Existence of a unique solution and invariant measures for the stochastic Landau–Lifshitz–Bloch equation

https://doi.org/10.1016/j.jde.2020.06.061Get rights and content

Abstract

The Landau–Lifshitz–Bloch equation perturbed by a space-dependent noise was proposed in [10] as a model for evolution of spins in ferromagnetic materials at the full range of temperatures, including the temperatures higher than the Curie temperature. In the case of a ferromagnet filling a bounded domain DRd, d=1,2,3, we show the existence of strong (in the sense of PDEs) martingale solutions. Furthermore, in cases d=1,2 we prove uniqueness of pathwise solutions and the existence of invariant measures.1

Introduction

The aim of this paper is to initiate the analysis of stochastic Landau-Lifshitz-Bloch equation (1.3). For the reader's convenience we recall here some background material introduced in [18].

A well-known model of ferromagnetic material leads to the Landau–Lifshitz–Gilbert equation (LLGE) for the evolution of magnetic moment, which is valid only for temperatures close to the Curie temperature Tc [13], [17]. Several recent technological applications such as heat-assisted magnetic recording [16], thermally assisted magnetic random access memories [22] or spincaloritronics have shown the need to generalise this theory to higher temperatures. For high temperatures, a thermodynamically consistent approach was introduced by Garanin [10], [11] who derived the Landau–Lifshitz–Bloch equation (LLBE) for ferromagnets. The LLBE essentially interpolates between the LLGE at low temperatures and the Ginzburg-Landau theory of phase transitions. It is valid not only below but also above the Curie temperature. Let u(t,x)R3 be the average spin polarisation for t>0 and xDRd, d=1,2,3. The LLBE takes the formut=γu×Heff+L11|u|2(uHeff)uL21|u|2u×(u×Heff), where the effective field Heff is given by (1.2) below. Here, || is the Euclidean norm in R3, γ>0 is the gyromagnetic ratio, and L1 and L2 are the longitudinal and transverse damping parameters, respectively.

Nevertheless, the deterministic LLBE is insufficient to capture the dispersion of individual trajectories at high temperatures. For example, when the magnetization is quenched it should describe the loss of magnetization correlations in different sites of the sample. In the laser-induced dynamics, this is responsible for the slowing down of the magnetization recovery at high laser fluency as the system temperature decreases [9]. Therefore, under these circumstances and according to Brown [2], [3], stochastic forms of the LLBE are discussed in [9], [12] where the LLBE is modified in order to incorporate random fluctuations into the dynamics of the magnetisation and to describe noise-induced transitions between equilibrium states of the ferromagnet.

In this paper, we consider the stochastic LLBE, introduced in [12], perturbing the effective field Heff in (1.1) by a Gaussian noise. Furthermore, we focus on a case in which the temperature T is raised higher than Tc, and as a consequence the longitudinal L1 and transverse L2 damping parameters are equal. The effective field Heff is given byHeff=Δu1χ||(1+35TTTc|u|2)u, where χ|| is the longitudinal susceptibility. Using the vector product identity a×(b×c)=b(ac)c(ab) where (,) is the scalar product in R3, we obtainu×(u×Heff)=(uHeff)u|u|2Heff, and from property L1=L2=:κ1, the stochastic LLBE takes the formdu=(κ1Δu+γu×Δuκ2(1+μ|u|2)u)dt+k=1(γu×hk+κ1hk)dWk(t), with κ2:=κ1χ|| and μ:=3T5(TTc). Here, we assume thatk=1hkW1,(D)2h<, and {Wk:k1} is a family of independent real-valued Wiener processes. Finally, the stochastic LLBE being studied in this paper is equation (1.3) with real positive coefficients κ1,κ2,γ,μ, initial data u(0,x)=u0(x) and subject to homogeneous Neumann boundary conditions.

We emphasise that introducing two kinds of noise, multiplicative and additive, seems necessary to capture important features of the physical system. Namely, it is argued in [9] that only then the model may lead to a Boltzmann distribution valid for the full range of temperatures.

Despite its importance, very little is known about solutions to the deterministic and stochastic LLBE. A pioneering work on the existence of weak solutions to the deterministic LLBE (1.1) in a bounded domain is carried out in [18]. In this paper a Faedo–Galerkin approximation was introduced and the method of compactness was used to prove the existence of a weak solution for the LLBE and its regularity properties. In this work we built on the theory developed in [18] and initiated the theory of stochastic LLBE. While preparing its final version we learnt about the paper [15]. In their work the authors, starting from the formulation in [18], prove the existence of weak (in PDE sense) martingale solutions to equation (1.3). In our work we show that martingale solutions are strong in PDE sense for d=1,2,3 and prove pathwise uniqueness in dimensions d=1,2 and this fact by the Yamada-Watanabe theorem implies uniqueness of martingale solutions. Finally, we prove the existence of an invariant measure which is an important step towards thermodynamic justification of the stochastic LLBE. The results of this paper have been presented at a number of international meetings (see footnote on p. 9471).

The paper is organized as follows. Section 2 contains Theorem 2.2 and Theorem 2.3 on the existence and uniqueness strong solution of (1.3) as well as its regularity properties. In Section 3 we introduce the Faedo–Galerkin approximations and prove for their solutions some uniform bounds in various norms. Sections 4 and 5 are devoted to the proof of Theorem 2.2. The existence of an invariant measure stated in Theorem 6.4 is proved in Section 6. Finally, in the Appendix we collect, for the reader's convenience, some facts scattered in the literature that are used in the course of the proof.

