Ginzburg-Landau Approximation Research Articles

Partial regular solution to the coupled spin polarized system in three dimensions

We consider the partial regular solution to the coupled spin polarized transport equations in 3D. This coupled system includes the Landau-Lifshitz equation and a quasi-linear parabolic equation describing the diffusion of spin accumulation. Our proof is strictly based on Ginzburg-Landau approximation theory and Moser iterative computation.

Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary

We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.

Mirror Mode Junctions as Sources of Radiation

Mirror modes in collisionless high-temperature plasmas represent macroscopic high-temperature quasi-superconductors with bouncing electrons in discrete-particle resonance with thermal ion-sound noise contributing to the ion-mode growth beyond quasilinear stability. In the semi-classical Ginzburg-Landau approximation the conditions for phase transition are reviewed. The quasi-superconducting state is of second kind causing a magnetically perforated plasma texture. Focussing on the interaction of mirror bubbles we apply semi-classical Josephson conditions and show that a mirror perforated plasma emits weak electromagnetic radiation which in the magnetosheath should be in the sub-millimeter, respectively, infrared range. This effect might be of astrophysical importance.

Concentration-cancellation in the Ericksen\u2013Leslie model

We establish the subconvergence of weak solutions to the Ginzburg–Landau approximation to global-in-time weak solutions of the Ericksen–Leslie model for nematic liquid crystals on the torus {mathbb {T}^2}. The key argument is a variation of concentration-cancellation methods originally introduced by DiPerna and Majda to investigate the weak stability of solutions to the (steady-state) Euler equations.

Convergence of the Ginzburg-Landau Approximation for the Ericksen-Leslie System

We establish the local well-posedness of the general Ericksen-Leslie system in liquid crystals with the initial velocity and director field in $H^1 \times H_b^2$. In particular, we prove that the solutions of the Ginzburg-Landau approximation system converge smoothly to the solution of the Ericksen-Leslie system for any $t \in (0,T^\ast)$ with a maximal existence time $T^\ast$ of the Ericksen- Leslie system.

Superconductivity Localized at an Edge Dislocation

A spatially inhomogeneous superconductor is considered in the Ginzburg—Landau approximation. A spatial inhomogeneity (edge dislocation) is described in terms of the local shift of the superconducting transition temperature in its neighborhood. It is assumed that shift δTc(R) is proportional to the elastic stress produced by the dislocation. It is shown that in this case, localized superconducting states exist at temperatures exceeding Tc of a homogeneous superconductor. The main characteristics of such states are obtained. The same approach is used for describing pinning of Abrikosov vortices at an edge dislocation.

Ginzburg-Landau approximation for self-sustained oscillators weakly coupled on complex directed graphs

A normal form approximation for the evolution of a reaction-diffusion system hosted on a directed graph is derived, in the vicinity of a supercritical Hopf bifurcation. Weak diffusive couplings are assumed to hold between adjacent nodes. Under this working assumption, a Complex Ginzburg–Landau equation (CGLE) is obtained, whose coefficients depend on the parameters of the model and the topological characteristics of the underlying network. The CGLE enables one to probe the stability of the synchronous oscillating solution, as displayed by the reaction-diffusion system above Hopf bifurcation. More specifically, conditions can be worked out for the onset of the symmetry breaking instability that eventually destroys the uniform oscillatory state. Numerical tests performed for the Brusselator model confirm the validity of the proposed theoretical scheme. Patterns recorded for the CGLE resemble closely those recovered upon integration of the original Brussellator dynamics.

Superconducting phase transitions in the milikelvin temperature range in splat-cooled U-Pt alloys**Invited talk at 8th Int. Workshop on Advanced Materials Science and Nanotechnology (Ha Long City, Vietnam, 8–12 November 2016).

We present the temperature and magnetic-field dependence of electrical resistivity (ρ(T, B) dependence) in the milikelvin range for U-Pt alloys with Pt concentration in the range of 0–15 atomic percent (at.%) prepared by splat-cooling technique. The high-temperature cubic γ-U phase can be stabilized down to room temperature in the samples with 15 at.% Pt. All samples exhibit superconducting transitions below 1.1 K revealed by abrupt resistivity drops with transition widths of several tens of milikelvins in zero magnetic field. The H–T diagrams for all investigated alloys are in good agreement with the Ginzburg-Landau approximation. The U-Pt alloys with cubic γ-U phase are very stable in air and hydrogen exposure at atmospheric conditions. They can absorb a large amount of hydrogen upon applying hydrogen pressure >2.5 bars and easily release it by heating in vacuum, showing the possibility of using U-Pt-based hydrides as a medium for hydrogen storage.

