Abstract
We study vortices for solutions of the perturbed Ginzburg-Landau equations Δu + (u/ε2)(1 -|u|2) = fε where fε is estimated in L2. We prove upper bounds for the Ginzburg-Landau energy in terms of ∥fε∥L2, and obtain lower bounds for ∥fε∥L2 in terms of the vortices when these form "unbalanced clusters" where σi di2 ≠ = (σid i)2. These results will serve in Part n 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.
Original language | English (US) |
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Pages (from-to) | 177-217 |
Number of pages | 41 |
Journal | Journal of the European Mathematical Society |
Volume | 9 |
Issue number | 2 |
DOIs | |
State | Published - 2007 |
Keywords
- Ginzburg-Landau equation
- Ginzburg-Landau vortices
- Vortex collisions
- Vortex dynamics
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics