no. 5
Global existence and collisions for symmetric configurations of nearly parallel vortex filaments
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 5, pp. 813-832.

We consider the Schrödinger system with Newton-type interactions that was derived by R. Klein, A. Majda and K. Damodaran (1995) [17] to modelize the dynamics of N nearly parallel vortex filaments in a 3-dimensional homogeneous incompressible fluid. The known large time existence results are due to C. Kenig, G. Ponce and L. Vega (2003) [16] and concern the interaction of two filaments and particular configurations of three filaments. In this article we prove large time existence results for particular configurations of four nearly parallel filaments and for a class of configurations of N nearly parallel filaments for any N⩾2. We also show the existence of travelling wave type dynamics. Finally we describe configurations leading to collision.

@article{AIHPC_2012__29_5_813_0,
     author = {Banica, Valeria and Miot, Evelyne},
     title = {Global existence and collisions for symmetric configurations of nearly parallel vortex filaments},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {813--832},
     publisher = {Elsevier},
     volume = {29},
     number = {5},
     year = {2012},
     doi = {10.1016/j.anihpc.2012.04.005},
     mrnumber = {2971032},
     zbl = {1283.35072},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.04.005/}
}
TY  - JOUR
AU  - Banica, Valeria
AU  - Miot, Evelyne
TI  - Global existence and collisions for symmetric configurations of nearly parallel vortex filaments
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2012
SP  - 813
EP  - 832
VL  - 29
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.04.005/
DO  - 10.1016/j.anihpc.2012.04.005
LA  - en
ID  - AIHPC_2012__29_5_813_0
ER  - 
%0 Journal Article
%A Banica, Valeria
%A Miot, Evelyne
%T Global existence and collisions for symmetric configurations of nearly parallel vortex filaments
%J Annales de l'I.H.P. Analyse non linéaire
%D 2012
%P 813-832
%V 29
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2012.04.005/
%R 10.1016/j.anihpc.2012.04.005
%G en
%F AIHPC_2012__29_5_813_0
Banica, Valeria; Miot, Evelyne. Global existence and collisions for symmetric configurations of nearly parallel vortex filaments. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 5, pp. 813-832. doi : 10.1016/j.anihpc.2012.04.005. http://www.numdam.org/articles/10.1016/j.anihpc.2012.04.005/

[1] H. Aref, Motion of three vortices, Phys. Fluids 22 (1979), 393-400 | Zbl

[2] F. Béthuel, H. Brezis, F. Helein, Ginzburg–Landau Vortices, Birkhäuser, Boston (1994) | MR | Zbl

[3] F. Béthuel, P. Gravejat, J.-C. Saut, D. Smets, On the Korteweg–de Vries long-wave approximation of the Gross–Pitaevskii equation I, http://arxiv.org/pdf/0810.4417 | MR | Zbl

[4] F. Béthuel, J.-C. Saut, Travelling waves for the Gross–Pitaevskii equation, Ann. Inst. Henri Poincaré, Phys. Théor. 70 no. 2 (1999), 147-238 | EuDML | Numdam | MR | Zbl

[5] J. Bona, J.-C. Saut, Dispersive blow-up. II: Schrödinger-type equations, optical and oceanic rogue waves, Chin. Ann. Math., Ser. B 31 no. 6 (2010), 793-818 | MR | Zbl

[6] T. Cazenave, F. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Math. Z. 228 no. 1 (1998), 83-120 | MR | Zbl

[7] C. Gallo, Growth rate of the Schrödinger group on Zhidkov spaces, C. R. Math. Acad. Sci. Paris 342 no. 5 (2006), 319-323 | MR | Zbl

[8] D. Chiron, F. Rousset, The KdV/KP-I limit of the nonlinear Schrödinger equation, SIAM Math. Anal. 42 no. 1 (2010), 64-96 | MR | Zbl

[9] A. De Bouard, Instability of stationary bubbles, SIAM J. Math. Anal. 26 no. 3 (1995), 566-582 | MR | Zbl

[10] L. Di Menza, C. Gallo, The black solitons of one-dimensional NLS equations, Nonlinearity 20 no. 2 (2007), 461-496 | MR | Zbl

[11] P. Gérard, The Cauchy problem for the Gross–Pitaevskii equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23 no. 5 (2006), 765-779 | EuDML | Numdam | MR | Zbl

[12] P. Gérard, The Gross–Pitaevskii equation in the energy space, Contemp. Math. 473 (2008), 129-148 | MR | Zbl

[13] P. Gravejat, Limit at infinity and nonexistence results for sonic travelling waves in the Gross–Pitaevskii equation, Differ. Integral Equ. 17 no. 11–12 (2004), 1213-1232 | MR | Zbl

[14] H. Hasimoto, A soliton in a vortex filament, J. Fluid Mech. 51 (1972), 477-485 | MR | Zbl

[15] E.J. Hopfinger, F.K. Browand, Vortex solitary waves in a rotating turbulent flow, Nature 295 (1981), 393-395

[16] C.E. Kenig, G. Ponce, L. Vega, On the interaction of nearly parallel vortex filaments, Commun. Math. Phys. 243 (2003), 471-483 | MR | Zbl

[17] R. Klein, A. Majda, K. Damodaran, Simplified equations for the interaction of nearly parallel vortex filaments, J. Fluid Mech. 288 (1995), 201-248 | MR | Zbl

[18] L.G. Kurakin, V.I. Yudovitch, The stability of stationary rotation of a regular vortex polygon, Chaos 12 no. 3 (2002), 574-595 | MR | Zbl

[19] P.-L. Lions, A. Majda, Equilibrium statistical theory for nearly parallel vortex filaments, Comm. Pure Appl. Math. 53 (2000), 76-142 | MR | Zbl

[20] A. Majda, A. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge (2002) | MR | Zbl

[21] M. Maris, Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity, http://arxiv.org/abs/0903.0354 | MR | Zbl

[22] E.A. Novikov, Dynamics and statistics of a system of vortices, Sov. Phys. JETP 41 (1975), 937-943

[23] G. Ponce, On the stability of nearly parallel vortex filaments, J. Dyn. Differ. Equations 18 no. 3 (2006), 551-575 | MR | Zbl

[24] P.E. Zhidkov, Korteweg–de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Math. vol. 1756, Springer-Verlag (2001) | MR | Zbl

[25] Z. Lin, Stability and instability of travelling solitonic bubbles, Adv. Differential Equations 7 (2002), 897-918 | MR | Zbl

Cité par Sources :