Dans cet article de comptes rendus on présente une série de résultats sur la stabilité des solutions auto-similaires de l’équation du tourbillon filamentaire. Cette équation décrit un flot de courbes de et est utilisée comme modèle pour l’évolution d’un tourbillon filamentaire dans un fluide. Le théorème principal donne, sous des hypothèses appropriées, l’existence et la description des solution engendrées par des courbes à un coin, sur temps positifs et négatifs. Le théorème compagnon décrit l’évolution des perturbations des solutions auto-similaires jusque’à formation d’une singularité en temps fini, et au-delà de ce temps. On va donner une esquisse des preuves. Ces résultats on été obtenus en collaboration avec Luis Vega.
In this proceedings article we shall survey a series of results on the stability of self-similar solutions of the vortex filament equation. This equation is a geometric flow for curves in and it is used as a model for the evolution of a vortex filament in fluid mechanics. The main theorem give, under suitable assumptions, the existence and description of solutions generated by curves with a corner, for positive and negative times. Its companion theorem describes the evolution of perturbations of self-similar solutions up to a singularity formation in finite time, and beyond this time. We shall give a sketch of the proof. These results were obtained in collaboration with Luis Vega.
@article{JEDP_2013____A1_0, author = {Banica, Valeria}, title = {Evolution by the vortex filament equation of curves with a corner}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, year = {2013}, pages = {1-18}, doi = {10.5802/jedp.97}, language = {en}, url = {http://dml.mathdoc.fr/item/JEDP_2013____A1_0} }
Banica, Valeria. Evolution by the vortex filament equation of curves with a corner. Journées équations aux dérivées partielles, (2013), pp. 1-18. doi : 10.5802/jedp.97. http://gdmltest.u-ga.fr/item/JEDP_2013____A1_0/
[1] R.J. Arms and F.R. Hama, Localized-induction concept on a curved vortex and motion of an elliptic vortex ring, Phys. Fluids, (1965), 553.
[2] V. Banica and L. Vega, On the stability of a singular vortex dynamics, Comm. Math. Phys. 286 (2009), 593–627. | MR 2472037 | Zbl 1183.35029
[3] V. Banica and L. Vega, Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc. 14 (2012), 209–253. | MR 2862038 | Zbl pre05995802
[4] V. Banica and L. Vega, Stability of the self-similar dynamics of a vortex filament, to appear in Arch. Ration. Mech. Anal. | MR 3116002 | Zbl pre06260948
[5] V. Banica and L. Vega, The initial value problem for the binormal flow with rough data, ArXiv:1304.0996.
[6] J Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), 107–156. | MR 1209299 | Zbl 0787.35097
[7] T. F. Buttke, A numerical study of superfluid turbulence in the Self Induction Approximation, J. of Compt. Physics 76 (1988), 301–326 | Zbl 0639.76136
[8] A. Calini and T. Ivey, Stability of Small-amplitude Torus Knot Solutions of the Localized Induction Approximation, J. Phys. A: Math. Theor. 44 (2011) 335204. | MR 2822117 | Zbl 1223.35286
[9] R. Carles, Geometric Optics and Long Range Scattering for One-Dimensional Nonlinear Schrödinger Equations, Comm. Math. Phys. 220 (2001), 41–67. | MR 1882399 | Zbl 1029.35211
[10] T. Cazenave and F.B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation, Non. Anal. TMA 14 (1990), 807–836. | MR 1055532 | Zbl 0706.35127
[11] M. Christ, J. Colliander, and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, ArXiv:0311048.
[12] 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.
[13] P. Germain, N. Masmoudi and J. Shatah, Global solutions for 2D quadratic Schrödinger equations, J. Math. Pures Appl. 97(2012), 505–543. | MR 2914945 | Zbl 1244.35134
[14] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. II Scattering theory, general case, J. Funct. Anal. 32 (1979), 33–71. | MR 533219 | Zbl 0396.35029
[15] A. Grünrock, Bi- and trilinear Schr?dinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Not. 41 (2005), 2525–2558. | MR 2181058 | Zbl 1088.35063
[16] S. Gustafson, K. Nakanishi, T.-P. Tsai, Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, Ann. Henri Poincaré 8 (2007), no. 7, 1303–1331. | MR 2360438 | Zbl pre05218113
[17] 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. | MR 1986056 | Zbl 1044.35089
[18] H. Hasimoto, A soliton in a vortex filament, J. Fluid Mech. 51 (1972), 477–485. | Zbl 0237.76010
[19] N. Hayashi and P. Naumkin, Domain and range of the modified wave operator for Schr?odinger equations with critical nonlinearity, Comm. Math. Phys. 267 (2006), 477–492. | MR 2249776 | Zbl 1113.81121
[20] E.J. Hopfinger, F.K. Browand, Vortex solitary waves in a rotating, turbulent flow, Nature 295, (1981), 393–395.
