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

Journal of the European Mathematical Society (2007)

- Volume: 009, Issue: 2, page 177-217
- ISSN: 1435-9855

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topSerfaty, Sylvia. "Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. Part I: Study of the perturbed Ginzburg–Landau equation." Journal of the European Mathematical Society 009.2 (2007): 177-217. <http://eudml.org/doc/277783>.

@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 -

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