Abstract
We adapt Lyon’s rough path theory to study Landau–Lifshitz–Gilbert equations (LLGEs) driven by geometric rough paths in one dimension, with non-zero exchange energy only. We convert the LLGEs to a fully nonlinear time-dependent partial differential equation without rough paths term by a suitable transformation. Our point of interest is the regular approximation of the geometric rough path. We investigate the limit equation, the form of the correction term, and its convergence rate in controlled rough path spaces. The key ingredients for constructing the solution and its corresponding convergence results are the Doss–Sussmann transformation, maximal regularity property, and the geometric rough path theory.
Similar content being viewed by others
1 Introduction
The stochastic Landau–Lifshitz–Gilbert equations (SLLGEs) describe the behaviour of the magnetisation under the influence of the randomly fluctuating effective field. In this work, we consider the SLLGEs with solutions taking values in the two-dimensional sphere \(\mathbb S^2\) in \(\mathbb R^3\). Let D be an open bounded interval of the real line. For simplicity, let D be the interval (0, 1). Let \({\mathcal {H}}_{\mathrm{{eff}}}\) denote the \(L^2\)-gradient of the energy functional \({\mathcal {E}}\) and is called the effective field, i.e. \({\mathcal {H}}_{\mathrm{{eff}}}(u)=-\nabla _u{\mathcal {E}}(u).\) Let \({\mathbf {M}}:[0, T]\times D\rightarrow \mathbb R^3\) denote the magnetisation of a ferromagnetic material occupying the domain D. The deterministic Landau–Lifshitz–Gilbert equations take the form
The parameters \(\lambda _1 \ne 0,\lambda _2>0\) are real constants. We assume that the material is saturated at the initial time, i.e.
If both anisotropy and the exchange energies are present (see Visintin [30] and Cimrák [11]), the total magnetic energy \({\mathcal {E}}\) of the LLGEs is given by
where \({\mathcal {E}}_{an}({\mathbf {M}}) :=\int _D \psi ({\mathbf {M}}(x)) dx\) is the anisotropy energy and \({\mathcal {E}}_{ex}({\mathbf {M}}) := \int _D |\nabla {\mathbf {M}}(x)|^2 dx\) is the exchange energy. The effective field \({\mathcal {H}}_{\mathrm{{eff}}}({\mathbf {M}})\) takes the form \(\varDelta {\mathbf {M}}- \nabla \psi ({\mathbf {M}}).\) In such case, the total magnetic energy is given by
and we obtain \({\mathcal {H}}_{\mathrm{{eff}}}({\mathbf {M}}) = \varDelta {\mathbf {M}}\). In particular, the LLGEs is given by
where \(\mathbf {{\varvec{n}}}\) is the unit outward normal vector field on the boundary \(\partial D\) and \(\frac{\partial {\varvec{M}}}{\partial \mathbf {{\varvec{n}}}}\) is the directional derivative of \({\varvec{M}}\) in the direction \(\mathbf {{\varvec{n}}}\). Here we assume that the material is saturated at the initial time.
It is well-known that the stationary solutions of system (1) correspond to the equilibrium states of the ferromagnet and are not unique in general. An interesting question in the theory of ferromagnetism is to describe phase transitions between different equilibrium states induced by thermal fluctuations of the field \({\mathcal {H}}_{\mathrm{{eff}}}\). Thus, conventionally, randomly fluctuating fields act responsible for magnetization fluctuations, see Neel [29]. It essentially characterizes deviations from the average magnetization trajectory in an ensemble of noninteracting nanoparticles, see Brown in [2]. According to Brown, the magnetization \({\mathbf {M}}\) evolves randomly, and for the stochastic version of system (1), one needs to modify the system in order to incorporate random fluctuations of the field \({\mathcal {H}}_{\mathrm{{eff}}}\) into the dynamics of \({\mathbf {M}}\) and to describe noise-induced transitions between equilibrium states of the ferromagnet. In this context, we refer to [3, 4] for further clarification of adding noise to the effective field \({\mathcal {H}}_{\mathrm{{eff}}}\).
The theory of rough paths was initiated by Terry Lyons in his seminal work [26] as an extension of the classical theory of controlled differential equations. Since its introduction, the theory of ordinary and partial differential equations driven by rough paths has developed intensively. We refer the reader to study the papers of Gubinelli et al. [13, 16,17,18], Friz et al. [9, 10, 14]. We quote the recent preprint [20], where using rough path formulation, existence, uniqueness and regularity for the SLLGE with Stratonovich noise on the one-dimensional torus has been studied.
In this paper, we are interested in the following form of the LLGEs equation:
where \({\varvec{X}}\) is a geometric rough path. We adapt Lyons’ rough paths theory to study system (1) driven by an irregular noise. By proposing a suitable transformation, we convert system (1) to a fully nonlinear time-dependent partial differential equation without rough paths term. The primary interest is the regular approximation of the geometric rough path.
The rough path theory is informally connected to Wong–Zakai results as it necessarily allows to construct solutions as limits of Wong–Zakai type approximations, for references, see, e.g. the seminal paper by Lyons [26], and the recent monograph by Lyons et al. [27]. There are various works on rough PDEs where weak solutions are constructed by exploiting energy arguments. Here in our work, we applied this way of constructing a solution. For a quick survey, we refer to Bailleul et al. [1], Deya et al. [12], Hocquet et al. [21], Hofmanová et al. [22]. We refer the readers to Section 1.5 in [8] for detailed discussion.
In [19], the authors have studied equations similar to SLLGEs in \({{\mathbb {R}}}^d\), for any \(d>0\). However, the noise corresponds to a choice of \({\varvec{g}}\) in equation (4) to be constant across the domain D, i.e., it can be independent of space and time variables. It is not clear how the concept of solution is defined. Brzeźniak et al. in [3] have proved existence of weak martingale solutions of SLLGEs taking values in a sphere \({\mathbb {S}}^2\). Furthermore, Li et al. in [5] have generalized the results in [3] with non-zero anisotropic energy \({\mathcal {E}}_{an}\) and multidimensional noise. Finite dimensional analysis of this problem has been discussed in [23, 25]. Brzeźniak et al. have studied in [4] the one-dimensional case and prove the large deviations principle to the SLLGEs for small noise asymptotic. Brzeźniak et al. have proved in [6, 7] existence of weak martingale solution for SLLGEs in three dimensions perturbed by jump noise in the Marcus canonical form with non-zero anisotropic energy \({\mathcal {E}}_{an}\), see [6] and non-zero exchange energy \({\mathcal {E}}_{ex}\) only, see [7]. Recently, in [8, 28], the authors have employed Wong–Zakai approximation technique to obtain the solvability and convergence of the time dependent transformed PDEs. The open questions framed in [8] have motivated us, as a first step to adapt Lyons’ rough paths theory to study LLGEs driven by geometric rough paths in one dimension.
As a particular case, we can take Stratonovich Brownian rough paths as geometric rough paths. Furthermore, we plan to extend our techniques to a more general setting when the stochastic process is no longer one-dimensional (and nor the corresponding vector fields commute). The motivation to use geometric rough path comes from [24], where they discuss the heat equation with a geometric rough path.
1.1 Problem Description
We introduce some notations and summarize the most important definition of rough paths which are taken from [14].
Notations For a domain \(D\subset {{\mathbb {R}}}^d\) with \(d=1,2,3\), we will use the notation \({\mathbb {L}}^p\) for the space \(L^p(D; {\mathbb {R}}^3)\) and \({\mathbb {W}}^{m,p}\) for the Sobolev space \(W^{m.p}(D; {\mathbb {R}}^3)\). We will often write \({\mathbb {H}}^m\) instead of \({\mathbb {W}}^{m,2}\). We will also denote, for a Banach space V, by \({{ {\mathcal {C}} }}^{\alpha }([0,T],V)\) the space of all \(\alpha \)–Hölder continuous functions \(f:[0,T]\rightarrow V\) with \(\alpha \in [0,1]\). We define the path increment \(f_{s,t}=f_t-f_s\) where \(f_t:=f(t)\) (we will write \({\dot{f}}\) for \(\frac{df}{dt}\)). For \(f\in {{ {\mathcal {C}} }}^{\alpha }([0,T],V)\), the usual \(\alpha \)-Hölder semi-norm is given by
For simplicity we write the semi-norm as \(|f|_{\alpha ,[0,T]}\) or \(|f|_{\alpha }\), if there is no danger confusion.
A rough path on an interval [0, T] with values in a Banach space V then consists of a continuous function \(X:[0,T]\rightarrow V\), as well as a continuous “second order process” \({\mathbb {X}}:[0,T]^2\rightarrow V\times V\), subjected to certain conditions which are given by Definition 3. Generically, we write \({\mathscr {C}}^{\alpha }([0,T],V)\) for the space of \(\alpha \)-Hölder rough paths and \({\mathscr {C}}_g^{\alpha }([0,T],V)\) for the space of \(\alpha \)–Hölder geometric rough paths over a Banach space V. For a Banach space H and \(\alpha \in (\frac{1}{3},\frac{1}{2}]\), we denote the space of controlled rough paths by \({\mathcal {D}}_{X}^ {2\alpha }([0,T],H)\). We note that \({\mathscr {C}}_g^{\alpha }([0,T],V)\) is a closed subset of \({\mathscr {C}}^{\alpha }([0,T],V)\). Hence, we can define \({\mathscr {C}}_g^{0,\alpha }([0,T],V)\) as the closure of lifts of smooth paths in \({\mathscr {C}}^{\alpha }([0,T],V)\). One has the obvious inclusion \({\mathscr {C}}_g^{0,\alpha }([0,T],V)\subset {\mathscr {C}}_g^{\alpha }([0,T],V)\). Furthermore, we recall that for any \(\alpha \in (1/3,1/2)\) with probability one, the corresponding Stratonovich lift is \({\varvec{B}}^{\text {strat}}=(B,{\mathbb {B}})\in {\mathscr {C}}_g^{0,\alpha }([0,T],{{\mathbb {R}}})\) where \({\mathbb {B}}_{s,t}:=\int _{s}^{t} B_{s,r}\circ dB_r\). For more details about rough paths and their integration, we refer to Appendix 2 and the book [14].
