The improved decay rate for the heat semigroup with local magnetic field in the plane

We consider the heat equation in the presence of compactly supported magnetic field in the plane. We show that the magnetic field leads to an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta. The proof employs Hardy-type inequalities due to Laptev and Weidl for the two-dimensional magnetic Schrödinger operator and the method of self-similar variables and weighted Sobolev spaces for the heat equation. A careful analysis of the asymptotic behaviour of the heat equation in the similarity variables shows that the magnetic field asymptotically degenerates to an Aharonov–Bohm magnetic field with the same total magnetic flux, which leads asymptotically to the gain on the polynomial decay rate in the original physical variables. Since no assumptions about the symmetry of the magnetic field are made in the present work, it gives a normwise variant of the recent pointwise results of Kova?ík (Calc Var doi:10.1007/s00526-011-0437-4) about large-time asymptotics of the heat kernel of magnetic Schrödinger operators with radially symmetric field in a more general setting.     

Catastrophe of coronal magnetic flux ropes in fully open magnetic field

The catastrophe of coronal magnetic flux ropes is closely related to solar explosive phenomena, such as prominence eruptions, coronal mass ejections, and two-ribbon solar flares. Using a 2-dimensional, 3-component ideal MHD model in Cartesian coordinates, numerical simulations are carried out to investigate the equilibrium property of a coronal magnetic flux rope which is embedded in a fully open background magnetic field. The flux rope emerges from the photosphere and enters the corona with its axial and annular magnetic fluxes controlled by a single “emergence parameter”. For a flux rope that has entered the corona, we may change its axial and annular fluxes artificially and let the whole system reach a new equilibrium through numerical simulations. The results obtained show that when the emergence parameter, the axial flux, or the annular flux is smaller than a certain critical value, the flux rope is in equilibrium and adheres to the photosphere. On the other hand, if the critical value is exceeded, the flux rope loses equilibrium and erupts freely upward, namely, a catastrophe takes place. In contrast with the partly-opened background field, the catastrophic amplitude is infinite for the case of fully-opened background field     

Catastrophe of coronal magnetic flux ropes in fully open magnetic field

The catastrophe of coronal magnetic flux ropes is closely related to solar explosive phenomena, such as prominence eruptions, coronal mass ejections, and two-ribbon solar flares. Using a 2-dimensional, 3-component ideal MHD model in Cartesian coordinates, numerical simulations are carried out to investigate the equilibrium property of a coronal magnetic flux rope which is embedded in a fully open background magnetic field. The flux rope emerges from the photosphere and enters the corona with its axial and annular magnetic fluxes controlled by a single "emergence parameter". For a flux rope that has entered the corona, we may change its axial and annular fluxes artificially and let the whole system reach a new equilibrium through numerical simulations. The results obtained show that when the emergence parameter, the axial flux, or the annular flux is smaller than a certain critical value, the flux rope is in equilibrium and adheres to the photosphere. On the other hand, if the critical value is exceeded, the flux rope loses equilibrium and erupts freely upward, namely, a catastrophe takes place. In contrast with the partly-opened background field, the catastrophic amplitude is infinite for the case of fully-opened background field.     

Stabilized nodal‐based finite element methods for the magnetic induction problem

We consider the time‐dependent magnetic induction model as a step towards the resistive magnetohydrodynamics model in incompressible media. Conforming nodal‐based finite element approximations of the induction model with inf‐sup stable finite elements for the magnetic field and the magnetic pseudo‐pressure are investigated. Based on a residual‐based stabilization technique proposed by Badia and Codina, SIAM J. Numer. Anal. 50 (2012), pp. 398–417, we consider a stabilized nodal‐based finite element method for the numerical solution. Error estimates are given for the semi‐discrete model in space. Finally, we present some examples, for example, for the magnetic flux expulsion problem, Shercliff's test case and singular solutions of the Maxwell problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Hamiltonian reduced fluid model for plasmas with temperature and heat flux anisotropies

