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We study the asymptotic behavior of a two dimensional linear PDE with a degenerate diffusion and a drift term. The structure of this equation typically arises in some mathematical mean-field models of neural network, and the investigation of the qualitative properties of this equation is still open, and a challenging question. We prove, via a Doeblin-Harris type method, that the solutions converge exponentially fast to the unique stationary state in a L1-weighted norm. [ABSTRACT FROM AUTHOR]

We introduce a functional framework taylored to investigate the minimality and stability properties of the Ginzburg–Landau vortex of degree one on the whole plane. We prove that a renormalized Ginzburg–Landau energy is well defined in that framework and that the vortex is its unique global minimizer up to the invariances by translation and phase shift. Our main result is a nonlinear coercivity estimate for the renormalized energy around the vortex, from which we can deduce its orbital stability as a solution to the Gross–Pitaevskii equation, the natural Hamiltonian evolution equation associated to the Ginzburg–Landau energy. [ABSTRACT FROM AUTHOR]

We construct families of smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation (SQG). These solutions can be viewed as the equivalents for this equation of the vortex anti-vortex pairs in the context of the incompressible Euler equation. Our argument relies on the stream function formulation and eventually amounts to solving a fractional nonlinear elliptic equation by variational methods. [ABSTRACT FROM AUTHOR]

We introduce a new framework for the analysis of the stability of solitons for the one-dimensional Gross–Pitaevskii equation. In particular, we establish the asymptotic stability of the black soliton with zero speed. [ABSTRACT FROM AUTHOR]

We propose a weak formulation for the binormal curvature flow of curves in R³: 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. [ABSTRACT FROM AUTHOR]

In this paper, we proceed along our analysis of the Korteweg-de Vries approximation of the Gross-Pitaevskii equation initiated in [6]. At the long-wave limit, we establish that solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation split into two waves with opposite constant speeds [image omitted], each of which are solutions to a Korteweg-de Vries equation. We also compute an estimate of the error term which is somewhat optimal as long as travelling waves are considered. At the cost of higher regularity of the initial data, this improves our previous estimate in [6]. [ABSTRACT FROM AUTHOR]

The fact that the Korteweg–de Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross–Pitaevskii equation was derived several years ago in the physical literature (see e.g. [17]). In this paper, we provide a rigorous proof of this fact, and compute a precise estimate for the error term. Our proof relies on the integrability of both the equations. In particular, we give a relation between the invariants of the two equations, which, we hope, is of independent interest. [ABSTRACT FROM PUBLISHER]

We analyse symmetry breaking for ground states of the Hénon equation [7] in a ball. Asymptotic estimates of the transition are also given when p is close to either 2 or 2[sup *]. [ABSTRACT FROM AUTHOR]

We look for explicit image segmentations in the framework of the variational model proposed by Mumford and Shah. We first treat the symmetric case when the "screen" is a disk Ω and the image is a concentric disk D ⊂ Ω. We prove the optimal segmentation is either the given disk D or the solution of the associated Neumann problem, depending on both the difference of intensity between the background and the disk, and the distance separating ∂Ω and ∂D. Both segmentations are optimal in some critical cases which we characterize. Our main result is a first step towards a generalization of this behaviour. In case Ω and D are convex, we prove the following for an optimal segmentation (u, K) such that K ⊂ D: K tends to ∂D (in the Hausdorff distance) when the difference of intensity between Ω and D goes to infinity. [ABSTRACT FROM AUTHOR]

Abstract: For 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. [Copyright &y& Elsevier]

Abstract: For the two-dimensional complex parabolic Ginzburg–Landau equation we prove that, asymptotically, vortices evolve according to a simple ordinary differential equation, which is a gradient flow of the Kirchhoff–Onsager functional. This convergence holds except for a finite number of times, corresponding to vortex collisions and splittings, which we describe carefully. The only assumption is a natural energy bound on the initial data. To cite this article: F. Bethuel et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]