In this paper, we consider the case \(d=1\) and \(\alpha \in (\frac{1}{3},\frac{1}{2})\) and assume that D to be a bounded open interval in \({\mathbb {R}}\). In particular, we take \(D=(0, 1)\). The LLGEs in consideration in this paper is of the form
where \(T>0\) is fixed, \({\varvec{g}}:D\rightarrow {\mathbb {R}}^3\) is given function such that \({\varvec{g}}\in {\mathbb {W}}^{4,\infty }\), and \({\varvec{X}}=(X,{\mathbb {X}})\in {\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\). We recall the fact that if \({\tilde{{\varvec{X}}}}=(X-X_0,{\mathbb {X}})\) then \({{\tilde{{\varvec{X}}}}}\in {\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\) and \(\int _{0}^{t}f~d{{\tilde{{\varvec{X}}}}}=\int _{0}^{t}f~d{\varvec{X}}\) for any \(f\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{{\mathbb {R}}}^3)\). Without loss of generality we assume that \(X_0=0\).
One of the most fundamental questions related to problems similar to the above is whether the solutions depend in a continuous way of the coefficients (the geometric rough path in our case). Let us describe our approach to this question. We first recall that \({\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\) is a closed subset of \({\mathscr {C}}^{\alpha }([0,T],{{\mathbb {R}}})\). Thus we can choose \({\varvec{X}}^{(n)}=\big (X^{(n)},{\mathbb {X}}^{(n)}\big )\), \(n\in {\mathbb {N}}\) be a sequence in \({\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\) such that
Let us now consider the corresponding system (7) approximating (5)
We note that if \({\varvec{X}}\in {\mathscr {C}}_g^{0,\alpha }([0,T],{{\mathbb {R}}})\subset {\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\) then one can choose \(X^{(n)}\) to be a sequence of piecewise smooth path or (piecewise) \({{ {\mathcal {C}} }}^{1}\) paths and \({\mathbb {X}}^{(n)}_{s,t}=\int _{s}^{t} X^{(n)}_{s,r} dX_{r}^{(n)}\), see page 17 of [14]. Moreover, in the stochastic case one can take Stratonovich Brownian rough paths \({\varvec{B}}^{\text {strat}}\) as \({\varvec{X}}\) and also sequence \({\varvec{B}}^{(n)}\) defined in Proposition 3.6 of [14] as \({\varvec{X}}^{(n)}\). For more details we refer to Sect. 4.1 and the book [14].
Our primary goal is to prove that the solution to the system (5) exists and is a unique strong solution in controlled rough path spaces, which indeed is a limit as \(n\rightarrow \infty \) of the solutions of sequence of the corresponding system (7) approximating (5). In particular, we show the convergence with respect to the distance \(d_{X,X^{(n)},2\alpha ,[0,T]}\) (see Definition 4 in Appendix 2).
We now briefly describe the content of the paper. In Sect. 2, we introduce an auxiliary ODE and state some auxiliary facts necessary for the transformation of system (7) to a deterministic PDE without rough paths term. Sections 3 and 4 are devoted to the proof of the main results. In Sect. 3, we show how a unique weak solution to (7) can be obtained from a unique weak solution of the reformulated equation in controlled rough path spaces; see Theorem 1. Section 4 is devoted to the proof of the convergence of solutions to the weak solution of the reformulated form; see Theorem 4 . Finally, in the Appendix, for the reader’s ease, we list several facts that are used in the course of the analysis. We split the Appendix into four subsections. First, in Appendix 1, we present several auxiliary lemmata which are essential to prove Theorem 1. Next, in Appendix 2 we handle some technical issues required in this paper, i.e., we introduce few results (see Lemmas 10, 11 and Corollary 5). These results, another contribution of the paper and have their own interest. In the last two subsections, we recall some simple results and algebraic identities used in this paper.
2 The Auxiliary Equations
In this section, we present some basic results on the the spaces and operators involved in the course of analysis. We also introduce new processes \({\varvec{m}}\) and \({\varvec{m}}^{(n)}\) gained by the Doss-Sussman transformation from the corresponding processes \({\varvec{M}}\) and \({\varvec{M}}^{(n)}\). We refer to [4, 8, 15] for further details about the properties of these processes.
2.1 Preliminaries
We define the Laplacian with the Neumann boundary conditions acting on \({{\mathbb {R}}}^3\)-valued functions by
We know that the unbounded operator A is self-adjoint in \({\mathbb {L}}^2\) and \( A_1^{-1}\) is compact for \(A_1:=I+A\). Therefore, there exists an orthonormal basis \(\{e_n\}_{n=1}^{\infty }\) of \({\mathbb {L}}^2\) consisting of eigenvectors of A. Furthermore, we know that, if \(V:=D(A_1^{1/2})\) is endowed with the graph norm, then V coincides with \({\mathbb {H}}^1\). Later on, we will write \(V\hookrightarrow {\mathbb {L}}^2\hookrightarrow V'\) which is a Gelfand triple.
We now present the following interpolation inequality which will be used in later sections.
Next, let us we recall an elementary result from [4].
Lemma 1
Let \({\mathbf {v}}\in {\mathbb {H}}^1\) such that \( |{\mathbf {v}}(x)|_{{\mathbb {R}}^3}=1\) for all \(x \in D\). Then in \(({\mathbb {H}}^1)'\), we have
Let us define the map \(G:{\mathbb {L}}^2 \rightarrow {\mathbb {L}}^2\) by
Lemma 2
Suppose \({\varvec{g}}\in {\mathbb {L}}^{\infty }\). Then, G is a bounded linear map and well defined.
In Appendix 3, Lemma 12, we list the properties of G. Let \({\mathbb {H}}^2(D;{\mathbb {S}}^2)\) be the set of all \({\mathbb {R}}^3\)-valued functions defined on the domain D belonging to the Sobolev space \({\mathbb {H}}^1:=H^1(D; \mathbb R^3)\) and satisfy the saturation condition (2). In particular, let
In other words, \({\mathbb {H}}^2(D;{\mathbb {S}}^2)\) is the set of all functions belonging to the Sobolev space \({\mathbb {H}}^2\) whose values are in the sphere. Since D is one-dimensional, \({\mathbb {H}}^2\) is embedded in \(C(D;{\mathbb {R}}^3)\) and the ‘a.e.’ condition in (11) can be substituted by ‘all’.
2.2 The Doss–Sussmann Transformation and the Corresponding New Processes \({\mathbf {m}}\) and \({\mathbf {m}}^{(n)}\)
Following the discussions in [8, 15], we define a new process \({\varvec{m}}\) from \({\varvec{M}}\) by
where the operator G is introduced in (10). We know, by identity (91) and Lemma 12, that \(e^{\cdot G},~Ge^{\cdot G}\) and \(G^2e^{\cdot G}\) are bounded functions. Therefore, by Proposition 7.6 of [14], we obtain
and \(e^{X(\cdot )G}\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathcal {L}}({\mathbb {L}}^2))\). In Lemma 5, we will show that if a process \({\varvec{M}}\) solves (5), then \({\varvec{m}}\) defined by (12) solves the following non-linear time dependent PDE with random coefficients given below. Later on, in Sect. 3, we will show that there exists indeed a process \({\varvec{m}}\) solving (14). Let us consider
where F is given by
Here, \({\tilde{C}}\) is an operator coming up in Lemma 15 and is given by
where \(s\in {{\mathbb {R}}}\), \(\varvec{v}\in {\mathbb {W}}_0^{1,\infty }\) and C is an operator coming up in Lemma 14 and is given by
From the assumption \(X(0)=0\), we note that the initial condition (2) is equivalent to an analogous one for \({\varvec{m}}_0\), i.e.
Note that, we could consider the transformation (12) for all \(t\ge 0\). Moreover, as one can easily prove \(\vert [e^{tG}(u)](x) \vert _{{\mathbb {R}}^3}= \vert u(x) \vert _{{\mathbb {R}}^3}\) for all \(t\in [0, T]\), for a.a. \(x\in D\), we see that the following saturation conditions for \({\varvec{M}}\) and \({\varvec{m}}\) are equivalent:
For proof of the saturation condition (20), we refer to Lemma 3.15 of [8]. Repeating the same algebraic calculations as done in [15], i.e. using the definition of G and the identity (17), we obtain
Substituting (21) in (16), we get
where \({\tilde{C}}(X(t),{\mathbf {m}})\) has now the following form
Applying identity (91), we obtain the following representation
where we define the following abbreviations
Remark 1
Using the representation in (23) and the fact that \( {\varvec{g}}\in {\mathbb {W}}^{2,\infty }(D)\), we know that there exists a constant \(K>0\) such that
Using the following inequalities
we obtain
Finally, by straightforward calculations we can estimate the nonlinearity F defined in (22). In particular, there exists a constant \(K>0\) such that using (20), we have
Let us now consider the corresponding system (36) approximating (14). Again, let us define a new auxiliary process \({\varvec{m}}^{(n)}\) from \({\varvec{M}}^{(n)}\) and \(X^{(n)}\) by
By the same calculations as above and Lemma 7, one can show that if \({\varvec{M}}^{(n)}\) is a solution to (7), then \({\varvec{m}}^{(n)}\) is a solution to
and vice versa. Here,
Moreover, in Theorem 3, we will show that system (36) has indeed a unique solution.