From the Q-Tensor Flow for the Liquid Crystal to the Harmonic Map Flow

In this paper, we consider the solutions of the relaxed Q-tensor flow in $${\mathbb{R}^3}$$ with small parameter $${\epsilon}$$ . We show that the limiting map is the so-called harmonic map flow. As a consequence, we present a new proof for the global existence of a weak solution for the harmonic map flow in three dimensions as in [18, 23], where the Ginzburg–Landau approximation approach was used.

Analysis of a variational model for nematic shells

We analyze an elastic surface energy which was recently introduced by G. Napoli and L. Vergori to model thin films of nematic liquid crystals. We show how a novel approach in modeling the surface’s extrinsic geometry leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: (i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; (ii) we prove, by a Ginzburg–Landau approximation, the well-posedness of the gradient flow of the energy; (iii) in the case of a parametrized torus we obtain a stronger characterization of global and local minimizers, which we supplement with numerical experiments.

Global existence of weak solutions to the non-isothermal nematic liquid crystals in 2D

In this article, we prove the global existence of weak solutions to the non-isothermal nematic liquid crystal system on ▪2, on the basis of a new approximate system which is different from the classical Ginzburg-Landau approximation. Local in space energy inequalities are employed to recover the estimates on the second order spatial derivatives of the director fields locally in time, which cannot be derived from the basic energy balance. It is shown that these weak solutions satisfy the temperature equation, and also the total energy equation but away from at most finite many “singular” times, at which the energy concentration occurs and the director field losses its second order derivatives.

Attractivity of the Ginzburg–Landau mode distribution for a pattern forming system with marginally stable long modes

The mathematical theory of the description of pattern forming systems close to the first instability via the Ginzburg–Landau equation is based on approximation and attractivity results. This theory allowed to prove global existence results and upper semicontinuity of attractors for classical hydrodynamical stability problems such as the Couette–Taylor problem. Recently, approximation results for the Ginzburg–Landau approximation for pattern forming systems with marginally stable long modes, such as the Bénard–Marangoni system, have been shown. It is the purpose of this paper to prove the second fundamental property, namely the attractivity of the Ginzburg–Landau mode distribution, for such systems.

Blow-up Criteria of Strong Solutions to the Ericksen-Leslie System in ℝ3

In this paper, we establish the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in ℝ3 for the well-known Oseen-Frank model. The local existence of strong solutions to liquid crystal flows is obtained by using the Ginzburg-Landau approximation approach to guarantee the constraint that the direction vector of the fluid is of length one. We establish four kinds of blow-up criteria, including (i) the Serrin type; (ii) the Beal-Kato-Majda type; (iii) the mixed type, i.e., Serrin type condition for one field and Beal-Kato-Majda type condition on the other one; (iv) a new one, which characterizes the maximal existence time of the strong solutions to the Ericksen-Leslie system in terms of Serrin type norms of the strong solutions to the Ginzburg-Landau approximate system. Furthermore, we also prove that the strong solutions of the Ginzburg-Landau approximate system converge to the strong solution of the Ericksen-Leslie system up to the maximal existence time.

Oscillations of critical superconducting current in thin doubly-connected Sn films in an external perpendicular magnetic field

We report the results of experimental and theoretical studies of critical current oscillations in thin doubly-connected Sn films in an external perpendicular magnetic field. The experiments were performed on samples that consisted of two wide electrodes joined together by two narrow channels. The length of the channels l satisfied the condition l ≫ ξ (ξ is the Ginzburg–Landau coherence length). At temperatures close to the critical temperature Tc, the dependence of the critical current Ic on average external magnetic flux Φ¯e has the form of a piecewise linear function, periodic with respect to the flux quantum Φ0. The amplitude of the Ic oscillation at a given temperature is proportional to the factor ξ/l. Moreover, the dependence Ic=Ic(Φ¯e) is found to be multivalued, hence indicating the presence of metastable states. Based on the Ginzburg–Landau approximation, a theory was constructed that explains the above features of the oscillation phenomenon taking a perfectly symmetric system as an example. Further, the experiments displayed the effects related to the critical currents imbalance between the superconducting channels, i.e., shift of the maxima of the dependence Ic=Ic(Φ¯e) accompanied by an asymmetry with respect to the transport current direction.

