MR2976046 Eiji ONODERA Eiji ONODERA A Curve Flow on an Almost Hermitian Manifold Evolved by a Third Order Dispersive Equation Funkcialaj Ekvacioj. Serio Internacia 55 2012 137--156 http://fe.math.kobe-u.ac.jp/FE/FullPapers/55-1/55_137.pdf http://www.ams.org/mathscinet-getitem?mr=MR2976046 We consider a curve flow for maps from a real line into a compact almost Hermitian manifold, which is governed by a third order nonlinear dispersive equation. This article shows short-time existence of a solution to the initial value problem for the equation. The difficulty comes from the lack of the Kähler condition on the target manifold, since the covariant derivative of the almost complex structure causes a loss of one derivative in our equation and thus the classical energy method breaks down in general. In the present article, we can overcome the difficulty by constructing a gauge transformation on the pull-back bundle for the map to eliminate the derivative loss essentially, which is based on the local smoothing effect of third order dispersive equations on the real line. Nonlinear dispersive partial differential equations, Vortex filament, Derivative loss, Smoothing effect, Energy method, Geometric analysis. 35Q53, 35Q55, 53C44. 55-137 2012 A Curve Flow on an Almost Hermitian Manifold Evolved by a Third Order Dispersive Equation Eiji ONODERA Eiji ONODERA
1 Chang, N. H.; Shatah, J.; Uhlenbeck, K. Schrödinger maps Comm. Pure Appl. Math. 53 2000 590-602 MR1737504
2 Chihara, H., Schrödinger flow into almost Hermitian manifolds, arXiv:0807.3395
3 Chihara, H.; Onodera, E. A third order dispersive flow for closed curves into almost Hermitian manifolds J. Funct. Anal. 257 2009 388-404 MR2527022
4 Da Rios, L. S. On the motion of an unbounded fluid with a vortex filament of any shape Rend. Circ. Mat. Palermo 22 1906 117-135 (in Italian)
5 Ding, W. Y. On the Schrödinger flows Proceedings of the ICM, Vol. II 2002 283-291 MR1957040
6 Doi, S.-I. Smoothing effects of Schrödinger evolution groups on Riemannian manifolds Duke Math. J. 82 1996 679-706 MR1387689
7 Fukumoto, Y.; Miyazaki, T. Three-dimensional distortions of a vortex filament with axial velocity J. Fluid Mech. 222 1991 369-416 MR1090719
8 Hama, F. R. Progressive deformation of a curved vortex filament by its own induction Phys. Fluids 5 1962 1156-1162
9 Hasimoto, H. A soliton on a vortex filament J. Fluid Mech. 51 1972 477-485
10 Kenig, C.; Lamm, T.; Pollack, D.; Staffilani, G.; Toro, T. The Cauchy problem for Schrödinger flows into Kähler manifolds Discrete Contin. Dyn. Syst. 27 2010 389-439 MR2600676
11 Kobayashi, S.; Nomizu, K. Foundations of Differential Geometry. Vol. II Wiley Classics Library, Wiley Inc., Princeton 1996 MR1393941
12 Koiso, N. The vortex filament equation and a semilinear Schrödinger equation in a Hermitian symmetric space Osaka J. Math. 34 1997 199-214 MR1439006
13 Koiso, N. Vortex filament equation and semilinear Schrödinger equation Nonlinear waves (Sapporo, 1995), GAKUTO Internat. Ser. Math. Sci. Appl., 10, Gakko-tosho, Tokyo 1997 pp. 231-236 MR1602745
14 Lakshmanan, M.; Bullough. R. K. Geometry of generalised nonlinear Schrödinger and Heisenberg ferromagnetic spin equations with linearly $x$-dependent coefficients Phys. Lett. A 80 1980 287-292 MR0594744
15 Landau, L. D.; Lifschitz, E. M. On the theory of the dispersion of magnetic permeability in ferro-aquatic bodies Phys. Zeitsch. der Sow. 8 1935 153-169
16 Lakshmanan, M.; Rujigrok, Th. W.; Thompson, C. J. On the dynamics of a continuum spin system Phys. A 84 1976 577-590 MR0449262
17 McGahagan, H. An approximation scheme for Schrödinger maps Comm. Partial Differential Equations 32 2007 375-400 MR2304153
18 Mizuhara, R. The initial value problem for third and fourth order dispersive equations in one space dimension Funkcial. Ekvac. 49 2006 1-38 MR2239909 2239909
19 Nahmod, A.; Shatah, J.; Vega, L.; Zeng, C. Schrödinger maps and their associated frame systems Int. Math. Res. Not. IMRN 21 2007 Art. ID rnm088, 29pp MR2352219
20 Nahmod, A.; Stefanov, A.; Uhlenbeck, K. On Schröodinger maps Comm. Pure Appl. Math. 56 2003 114-151 MR1929444
21 Nahmod, A.; Stefanov, A.; Uhlenbeck, K. Erratum: "On Schrödinger maps" [Comm. Pure Appl. Math. 56 (2003), 114-151] Comm. Pure Appl. Math. 57 2004 833-839 MR2038118
22 Nishikawa, S. Variational Problems in Geometry Translations of Mathematical Monographs, vol 205, American Mathematical Society 2002 MR1896863
23 Nishiyama, T.; Tani, A. Initial and initial-boundary value problems for a vortex filament with or without axial flow SIAM J. Math. Anal. 27 1996 1015-1023 MR1393421
24 Onodera, E. A third-order dispersive flow for closed curves into Kähler manifolds J. Geom. Anal. 18 2008 889-918 MR2420768
25 Onodera, E. Generalized Hasimoto transform of one-dimensional dispersive flows into compact Riemann surfaces SIGMA Symmetry Integrability Geom. Methods Appl. 4 2008 044, 10pp. MR2425648
26 Onodera, E. A remark on the global existence of a third order dispersive flow into locally Hermitian symmetric spaces Comm. Partial Differential Equations 35 2010 1130-1144 MR2753630
27 Pang, P. Y. Y.; Wang, H. Y.; Wang, Y. D. Schrödinger flow on Hermitian locally symmetric spaces Comm. Anal. Geom. 10 2002 653-681 MR1925498
28 Ricca, R. L. Rediscovery of Da Rios equations Nature 352 1991 561-562
29 Rodnianski, I.; Rubinstein, Y. A.; Staffilani, G. On the global well-posedness of the one-dimensional Schrödinger map flow Anal. PDE 2 2009 187-209 MR2547134
30 Sulem, P.-L.; Sulem, C.; Bardos, C. On the continuous limit for a system of classical spins Comm. Math. Phys. 107 1986 431-454 MR0866199
31 Tani, A.; Nishiyama, T. Solvability of equations for motion of a vortex filament with or without axial flow Publ.Res. Inst. Math. Sci. 33 1997 509-526 MR1489989
32 Tarama, S. Remarks on $L^2$-wellposed Cauchy problem for some dispersive equations J. Math. Kyoto Univ. 37 1998 757-765 MR1625936
33 Terng, C-L.; Uhlenbeck, K. Schrödinger flows on Grassmannians Integrable systems, geometry, and topology, AMS/IP Stud. Adv. Math., 36, Amer. Math. Soc., Providence, RI 2006 pp. 235-256 MR2222517