3 Existence and Uniqueness of the Solutions to the Systems (5) and (7)
We begin with the definition of (strong) solution to the system (5).
Definition 1
Let \(T>0\) and \({\varvec{X}}=(X,{\mathbb {X}})\in {\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\) with \(\alpha \in (\frac{1}{3},\frac{1}{2})\). A path \({\varvec{M}}\in {{ {\mathcal {C}} }}^{\alpha }([0,T],{\mathbb {L}}^2)\) is said to be a (strong) solution of the system (5) if the following properties hold:
-
(i)
\({\varvec{M}}\) satisfies the following saturation condition
$$\begin{aligned} |{\varvec{M}}(t,x)|_{{{\mathbb {R}}}^3}=1\text { for a.e. }x\in D,\text { for all }t\in [0,T]; \end{aligned}$$ -
(ii)
\({\varvec{M}}\) satisfies the second equation of (5) and \(\sup _{t \in [0,T]}|{\varvec{M}}(t)|_{{\mathbb {H}}^2}<\infty ;\)
-
(iii)
\({\varvec{M}}\) is controlled by \({\varvec{X}}\), i.e. there exists a path \({\varvec{M}}'\in {{ {\mathcal {C}} }}^{\alpha }([0,T],{\mathbb {L}}^2)\) (being called Gubinelli derivative) such that the reminder term \({\mathscr {R}}^Y\) is given implicitly through the relation \( Y_{s,t}=Y_s'X_{s,t} + {\mathscr {R}}_{s,t}^ Y, \) and satisfies \(|{\mathscr {R}}^Y |_{2\alpha ,H}<\infty \). Moreover,
$$\begin{aligned} ({\varvec{M}},{\varvec{M}}')\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathbb {L}}^2); \end{aligned}$$ -
(iv)
\({\varvec{M}}\) satisfies the rough differential equation given by
$$\begin{aligned} {\varvec{M}}(t,x)= & {} {\varvec{M}}_0(x)+ \lambda _1 \int _{0}^{t} {\varvec{M}}(s,x)\times {\varvec{M}}_{xx}(s,x)~ds\\&-\lambda _2 \int _{0}^{t} {\varvec{M}}(s,x) \times ({\varvec{M}}(s,x)\times {\varvec{M}}_{xx}(s,x))~ ds\\&+ \int _{0}^{t}({\varvec{M}}(s,x)\times {\varvec{g}}(x))~d{\varvec{X}}_s, \end{aligned}$$where the third integral is interpreted in the sense of (77).
In this section, we prove the following theorem which shows that the solution to the system (5) (resp. (7)) exists and is unique.
Theorem 1
Let \({\varvec{X}}=(X,{\mathbb {X}}) \in {\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\). Let \({\varvec{X}}^{(n)}=\big (X^{(n)},{\mathbb {X}}^{(n)}\big )\), \(n\in {\mathbb {N}}\) be a sequence in \({\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\) such that
If \({\varvec{M}}_0,{\varvec{M}}_0^{(n)}\in {\mathbb {H}}^2(D;{\mathbb {S}}^2)\), then there exist unique (strong) solutions \({\varvec{M}}\) and \({\varvec{M}}^{(n)}\) to the respective systems (5) and (7). In addition, we have
-
(i)
\(({\varvec{M}},{\varvec{M}}\times {\varvec{g}})\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathbb {L}}^2)\),
-
(ii)
\(({\varvec{M}}^{(n)},{\varvec{M}}^{(n)}\times {\varvec{g}})\in {\mathcal {D}}_{X^{(n)}}^ {2\alpha }([0,T],{\mathbb {L}}^2)\),
-
(iii)
and \({\varvec{M}},{\varvec{M}}^{(n)}\in L^{\infty }(0,T;{\mathbb {H}}^2)\).
Before we present the proof of Theorem 1, we introduce Theorem 2 and Theorem 3. In the first theorem, we show the existence, uniqueness and regularity of the solution to the system (14). In the second theorem, we obtain the same for system (36). Then, using the Doss–Sussmann transformation, we show that from the existence of a unique solution to systems (14) and (36), respectively, follows the existence of the solution of systems (5) and (7), respectively. Similarly, the convergence is proven.
Let us start with introducing the definition of (weak) solution \({\varvec{m}}\) to the system (14) and the theorem for existence and regularity of \({\varvec{m}}\).
Definition 2
Let \(T>0\). A path \({\varvec{m}}\in {{ {\mathcal {C}} }}^{2\alpha }([0,T],{\mathbb {L}}^2)\) is said to be a (weak) solution to the system (14) if the following properties hold:
-
(i)
\({\varvec{m}}\) satisfies the following saturation condition
$$\begin{aligned} |{\varvec{m}}(t,x)|_{{{\mathbb {R}}}^3}=1\text { for a.e. }x\in D,\text { for all }t\in [0,T]; \end{aligned}$$ -
(ii)
\({\varvec{m}}\) satisfies the second equation of (14) and \(\sup _{t \in [0,T]}|{\varvec{m}}(t)|_{{\mathbb {H}}^2}<\infty ;\)
-
(iii)
\({\varvec{m}}\) satisfies the time dependent differential equation for all \(\phi \in {\mathbb {H}}^1,\) \(t \in [0,T]\)
$$\begin{aligned} \langle {\varvec{m}}(t), \phi \rangle _{{\mathbb {L}}^2}&=\langle {\mathbf {m}}(0), \phi \rangle _{{\mathbb {L}}^2}-\lambda _1 \int _0^t \int _D \langle {\varvec{m}}_x(s,x), \phi _x(x) \times {\varvec{m}}(s,x) \rangle _{{\mathbb {R}}^3} dx\, ds \nonumber \\&\quad -\lambda _2 \int _0^t \int _D \langle {\varvec{m}}_x(s,x), ({\mathbf {m}}\times \phi )_x(s,x) \times {\varvec{m}}(s,x) \rangle _{{\mathbb {R}}^3} dx\, ds\nonumber \\&\quad +\int _0^t \int _D \langle F(s,{\mathbf {m}}(s,x)),\phi (x)\rangle _{{\mathbb {R}}^3} dx\, ds. \end{aligned}$$(38)
Theorem 2
Let \({\mathbf {m}}_0 \in {\mathbb {H}}^2(D;{\mathbb {S}}^2).\) Then there exists a weak solution \({\mathbf {m}}\) to the system (14) in the sense of Definition 2 satisfying the following:
-
(i)
there exists a positive constant C, depending on \(T, \lambda _1,\lambda _2, |{\mathbf {m}}_0|_{{\mathbb {H}}^2}\), such that
$$\begin{aligned} \sup _{t \in [0,T]} |{\mathbf {m}}(t)|_{{\mathbb {H}}^2} \le C; \end{aligned}$$ -
(ii)
for almost every \(t \in [0,\infty ),\quad {\mathbf {m}}(t) \times {\mathbf {m}}_{xx}(t) \in {\mathbb {L}}^2\) and for every \(T>0,\) there exists a positive constant C, depending on \(T, \lambda _1,\lambda _2, |{\mathbf {m}}_0|_{{\mathbb {H}}^2}\), such that we have
$$\begin{aligned} \int _0^T |{\mathbf {m}}(t) \times {\mathbf {m}}_{xx}(t)|_{{\mathbb {L}}^2}^2 dt \le C. \end{aligned}$$(39)
Furthermore, we obtain that \({\varvec{m}}\in H^1(0,T;{\mathbb {L}}^2)\cap L^{\infty }(0,T;{\mathbb {H}}^2)\) and \({\varvec{m}}\) is also a strong solution of the system (14).
Proof
For proof, we refer to Theorem 3.2 and Lemma 3.13 of [8]. In addition, for the regularity property of \({\varvec{m}}\), we refer to Lemma 8 and Lemma 9 in Appendix 2. \(\square \)
In the same way as above (see Definition 2 and Theorem 2), we can define weak solutions for the system of equations (36) and obtain the following theorem.
