Existence of Partially Regular Solutions for Landau–Lifshitz Equations in ℝ3

Abstract

We establish the existence of partially regular weak solutions for the Landau–Lifshitz equation in three space dimensions for smooth initial data of finite Dirichlet energy. The construction is based on Ginzburg–Landau approximation. The new key ingredient is a nonlocal representation formula for the penalty term that permits us to take advantage of the special trilinear structure of the limiting nonlinearity.     

Weak martingale solutions to the stochastic Landau–Lifshitz–Gilbert equation with multi-dimensional noise via a convergent finite-element scheme

We propose an unconditionally convergent linear finite element scheme for the stochastic Landau–Lifshitz–Gilbert (LLG) equation with multi-dimensional noise. By using the Doss–Sussmann technique, we first transform the stochastic LLG equation into a partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent $\theta$-linear scheme for the numerical solution of the reformulated equation. As a consequence, we are able to show the existence of weak martingale solutions to the stochastic LLG equation.     

Well-posedness for magnetoviscoelastic fluids in 3D

We show that the system of equations describing a magnetoviscoelastic fluid in three dimensions can be cast as a quasilinear parabolic system. Using the theory of maximal $L_p$-regularity, we establish existence and uniqueness of local strong solutions and we show that each solution is smooth (in fact analytic) in space and time. Moreover, we give a complete characterization of the set of equilibria and show that solutions that start out close to a constant equilibrium exist globally and converge to a (possibly different) constant equilibrium. Finally, we show that every solution that is eventually bounded in the topology of the state space exists globally and converges to the set of equilibria.     

A stable numerical method for space fractional Landau–Lifshitz equations

In this paper, we proposed a simple and unconditional stable time-split Gauss–Seidel projection (GSP) method for the space fractional Landau–Lifshitz (FLL) equations. Numerical results are presented to demonstrate the effectiveness and stability of this method.     

A Note on the Solution of a Stochastic Partial Differential Equation

Abstract

A physical model is described which justifies the appearance of a stochastic term in the two-dimensional Navier–Stokes equations. In this model, a linear oppositional control term accrues as well. The resulting stochastic partial differential equation is shown to have a unique stationary solution.     

Martingale weak solutions of the stochastic Landau–Lifshitz–Bloch equation

The present paper studies the stochastic Landau–Lifshitz–Bloch equation which is recommended as the only valid model at temperature around the Curie temperature and is especially important for the simulation of heat-assisted magnetic recording. We study the stochastic Landau–Lifshitz–Bloch equation in the case that the temperature is raised higher than the Curie temperature. The global existence of martingale weak solutions is proved by using a new argument and regularity properties of the weak solutions are discussed.     

A blowup criterion for nonhomogeneous incompressible Navier–Stokes–Landau–Lifshitz system in 2-D

In this paper, we investigate nonhomogeneous incompressible Navier–Stokes–Landau–Lifshitz system in two-dimensional (2-D). This system consists of Navier–Stokes equations coupled with Landau–Lifshitz–Gilbert equation, an evolutionary equation for the magnetization vector. We establish a blowup criterion for the 2-D incompressible Navier–Stokes–Landau–Lifshitz system with finite positive initial density.     

The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy

It is well-known that the dynamics of biaxial ferromagnets with a strong easy-plane anisotropy is essentially governed by the Sine-Gordon equation. In this paper, we provide a rigorous justification to this observation. More precisely, we show the convergence of the solutions to the Landau–Lifshitz equation for biaxial ferromagnets towards the solutions to the Sine-Gordon equation in the regime of a strong easy-plane anisotropy. Moreover, we establish the sharpness of our convergence result.This result holds for solutions to the Landau–Lifshitz equation in high order Sobolev spaces. We first provide an alternative proof for local well-posedness in this setting by introducing high order energy quantities with better symmetrization properties. We then derive the convergence from the consistency of the Landau–Lifshitz equation with the Sine-Gordon equation by using well-tailored energy estimates. As a by-product, we also obtain a further derivation of the free wave regime of the Landau–Lifshitz equation.     

Analysis of an eddy current and micromagnetic model

We consider a coupled eddy current and micromagnetic model describing the behaviour of dynamic electromagnetic phenomena in applications such as disk write heads. We first prove the existence of a weak solution to this nonlinear problem. Then we outline a numerical time-stepping scheme. Since the numerical method requires a nonstandard mixed boundary value eddy current problem to be solved at each time step, we show the existence and uniqueness of a solution for the corresponding eddy current problem. This is accomplished using an image principle and the verification of a suitable Babu?ka–Brezzi condition.     

Weak–strong uniqueness for the Landau–Lifshitz–Gilbert equation in micromagnetics

We consider the time-dependent Landau–Lifshitz–Gilbert equation. We prove that each weak solution coincides with the (unique) strong solution, as long as the latter exists in time. Unlike available results in the literature, our analysis also includes the physically relevant lower-order terms like Zeeman contribution, anisotropy, stray field, and the Dzyaloshinskii–Moriya interaction (which accounts for the emergence of magnetic Skyrmions). Moreover, our proof gives a template on how to approach weak–strong uniqueness for even more complicated problems, where LLG is (nonlinearly) coupled to other (nonlinear) PDE systems.     