For an arbitrary number of species, we derive a Hamiltonian fluid model for strongly magnetized plasmas describing the evolution of the density, velocity, and electromagnetic fluctuations and also of the temperature and heat flux fluctuations associated with motions parallel and perpendicular to the direction of a background magnetic field. We derive the model as a reduction of the infinite hierarchy of equations obtained by taking moments of a Hamiltonian drift-kinetic system with respect to Hermite–Laguerre polynomials in velocity–magnetic-moment coordinates. We show that a closure relation directly coupling the heat flux fluctuations in the directions parallel and perpendicular to the background magnetic field provides a fluid reduction that preserves the Hamiltonian character of the parent drift-kinetic model. We find an alternative set of dynamical variables in terms of which the Poisson bracket of the fluid model takes a structure of a simple direct sum and permits an easy identification of the Casimir invariants. Such invariants in the limit of translational symmetry with respect to the direction of the background magnetic field turn out to be associated with Lagrangian invariants of the fluid model. We show that the coupling between the parallel and perpendicular heat flux evolutions introduced by the closure is necessary for ensuring the existence of a Hamiltonian structure with a Poisson bracket obtained as an extension of a Lie–Poisson bracket.     

Numerical simulation of particles movement in cellular flows under the influence of magnetic forces

A numerical model of particle motion in fluid flow under the influence of hydrodynamic and magnetic forces is presented. As computational tool, a flow solver based on the Boundary Element Method is used. The Euler-Lagrange formulation of multiphase flow is considered. In the case of a particle with a magnetic moment in a nonuniform external magnetic field, the Kelvin body force acts on a single particle. The derived Lagrangian particle tracking algorithm is used for simulation of dilute suspensions of particles in viscous flows taking into account gravity, buoyancy, drag, pressure gradient, added mass and magnetophoretic force. As a benchmark test case the magnetite particle motion in cellular flow field of water is computed with and without the action of the magnetic force. The effect of the Kelvin force on particle motion and separation from the main flow is studied for a predefined magnetic field and different values of magnetic flux density. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Superconductivity and the Aharonov–Bohm effect

We consider the influence of the Aharonov–Bohm magnetic potential on the onset of superconductivity within the Ginzburg–Landau model. As the flux of the magnetic potential varies, we obtain a relation with the Little–Parks effect.     

Integrable Discrete Model for One‐Dimensional Soil Water Infiltration

We propose an integrable discrete model of one‐dimensional soil water infiltration. This model is based on the continuum model by Broadbridge and White, which takes the form of nonlinear convection–diffusion equation with a nonlinear flux boundary condition at the surface. It is transformed to the Burgers equation with a time‐dependent flux term by the hodograph transformation. We construct a discrete model preserving the underlying integrability, which is formulated as the self‐adaptive moving mesh scheme. The discretization is based on linearizability of the Burgers equation to the linear diffusion equation, but the naïve discretization based on the Euler scheme which is often used in the theory of discrete integrable systems does not necessarily give a good numerical scheme. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model that produces solutions with similar accuracy to direct computation on the original nonlinear equation, but with clear benefits regarding computational cost.     

Numerical investigation of a three-dimensional four field model for collisionless magnetic reconnection

In this paper we present the numerical investigation of a three-dimensional four field model for magnetic reconnection in collisionless regimes. The model describes the evolution of the magnetic flux and vorticity together with the perturbations of the parallel magnetic and velocity fields. We explored the different behavior of vorticity and current density structures in low and high β regimes, β being the ratio between the plasma and magnetic pressure. A detailed analysis of the velocity field advecting the relevant physical quantities is presented. We show that, as the reconnection process evolves, velocity layers develop and become more and more localized. The shear of these layers increases with time ending up with the occurrence of secondary instabilities of the Kelvin-Helmholtz type. We also show how the β parameter influences the different evolution of the current density structures, that preserve for longer time a laminar behavior at smaller β values. A qualitative explanation of the structures formation on the different z-sections is also presented.     

