[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.2005.42.1.075

MOTION OF VORTEX FILAMENTS IN 3-MANIFOLDS

PAK, HEE-CHUL (Department of Applied Mathematics, Dankook University)

Publication Information

Bulletin of the Korean Mathematical Society / v.42, no.1, 2005 , pp. 75-85 More about this Journal

Abstract

In this paper, the visco-Da-Rios equation; (0.1) ( $\frac{{\partial}{\gamma}}{{\partial}t}=\frac{{\partial}{\gamma}}{{\partial}s}{\bigwedge}\frac{D}{ds}\frac{{\partial}{\gamma}}{{\partial}s}+{\nu}\frac{{\partial}{\gamma}}{{\partial}s}$ ) is investigated on 3-dimensional complete orientable Riemannian manifolds. The global existence of solution is discussed by trans-forming (0.1) into a cubic nonlinear Schrodinger equation for complete orient able Riemannian 3-manifolds of constant curvature.

Keywords

nonlinear Schrodinger equation; Da-Rios equation; incompressible fluid; Hasimoto transform; Schrodinger maps;

Citations & Related Records

Reference

1	T. C. Cazenave and A. Haraux, An introduction to semilinear evolution equations, Oxford University Press Inc., New York, 1998
2	N. Chang, J. Shatah, and K. Uhlenbeck, Shrodinger maps, Commun. Pure. Appl. Anal. 53 (2000), 590-602 DOI ScienceOn
3	S. Gallot, D. HuIi, and J. Lfontaine, Riemannian Geometry, 2nd edition, Springer-Verlag, 1993
4	C. L. Terng and K. Uhlenbeck, Schrodinqer flows on Grassmannians, Preprint math. DG/9901086
5	T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer, 1998
6	J. Bourgain, Global solutions of nonlinear Shriidinqer equations, Amer. Math. Soc. Colloq. Publ. 46 (1999)
7	H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1972), 477-485 DOI
8	J. Jost, Riemannian Geometry and Geometric Analysis, Springer, 1995
9	H. C. Pak, Flow via binormal in 3-manifolds, Preprint
10	J. Langer and R. PerIine, Geometric Realizations of Fordy-Kulish nonlinear Schrddinqer systems, Pacific J. Math. 195 (2000), 157-178 DOI
11	M. Spivak, A Comprehensive Introduction to Riemannian Geometry, vol. 2, 2nd edition, Publish or Perish, Inc., Boston, 1979