We study the Landau-Lifshitz system associated with Maxwell equations in a bilayered ferromagnetic body when super-exchange and surface anisotropy interactions are present in the spacer in-between the layers. In the presence of these surface energies, the Neumann boundary condition becomes nonlinear. We prove, in three dimensions, the existence of global weak solutions to the Landau-Lifshitz-Maxwell system with nonlinear Neumann boundary conditions.
We study semilinear Maxwell-Landau-Lifshitz systems in one space dimension. For highly oscillatory and prepared initial data, we construct WKB approximate solutions over long times $O(1/\e)$. The leading terms of the WKB solutions solve cubic Schrödinger equations. We show that the nonlinear normal form method of Joly, Métivier and Rauch applies to this context. This implies that the Schrödinger approximation stays close to the exact solution of Maxwell-Landau-Lifshitz over its existence time. In the context of Maxwell-Landau-Lifshitz, this extends the analysis of Colin and Lannes from times $…
The existence of global weak solutions to a coupled spin drift-diffusion and Maxwell-Landau-Lifshitz system is proved. The equations are considered in a two-dimensional magnetic layer structure and are supplemented with Dirichlet-Neumann boundary conditions. The spin drift-diffusion model for the charge density and spin density vector is the diffusion limit of a spinorial Boltzmann equation for a vanishing spin polarization constant. The Maxwell-Landau-Lifshitz system consists of the time-dependent Maxwell equations for the electric and magnetic fields and of the Landau-Lifshitz-Gilbert equati…
Motivated by recent models of current driven magnetization dynamics, we examine the coupling of the Landau-Lifshitz-Gilbert equation and classical electron transport governed by the Vlasov-Maxwell system. The interaction is based on space-time gyro-coupling in the form of emergent electromagnetic fields of quantized helicity that add up to the conventional Maxwell fields. We construct global weak solutions of the coupled system in the framework of frustrated magnets with competing first and second order gradient interactions known to host topological solitons such as magnetic skyrmions and hop…
In this paper we deal with weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations. First we prove that these solutions satisfy some weak-strong uniqueness property. Then we investigate the validity of energy identities. In particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation. In the case of the Hall-Magneto-Hydrodynamic equations we also give a sufficient condition to guarantee the magneto-helicity identity. Our conditions correspond to the same heuristic scaling as the one introduced by …
We prove the existence and uniqueness of non-trivial stable solutions to Landau-Lifshitz-Maxwell equations with Dirichlet boundary condition for large anisotropies and small domains, where the domains are non-simply connected.
In this paper, we provide a precise description of the compatibility conditions for the initial data so that one can show the existence and uniqueness of regular short-time solution to the Neumann initial-boundary problem of a class of Landau-Lifshitz-Gilbert system with spin-polarized transport, which is a strong nonlinear coupled parabolic system with non-local energy.
Symmetrization of the Maxwell--Neumann--Poincar'e operator, spectral decomposition in $\mathbf{H}(\mathrm{curl},D)$ traces, and boundary localisation of SPRs
Northeast Normal University（东北师范大学）；City University of Hong Kong（香港城市大学）
This paper deals with the Klein-Gordon-Maxwell system in a bounded spatial domain. We study the existence of solutions having a specific form, namely standing waves in equilibrium with a purely electrostatic field. We prescribe Dirichlet boundary conditions on the matter field, and either Dirichlet or Neumann boundary conditions on the electric potential.
We consider the initial boundary value problem of Landau-Lifshitz-Bloch equation on three-dimensional ferromagnetic films, where the effective field contains the stray field controlled by Maxwell equation and the exchange field contains exchange constant. In this paper, we establish the existence of weak solutions of the equation by using the Faedo-Galerkin approximation method. We also derive its two-dimensional limit equation in a mathematically rigorous way when the film thickness tends to zero under appropriate compactness conditions. Moreover, we obtain an equation that can better describ…
Let (M,g) be a smooth compact, n dimensional Riemannian manifold, n=3,4 with smooth n-1 dimensional boundary. We search the positive solutions of the singularly perturbed Klein Gordon Maxwell Proca system with homogeneous Neumann boundary conditions or for the singularly perturbed Klein Gordon Maxwell system with mixed Dirichlet Neumann homogeneous boundary conditions. We prove that stable critical points of the mean curvature of the boundary generates solutions when the perturbation parameter is sufficiently small.
We study a Klein-Gordon-Maxwell system, in a bounded spatial domain, under Neumann boundary conditions on the electric potential. We allow a nonconstant coupling coefficient. For sufficiently small data, we find infinitely many standing waves.
