On Schrödinger maps from $T^1$ to $S^2$

Jerrard, Robert L.; Smets, Didier

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 4, pp. 637-680.

Nous obtenons une estimation de la différence entre deux solutions de l’équation des Schrödinger maps de $T^1$ dans $S^2.$ Cette estimation fournit une propriété de continuité du flot associé pour la topologie de $L^2(T^1,S^2)$, quotientée par l’action continue du groupe $T^1$ via les translations. Nous démontrons également que sans cette prise de quotient, quel que soit $t>0$ l’application flot au temps $t$ est discontinue de ${𝒞}^{\infty }(T^1,S^2)$, équipé de la topologie faible de $H^{1/2},$ vers l’espace des distributions périodiques ${\left({𝒞}^{\infty }(T^1,{ℝ}^3)\right)}^{*}.$ L’argument repose de manière essentielle sur le lien étroit entre l’équation des Schrödinger maps et celle du flot par courbure binormale pour une courbe dans l’espace, et sur une nouvelle estimation concernant ce dernier.

We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from $T^1$ to $S^2.$ This estimate yields some continuity properties of the flow map for the topology of $L^2(T^1,S^2)$, provided one takes its quotient by the continuous group action of $T^1$ given by translations. We also prove that without taking this quotient, for any $t>0$ the flow map at time $t$ is discontinuous as a map from ${𝒞}^{\infty }(T^1,S^2)$, equipped with the weak topology of $H^{1/2},$ to the space of distributions ${\left({𝒞}^{\infty }(T^1,{ℝ}^3)\right)}^{*}.$ The argument relies in an essential way on the link between the Schrödinger map equation and the binormal curvature flow for curves in the euclidean space, and on a new estimate for the latter.

[1] M. Agueh, Gagliardo-Nirenberg inequalities involving the gradient $L^2$-norm, C. R. Math. Acad. Sci. Paris 346 (2008), 757-762. | MR | Zbl

[2] 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 | Zbl

[3] J. Bourgain, Periodic Korteweg-de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997), 115-159. | MR | Zbl

[4] N. Burq, P. Gérard & N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Math. Res. Lett. 9 (2002), 323-335. | MR | Zbl

[5] N.-H. Chang, J. Shatah & K. Uhlenbeck, Schrödinger maps, Comm. Pure Appl. Math. 53 (2000), 590-602. | MR | Zbl

[6] M. Christ, J. Colliander & T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003), 1235-1293. | MR | Zbl

[7] W. Ding & Y. Wang, Local Schrödinger flow into Kähler manifolds, Sci. China Ser. A 44 (2001), 1446-1464. | MR | Zbl

[8] J. Hadamard, Sur quelques applications de l'indice de Kronecker, in Introduction à la théorie des fonctions d'une variable (J. Tannery, éd.), Hermann, 1910.

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

[10] R. L. Jerrard, Vortex filament dynamics for Gross-Pitaevsky type equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002), 733-768. | Numdam | MR | Zbl

[11] R. L. Jerrard & D. Smets, On the motion of a curve by its binormal curvature, preprint arXiv:1109.5483.

[12] C. E. Kenig, G. Ponce & L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), 617-633. | MR | Zbl

[13] S. Kida, A vortex filament moving without change of form, J. Fluid Mech. 112 (1981), 397-409. | MR | Zbl

[14] Y. C. Ma & M. J. Ablowitz, The periodic cubic Schrödinger equation, Stud. Appl. Math. 65 (1981), 113-158. | MR | Zbl

[15] H. Mcgahagan, Some existence and uniqueness results for Schroedinger maps and Landau-Lifshitz-Maxwell equations, Thèse, New York University, 2004. | MR

[16] H. Mcgahagan, An approximation scheme for Schrödinger maps, Comm. Partial Differential Equations 32 (2007), 375-400. | MR | Zbl

[17] L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation, Math. Res. Lett. 16 (2009), 111-120. | MR | Zbl

[18] A. Nahmod, J. Shatah, L. Vega & C. Zeng, Schrödinger maps and their associated frame systems, Int. Math. Res. Not. 2007 (2007), Art. ID rnm088, 1-29. | Zbl

[19] J. Shatah, Regularity results for semilinear and geometric wave equations, in Mathematics of gravitation, Part I (Warsaw, 1996), Banach Center Publ. 41, Polish Acad. Sci., 1997, 69-90. | MR | Zbl

[20] P.-L. Sulem, C. Sulem & C. Bardos, On the continuous limit for a system of classical spins, Comm. Math. Phys. 107 (1986), 431-454. | MR | Zbl

[21] V. E. Zakharov & A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz. 61 (1971), 118-134; English translation: Soviet Physics JETP 34 (1972), 62-69. | MR

[22] Y. L. Zhou & B. L. Guo, Existence of weak solution for boundary problems of systems of ferro-magnetic chain, Sci. Sinica Ser. A 27 (1984), 799-811. | MR | Zbl

[23] Y. L. Zhou, B. L. Guo & S. B. Tan, Existence and uniqueness of smooth solution for system of ferro-magnetic chain, Sci. China Ser. A 34 (1991), 257-266. | MR | Zbl