Gauged harmonic maps,Born‐Infeld electromagnetism,and magnetic vortices

We study maps from a 2‐surface into the standard 2‐sphere coupled with Born‐Infeld geometric electromagnetism through an Abelian gauge field. Such a formalism extends the classical harmonic map model, known as the σ‐model, governing the spin vector orientation in a ferromagnet allows us to obtain the coexistence of vortices and antivortices characterized by opposite, self‐excited, magnetic flux lines. We show that the Born‐Infeld free parameter may be used to achieve arbitrarily high local concentration of magnetic flux lines that the total minimum energy is an additive function of these quantized flux lines realized as the numbers of vortices antivortices. In the case where the underlying surface, or the domain, is compact, we obtain a necessary sufficient condition for the existence of a unique solution representing a prescribed distribution of vortices antivortices. In the case where the domain is the full plane, we prove the existence of a unique solution representing an arbitrary distribution of vortices and antivortices. Furthermore, we also consider the Einstein gravitation induced by these vortices, known as cosmic strings, establish the existence of a solution representing a prescribed distribution of cosmic strings cosmic antistrings under a necessary sufficient condition that makes the underlying surface a complete surface with respect to the induced gravitational metric. © 2003 Wiley Periodicals, Inc.     

Zero level of a purely magnetic two-dimensional nonrelativistic Pauli operator for SPIN-1/2 particles

We study the manifold of complex Bloch-Floquet eigenfunctions for the zero level of a two-dimensional nonrelativistic Pauli operator describing the propagation of a charged particle in a periodic magnetic field with zero flux through the elementary cell and a zero electric field. We study this manifold in full detail for a wide class of algebraic-geometric operators. In the nonzero flux case, the Pauli operator ground state was found by Aharonov and Casher for fields rapidly decreasing at infinity and by Dubrovin and Novikov for periodic fields. Algebraic-geometric operators were not previously known for fields with nonzero flux because the complex continuation of “magnetic” Bloch-Floquet eigenfunctions behaves wildly at infinity. We construct several nonsingular algebraic-geometric periodic fields (with zero flux through the elementary cell) corresponding to complex Riemann surfaces of genus zero. For higher genera, we construct periodic operators with interesting magnetic fields and with the Aharonov-Bohm phenomenon. Algebraic-geometric solutions of genus zero also generate soliton-like nonsingular magnetic fields whose flux through a disc of radius R is proportional to R (and diverges slowly as R → ∞). In this case, we find the most interesting ground states in the Hilbert space L ₂ (ℝ ² ).     

The Hardy inequality and the heat equation with magnetic field in any dimension

In the Euclidean space of any dimension d, we consider the heat semigroup generated by the magnetic Schrödinger operator from which an inverse-square potential is subtracted to make the operator critical in the magnetic-free case. Assuming that the magnetic field is compactly supported, we show that the polynomial large-time behavior of the heat semigroup is determined by the eigenvalue problem for a magnetic Schrödinger operator on the (d ? 1)-dimensional sphere whose vector potential reflects the behavior of the magnetic field at the space infinity. From the spectral problem on the sphere, we deduce that in d = 2 there is an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta, while there is no extra polynomial decay rate in higher dimensions. To prove the results, we establish new magnetic Hardy-type inequalities for the Schrödinger operator and develop the method of self-similar variables and weighted Sobolev spaces for the associated heat equation.     

Basic solutions of a 3D rectangular limited-permeable crack or two 3D rectangular limited-permeable cracks in the piezoelectric/piezomagnetic composite materials

The solutions of a three-dimensional rectangular limited-permeable crack or two three-dimensional rectangular limited-permeable cracks in the piezoelectric/piezomagnetic composite materials were investigated by using the generalized Almansi’s theorem and the Schmidt method. Finally, the relations among the electric field, the magnetic flux field and the stress field near the crack tips were obtained and the effects of the electric permittivity, the magnetic permeability of the air inside the crack, the shape of the rectangular crack on the stress, the electric displacement and magnetic flux intensity factors in the piezoelectric/piezomagnetic composite materials were analyzed.     