Theorem 3
Let \({\mathbf {m}}_0 \in {\mathbb {H}}^2(D;{\mathbb {S}}^2).\) Then there exists a weak solution \({\mathbf {m}}^{(n)}\) to the system (36), i.e., it satisfies the following:
-
(i)
for every \(T>0,\)
$$\begin{aligned} \sup _{t \in [0,T]} |{\mathbf {m}}^{(n)}(t)|_{{\mathbb {H}}^2} \le C(T, \lambda _2, |{\mathbf {m}}_0|_{{\mathbb {H}}^2}); \end{aligned}$$(40) -
(ii)
for almost every \(t \in [0,\infty ),\quad {\mathbf {m}}^{(n)}(t) \times {\mathbf {m}}^{(n)}_{xx}(t) \in {\mathbb {L}}^2\) and every \(T>0\) we have
$$\begin{aligned} \int _0^T |{\mathbf {m}}^{(n)}(t) \times {\mathbf {m}}^{(n)}_{xx}(t)|_{{\mathbb {L}}^2}^2 dt \le C(T,\lambda _2,|{\mathbf {m}}_0|_{{\mathbb {H}}^2}); \end{aligned}$$(41) -
(iii)
\(|{\mathbf {m}}^{(n)}(t,x)|_{{\mathbb {R}}^3}=1,\quad \) a.e. \(x \in D\) and for all \(t \in [0, T]\);
-
(iv)
for all \(\phi \in {\mathbb {H}}^1,\)
$$\begin{aligned} \langle {\mathbf {m}}^{(n)}(t), \phi \rangle _{{\mathbb {L}}^2}&=\langle {\mathbf {m}}(0), \phi \rangle _{{\mathbb {L}}^2}\\&\quad -\lambda _1 \int _0^t \int _D \langle {\mathbf {m}}^{(n)}_x(s,x), \phi _x(x) \times {\mathbf {m}}^{(n)}(s,x) \rangle _{{\mathbb {R}}^3} dx\, ds \\&\quad -\lambda _2 \int _0^t \int _D \langle {\mathbf {m}}^{(n)}_x(s,x), ({\mathbf {m}}^{(n)} \times \phi )_x(s,x) \times {\mathbf {m}}^{(n)}(s,x) \rangle _{{\mathbb {R}}^3} dx\, ds\\&\quad +\int _0^t \int _D \langle F(s,{\mathbf {m}}^{(n)}(s,x)),\phi (x)\rangle _{{\mathbb {R}}^3} dx\, ds, \end{aligned}$$holds for all \(t \in [0,T].\)
Moreover, we obtain that \({\varvec{m}}^{(n)}\in H^1(0,T;{\mathbb {L}}^2)\cap L^{\infty }(0,T;{\mathbb {H}}^2)\) and \({\varvec{m}}^{(n)}\) is the strong solution of the system (36).
Proof of Theorem 1
Thanks to Theorem 2, there exists a unique strong solution \({\varvec{m}}\in H^1(0,T;{\mathbb {L}}^2)\cap L^{\infty }(0,T;{\mathbb {H}}^2)\) to the system (14). Hence, applying Lemma 5, we conclude that there exists a unique strong solution \({\varvec{M}}\in L^{\infty }(0,T;{\mathbb {H}}^2)\) of the system (5) and \(({\varvec{M}},{\varvec{M}}\times {\varvec{g}})\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathbb {L}}^2)\). Finally, using Lemma 10, we conclude that the integration against rough paths in (5) is well defined. Proceeding in similar lines, one can observe that there exists a unique strong solution \({\varvec{M}}^{(n)}\in L^{\infty }(0,T;{\mathbb {H}}^2)\) to the system (7) and \(({\varvec{M}}^{(n)},{\varvec{M}}^{(n)}\times {\varvec{g}})\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathbb {L}}^2)\), thus completing the proof. \(\square \)
4 Convergence of Solution in Controlled Rough Path Spaces
In this section we prove the convergence result in the space \({\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathbb {L}}^2)\).
Theorem 4
Let \({\varvec{M}}\) and \({\varvec{M}}^{(n)}\) be the solutions to the systems (5) and (7), respectively. Then we have the following convergence
Before going into the proof of Theorem 4, we introduce and prove some lemmata and corollaries. These lemmata are essential to show Theorem 4.
Lemma 3
Let \({\varvec{m}}\) and \({\varvec{m}}^{(n)}\) be the solutions to the systems (14) and (36), respectively. Then we have the following convergence
Proof
By similar arguments as with the proof of Lemma 5.2 in [8], one can show that there exists a constant \(C>0\) and an integrable function \(\varphi _C\), so that we have the following estimate
for \(t\in [0,T]\). Using (6), we obtain
\(\square \)
Corollary 1
Let \({\varvec{M}}\) and \({\varvec{M}}^{(n)}\) be the solutions to the systems (5) and (7), respectively. Then we have the following convergence
Proof
We note that
Since \(\varvec{X}^{(n)}\rightarrow \varvec{X}\) in \({\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\) (compare (6)), we can observe that \(X^{(n)}\rightarrow X\) in \(L^{\infty }(0,T;{{\mathbb {R}}})\). Therefore, using the identity (91) we have \(e^{X^{(n)}G}\rightarrow e^{XG}\) in \(L^{\infty }(0,T;{\mathcal {L}}({\mathbb {L}}^2))\), where \({\mathcal {L}}({\mathbb {L}}^2)\) denote space of bounded linear operators on \({\mathbb {L}}^2\). Finally, from Lemma 3, we obtain
which is the assertion. \(\square \)
In the next lemma, we investigate the convergence of sequence \({\varvec{m}}_{xx}^{(n)}\) in \(L^{\infty }(0,T;{\mathbb {L}}^2)\).
Lemma 4
Let \({\varvec{m}}\) and \({\varvec{m}}^{(n)}\) be the solutions to the systems (14) and (36), respectively. Then we have the following convergence
Proof
Let \(u={\varvec{m}}_{xx}\), \(u^{(n)}={\varvec{m}}_{xx}^{(n)}\) and \(z=u^{(n)}-u\). We proceed in the same way as we advance in the proof of Lemma 9 in Appendix 2. Substituting z in system (14), we get the following identity
Substituting \(z=u^{(n)}-u\), we obtain
Testing with z and integrating over the interval (0, 1), we obtain
Now we estimate each \(I_j\)’s for \(j=1, 2, 3\).
Estimate of \(I_1\): Integrating by parts and using the fact that \(z_x=0\) and \(z^{(n)}_x=0\) at \(x=0\) and \(x=1\), we obtain
Using Young’s inequality, we obtain for any \(\varepsilon >0\),
Above, we introduced the abbreviation \(\psi _1:=\varepsilon |\lambda _1|\), \(\varphi _1:=|\lambda _1|\big [{|{\varvec{m}}_x^{(n)}|_{{\mathbb {L}}^{\infty }}^2}/({2\varepsilon })+{1}/({2\varepsilon }) \big ]\) and \(\chi _1:= |\lambda _1|\big [\varepsilon /{2}\,|{\varvec{m}}_x^{(n)}-{\varvec{m}}_x|_{{\mathbb {L}}^{\infty }}^2+{|{\varvec{m}}^{(n)}-{\varvec{m}}|_{{\mathbb {L}}^{\infty }}^2}/({2\varepsilon })\big ]|u_x|_{{\mathbb {L}}^2}^2\).
Estimate of \(I_2\): By a simple vector algebraic identity and equation (20), we obtain
We now derive estimates \(I_{2,j}\) with \(j=1,2,\cdots ,5\) for a fixed \(\varepsilon \). Proceeding in a similar manner as in \(I_1(t)\), we get
Now we estimate \(I_{2,2}\). Applying Young’s inequality, we have
Proceeding in similar manner as in \(I_{2,2}\), we get for \(I_{2,3}\) and \(I_{2,4}\)
Now, by similar argument as applied to \(I_{2,1}\), we obtain
where
Combining the estimates of \(I_{2,j}\), \(j=1,2,\cdots ,5\) and substituting back in (44), we get
Estimate of \(I_3\): To simplify the notation, we introduce \(y={\varvec{m}}^{(n)}-{\varvec{m}}\). By the identities (24) and (26), it can be observed that \({\mathcal {C}},{\mathcal {C}}_{x},{\mathcal {C}}_{xx}, S , S _{x}\) and \( S _{xx}\) are Lipschitz continuous functions. In particular, there exists a constant \(K=K(|{\varvec{g}}|_{W^{4,\infty }})>0\) such that the following holds:
Furthermore, an elementary calculation using the identities (25) and (27) yields that
Hence using the estimates (46), (47) and the identity (28) we have
Now we are ready to estimate \(I_3\). Using the identity (22) and a simple vector algebraic identity, we obtain:
We now derive estimates \(I_{3,j}\) with \(j=1,2,\cdots ,9\) for fixed \(\varepsilon \). Using (48a), (29) and Young’s inequality, we have
Again using (48a), (29) and Young’s inequality, we get
Proceeding in similar manner as in \(I_{3,4}\), we have
Again, proceeding as in \(I_{3,4}\), we obtain
Now applying (48b), (30) and Young’s inequality, we obtain
Again using (48b), (30) and Young’s inequality, we get
Proceeding in similar manner as in \(I_{3,7}\), we have
Now using Young’s inequality, (48b) and (30), we get
Again using (48c), (31) and Young’s inequality, we get
Thus combining the estimates of \(I_{3,j}\), \(j=1,2,\cdots ,9\) and substituting back in (49), we have
Substituting (43), (45) and (50) in (42) we have
We note by (20) and Corollary 4, we have \({\varvec{m}}^{(n)},{\varvec{m}}_x^{(n)}\in L^{\infty }(0,T;{\mathbb {L}}^{\infty })\) for all \(n\in {\mathbb {N}}\). Therefore we can choose \(\varepsilon \) such that
By Lemma 8, we note that \({\varvec{m}},{\varvec{m}}^{n}\in L^{\infty }(0,T;{\mathbb {H}}^1)\cap L^2(0,T;D(A))\). From Lemma 9 and Corollary 4, we obtain \({\varvec{m}}^{(n)},{\varvec{m}},{\varvec{m}}_x^{(n)},{\varvec{m}}_{x}\in L^{\infty }(0,T;{\mathbb {L}}^{\infty })\), \({\varvec{m}}_{xx},{\varvec{m}}_{xx}^{(n)}\in L^{\infty }(0,T;{\mathbb {L}}^{2})\) and \(u_{x},u_{x}^{(n)}\in L^{2}(0,T;{\mathbb {L}}^{2})\). Thus \(\varphi \) and \(\chi \) are integrable on [0, T]. Using Gronwall’s inequality we get
Now by Lemma 3 and (8), we note that
As \({\varvec{m}}_{xx},{\varvec{m}}_{xx}^{(n)}\in L^{\infty }(0,T;{\mathbb {L}}^{2})\), we obtain \({\varvec{m}}_x^{(n)}\rightarrow {\varvec{m}}_x\) in \(L^{\infty }(0,T;{\mathbb {L}}^{\infty })\). Moreover, we also have \({\varvec{X}}^{(n)} \rightarrow {\varvec{X}}\text { in }{\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}}).\) This implies that \(z \rightarrow 0\) in \(L^{\infty }(0,T;{\mathbb {L}}^2)\). Hence
This completes the proof of Lemma 4. \(\square \)
Corollary 2
Let \({\varvec{M}}\) and \({\varvec{M}}^{(n)}\) be the solutions to the systems (5) and (7), respectively. Then we have the following convergence
Proof
Using the identity (91), we obtain
Thus, we have
Using \(|\cos (x_1)-\cos (x_2)|,|\sin (x_1)-\sin (x_2)|\le c|x_1-x_2|\), and the boundedness of \(\sin \) and \(\cos \), we get
Using Lemma 4, Lemma 3 and that \({\varvec{X}}^{(n)} \rightarrow {\varvec{X}}\) in \({\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\), we obtain
\(\square \)
Corollary 3
Let \({\varvec{M}}\) and \({\varvec{M}}^{(n)}\) be the solutions to the systems (5) and (7), respectively. Then, we have
in \(L^{\infty }(0,T;{\mathbb {L}}^2)\) as \(n\rightarrow \infty \).