Section snippets

Notation and the formulation of the main results

Let DRd, d=1,2,3, be an open bounded domain with uniformly C2 boundary. The function space H1:=H1(D,R3) is defined as follows:H1(D,R3)={uL2(D,R3):uxiL2(D,R3)for i=1,2,3}. Here, Lp:=Lp(D,R3) with p1 is the usual space of pth-power Lebesgue integrable functions defined on D and taking values in R3. Throughout this paper, we denote a scalar product in a Hilbert space H by ,H and its associated norm by H. The duality between a space X and its dual X will be denoted by ,XX.

Let Xw

Faedo-Galerkin approximation

Let A=Δ be the negative Neumann Laplacian in D. Then [8, Theorem 1, p. 335], there exists an orthonormal basis {ei}i=1 of L2, consisting of eigenvectors of A, such that for all i=1,2,Δei=λiei,ein=0 on D,eiL, where n is the outward normal on the boundary ∂D; and λi>0 for i=1,2, are eigenvalues of A. For β>0 we define the Hilbert space Xβ=dom(Aβ) endowed with the normvXβ=(I+A)βvL2. The dual space will be denoted by Xβ.

Let Sn:=span{e1,,en} and Πn be the orthogonal projection from L2

Tightness and construction of new probability space and processes

Equation (3.6) can be written in the following way as an approximation of equation (2.1)un(t)=un(0)+0tFn1(un(s))Fn3(un(s))ds+0tFn2(un(s))ds+12k=1n0tΠn(Gnk(un(s))×hk)ds+k=1n0tGnk(un(s))dWk(s). We will write shortlyun(t)=un(0)+i=13Bn,i(un)(t)+Bn,4(un,W)(t),t[0,T]. We now prove a uniform bound for un.

Lemma 4.1

Let DRd be an open bounded domain. Let r[1,43), q[1,), p>2 and α(0,12) with pα>1. Then there exists a constant c depending on p, C1 and h, such that for all n1EBn,1(un)W1,2(0,T;L2)qc

Existence of a weak solution

Our aim is to prove that u from Proposition 4.3 is a weak solution of the stochastic LLBEs according to the Definition 2.1. We first find an equation satisfied by the new process (un(t),Wn(t))t[0,T] in Subsection 5.1. Then in Subsection 5.2 we prove the convergence of that equation.

Existence of an invariant measure for the stochastic LLBE on 1 or 2-dimensional domains

In this section we will show the existence of invariant measure for equation (2.4). In our proof we modify the ideas from [7] and [4], where different type of difficulties had to be dealt with.

We start with the following result.

Lemma 6.1

Let u be a weak solution to equation (2.4) with properties listed in Theorem 2.2. Then there exists a positive constant c depending on C1 and h such that for all t0 we have0tEu(s)H22dsc(1+t).

Proof

We will use a version of the Itô Lemma proved in [21]. By Theorem 2.2 and

Appendix

Lemma 7.1

Assume that E is a separable Hilbert space, p[2,) and α(0,12). Then there exists a constant c depending on T and α such that for any progressively measurable process ξ=(ξj)j=1 there holdsEj=1I(ξj)Wα,p(0,T;E)pcE0T(j=1|ξj(t)|E2)p2dt, where I(ξj) is defined byI(ξj):=0tξj(s)dWj(s),t0. In particular, P–a.s. the trajectories of the process I(ξj) belong to Wα,2(0,T;E).

Lemma 7.2

[23, Corollary 19] Suppose sr, pq and s1/pr1/q (0<rs<1, 1pq). Let E be a Banach space and I be an interval of R

References (24)

  • D. Garanin

    Generalized equation of motion for a ferromagnet

    Phys. A, Stat. Mech. Appl.

    (1991)
  • S. Jiang et al.

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

    J. Differ. Equ.

    (2019)
  • K.N. Le

    Weak solutions of the Landau–Lifshitz–Bloch equation

    J. Differ. Equ.

    (2016)
  • H. Brezis

    Analyse Fonctionnelle

    (1983)
  • W. Brown

    Thermal fluctuation of fine ferromagnetic particles

    IEEE Trans. Magn.

    (Sep 1979)
  • W.F. Brown

    Thermal fluctuations of a single-domain particle

    Phys. Rev.

    (Jun 1963)
  • Z. Brzeźniak et al.

    Stationary solutions for stochastic damped Navier-Stokes equations in Rd

    Indiana Univ. Math. J.

    (2019)
  • Z. Brzeźniak et al.

    Weak solutions of a stochastic Landau–Lifshitz–Gilbert equation

    Appl. Math. Res. Express

    (2013)
  • Z. Brzeźniak et al.

    Weak solutions of the stochastic Landau–Lifshitz–Gilbert equation with non–zero anisotrophy energy

    Appl. Math. Res. Express

    (2016)
  • Z. Brzeźniak et al.

    Invariant measure for the stochastic Navier–Stokes equations in unbounded 2d domains

    Ann. Probab.

    (2017)
  • L.C. Evans

    Partial Differential Equations

    (1998)
  • R.F.L. Evans et al.

    Stochastic form of the Landau-Lifshitz-Bloch equation

    Phys. Rev. B

    (Jan 2012)
  • Cited by (0)

    The authors acknowledge financial support through the ARC Discovery projects DP160101755 and DP180100506.

    View full text