Uniform estimates for the parabolic Ginzburg–Landau equation

F. Bethuel; G. Orlandi

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The gradient flow motion of boundary vortices

Matthias Kurzke (2007)

Annales de l'I.H.P. Analyse non linéaire

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Vortex analysis of the periodic Ginzburg-Landau model

Hassen Aydi, Etienne Sandier (2009)

Annales de l'I.H.P. Analyse non linéaire

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Morse index and blow-up points of solutions of some nonlinear problems

Khalil El Mehdi, Filomena Pacella (2002)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this Note we consider the following problem $\{\begin{matrix} - △ u = N (N - 2) u^{p_{ϵ}} - λ u & in Ω \\ u > 0 & in Ω \\ u = 0 & on \partial Ω . \end{matrix}$ where $Ω$ is a bounded smooth starshaped domain in $R^{N}$ , $N \geq 3$ , $p_{ϵ} = \frac{N + 2}{N - 2} - ϵ$ , $ϵ > 0$ , and $λ \geq 0$ . We prove that if $u_{ϵ}$ is a solution of Morse index $m > 0$ than $u_{ϵ}$ cannot have more than $m$ maximum points in $Ω$ for $ϵ$ sufficiently small. Moreover if $Ω$ is convex we prove that any solution of index one has only one critical point and the level sets are starshaped for $ϵ$ sufficiently small.

Vortex motion and phase-vortex interaction in dissipative Ginzburg-Landau dynamics

F. Bethuel, G. Orlandi, D. Smets (2004)

Journées Équations aux dérivées partielles

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We discuss the asymptotics of the parabolic Ginzburg-Landau equation in dimension $N \geq 2 .$ Our only asumption on the initial datum is a natural energy bound. Compared to the case of “well-prepared” initial datum, this induces possible new energy modes which we analyze, and in particular their mutual interaction. The two dimensional case is qualitatively different and requires a separate treatment.

Homogenization and localization in locally periodic transport

Grégoire Allaire, Guillaume Bal, Vincent Siess (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper, we study the homogenization and localization of a spectral transport equation posed in a locally periodic heterogeneous domain. This equation models the equilibrium of particles interacting with an underlying medium in the presence of a creation mechanism such as, for instance, neutrons in nuclear reactors. The physical coefficients of the domain are $ε$ -periodic functions modulated by a macroscopic variable, where $ε$ is a small parameter. The mean free path of the particles...

Limiting behavior of global attractors for singularly perturbed beam equations with strong damping

Daniel Ševčovič (1991)

Commentationes Mathematicae Universitatis Carolinae

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The limiting behavior of global attractors $𝒜_{ε}$ for singularly perturbed beam equations $ε^{2} \frac{\partial^{2} u}{\partial t^{2}} + ε δ \frac{\partial u}{\partial t} + A \frac{\partial u}{\partial t} + α A u + {g (∥ u ∥}_{1 / 4}^{2}) A^{1 / 2} u = 0$ is investigated. It is shown that for any neighborhood $𝒰$ of $𝒜_{0}$ the set $𝒜_{ε}$ is included in $𝒰$ for $ε$ small.