Motion of vortex filaments in 3-manifolds
Bull. Korean Math. Soc. 2005 Vol. 42, No. 1, 75-85
Published online March 1, 2005
Hee Chul Pak
Dankook University
Abstract : In this paper, the visco-Da-Rios equation; \begin{align} \frac{\partial \gamma }{\partial t} = \frac{\partial \gamma }{\partial s} \wedge \frac{D}{d s} \frac{\partial \gamma }{\partial s} + \nu \frac{\partial \gamma }{\partial s} \label{LIE} \end{align} is investigated on 3-dimensional complete orientable Riemannian manifolds. The {\it global existence} of solution is discussed by transforming (\ref{LIE}) into a cubic nonlinear Schr\"{o}dinger equation for complete orientable Riemannian 3-manifolds of constant curvature.
Keywords : nonlinear Schrodinger equation, Da-Rios equation, incompressible fluid, Hasimoto transform, Schrodinger maps
MSC numbers : 58J72, 22E30, 76B03, 35Q55
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