgrok and C. J. Thompson, On the the dynamics of a continuum spin system, Physica A, 84 (1976), 577-590.

[23]

T, Lipniacki, Quasi-static solutions for quantum vortex motion under the localized induction approximation, J. Fluid Mech., 477 (2003), 321-337. doi: 10.1017/S0022112002003282.

[24]

K. I. Oskolkov, A class of I. M. Vinogradov's series and its applications in harmonic analysis, in Progress in Approximation Theory (Tampa, FL, 1990), Springer Ser. Comput. Math. 19, Springer, New York, 1992, 353-402. doi: 10.1007/978-1-4612-2966-7_16.

[25]

C. S. Peskin and D. M. McQueen, Mechanical equilibrium determines the fractal fiber architecture of aortic heart valve leaflets, Am. J. Physiol. 266 (Heart Circ. Physiol. 35), (1994), H319-H328.

[26]

R. L. Ricca, The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics, Fluid Dynam. Res., 18 (1996), 245-268. doi: 10.1016/0169-5983(96)82495-6.

[27]

R. L. Ricca, Rediscovery of Da Rios equations, Nature, 352 (1991), 561-562.

[28]

P. G. Saffman, Vortex dynamics, in Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992.

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