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Problèmes d’évolution liés à l’énergie de Ginzburg-Landau

Séminaire Équations aux dérivées partielles

Improved estimates for the Ginzburg-Landau equation : the elliptic case

Fabrice Bethuel; Giandomenico Orlandi; Didier Smets — 2005

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the $G L$ -energy $E_{ε}$ and the parameter $ε$ . These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.

On Schrödinger maps from $T^{1}$ to $S^{2}$

Robert L. Jerrard; Didier Smets — 2012

Annales scientifiques de l'École Normale Supérieure

We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from $T^{1}$ to $S^{2} .$ This estimate yields some continuity properties of the flow map for the topology of $L^{2} (T^{1}, S^{2})$ , provided one takes its quotient by the continuous group action of $T^{1}$ given by translations. We also prove that without taking this quotient, for any $t > 0$ the flow map at time $t$ is discontinuous as a map from $𝒞^{\infty} (T^{1}, S^{2})$ , equipped with the weak topology of $H^{1 / 2},$ to the space of distributions ${(𝒞^{\infty} (T^{1}, ℝ^{3}))}^{*} .$ The argument relies in an essential...

Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation

Fabrice Béthuel; Philippe Gravejat; Didier Smets — 2014

Annales de l’institut Fourier

We establish the stability in the energy space for sums of solitons of the one-dimensional Gross-Pitaevskii equation when their speeds are mutually distinct and distinct from zero, and when the solitons are initially well-separated and spatially ordered according to their speeds.

Dynamique des tourbillons de vorticité pour l’équation de Ginzburg-Landau parabolique

Fabrice Bethuel; Giandomenico Orlandi; Didier Smets

Séminaire Équations aux dérivées partielles

Motion of concentration sets in Ginzburg-Landau equations

Fabrice Bethuel; Giandomenico Orlandi; Didier Smets — 2004

Annales de la Faculté des sciences de Toulouse : Mathématiques

On the motion of a curve by its binormal curvature

Jerrard, Robert L.; Didier Smets — 2015

Journal of the European Mathematical Society

We propose a weak formulation for the binormal curvature flow of curves in $ℝ^{3}$ . This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence theorem in that framework.

Vortex rings for the Gross-Pitaevskii equation

Fabrice Bethuel; G. Orlandi; Didier Smets — 2004

Journal of the European Mathematical Society

We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension $N \geq 3$ . We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if $N = 3$ ).