Gauges for the Ginzburg-Landau equations of superconductivity

This note is concerned with gauge choices for the time-dependent Ginzburg-Landau equations of superconductivity. The requiations model the state of a superconducting sample in a magnetic field near the critical tempeature. Any two solutions related through a ``gauge transformation`` describe the same state and are physically indistinquishable. This ``gauge invariance`` can be exploited for analtyical and numerical purposes. A new gauge is proposed, which reduces the equations to a particularly attractive form

Semilinear cooperative elliptic systems on Rn

We study here the following semilinear cooperative elliptic system defined on IRn , n > 2 : (1 – a) −∆u = aρ(x)u + bρ(x)v + f(x, u, v) x ∈ IRn , (1 – b) −∆v = cρ(x)u + dρ(x)v + g(x, u, v) x ∈ IRn , (1 – c) u −→ 0 , v −→ 0 as |x| −→ +∞. Here a, b, c, d are constants such that b, c > 0 ; ρ, f and g are given functions; ρ is nonnegative and tends to 0 at ∞. We first establish necessary and sufficient conditions on the coefficients for having a Maximum Principle for the linear System. Then we show that these conditions ensure existence of solutions for the linear System and for the semilinear System when f and g satisfy some ”sublinear” condition. Under some additional assumption we also derive uniqueness of the solutions. Finally we show that our results can be extended to N × N systems, N > 2

Global and exponential attractors for a Ginzburg-Landau model of superfluidity

The long-time behavior of the solutions for a non-isothermal model in superfluidity is investigated. The model describes the transition between the normal and the superfluid phase in liquid 4He by means of a non-linear differential system, where the concentration of the superfluid phase satisfies a non-isothermal Ginzburg-Landau equation. This system, which turns out to be consistent with thermodynamical principles and whose well-posedness has been recently proved, has been shown to admit a Lyapunov functional. This allows to prove existence of the global attractor which consists of the unstable manifold of the stationary solutions. Finally, by exploiting recent techniques of semigroups theory, we prove the existence of an exponential attractor of finite fractal dimension which contains the global attractor.Comment: 39 page

Non-existence and uniqueness results for supercritical semilinear elliptic equations

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.Comment: Annales Henri Poincar\'e (2009) to appea

Marking their own homework: The pragmatic and moral legitimacy of industry self-regulation

When is industry self-regulation (ISR) a legitimate form of governance? In principle, ISR can serve the interests of participating companies, regulators and other stakeholders. However, in practice, empirical evidence shows that ISR schemes often under-perform, leading to criticism that such schemes are tantamount to firms marking their own homework. In response, this paper explains how current management theory on ISR has failed to separate the pragmatic legitimacy of ISR based on self-interested calculations, from moral legitimacy based on normative approval. The paper traces three families of management theory on ISR and uses these to map the pragmatic and moral legitimacy of ISR schemes. It identifies tensions between the pragmatic and moral legitimacy of ISR schemes, which the current ISR literature does not address, and draws implications for the future theory and practice of ISR