Mathematical model of the Galathea-belt toroidal magnetic trap

We consider a mathematical model of equilibrium configurations of plasma, magnetic field, and electric field in a toroidal trap with two ring conductors with current loaded into plasma. We present the mathematical apparatus of the model based on the numerical solution of boundary value problems for the Grad–Shafranov equation (a differential equation of elliptic type for the magnetic flux function), solution methods for these problems, and numerically obtained properties of equilibrium configurations. We indicate the differences in configurations in the toroidal trap and in its analog straightened into a cylinder.     

Conservation theorems for magnetic field and vorticity in two-dimensional conducting fluid flows

The classical conservation theorems for magnetic force lines, magnetic flux through a fluid surface, and intensity of magnetic vector tubes are generalized to plane flows of a finitely conducting fluid in an orthogonal magnetic field. The Helmholtz and Kelvin vorticity conservation theorems are generalized for plane motion of a viscous conducting fluid in an orthogonal magnetic field and the Bernoulli integral is derived. The Bernoulli integral is also generalized for plane motion of viscous ideally conducting fluid in a longitudinal magnetic field. Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 2, pp. 46–49, 1994.     

Spectrum of three-dimensional landau operator perturbed by a periodic point potential

A study is made of a three-dimensional Schrödinger operator with magnetic field and perturbed by a periodic sum of zero-range potentials. In the case of a rational flux, the explicit form of the decomposition of the resolvent of this operator with respect to the spectrum of irreducible representations of the group of magnetic translations is found. In the case of integer flux, the explicit form of the dispersion laws is found, the spectrum is described, and a qualitative investigation of it is made (in particular, it is established that not more than one gap exists).Mordovian State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 2, pp. 283–294, May, 1995.     

Quantum flux-based resonances in solitonic vortices

Working with the Nielsen–Olesen Lagrangian in static cylindric coordinates, we derive the system of coupled field equations and perform a first-order perturbative approach, pointing out an interesting contribution connected to the London–Heitler current. For an r,θ-depending scalar boson, evolving in a constant or zero magnetic field we get, besides the flux quantization and the Landau energy levels, a less expected structure of the scalar modes whose radial and azimuthal parts are decoupled by the presence of the quantized magnetic flux.     

Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects

We consider the problem of influencing the motion of an electrically conducting fluid with an applied steady magnetic field. Since the flow is originating from buoyancy, heat transfer has to be included in the model. The stationary system of magnetohydrodynamics is considered, and an approximation of Boussinesq type is used to describe the buoyancy. The heat sources given by the dissipation of current and the viscous friction are not neglected in the fluid. The vessel containing the fluid is embedded in a larger domain, relevant for the global temperature- and magnetic field- distributions. Material inhomogeneities in this larger region lead to transmission relations for the electromagnetic fields and the heat flux on inner boundaries. In the presence of transparent materials, the radiative heat transfer is important and leads to a nonlocal and nonlinear jump relation for the heat flux. We prove the existence of weak solutions, under the assumption that the imposed velocity at the boundary of the fluid remains sufficiently small.     

Asymmetric fermi surfaces for magnetic schrödinger operators

We consider Schrödinger operators with periodic magnetic field having zero flux through a fundamental cell of the period lattice. We show that, for a generic small magnetic field and a generic small Fermi energy, the corresponding Fermi surface is convex and not invariant under inversion in any point.     

非线性Schrdinger方程新混合元方法的高精度分析

基于双线性元及其梯度所属空间,建立了非线性Schrdinger方程的自由度少且易满足B-B条件的新混合元格式.首先,利用双线性元的高精度分析和导数转移技巧,在半离散格式下,导出了原始变量在H~1模及流量在L~2模意义下的超逼近性质,进而,借助于插值后处理算子,得到了整体超收敛结果.最后,对向后:Euler和Crank-Nicolson-Galerkin全离散格式分别给出了原始变量的H~1模及L~2模和流量的L~2模误差分析,并通过数值算例,表明逼近格式是高效的.