Existence of a unique solution and invariant measures for the stochastic Landau–Lifshitz–Bloch equation☆
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 [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 be the average spin polarisation for and , . The LLBE takes the form where the effective field is given by (1.2) below. Here, is the Euclidean norm in , is the gyromagnetic ratio, and and 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 in (1.1) by a Gaussian noise. Furthermore, we focus on a case in which the temperature T is raised higher than , and as a consequence the longitudinal and transverse damping parameters are equal. The effective field is given by where is the longitudinal susceptibility. Using the vector product identity where is the scalar product in , we obtain and from property , the stochastic LLBE takes the form with and . Here, we assume that and 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 , initial data 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 and prove pathwise uniqueness in dimensions 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 , , be an open bounded domain with uniformly boundary. The function space is defined as follows: Here, with is the usual space of pth-power Lebesgue integrable functions defined on D and taking values in . Throughout this paper, we denote a scalar product in a Hilbert space H by and its associated norm by . The duality between a space X and its dual will be denoted by .
Let
Faedo-Galerkin approximation
Let be the negative Neumann Laplacian in D. Then [8, Theorem 1, p. 335], there exists an orthonormal basis of , consisting of eigenvectors of A, such that for all where n is the outward normal on the boundary ∂D; and for are eigenvalues of A. For we define the Hilbert space endowed with the norm The dual space will be denoted by .
Let and be the orthogonal projection from
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) We will write shortly We now prove a uniform bound for . Lemma 4.1 Let be an open bounded domain. Let , , and with . Then there exists a constant c depending on p, and h, such that for all
Existence of a weak solution
Our aim is to prove that 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 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 and h such that for all we have 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, and . Then there exists a constant c depending on T and α such that for any progressively measurable process there holds where is defined by In particular, –a.s. the trajectories of the process belong to . Lemma 7.2 [23, Corollary 19] Suppose , and (, ). Let E be a Banach space and I be an interval of
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The authors acknowledge financial support through the ARC Discovery projects DP160101755 and DP180100506.