1,720,986 research outputs found
Uniform estimates for the parabolic Ginzburg-Landau equation. A tribute to J. L. Lions.
No full textWe consider complex-valued solutions ue of the Ginzburg-Landau equation on a smooth bounded simply connected domain W of RN, N 3 2, where e > 0 is a small parameter. We assume that the Ginzburg-Landau energy Ee(ue) verifies the bound (natural in the context) Ee(ue) M|log e|, where M0 is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of ue, as e® 0, is to establish uniform Lp bounds for the gradient, for some p > 1. We review some recent techniques developed in the elliptic case in [7], discuss some variants, and extend the methods to the associated parabolic equatio
Asymptotics for the Ginzburg–Landau Equation in Arbitrary Dimensions
No full textAbstractLet Ω be a bounded, simply connected, regular domain of RN, N⩾2. For 0<ε<1, let uε:Ω→C be a smooth solution of the Ginzburg–Landau equation in Ω with Dirichlet boundary condition gε, i.e.,[formula] We are interested in the asymptotic behavior of uε as ε goes to zero under the assumption that Eε(uε)⩽M0|logε| and some conditions on gε which allow singularities of dimension N−3 on ∂Ω
Aspects of vortex dynamics in Ginzburg-Landau models
No full textWe survey some recent work concerning the asymptotic dynamics of vortices in the 2-dimensional parabolic Ginzburg-Landau equation, the interaction of vortices with the phase field and the limiting initial value problem for both vortices and phase
Slow motion for gradient systems with equal depth multiple-well potentials
No full textAbstractFor scalar reaction–diffusion in one space dimension, it has been known for a long time that fronts move with an exponentially small speed for potentials with several distinct minimizers. The purpose of this paper is to provide a similar result in the case of systems. Our method relies on a careful study of the evolution of localized energy. This approach also has the advantage of relaxing the preparedness assumptions on the initial datum
Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature
No full textFor the complex parabolic Ginzburg-Landau equation, we prove that, asymptotically, vorticity evolves according to motion by mean curvature in Brakke¿s weak formulation. The only assumption is a natural energy bound on the initial data. In some cases, we also prove convergence to enhanced motion in the sense of Ilmanen
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