Proof
This is a direct consequence of Corollaries 1 and 2. \(\square \)
Now we are ready to prove Theorem 4.
Proof of Theorem 4
Since \(({\varvec{M}},{\varvec{M}}\times {\varvec{g}})\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathbb {L}}^2)\) and \(({\varvec{M}}^{(n)},{\varvec{M}}^{(n)}\times {\varvec{g}})\in {\mathcal {D}}_{X^{(n)}}^ {2\alpha }([0,T],{\mathbb {L}}^2)\), there exists a number \(N\in {{\mathbb {R}}}^{+}\) such that
Let \(0<T_1\le \min \{1,T\}\). In particular, we assume that \(C_{\alpha ,N,g}T_1^{\alpha }\le \frac{1}{2}\), where \(C_{\alpha ,N,g}>0\) is a generic constant depending only on \(\alpha \), N, and g and popping up in estimate (59). In the next lines we proof the following estimate
Let us compute \(|{\varvec{M}}\times {\varvec{g}}-{\varvec{M}}^{(n)}\times {\varvec{g}}|_{\alpha ,[0,T_1],{\mathbb {L}}^2}\). Due to the fact that \({\varvec{g}}\in {\mathbb {W}}^{2,\infty }\), we can write for \(0\le s<t\le T_1\)
Next, using the identities \({\varvec{M}}_{s,t}={\varvec{M}}'_sX_{s,t}+{\mathscr {R}}_{s,t}^{{\varvec{M}}}\) and \({\varvec{M}}^{(n)}_{s,t}=({\varvec{M}}^{(n)})_s'X^{(n)}_{s,t}+{\mathscr {R}}_{s,t}^{{\varvec{M}}^{(n)}}\), where \({\varvec{M}}'={\varvec{M}}\times {\varvec{g}}\) and \(({\varvec{M}}^{(n)})'={\varvec{M}}^{(n)}\times {\varvec{g}}\), we obtain
This implies
We now start to estimate \(|{\mathscr {R}}^M-{\mathscr {R}}^{M^{(n)}}|_{2\alpha ,[0,T_1],{\mathbb {L}}^2}\). Using system (5), we obtain
where \(\varXi _{s,t}:=({\varvec{M}}_s\times {\varvec{g}})X_{s,t}+({\varvec{M}}_s'\times {\varvec{g}}) {\mathbb {X}}_{s,t}\) and
In similar way, we also obtain the same identity as above for \({\mathscr {R}}_{s,t}^{M^{(n)}}\), i.e.
where \(\varXi ^{(n)}_{s,t}:=({\varvec{M}}^{(n)}_s\times {\varvec{g}})X^{(n)}_{s,t}+(({\varvec{M}}^{(n)}_s)'\times {\varvec{g}}) {\mathbb {X}}^{(n)}_{s,t}\) and
Setting \(\varPsi =\varXi -\varXi ^{(n)}\), we use equation (4.11) of [14] with \(\beta =3\alpha \) and replaced \(\varXi \) by \(\varPsi \), so that we can write for \(0\le s<t\le T_1\)
where
For \(0\le s<u<t\le T_1\), we get now
Continuing, we obtain
This implies
Finally, using (58) and (59), we obtain the estimate
Observe, we have taken \(C_{\alpha ,N,g}T_1^{\alpha }\le \frac{1}{2} \), from which it follows that
Noting that the choice \(T_1\) does not depend on the initial condition, we can extend the estimate by gluing techniques to the whole time interval [0, T] and get
for \(j=1,\cdots ,K\) where \(K\in {\mathbb {N}}\) and \(K<\infty \) such that \(KT_1< T\) and \((K+1)T_1\ge T\). Also, we obtain
Therefore, combining (61) and (62), we achieve
Using Corollary 3 and (6), we obtain
\(\square \)
4.1 Application to Stochastic Landau–Lifshitz–Gilbert Equations (SLLGEs)
In [8, 28], the authors used the Wong–Zakai approximation to obtain a solution and the convergence of (63). In this section, we show the same results using the results from the previous section and by taking as geometric rough path the rough path generated by the Brownian motion where the stochastic integral is interpreted in the Stratonovich sense. To be more precise, given a real-valued Brownian motion \(B=\{B_t, t\in [0,T]\}\) defined on a complete probability space \((\varOmega ,{\mathcal {F}},({\mathcal {F}}_t)_{t\ge 0},{\mathbb {P}})\), where \({\mathcal {F}}_t=\sigma \{B_s:s\in [0,t]\}\), the corresponding geometric rough path is defined by \({\varvec{B}}^{\text {strat}}=(B,{\mathbb {B}})\), where \({\mathbb {B}}_{s,t}:=\int _s^t B_{s,u}\otimes \circ dB_u\), \(B_{s,u}:=B_u-B_s\), \(0\le s\le t\le T\) (compare [14, p. 69]). In this way, we arrive at the SLLGEs of the following form:
Since we know that for any \(\alpha \in (1/3,1/2)\), we have with probability one \({\varvec{B}}^{\text {strat}}=(B,{\mathbb {B}})\in {\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\) (compare [14, Proposition 3.4]), the rough path version of (63) becomes
Let \(B^{(n)}\) be the dyadic piecewise-linear approximations of B, let
and \({\varvec{B}}^{(n)}=\big (B^{(n)},{\mathbb {B}}^{(n)}\big )\) be the corresponding geometric rough path. Here, the integral \(\int _{0}^{t} B^{(n)} dB^{(n)}\) is understood as classical Riemann-Stieltjes. Since \(B^{(n)}\) is a piecewise linear function and continuous function, the Riemann-Stieltjes is well defined. From Proposition 3.6 of [14], we know that \({\varvec{B}}^{(n)}\) is a sequence in \({\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\) and we have with probability one
Let us now consider the following approximation of system (64)
From Theorem 1, one can conclude that there exist unique solutions \({\varvec{M}}\) and \({\varvec{M}}^{(n)}\) to the systems (64) and (66), respectively. Furthermore, we have
and
for \({\mathbb {P}}\)-a.s. \(\omega \in \varOmega \). Moreover, from Theorem 4, we know
References
Bailleul, I., Gubinelli, M.: Unbounded rough drivers. Annales de la Faculté des sciences de Toulouse : Mathématiques Ser. 6 26(4), 795–830 (2017)
Brown, W.F.: Thermal fluctuations of a single-domain particle. Phys. Rev. 130(5), 1677–1686 (1963)
Brzeźniak, Z., Goldys, B., Jegaraj, T.: Weak solutions of a stochastic Landau-Lifshitz-Gilbert Equation. Appl. Math. Res. eXpress 2013(1), 1–33 (2012)
Brzeźniak, Z., Goldys, B., Jegaraj, T.: Large deviations and transitions between equilibria for stochastic Landau-Lifshitz-Gilbert equation. Arch. Ration. Mech. Anal. 226(2), 497–558 (2017)
Brzeźniak, Z., Liang, L.: Weak solutions of the stochastic Landau-Lifshitz-Gilbert equations with nonzero anisotrophy energy. Appl. Math. Res. Express. AMRX 2016(2), 334–375 (2016)
Brzeźniak, Z., Manna, U.: Stochastic Landau-Lifshitz-Gilbert equation with anisotropy energy driven by pure jump noise. Comput. Math. Appl. 77(2), 1503–1512 (2019)
Brzeźniak, Z., Manna, U.: Weak solutions of a stochastic Landau-Lifshitz-Gilbert equation driven by pure jump noise. Commun. Math. Phys. 371, 1071–1129 (2019)
Brzeźniak, Z., Manna, U., Mukherjee, D.: Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations. J. Differ. Equ. 267(2), 776–825 (2019)
Caruana, M., Friz, P.: Partial differential equations driven by rough paths. J. Differ. Equ. 247(1), 140–173 (2009). https://doi.org/10.1016/j.jde.2009.01.026
Caruana, M., Friz, P.K., Oberhauser, H.: A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 28, 27–46 (2011). https://doi.org/10.1016/j.anihpc.2010.11.002
Cimrák, I.: A survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism. Arch. Comput. Methods Eng. 15, 277–309 (2008)
Deya, A., Gubinelli, M., Hofmanová, M., Tindel, S.: A priori estimates for rough PDEs with application to rough conservation laws. J. Funct. Anal. 276(12), 3577–3645 (2019)
Deya, A., Gubinelli, M., Tindel, S.: Non-linear rough heat equations. Probab. Theory Relat. Fields 153(1), 97–147 (2012). https://doi.org/10.1007/s00440-011-0341-z
Friz, P., Hairer, M.: A Course on Rough Paths: With an Introduction to Regularity Structures. Springer (2014). https://doi.org/10.1007/978-3-319-08332-2
Goldys, B., Le, K.N., Tran, T.: A finite element approximation for the stochastic Landau-Lifshitz-Gilbert equation. J. Differ. Equ. 260(2), 937–970 (2016)
Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3, e6 (2015). https://doi.org/10.1017/fmp.2015.2
Gubinelli, M., Lejay, A., Tindell, S.: Young integrals and SPDEs. Potential Anal. 35(4), 307–326 (2006). https://doi.org/10.1007/s11118-006-9013-5
Gubinelli, M., Tindel, S.: Rough evolution equations. Ann. Probab. 38(1), 1–75 (2010)
Guo, B., Pu, X.: Stochastic Landau-Lifshitz equation. Differ. Integr. Equ. 22, 251–274 (2009)
Gussetti, E., Hocquet, A.: A pathwise stochastic Landau-Lifshitz-Gilbert equation with application to large deviations (2021)
