Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. Part I: Study of the perturbed Ginzburg–Landau equation

@article{Serfaty2007,
abstract = {We study vortices for solutions of the perturbed Ginzburg–Landau equations $\Delta u+(u/\varepsilon ^2)(1−|u|^2)=f_\varepsilon $ where $f_\varepsilon $ is estimated in $L^2$. We prove upper bounds for the Ginzburg–Landau energy in terms of $\Vert f_\varepsilon \Vert _\{L^2\}$, and obtain lower bounds for $\Vert f_\varepsilon \Vert _\{L^2\}$ in terms of the vortices when these form “unbalanced clusters” where $\sum _i d^2_i\ne (\sum _i d_i)^2$. These results will serve in Part II of this paper to provide estimates on the energy-dissipation rates for solutions of the Ginzburg–Landau heat flow, which allow one to study various phenomena occurring in this flow, including vortex collisions; they allow in particular extending the dynamical law of vortices beyond collision times.},
author = {Serfaty, Sylvia},
journal = {Journal of the European Mathematical Society},
keywords = {Ginzburg-Landau equation; Ginzburg-Landau vortices; vortex dynamics; vortex collisions; vortices; Pohožaev identity},
language = {eng},
number = {2},
pages = {177-217},
publisher = {European Mathematical Society Publishing House},
title = {Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. Part I: Study of the perturbed Ginzburg–Landau equation},
url = {http://eudml.org/doc/277783},
volume = {009},
year = {2007},
}

TY - JOUR
AU - Serfaty, Sylvia
TI - Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. Part I: Study of the perturbed Ginzburg–Landau equation
JO - Journal of the European Mathematical Society
PY - 2007
PB - European Mathematical Society Publishing House
VL - 009
IS - 2
SP - 177
EP - 217
AB - We study vortices for solutions of the perturbed Ginzburg–Landau equations $\Delta u+(u/\varepsilon ^2)(1−|u|^2)=f_\varepsilon $ where $f_\varepsilon $ is estimated in $L^2$. We prove upper bounds for the Ginzburg–Landau energy in terms of $\Vert f_\varepsilon \Vert _{L^2}$, and obtain lower bounds for $\Vert f_\varepsilon \Vert _{L^2}$ in terms of the vortices when these form “unbalanced clusters” where $\sum _i d^2_i\ne (\sum _i d_i)^2$. These results will serve in Part II of this paper to provide estimates on the energy-dissipation rates for solutions of the Ginzburg–Landau heat flow, which allow one to study various phenomena occurring in this flow, including vortex collisions; they allow in particular extending the dynamical law of vortices beyond collision times.
LA - eng
KW - Ginzburg-Landau equation; Ginzburg-Landau vortices; vortex dynamics; vortex collisions; vortices; Pohožaev identity
UR - http://eudml.org/doc/277783
ER -