We study a Klein-Gordon-Maxwell system, in a bounded spatial domain, under Neumann boundary conditions on the electric potential. We allow a nonconstant coupling coefficient. For sufficiently small data, we find infinitely many static solutions.
In this chapter of the book entitled, "Extending the Theory of Composites to Other Areas of Science" [edited by Graeme W. Milton, 2016] we derive the analyticity properties of the electromagnetic Dirichlet-to-Neumann map for the time-harmonic Maxwell's equations for passive linear multicomponent media. Moreover, we discuss the connection of this map to Herglotz functions for isotropic and anisotropic multicomponent composites.
The full Landau-Lifshitz-Gilbert equation with periodic material coefficients and natural boundary condition is employed to model the magnetization dynamics in composite ferromagnets. In this work, we establish the convergence between the homogenized solution and the original solution via a Lax equivalence theorem kind of argument. There are a few technical difficulties, including: 1) it is proven the classic choice of corrector to homogenization cannot provide the convergence result in the $H^1$ norm; 2) a boundary layer is induced due to the natural boundary condition; 3) the presence of str…
We prove that the Landau-Lifshitz-Gilbert equation in three space dimensions with homogeneous Neumann boundary conditions admits arbitrarily smooth solutions, given that the initial data is sufficiently close to a constant function.
We consider Calderón's inverse boundary value problems for a class of nonlinear Helmholtz Schrödinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the Dirichlet-to-Neumann map of the corresponding equations. The local uniqueness of the linearized partial data Calderón's inverse problem is obtained following \cite{DKSU}. The Runge approximation properties and unique continuation principle allow us to extend to global situations. Simultaneous recovery of some unknown cavity$/$boundary and coefficients are given as some appl…
We discuss a Steklov-type problem for Maxwell's equations which is related to an interior Calderón operator and an appropriate Dirichlet-to-Neumann type map. The corresponding Neumann-to-Dirichlet map turns out to be compact and this provides a Fourier basis of Steklov eigenfunctions for the associated energy spaces. With an approach similar to that developed by Auchmuty for the Laplace operator, we provide natural spectral representations for the appropriate trace spaces, for the Calderón operator itself and for the solutions of the corresponding boundary value problems subject to electric or…
Maxwell-Stefan systems describing the dynamics of the molar concentrations of a gas mixture with an arbitrary number of components are analyzed in a bounded domain under isobaric, isothermal conditions. The systems consist of mass balance equations and equations for the chemical potentials, depending on the relative velocities, supplemented with initial and homogeneous Neumann boundary conditions. Global-in-time existence of bounded weak solutions to the quasilinear parabolic system and their exponential decay to the homogeneous steady state are proved. The mathematical difficulties are due to…
We introduce an analog of the Dirichlet-to-Neumann map for the Maxwell equation in a bounded domain. We show that it can be approximated by a pseudodifferential operator on the boundary with a matrix-valued symbol and we compute the principal symbol. As a consequence, we obtain a parabolic region free of the transmission eigenvalues associated to the Maxwell equation.
This paper is concerned with the Klein-Gordon-Maxwell system in a bounded spatial domain. We discuss the existence of standing waves $ψ=u(x)e^{-iωt}$ in equilibrium with a purely electrostatic field $\mathbf{E}=-\nablaϕ(x)$. We assume an homogeneous Dirichlet boundary condition on $u$ and an inhomogeneous Neumann boundary condition on $ϕ$. In the "linear" case we characterize the existence of nontrivial solutions for small boundary data. With a suitable nonlinear perturbation in the matter equation, we get the existence of infinitely many solutions.
We prove that the time-harmonic solutions to Maxwell's equations in a 3D exterior domain converge to a certain static solution as the frequency tends to zero. We work in weighted Sobolev spaces and construct new compactly supported replacements for Dirichlet-Neumann fields. Moreover, we even show convergence in operator norm.
For Maxwell operators $(E,H) \to (i ε^{-1} \nabla\times H, -i μ^{-1} \nabla \times E)$ in Lipschitz domains, we describe all m-dissipative boundary conditions and apply this result to generalized impedance and Leontovich boundary conditions including the cases of singular, degenerate, and randomized impedance coefficients. To this end we construct Riesz bases in the trace spaces associated with the curl-operator and introduce a modified version of boundary triple adapted for the specifics of Maxwell equations, namely, to the mixed-order duality of the related trace spaces. This provides a tran…