Hocquet, A., Hofmanová, M.: An energy method for rough partial differential equations. J. Differ. Equ. 265(4), 1407–1466 (2018)
Hofmanová, M., Leahy, J., Nilssen, T.: On the Navier-Stokes equation perturbed by rough transport noise. J. Evol. Equ. 19, 203–247 (2019)
Kamppeter, T., Mertens, F.G., Moro, E., Sánchez, A., Bishop, A.R.: Magnetic elements at finite temperature and large deviation theory. Kohn, R. V. and Reznikoff, M. G. and Vanden-Eijnden, E. 15(4), 223–253 (2005)
Kim, H.J., Lototsky, S.V.: Heat equation with geometric rough path potential in one space dimension: existence and regularity of solution. Commun. Korean Math. Soc. 34(3), 757–769 (2019)
Leliévre, T., Le Bris, C., Vanden-Eijnden, E.: Analyse de certains schémas de discrétisation pour des équations différentielles stochastiques contraintes. C. R. Math. Acad. Sci. Paris 346(7–8), 471–476 (2008)
Lyons, T.J.: Differential equations driven by rough signals. Revista Matematica Iberoamericana 14(2), 215–310 (1998)
Lyons, T.J., Caruana, M., Lévy, T.: Differential Equations Driven by Rough Paths, Lecture Notes in Mathematics edn. Ecole d’te de Probabilities de Saint-Flour XXXIV - 2004. Springer (2007)
Manna, U., Mukherjee, D., Panda, A.A.: Wong-Zakai approximation for the stochastic Landau-Lifshitz-Gilbert equations with anisotropy energy. J. Math. Anal. Appl. 480(1), 123384 (2019)
Neel, L.: Influence of thermal fluctuations on the magnetization of ferromagnetic small particles. Phys. Status Solidi (a) 122, 377–389 (1990)
Visintin, A.: On Landau-Lifshitz equations for ferromagnetism. Jpn. J. Appl. Math. 2(1), 69–84 (1985)
Acknowledgements
The first author of this paper is partially supported by Austrian Agency for International Cooperation in Education and Research (OeAD), Centre for International Cooperation and Mobility (ICM): ICM-2019-13458. The second author is supported by Austrian Science Foundation, project number P 32295 and the third author is supported by Marie Skłodowska-Curie Individual Fellowships H2020-MSCA-IF-2020, P 888255
Funding
Open access funding provided by Montanuniversität Leoben.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Equivalence of Systems (5) and (14) and Systems (7) and (36)
In this section, we present several auxiliary lemmas to deal with the equivalence and the regularity properties between \({\varvec{m}}\) (\({\varvec{m}}^n\) resp.) and \({\varvec{M}}\) (\({\varvec{M}}^n\) resp.). These results are essential to prove Theorem 1.
1.1 Technical Lemmas
Now we present the following lemma, which states that by using the transformation (12) we obtain that if \({\varvec{M}}\) is a solution to the system (5), then \({\varvec{m}}\) is a solution to the system (14) and vice versa.
Lemma 5
Suppose \(\varvec{M}(t)=e^{X(t)G}{\varvec{m}}(t)\), \({\varvec{m}}_0\in {\mathbb {H}}^2(D;{\mathbb {S}}^2)\) and assume that \(g\in {\mathbb {W}}^{2,\infty }(D)\). Then the following statements are equivalent
-
(i)
\({\varvec{m}}\in H^1(0,T;{\mathbb {L}}^2)\cap L^{\infty }(0,T;{\mathbb {H}}^2)\) satisfies system (14);
-
(ii)
\({\varvec{M}}\in L^{\infty }(0,T;{\mathbb {H}}^2)\) satisfies system (5) and \(({\varvec{M}},{\varvec{M}}\times {\varvec{g}})\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathbb {L}}^2)\).
Before going to the proof of Lemma 5, we give the following lemma which is essential to the proof of Lemma 5.
Lemma 6
Let X be any continuous function on [0, T] and
Assume that \({\varvec{g}}\in {\mathbb {W}}^{2,\infty }(D)\). If \({\varvec{m}}\) belongs to \(L^{\infty }(0,T;{\mathbb {H}}^2)\), then \({\varvec{M}}\) belongs to \(L^{\infty }(0,T;{\mathbb {H}}^2)\) and vice versa.
Proof
Assume that \({\varvec{m}}\) belongs to \(L^{\infty }(0,T;{\mathbb {H}}^2)\). It suffices to prove that \({\varvec{M}}_{xx}\) belongs to \(L^{\infty }(0,T;{\mathbb {L}}^2)\). Using the identity (91) we obtain
We know that
So we get
Since \({\varvec{m}}\in L^{\infty }(0,T;{\mathbb {H}}^2)\) and \({\varvec{g}}\in {\mathbb {W}}^{2,\infty }\), we have \( ({\varvec{m}}(t)\times {\varvec{g}})_{xx}\in L^{\infty }(0,T;{\mathbb {L}}^2)\). Furthermore, using
we obtain
Since \({\varvec{m}}\in L^{\infty }(0,T;{\mathbb {H}}^2)\) and \({\varvec{g}}\in {\mathbb {W}}^{2,\infty }\), then \(\frac{\partial }{\partial x}\big ({\varvec{m}}(t)\times {\varvec{g}}\big )\in L^{\infty }(0,T;{\mathbb {L}}^2)\). Therefore, we obtain \(\big (({\varvec{m}}(t)\times {\varvec{g}})\times {\varvec{g}}\big )_{xx}\in L^{\infty }(0,T;{\mathbb {L}}^2)\). Finally, \({\varvec{M}}_{xx}\) belongs to \(L^{\infty }(0,T;{\mathbb {L}}^2)\).
By similar arguments we also have the converse. \(\square \)
We are now ready to prove Lemma 5.
Proof of Lemma 5
We first assume that the statement (i) is correct. Using Lemma 6, we know \({\varvec{M}}\in L^{\infty }(0,T;{\mathbb {H}}^2)\). Using (8) and the fact that \({\varvec{m}}\in L^{\infty }(0,T;{\mathbb {H}}^2)\) and \(\displaystyle \frac{d{\varvec{m}}}{dt}\in L^2(0,T;{\mathbb {L}}^2)\), we obtain
Thus by (32), we have \(F(\cdot ,{\mathbf {m}}(\cdot ))\in L^{\infty }(0,T;{\mathbb {L}}^2)\). Therefore, it follows \({\varvec{m}}\in {{ {\mathcal {C}} }}^{2\alpha }([0,T],{\mathbb {L}}^2)\) and \(({\varvec{M}},{\varvec{M}}\times {\varvec{g}})\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathbb {L}}^2)\). Indeed,
Since \({\varvec{m}}\in {{ {\mathcal {C}} }}^{2\alpha }([0,T],{\mathbb {L}}^2)\) and \(e^{XG}\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathcal {L}}({\mathbb {L}}^2))\) we have
Furthermore, if \(Z(t):=e^{X(t)G}\) then \({\mathscr {R}}^{M}\) is given by
Therefore, we conclude
It follows that \({\mathscr {R}}^M\in {{ {\mathcal {C}} }}^{2\alpha }([0,T],{\mathbb {L}}^2)\) and we have
We now prove that \({\varvec{M}}\) satisfies system (5). It can be clearly observed that \({\varvec{M}}\) satisfies the last two equations of (5). Next, using (13a), Corollary 5 and the fact that \({\varvec{m}}\in {{ {\mathcal {C}} }}^{2\alpha }([0,T],{\mathbb {L}}^2)\cap L^2(0,T;{\mathbb {H}}^1)\), we deduce
Multiplying both sides by a test function \(\varvec{\psi }\in C_c^{\infty }(D)\) and integrating over D we obtain
Following the similar calculations in Lemma 4.1 of [15], we infer the following
Since (68) is satisfied for all \(\varvec{\psi }\in C_c^{\infty }(D)\), using integration by parts, we have for all \(t\in [0,T]\)
which completes the statement (ii). Conversely, assume statement (ii) is correct. Using Lemma 6, we have \({\varvec{m}}\in L^{\infty }(0,T;{\mathbb {H}}^2)\). Now, we will prove that \({\varvec{m}}\) satisfies system (14). It can be clearly observed that \({\varvec{m}}\) satisfies the last two equations of (14). Next, using (13b), Lemma 11, (5) and the fact that \(({\varvec{M}},{\varvec{M}}\times {\varvec{g}})\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathbb {L}}^2)\), we deduce
We note that \({\mathbb {X}}:[0,T]\rightarrow {{\mathbb {R}}}\otimes {{\mathbb {R}}}\cong {{\mathbb {R}}}\), thus we obtain
Using the property of geometric rough paths \(Sym({\mathbb {X}})_{u,v}=\frac{1}{2}X_{u,v}\otimes X_{u,v}\) we have
This implies
Multiplying both sides by a test function \(\varvec{\psi }\in C_c^{\infty }(D)\) and integrating over D we obtain
Recalling \({\varvec{m}}=e^{-X(r)G} {\varvec{M}}\) and using (96), we obtain
Thus by (104) and (93), we get
Integrating by part and applying (100), we have
Again using \({\varvec{m}}=e^{-X(r)G} {\varvec{M}}\), (104) and integrating by part, we obtain
Since it is satisfied for all \(\varvec{\psi }\in C_c^{\infty }(D)\), using integration by parts, we have for all \(t\in [0,T]\)
where \(F(t,{\varvec{m}})\) is given by (15). By standard argument, one can conclude that \({\varvec{m}}\in H^1(0,T;{\mathbb {L}}^2)\), which completes the proof.
