Existence results for the Landau–Lifshitz

Abstract

In this paper, the Landau–Lifshitz–Baryakhtar (LLBar) equation for magnetization dynamics in ferrimagnets is considered. We prove global existence of a periodic solutions as well as local existence and uniqueness of regular solutions. We also study the relationships between the Landau–Lifshitz–Baryakhtar equation and both Landau–Lifshitz–Bloch and harmonic map equations.

Keywords

Landau–Lifshitz–Baryakhtar equation harmonic map equation local well-posedness asymptotic behaviour

1. Introduction

Ferrimagnetism is one of the many manifestations of magnetism. It takes its name from ferrites, crystalline bodies based on iron oxide, the magnetic properties of which lie between those of antiferromagnetic materials and those of ferromagnetic materials. The magnetic properties of a material are the result of the sum of its magnetic moments. A ferrimagnetic material is a type of magnetic material where the individual magnetic moments of atoms or ions are aligned in slightly different directions, resulting in a net magnetic moment for the material even in the absence of the external magnetic field. Unlike ferromagnetic materials where the magnetic moments are fully aligned, in ferrimagnetic materials, there is a partial cancellation between the opposing magnetic moments, but the amplitudes of the magnetic moments are slightly different. A magnetization however is generally weaker than in the case of a ferromagnetic material. At Curie temperature, a ferrimagnetic material loses its spontaneous magnetization and gradually becomes paramagnetic. In other words, it can then acquire a magnetization under the effect of an external magnetic field.

For our purpose, we consider the model derived in [8,16,29] which consists on the following Landau–Lifshitz–Baryakhtar (LLBar) equation $\begin{matrix} (1) & \partial_{t} m = - γ m \times H_{eff} (m) + {\hat{λ}}_{r} (m) H_{eff} (m) - {\hat{λ}}_{i j}^{e} (m) \frac{\partial^{2} H_{eff} (m)}{\partial x_{i} x_{j}} \end{matrix}$ such that the symbol × represents the vector product in $R^{3}$ , $m$ , γ, $H_{eff} (m)$ are the magnetization field, the gyromagnetic ratio, and the effective magnetic field, respectively. The magnnetization-dependent parameters ${\hat{λ}}_{r}$ and ${\hat{λ}}^{e}$ are the relativistic and exchange relaxation tensors, respectively. We know that LLBar equation does not preserve the magnitude of the magnetization vector, while Landau–Lifshitz model guarantees this property. The effective field $H_{eff} (m)$ is expressed as $\begin{matrix} (2) & H_{eff} (m) = △ m - \frac{1}{χ_{‖}} (1 + \frac{3}{5} \frac{θ}{θ - θ_{c}} | m |^{2}) m, \end{matrix}$ where $χ_{‖}$ is the longitudinal susceptibility. The polycristalline anisotropy field which is linear on $m$ is neglected. Then, we denote the relaxation tensors as ${\hat{λ}}_{r} = λ_{r} I$ and ${\hat{λ}}^{e} = λ_{e} I$ ( $I$ is the unit tensor) where the parameters $λ_{r}$ and $λ_{e}$ are constants. Equation (1) reduces to $\begin{matrix} (3) & \partial_{t} m = - γ m \times H_{eff} (m) + λ_{r} H_{eff} (m) - λ_{e} △ H_{eff} (m) . \end{matrix}$ We replace $H_{eff} (m)$ by its value in (3) (we assume that $m$ is regular enough) to obtain $\begin{matrix} (4) & \partial_{t} m = - γ m \times △ m + (λ_{r} + λ_{e} α) △ m - λ_{r} α (1 + β | m |^{2}) m - λ_{e} △^{2} m + λ_{e} α β △ (| m |^{2} m), \end{matrix}$ where $α = \frac{1}{χ_{‖}}$ , $β = \frac{3}{5} \frac{θ}{θ - θ_{c}}$ and $△^{2}$ denotes the bilaplacian operator. Here we consider a magnetic material, in which θ exceeds the value $θ_{c}$ (the Curie temperature), and as a consequence $β > 0$ .

Recall that above the Curie temperature, the material undergoes a phase transition from a ferromagnetic to a ferrimagnetic state. The modulus of magnetization, as described in the Landau–Lifshitz theory by Baryakhtar, typically decreases as the temperature increases beyond the Curie temperature due to the diminishing alignment of magnetic moments. The modulus of magnetization is no longer preserved due to the partial cancellation of opposing magnetic moments. This leads to a reduction in the net magnetization of the material.

To equation (4), we associate the initial conditions $\begin{matrix} (5) & m (0, x) = m_{0} (x) in Ω . \end{matrix}$ Throughout this paper, we assume that $m_{0}$ is 2D-periodic with $D > 0$ is a constant, i.e, $m_{0} (x + 2 D e_{i}) = m_{0} (x)$ , where $(e_{1}, e_{2}, e_{3})$ denotes the unit orthogonal basis of $R^{3}$ and $Ω = {(- D, D)}^{3}$ . We assume that the ferrimagnetic material is represented by the domain Ω. Then, we obtain the following LLBar problem $\begin{matrix} (6) & \{\begin{array}{l} \partial_{t} m = - γ m \times △ m + σ △ m - λ_{r} α (1 + β | m |^{2}) m \\ - λ_{e} △^{2} m + λ_{e} α β △ (| m |^{2} m) in D_{T} = (0, T) \times Ω, \\ m (0, \cdot) = m_{0} in Ω, \end{array} \end{matrix}$ where $σ = : λ_{r} + λ_{e} α$ .