The density evolution of the killed McKean–Vlasov process

ABSTRACT

The density evolution of McKean–Vlasov stochastic differential equations in the presence of an absorbing boundary is analysed where the solution to such equations corresponds to the dynamics of partially killed large populations. By using a fixed point theorem, we show that the density evolution is characterized as the solution of an integro-differential Fokker–Planck equation with Cauchy–Dirichlet data. This problem arises naturally within mean field game theory.     

Dirichlet problem for complex model partial differential equations

ABSTRACT

Complex model partial differential equations of arbitrary order are considered. The uniqueness of the Dirichlet problem is studied. It is proved that the Dirichlet problem for higher order complex partial differential equations with one complex variable has infinitely many solutions.     

Existence of standing waves for the complex Ginzburg–Landau equation

We study the existence of standing wave solutions of the complex Ginzburg–Landau equation

equation(GL)

_φt−e^iθ(ρI−Δ)φ−e^iγ|φ|^αφ=0${\phi }_t-e^{i\theta }(\rho I-\mathrm{\Delta })\phi -e^{i\gamma }{|\phi |}^{\alpha }\phi =0$

in R^N${\mathbb{R}}^N$, where α>0$\alpha >0$, (N−2)α<4$(N-2)\alpha <4$, ρ>0$\rho >0$ and θ,γ∈R$\theta ,\gamma \in \mathbb{R}$. We show that for any θ∈(−π/2,π/2)$\theta \in (-\pi /2,\pi /2)$ there exists ε>0$\epsilon >0$ such that (GL) has a non-trivial standing wave solution if |γ−θ|<ε$|\gamma -\theta |<\epsilon$. Analogous result is obtained in a ball Ω∈R^N$\Omega \in {\mathbb{R}}^N$ for ρ>−_λ1$\rho >-{\lambda }_1$, where _λ1${\lambda }_1$ is the first eigenvalue of the Laplace operator with Dirichlet boundary conditions.     

Justification of the Zakharov Model from Klein–Gordon-Wave Systems

Abstract

We study semilinear and quasilinear systems of the type Klein–Gordon-waves in the high-frequency limit. These systems are derived from the Euler–Maxwell system describing laser-plasma interactions. We prove the existence and the stability of high-amplitude WKB solutions for these systems. The leading terms of the solutions satisfy Zakharov-type equations. The key is the existence of transparency equalities for the Klein–Gordon-waves systems. These equalities are comparable to the transparency equalities exhibited by J.-L. Joly, G. Métivier and J. Rauch for Maxwell–Bloch systems.     

Ginzburg–Landau equations and their generalizations

The Ginzburg–Landau equations were proposed in the superconductivity theory to describe mathematically the intermediate state of superconductors in which the normal conductivity is mixed with the superconductivity. It turned out that these equations have interesting and non-trivial generalizations. First of all, they can be extended to arbitrary compact Riemann surfaces. Next, they can be generalized to dimension 3 as dynamical (or hyperbolic) Ginzburg–Landau equations. They also have a 4-dimensional extension provided by Seiberg–Witten equations. In this review we describe all these interesting topics together with some unsolved problems.     

SMOOTHING EFFECTS OF DISPERSIVE PSEUDODIFFERENTIAL EQUATIONS

ABSTRACT

We discuss smoothing effects of dispersive-type pseudodifferential equations whose principal part is not necessarily elliptic. For equations with constant coefficients, a restriction theorem and a smoothing estimate of the resolvent of the principal part obtain smoothing estimates of solutions in weighted Lebesgue spaces. Moreover, we discuss well-posedness of the initial value problem and an alternative approach to the smoothing effects of general dispersive equations with variable coefficients via pseudodifferential calculus. Our results are the natural generalization of smoothing effects of Schrödinger-type equations.     

Critical spaces for quasilinear parabolic evolution equations and applications

We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal $L_p$-regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier–Stokes problem, convection–diffusion equations, the Nernst–Planck–Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and some other well-known parabolic equations are given.     

Travelling Waves and Numerical Approximations in a Reaction Advection Diffusion Equation with Nonlocal Delayed Effects

Summary. In this paper, we consider the growth dynamics of a single-species population with two age classes and a fixed maturation period living in a spatial transport field. A Reaction Advection Diffusion Equation (RADE) model with time delay and nonlocal effect is derived if the mature death and diffusion rates are age independent. We discuss the existence of travelling waves for the delay model with three birth functions which appeared in the well-known Nicholson's blowflies equation, and we consider and analyze numerical solutions of the travelling wavefronts from the wave equations for the problems with nonlocal temporally delayed effects. In particular, we report our numerical observations about the change of the monotonicity and the possible occurrence of multihump waves. The stability of the travelling wavefront is numerically considered by computing the full time-dependent partial differential equations with nonlocal delay.