Lia Bronsard

Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics

in a bounded domain Q c W, n > 2, with appropriate initial and boundary data. Our interest is in the limiting behavior of u = U’ as E + 0. Formal analysis suggests the following picture: ~8 separates Q into two regions, where U&Z +l and ~8% 1, respectively, and the interface between them moves with normal velocity equal to the sum of its principal curvatures. Our goal here is to present two rigorous results which tend to confirm this picture. The first is a compactness theorem: we show that as E + 0, the solutions of (1.1) are in a certain sense compact as functions of space-time (see Theorem 2.3 and Remark 2.5). Thus it makes sense to discuss the limiting behavior. Our second result is a verification of the picture for certain radial solutions: we prove that lim,,, U’ exists and has the expected form if Q is a ball, U’ is radial with one transition sphere, and the boundary condition is of Dirichlet type (see Theorem 3.1).

On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation

We present a formal asymptotic analysis which suggests a model for three-phase boundary motion as a singular limit of a vector-valued Ginzburg-Landau equation. We prove short-time existence and uniqueness of solutions for this model, that is, for a system of three-phase boundaries undergoing curvature motion with assigned angle conditions at the meeting point. Such models pertain to grain-boundary motion in alloys. The method we use, based on linearization about the initial conditions, applies to a wide class of parabolic systems. We illustrate this further by its application to an eutectic solidification problem.

A MULTI-PHASE MULLINS-SEKERKA SYSTEM : MATCHED ASYMPTOTIC EXPANSIONS AND AN IMPLICIT TIME DISCRETISATION FOR THE GEOMETRIC EVOLUTION PROBLEM

We propose a generalisation of the Mullins–Sekerka problem to model phase separation in multi-component systems. The model includes equilibrium equations in bulk, the Gibbs–Thomson relation on the interfaces, Youngs law at triple junctions, together with a dynamic law of Stefan type. Using formal asymptotic expansions, we establish the relationship to a transition layer model known as the Cahn-Hilliard system. We introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds, together with measure theoretical tools. Through an implicit time discretisation, we construct approximate solutions by stepwise minimisation. Under the assumption that there is no loss of area as the time step tends to zero, we show the existence of a weak solution.

Front propagation for reaction-diffusion equations of bistable type

Abstract We present a direct PDE approach to study the behavior as e → 0 of the solution ue of the reaction-diffusion equation: u t e − e Δ u e = ( 1 / e ) f ( u e ) in ℝN × (0, ∞) in the case when f is the derivative of a bistable potential. Such singular perturbation problems arise in the study of large time wavefront propagations generated by such equations following a method introduced by M. Freidlin.

Pinning effects and their breakdown for a Ginzburg–Landau model with normal inclusions

We study a Ginzburg–Landau model for an inhomogeneous superconductor in the singular limit as the Ginzburg–Landau parameter κ=1∕ϵ→∞. The inhomogeneity is represented by a potential term V(ψ)=14(a(x)−∣ψ∣2)2, with a given smooth function a(x) which is assumed to become negative in finitely many smooth subdomains, the “normally included” regions. For bounded applied fields (independent of the Ginzburg–Landau parameter κ=1∕ϵ→∞) we show that the normal regions act as “giant vortices,” acquiring large vorticity for large (fixed) applied field hex. For hex=O(∣lnϵ∣) we show that this pinning effect eventually breaks down, and free vortices begin to appear in the superconducting region where a(x)>0, at a point set which is determined by solving an elliptic boundary-value problem. The associated operators are strictly but not uniformly elliptic, leading to some regularity questions to be resolved near the boundaries of the normal regions.

Domain Walls in the Coupled Gross–Pitaevskii Equations

A thorough study of domain wall solutions in coupled Gross–Pitaevskii equations on the real line is carried out including existence of these solutions; their spectral and nonlinear stability; their persistence and stability under a small localized potential. The proof of existence is variational and is presented in a general framework: we show that the domain wall solutions are energy minimizers within a class of vector-valued functions with nontrivial conditions at infinity. The admissible energy functionals include those corresponding to coupled Gross–Pitaevskii equations, arising in modeling of Bose–Einstein condensates. The results on spectral and nonlinear stability follow from properties of the linearized operator about the domain wall. The methods apply to many systems of interest and integrability is not germane to our analysis. Finally, sufficient conditions for persistence and stability of domain wall solutions are obtained to show that stable pinning occurs near maxima of the potential, thus giving rigorous justification to earlier results in the physics literature.

On the shape of interlayer vortices in the Lawrence-Doniach model

We consider the Lawrence-Doniach model for layered superconductors, in which stacks of parallel superconducting planes are coupled via the Josephson effect. To model experiments in which the superconductor is placed in an external magnetic field oriented parallel to the superconducting planes, we study the structure of isolated vortices for a doubly periodic problem. We consider a singular limit which simulates certain experimental regimes in which isolated vortices have been observed, corresponding to letting the interlayer spacing of the superconducting planes tend to zero and the Ginzburg-Landau parameter κ → ∞ simultaneously, but at a fixed relative rate.

FRACTIONAL DEGREE VORTICES FOR A SPINOR GINZBURG–LANDAU MODEL

Recent papers in the physics literature have introduced spin-coupled (or spinor) Ginzburg–Landau models for complex vector-valued order parameters in order to account for ferromagnetic or antiferromagnetic effects in high-temperature superconductors and in optically confined Bose–Einstein condensates. In this paper, we show that such models give rise to new types of vortices, with fractional degree and nontrivial core structure. We illustrate the various possibilites with some specific examples of Dirichlet problems in the unit disk.

Analytical description of the Saturn-ring defect in nematic colloids.

We derive an analytical formula for the Saturn-ring configuration around a small colloidal particle suspended in nematic liquid crystal. In particular we obtain an explicit expression for the ring radius and its dependence on the anchoring energy. We work within Landau-de Gennes theory: Nematic alignment is described by a tensorial order parameter. For nematic colloids this model had previously been used exclusively to perform numerical computations. Our method demonstrates that the tensorial theory can also be used to obtain analytical results, suggesting a different approach to the understanding of nematic colloidal interactions.

Thin Film Limits for Ginzburg-Landau with Strong Applied Magnetic Fields

In this work, we study thin film limits of the full three-dimensional Ginzburg–Landau model for a superconductor in an applied magnetic field oriented obliquely to the film surface. We obtain