\(\square \)
We consider the systems (7) and (36) and provide a lemma similar to Lemma 5 which establishes the equivalence of these two systems.
Lemma 7
Suppose \(\varvec{M}^{(n)}(t)=e^{X(t)G}{\varvec{m}}^{(n)}(t)\), \({\varvec{m}}_0\in {\mathbb {H}}^2(D;{\mathbb {S}}^2)\) and assume that \(g\in {\mathbb {W}}^{2,\infty }(D)\). Then the following statements are equivalent:
-
(i)
\({\varvec{m}}^{(n)}\in H^1(0,T;{\mathbb {L}}^2)\cap L^{\infty }(0,T;{\mathbb {H}}^2)\) satisfies system (36),
-
(ii)
\({\varvec{M}}^{(n)}\in L^{\infty }(0,T;{\mathbb {H}}^2)\) satisfies system (7) and \(({\varvec{M}}^{(n)},{\varvec{M}}^{(n)}{\times } {\varvec{g}}){\in } {\mathcal {D}}_{X^{(n)}}^ {2\alpha }([0,T],{\mathbb {L}}^2)\).
We note that the proof of Lemma 7 is completely analogous to the proof of Lemma 5. We use the fact that \({\varvec{X}}^{(n)}=\big (X^{(n)},~{\mathbb {X}}^{(n)}\big )\) is a geometric rough path, i.e. \(Sym({\mathbb {X}}^{(n)}_{s,t})=\frac{1}{2}X^{(n)}_{s,t}\otimes X^{(n)}_{s,t}\).
1.2 Regularity Properties
In this subsection, we first precisely give the following lemma. For proof we refer to Lemma 3.10, Lemma 3.13 and Corollary 6.2 of [8].
Lemma 8
Let \({\varvec{m}}_0\in {\mathbb {H}}^2(D;{\mathbb {S}}^2)\) and suppose that \({\varvec{m}}\) and \({\varvec{m}}^{(n)}\) satisfy the systems (14) and (36), respectively. Then, \({\varvec{m}}\) and \({\varvec{m}}^{(n)}\) belong to \(C([0,T];{\mathbb {L}}^2)\cap H^1(0,T;{\mathbb {L}}^2)\cap L^{\infty }(0,T;{\mathbb {H}}^1) \cap L^2(0,T;D(A))\).
We show \(L^{\infty }(0,T;{\mathbb {L}}^2)\)-regularity of \({\varvec{m}}_{xx}\) and \({\varvec{m}}_{xx}^{(n)}\).
Lemma 9
Let \({\varvec{m}}_0\in {\mathbb {H}}^2(D;{\mathbb {S}}^2)\) and suppose that \({\varvec{m}}\) and \({\varvec{m}}^{(n)}\) satisfy the systems (14) and (36), respectively. Then, \({\varvec{m}}_{xx}, {\varvec{m}}_{xx}^{(n)} \in L^{\infty }(0,T;{\mathbb {L}}^2)\).
Proof
We show that \({\varvec{m}}_{xx} \in L^{\infty }(0,T;{\mathbb {L}}^2)\). Denoting \(z:= {\varvec{m}}_{xx}\), system (14) can be written as
Thus one can obtain in the sense of distribution
Using the identity \(z\times z=0\) we have
By (107) we have
Our goal is to obtain the existence of the fourth derivative of \({\varvec{m}}\) in a.e. sense. Let us approximate z by a Faedo-Galerkin approximation. For \(k \in {\mathbb {N}}\), let \(z_k\) be the solution of
Note, since the system is finite dimensional, a unique solution exists by standard arguments. Taking the dot product with \(z_k\) in \({{\mathbb {R}}}^3\), we get
Using the saturation condition, i.e. (20), identity (105) and integrating over (0, 1), we obtain
Here, \(\big < \cdot ,\cdot \big>\) denotes the inner product in \({{\mathbb {R}}}^3\). Using standard calculation, we obtain
Integration by parts and the fact that that \(z_x=0\) at \(x=0\) and \(x=1\) gives
where
Choosing \(\varepsilon =\frac{\lambda _2}{5}\), we get
This yields
Lemma 8 yields \(|\varphi _2|_{L^{\infty }(0,T;{{\mathbb {R}}})}<\infty \), and by the interpolation inequality (8), we get \(\int _{0}^{t}\varphi _1(s)~ds<\infty \), for \(t\in [0,T]\). Using that \(z_0 \in {\mathbb {L}}^2\) and the estimate (73), we can show that (71) has a solution in \(L^2(0,T;{\mathbb {H}}^1) \cap L^2(0,T;{\mathbb {H}}^{-1})\). Therefore, the weak limit of \(z_k\), denoted by z, exists and satisfies (70). It follows the existence of \(z_{xx}\), i.e., that the fourth derivative of \({\mathbf {m}}\) exists. Passing to the limit as \(k \rightarrow \infty \) in (73), it can be clearly observed that z satisfies
This completes the proof. \(\square \)
Corollary 4
Let \({\varvec{m}}_0\in {\mathbb {H}}^2(D;{\mathbb {S}}^2)\) and suppose that \({\varvec{m}}\) and \({\varvec{m}}^{(n)}\) satisfy systems (14) and (36), respectively. Then \({\varvec{m}}\times {\varvec{m}}_{xx}\), \({\varvec{m}}\times ({\varvec{m}}\times {\varvec{m}}_{xx})\), \({\varvec{m}}^{(n)}\times {\varvec{m}}_{xx}^{(n)}\) and \({\varvec{m}}^{(n)}\times ({\varvec{m}}^{(n)}\times {\varvec{m}}_{xx}^{(n)})\) belong to \(L^{\infty }(0,T;{\mathbb {L}}^2)\). Furthermore, we also have \({\varvec{m}}_x\) and \({\varvec{m}}_x^{(n)}\) belong to \(L^{\infty }(0,T;{\mathbb {L}}^{\infty })\).
Proof
This is a direct consequence from Lemmas 8, 9 and inequality (8). \(\square \)
Rough Paths
In this section we summarize the most important definitions, lemma and corollary which are necessary for the existence theory and the convergence results. In order to prove the equivalence between \({\varvec{m}}\) and \({\varvec{M}}\) which is given in Lemma 5, some technical issues require us to introduce Lemmas 10, 11 and Corollary 5. These results, another contribution of the paper, have its own interest.
The following definitions (Definitions 3, 4), and Remark 2 are taken from [14].
Definition 3
For \(\alpha \in (\frac{1}{3},\frac{1}{2}]\), let \({\mathscr {C}}^{\alpha }([0,T],V)\) be the space of \(\alpha \)-Hölder continuous rough paths over V consisting of those pairs \({\mathbf {X}}:=(X,{\mathbb {X}})\) where \(X:[0,T]\rightarrow V\) and \({\mathbb {X}}:[0,T]^2\rightarrow V\otimes V\) such that
and \({\mathbb {X}}_{s,t}={\mathbb {X}}_{s,u}+{\mathbb {X}}_{u,t}+X_{s,u}\otimes X_{u,t}\) for all \(s,t.u\in [0,T]\) (Chen identity). Moreover, if \(\text {Sym}({\mathbb {X}}_{s,t})=\frac{1}{2}X_{s,t}\otimes X_{s,t}\) then we say that \({\varvec{X}}\) is \(\alpha \)–Hölder geometric rough paths and we denote the space by \({\mathscr {C}}_g^{\alpha }([0,T],V)\). Also, we define \({\mathscr {C}}_g^{0,\alpha }([0,T],V)\) as the closure of lifts of smooth paths in \({\mathscr {C}}^{\alpha }([0,T],V)\). In addition, given two rough paths \({ \mathbf{Y}}^1,{ \mathbf{Y}}^2\in {\mathscr {C}}^ \alpha ([0,T],V)\), we define the (inhomogenous) \(\alpha \)–Hölder rough path metric
In order to present an integration against rough paths, we also need to give the following definition.