[21] 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. | MR 2546352 | Zbl 1219.65139
[22] R. L. Jerrard and D. Smets, On Schrödinger maps from to , arXiv:1105.2736.
[23] R. L. Jerrard and D. Smets, On the motion of a curve by its binormal curvature, arXiv:1109.5483.
[24] C. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical non-linear dispersive equations, Duke Math. J. 106 (2001) 716–633. | MR 1813239 | Zbl 1034.35145
[25] N. Koiso, Vortex filament equation and semilinear Schrödinger equation, Nonlinear Waves, Hokkaido University Technical Report Series in Mathematics 43 (1996) 221–226. | Zbl 0968.35110
[26] S. Lafortune, Stability of solitons on vortex filaments, Phys. Lett. A bf 377 (2013), 766–769. | MR 3021944
[27] M. Lakshmanan and M. Daniel, On the evolution of higher dimensional Heisenberg continuum spin systems, Physica A (1981), 107, 533–552. | MR 624580
[28] M. Lakshmanan, T. W. Ruijgrok and C. J. Thompson, On the the dynamics of a continuum spin system, Physica A (1976), 84, 577–590. | MR 449262
[29] T. Levi-Civita, Attrazione Newtoniana dei Tubi Sottili e Vortici Filiformi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 1 (1932), 229–250 | Numdam | Zbl 0004.37305
[30] T. Lipniacki, Quasi-static solutions for quantum vortex motion under the localized induction approximation, J. Fluid Mech. 477 (2002), 321–337. | MR 2011430 | Zbl 1063.76521
[31] F. Maggioni, S. Z. Alamri, C. F. Barenghi, and R. L. Ricca, Velocity, energy and helicity of vortex knots and unknots, Phys. Rev. E 82 (2010), 26309–26317. | MR 2736443
[32] A. Majda and A. Bertozzi, Vorticity and incompressible flow., Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. | MR 1867882 | Zbl 0983.76001
[33] K. Moriyama, S. Tonegawa and Y. Tsutsumi, Wave operators for the nonlinear Schrödinger equation with a nonlinearity of low degree in one or two space dimensions, Commun. Contemp. Math. 5 (2003), 983–996 . | MR 2030566 | Zbl 1055.35112
[34] T. Nishiyama and A. Tani, Solvability of the localized induction equation for vortex motion, Comm. Math. Phys. 162 (1994), 433?-445. | MR 1277470 | Zbl 0811.35100
[35] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys. 139 (1991), 479–493. | MR 1121130 | Zbl 0742.35043
[36] 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.
[37] 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. | MR 1408546 | Zbl 1006.01505
[38] R.L. Ricca, Rediscovery of Da Rios equations, Nature 352 (1991), 561–562.
[39] K.W. Schwarz, Three-dimensional vortex dynamics in superfluid He: Line-line and line-boundary interactions, Phys. Rev B 31 (1985), 5782–5804.
[40] A. Shimomura and S. Tonegawa, Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions, Differ. Integral Equ. 17 (2004), 127–150. | MR 2035499 | Zbl 1164.35325
[41] A. Tani and T. Nishiyama, Solvability of equations for motion of a vortex filament with or without axial flow, Publ. Res. Inst. Math. Sci. 33 (1997), 509?-526. | MR 1489989 | Zbl 0905.35070
[42] A. Vargas and L. Vega, Global well-posedness for 1d non-linear Schrodinger equation for data with an infinite norm, J. Math. Pures Appl. 80 (2001), 1029–1044. | MR 1876762 | Zbl 1027.35134
[43] E.J. Vigmond, C. Clements, D.M. McQueen and C.S. Peskin, Effect of bundle branch block on cardiac output: A whole heart simulation study, Prog. Biophys. Mol. Biol. 97 (2008), 520–42.