Well-Posedness of the Ericksen–Leslie System

We prove the local well-posedness of the Ericksen–Leslie system, and the global well-posedness for small initial data under a physical constraint condition on the Leslie coefficients, which ensures that the energy of the system is dissipated. Instead of the Ginzburg–Landau approximation, we construct an approximate system with the dissipated energy based on a new formulation of the system.

Justification of the Ginzburg–Landau approximation for an instability as it appears for Marangoni convection

The Ginzburg–Landau equation appears as a universal amplitude equation for spatially extended pattern forming systems close to the first instability. It can be derived via multiple scaling analysis for the Marangoni convection problem that is driven by temperature‐dependent surface tension and is the subject of our interest. In this paper, we prove estimates between this formal approximation and true solutions of a scalar pattern forming model problem showing the same spectral picture as the Marangoni convection problem in case of a thin fluid. The new difficulties come from neutral modes touching the imaginary axis for the wave number k = 0 and from identical group velocities at the critical wave number k = kc and the wave number k = 0. The problem is solved by using the reflection symmetry of the system and by using the fact that the modes concentrate at integer multiples of the critical wave number k = kc. The paper presents a method that is applicable whenever this kind of instability occurs. Copyright © 2012 John Wiley & Sons, Ltd.

Long-time behavior of solutions to the compressible liquid crystals

In this paper, we consider the long-time behavior of the liquid crystals equations of compressible flow. We prove that if the static state ds minimizes the Ginzburg-Landau approximation energy, then the finite energy weak solutions (ρ, u, d)(t, x) converges to (ρs, 0, ds)(x) with unique density function ρs, which satisfies the conservation of mass.

Islands of instability for growth of spiral vortices in the Taylor-Couette system with and without axial through flow

We present numerical calculations of the linearized Navier-Stokes equations for axially extended and axially localized spiral structures in the Taylor-Couette system. The eigenvalue surface for spiral vortices with azimuthal wave number M=2 shows significantly more structure than that for vortices with M=0 and 1. Islands are found in parameter space where axially periodic vortex perturbations can grow. Bicriticality of different axial wave numbers is observed. Furthermore, parameter islands of absolute instability are found where wave packets consisting of near-critical extended perturbations can grow and expand via oppositely moving fronts. Some results are compared with those of the Ginzburg-Landau approximation.

The fractional Landau–Lifshitz–Gilbert equation and the heat flow of harmonic maps

In this paper, we prove the existence of global weak solutions to the periodic fractional Landau–Lifshitz–Gilbert equation through the Ginzburg–Landau approximation and the Galerkin approximation. Since the nonlinear term is nonlocal and of full order of the equation, some special structures of the equation, the commutator estimate and some calculus inequalities of fractional order are exploited to get the convergence of the approximating solutions. The equation considered in this paper can also be regarded as a generalization of the heat flow of harmonic maps to the fractional order.

Justification of the Ginzburg–Landau Approximation in Case of Marginally Stable Long Waves

The Ginzburg–Landau equation can be derived via multiple-scaling analysis as a universal amplitude equation for the description of bifurcating solutions in spatially extended pattern-forming systems close to the first instability. Here we are interested in approximation results showing that there are solutions of the pattern-forming system which behave as predicted by the Ginzburg–Landau equation. In the classical case the proof of the approximation result is based on the fact that the quadratic interaction of the critical modes, i.e., of the modes with positive or zero growth rates, gives only non-critical modes, i.e., modes which are damped with some exponential rate. It is the purpose of this paper to develop a method to handle a situation when this condition is violated by an additional curve of stable eigenvalues which possesses a vanishing real part at the Fourier wave number k=0 for all values of the bifurcation parameter. The investigations are motivated by the Benard–Marangoni problem and short-wave instabilities in the flow down an inclined plane.