Definition 4
Given a path \(X\in {{ {\mathcal {C}} }}^\alpha ([0,T],V)\), we say that \(Y\in {\mathcal {C}}^\alpha ([0,T],H)\) is controlled by X, if there exists a path \(Y'\in {{ {\mathcal {C}} }}^\alpha ([0,T],L(V,H))\) (we call Gubinelli derivative) so that the reminder term \({\mathscr {R}}^Y\) given implicitly through the relation \( Y_{s,t}=Y_s'X_{s,t} + {\mathscr {R}}_{s,t}^ Y, \) satisfies \(|{\mathscr {R}}^Y |_{2\alpha ,H}<\infty \). This defines the space of controlled rough paths denoted by \( {\mathcal {D}}_{X}^ {2\alpha }([0,T],H) \) which is a Banach space under the norm \((Y,Y') \mapsto |Y_0|+|Y_0'|+|Y,Y'|_{2 \alpha ,H}\) where
In addition, consider \({\mathbf {X}}:=(X,{\mathbb {X}}),{{\tilde{{\mathbf {X}}}}}:=({{\tilde{X}}},{{\tilde{{\mathbb {X}}}}})\in {\mathscr {C}}^{\alpha }([0,T],V)\) with \((Y,Y')\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],H)\) and \(({{\tilde{Y}}},{{\tilde{Y}}}')\in {\mathcal {D}}_{{{\tilde{X}}}}^ {2\alpha }([0,T],H)\), we define the distance
Remark 2
Let us recall that if \( (Y,Y')\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],H) \) then
and
i.e. \(\sup \limits _{s\in [0,T]}|Y_{s}|_{H}<\infty \) and \(\sup \limits _{s\in [0,T]}|Y'_{s}|_{L(V,H)}<\infty \).
We are now ready to extend Young’s integral to that of a path controlled by X against \({\varvec{X}}=(X,{\mathbb {X}})\). The definition of the rough integral \(\int Yd{\varvec{X}}\) in terms of compensated Riemann sums then immadeately suggests to define the integral of Y against \({\varvec{X}}\) by
Example 1
Let B(t), \(t\in [0,T]\) is the standard real-valued Brownian motion defined on a complete probability space \((\varOmega ,{\mathcal {F}},{\mathbb {P}})\), where \({\mathcal {F}}_t=\sigma \{B(s):s\in [0,t]\}\). For any \(\alpha \in (1/3,1/2)\) with probability one, \({\varvec{B}}^{\text {strat}}=(B,{\mathbb {B}})\in {\mathscr {C}}_g^{\alpha }([0,T],{{\mathbb {R}}})\) where \({\mathbb {B}}_{s,t}:=\int _{s}^{t} B_{s,r}\circ dB_r\) (cf. Proposition 3.4 of [14]). Moreover, if \((Y,Y')\in {\mathcal {D}}_{B(\omega )}^ {2\alpha }([0,T],{\mathbb {L}}^2) \), for \({\mathbb {P}}\)-a.s. \(\omega \in \varOmega \), then the rough integral \(\int _0^TY~d{\varvec{B}}^{\text {strat}}\) exists (cf. Corollary 5.2 of [14]).
Now we introduce the following lemma which gives us the required conditions for the integration against rough paths in (5) and (7) to be well defined.
Lemma 10
Let \({\varvec{g}}\) be as before and for any \(\alpha \in (1/3,1/2)\) we consider \({\mathbf {X}}=(X,{\mathbb {X}})\in {\mathscr {C}}^{\alpha }([0,T],{{\mathbb {R}}})\). If \((Y,Y')\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathbb {L}}^2) \), then \((Y\times {\varvec{g}},Y'\times {\varvec{g}})\in {\mathcal {D}}_{X}^ {2\alpha }([0,T],{\mathbb {L}}^2) \) and
Proof
We will prove that \(Y\times {\varvec{g}},Y'\times {\varvec{g}}\in {{ {\mathcal {C}} }}^{\alpha }([0,T],{\mathbb {L}}^2)\) and \({\mathscr {R}}_{s,t}^{Y\times g}=Y_{s,t}\times {\varvec{g}}-(Y_s\times {\varvec{g}})X_{s,t}\in {{ {\mathcal {C}} }}^{2\alpha }\). In order to prove first term we use (108), i.e.
Since \(Y\in {{ {\mathcal {C}} }}^{\alpha }([0,T],{\mathbb {L}}^2)\), we have \(Y\times {\varvec{g}}\in {{ {\mathcal {C}} }}^{\alpha }([0,T],{\mathbb {L}}^2)\). The similar way, we can prove that \(Y'\times {\varvec{g}}\in {{ {\mathcal {C}} }}^{\alpha }([0,T],{\mathbb {L}}^2)\) and \({\mathscr {R}}_{s,t}^{Y\times g}\in {{ {\mathcal {C}} }}^{2\alpha }\). Furthermore, we have
Therefore we have
\(\square \)
To prove the equivalence of weak solution between \({\varvec{m}}\) and \({\varvec{M}}\), we have to apply Itô formula, see [15]. Therefore, we present the following lemma to handle our case.
Lemma 11
Let \({\varvec{X}}=(X,{\mathbb {X}})\in \) \({\mathscr {C}}^{\alpha }([0,T],V)\) and let \((Y,Y'), (Z,Z')\in {\mathcal {D}}_{X}^ {2\alpha }\) are controlled rough paths of the form
for some controlled rough paths \((Y',Y''), (Z',Z'')\in {\mathcal {D}}_{X}^ {2\alpha }\) and some paths \(\varGamma ,\varLambda \in {{ {\mathcal {C}} }}^{2\alpha }\). Then,
which we interpret the integrals \(\int _{0}^{T} (d\varGamma _r~Z_r)\), \(\int _{0}^{T}Y_rd\varLambda _r\) and \(\int _{0}^{T}Y_r'Z_r'd{\mathbb {X}}_r\) as Young integrals. Furthermore, the last integral is given by
where the bracket \([\cdot ,\cdot ]\) is defined in [14, Definition 5.8, p. 73].
Proof
To show (80), note first that a consequence of (78), (79) and Theorem 4.10 of [14], the increments of Y and Z are of the form
Thanks to (81) and (82), we have
By taking the limit \(|{\mathcal {P}}|\rightarrow 0\), also nothing that \(\sum _{[u,v]\in {\mathcal {P}}}o(|v-u|)\rightarrow 0\), we have
Thanks to Remark 5.11 and Corollary 7.4 of [14], we obtain
By assumption \((Y,Y'), (Z,Z')\in {\mathcal {D}}_{X}^ {2\alpha }\) and \(\varGamma ,\varLambda \in {{ {\mathcal {C}} }}^{2\alpha }\), we have
Now using the Chen identity, we get
Applying Lemma 5.9 of [14], we obtain
The convergence to the Young integral in the last three integrals follows from \((Y,Y'), (Z,Z')\in {\mathcal {D}}_{X}^ {2\alpha }\) and \(\varGamma ,\varLambda \in {{ {\mathcal {C}} }}^{2\alpha }\). The proof is complete. \(\square \)
In the following corollary we write the specific assumption for Y and Z which satisfies the conditions in Lemma 5.
Corollary 5
Let \({\varvec{X}}=(X,{\mathbb {X}})\in \) \({\mathscr {C}}^{\alpha }([0,T],V)\) and let \((Y,Y')\in {\mathcal {D}}_{X}^ {2\alpha }\) be controlled rough paths of the form
for some controlled rough paths \((Y',Y'')\in {\mathcal {D}}_{X}^ {2\alpha }\) and some paths \(Z\in {{ {\mathcal {C}} }}^{2\alpha }\). Then,
which we interpret the last integrals as Young integrals.
Proof
The assertion of theorem is shown by Lemma 11. We omit the complete proof of the above Lemma. \(\square \)
A Few Properties of Operator G and \(e^{sG}\)
We recall some simple results from [4, 15].
Lemma 12
Assume that \({\varvec{g}}\in {\mathbb {L}}^{\infty }\). Let \(G:{\mathbb {L}}^2\rightarrow {\mathbb {L}}^2\) be defined by
Then the operator G is well defined and bounded linear map. Moreover, for any \(\varvec{u},~\varvec{v}\in {\mathbb {L}}^2\),
Lemma 13
For any \(s\in {\mathbb {R}}\) and \(\varvec{u},\varvec{v}\in {\mathbb {L}}^2\),
Lemma 14
Assume that \(\varvec{g}\in {\mathbb {H}}^2\). For any \(\varvec{u}\in {\mathbb {H}}^1\) and \(\varvec{v}\in {\mathbb {W}}_0^{1,\infty }\),
and
where
Lemma 15
Assume that \(\varvec{g}\in {\mathbb {H}}^2\). For any \(s\in {{\mathbb {R}}}\), \(\varvec{u}\in {\mathbb {H}}^1\) and \(\varvec{v}\in {\mathbb {W}}_0^{1,\infty }\),
where
and C is given by (99).
Some Algebraic Identities
Here, we present list all algebraic identities will be used in this paper. Suppose that \(a,b,c,d\in {{\mathbb {R}}}^3\). Then
In particular, if \(\langle a,b\rangle =0\), then
and
Corollary 6
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Fahim, K., Hausenblas, E. & Mukherjee, D. Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths. Appl Math Optim 84 (Suppl 2), 1685–1730 (2021). https://doi.org/10.1007/s00245-021-09808-1
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-021-09808-1
Keywords
- Rough paths theory
- Partial differential equation
- Landau–Lifshitz–Gilbert equations
- Wong–Zakai approximation
- Ferromagnetism