Motion of vortices for the extrinsic Ginzburg-Landau flow for vector fields on surfaces

Abstract

<p style='text-indent:20px;'>We consider the gradient flow of a Ginzburg-Landau functional of the type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ F_ \varepsilon^{ \mathrm{extr}}(u): = \frac{1}{2}\int_M \left| {D u} \right|_g^2 + \left| { \mathscr{S} u} \right|^2_g +\frac{1}{2 \varepsilon^2}\left(\left| {u} \right|^2_g-1\right)^2 \mathrm{vol}_g $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>which is defined for tangent vector fields (here <inline-formula><tex-math id="M1">\begin{document}$ D $\end{document}</tex-math></inline-formula> stands for the covariant derivative) on a closed surface <inline-formula><tex-math id="M2">\begin{document}$ M\subseteq \mathbb{R}^3 $\end{document}</tex-math></inline-formula> and includes extrinsic effects via the shape operator <inline-formula><tex-math id="M3">\begin{document}$ \mathscr{S} $\end{document}</tex-math></inline-formula> induced by the Euclidean embedding of <inline-formula><tex-math id="M4">\begin{document}$ M $\end{document}</tex-math></inline-formula>. The functional depends on the small parameter <inline-formula><tex-math id="M5">\begin{document}$ \varepsilon&gt;0 $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M6">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> is small it is clear from the structure of the Ginzburg-Landau functional that <inline-formula><tex-math id="M7">\begin{document}$ \left| {u} \right|_g $\end{document}</tex-math></inline-formula> "prefers" to be close to <inline-formula><tex-math id="M8">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>. However, due to the incompatibility for vector fields on <inline-formula><tex-math id="M9">\begin{document}$ M $\end{document}</tex-math></inline-formula> between the Sobolev regularity and the unit norm constraint, when <inline-formula><tex-math id="M10">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> is close to <inline-formula><tex-math id="M11">\begin{document}$ 0 $\end{document}</tex-math></inline-formula>, it is expected that a finite number of singular points (called vortices) having non-zero index emerges (when the Euler characteristic is non-zero). This intuitive picture has been made precise in the recent work by R. Ignat &amp; R. Jerrard [<xref ref-type="bibr" rid="b7">7</xref>]. In this paper we are interested the dynamics of vortices generated by <inline-formula><tex-math id="M12">\begin{document}$ F_ \varepsilon^{ \mathrm{extr}} $\end{document}</tex-math></inline-formula>. To this end we study the behavior when <inline-formula><tex-math id="M13">\begin{document}$ \varepsilon\to 0 $\end{document}</tex-math></inline-formula> of the solutions of the (properly rescaled) gradient flow of <inline-formula><tex-math id="M14">\begin{document}$ F_ \varepsilon^{ \mathrm{extr}} $\end{document}</tex-math></inline-formula>. In the limit <inline-formula><tex-math id="M15">\begin{document}$ \varepsilon\to 0 $\end{document}</tex-math></inline-formula> we obtain the effective dynamics of the vortices. The dynamics, as expected, is influenced by both the intrinsic and extrinsic properties of the surface <inline-formula><tex-math id="M16">\begin{document}$ M\subseteq \mathbb{R}^3 $\end{document}</tex-math></inline-formula>.</p>