In the case when $λ_{e} = 0$ , the first equation of (6) becomes the Landau–Lifshitz–Bloch (LLB) equation $\begin{matrix} (7) & \partial_{t} m = - γ m \times △ m + λ_{r} △ m - λ_{r} α (1 + β | m |^{2}) m . \end{matrix}$ Significant advancements have been made in the field of micromagnetics, particularly concerning the global existence of solutions within the framework LLB equations. Results such as the existence of weak solutions for (7) with homogeneous Neumann boundary condition can be found in [21], where the proof uses Faedo–Galerkin approximations. In [4], another result concerning a finite difference scheme for the fractional in time LLB model has been established. The paper [22] provides a comprehensive analysis of weak and strong solutions to LLB coupled to a Maxwell equations with polarization. The LLB equation including adiabatic and non-adiabatic torques is considered in [6] where some results on existence, uniqueness and asymptotic behavior are provided. The question of local existence and uniqueness of regular solutions to LLB equation is addressed in [7] while regular solutions for the compressible LLB equation is investigated in [2]. Some asymptotic behavior results for the LLB equation are presented in [20]. Finally, there has been a notable surge in interest regarding the stochastic versions of the LLB equations. These stochastic formulations introduce a crucial element of randomness into the dynamics of magnetization, reflecting the inherent uncertainty and thermal fluctuations present at the nanoscale regime. In this context, we refer for example to [9,18].

When $λ_{e} = λ_{r} = 0$ , we obtain the well known Heisenberg equation $\begin{matrix} (8) & \partial_{t} m = - γ m \times △ m . \end{matrix}$

This paper aims to build upon the insights gained from references above and contribute to the current research on the existence of solutions to (6). Thereafter, for the classical Banach spaces, we will use the following notations: $\begin{array}{c} L^{p} (Ω) : = {v = (v_{1}, v_{2}, v_{3}), v_{i} \in L^{p} (Ω), i = 1, 2, 3}, \\ W^{m, p} (Ω) : = {v = (v_{1}, v_{2}, v_{3}), v_{i} \in W^{m, p} (Ω), i = 1, 2, 3}, \end{array}$ and $\begin{matrix} W_{0}^{m, p} (Ω) : = {v = (v_{1}, v_{2}, v_{3}), v_{i} \in W_{0}^{m, p} (Ω), i = 1, 2, 3} . \end{matrix}$ We also define the 2D-periodic spaces $L_{per}^{p} (Ω)$ and $W_{per}^{m, p} (Ω)$ by $\begin{matrix} L_{per}^{p} (Ω) : = {v \in L^{p} (Ω), v is 2D-periodic}, \end{matrix}$ and $\begin{matrix} W_{per}^{m, p} (Ω) : = {v \in W^{m, p} (Ω), v is 2D-periodic} . \end{matrix}$ These spaces are equipped respectively with the usual norm denoted by $‖ \cdot ‖_{p}$ and $‖ \cdot ‖_{m, p}$ . For $p = 2$ , $n ⩾ 1$ , we put $H_{per}^{n} (Ω) = W_{per}^{n, 2} (Ω)$ and $H_{0}^{n} (Ω) = W_{0}^{n, 2} (Ω)$ . The dual spaces of $W_{0}^{m, p} (Ω)$ and $H_{0}^{n} (Ω)$ are denoted respectively by $W^{- m, p^{'}} (Ω)$ and $H^{- n} (Ω)$ where $\frac{1}{p} + \frac{1}{p^{'}} = 1$ . In the rest of this paper, for a real numbers m, p, q such that $p, q ⩾ 1$ , we use the notation $L_{T}^{q} (W^{m, p} (Ω)) : = L^{q} (0, T, W^{m, p} (Ω))$ with the convention that $W^{0, p} (Ω) = L^{p} (Ω)$ .

Before proceeding, let us give the definition of weak solutions to problem (6).

Definition 1.
Consider $m_{0} \in L_{per}^{2} (Ω)$ . We call a weak solution of problem (6) a function $m$ satisfying
For all $T > 0$ , $m \in L_{T}^{\infty} (L_{per}^{2} (Ω)) \cap L_{T}^{2} (H_{per}^{2} (Ω)) \cap L_{T}^{4} (L_{per}^{4} (Ω))$ and $\partial_{t} m \in L_{T}^{\frac{4}{3}} (H^{- 2} (Ω))$ ;

For all $ϕ \in C_{0}^{\infty} (D_{T})$ which is 2D-periodic in space, there holds $\begin{aligned} \int_{D_{T}} \partial_{t} m \cdot ϕ d x d t & = γ \int_{D_{T}} m \times \nabla m \cdot \nabla ϕ d x d t - σ \int_{D_{T}} \nabla m \cdot \nabla ϕ d x d t \\ - λ_{e} \int_{D_{T}} △ m \cdot △ ϕ d x d t - λ_{r} α \int_{D_{T}} (1 + β | m |^{2}) m \cdot ϕ d x d t \\ - λ_{e} α β \int_{D_{T}} | m |^{2} \nabla m \cdot \nabla ϕ d x d t - λ_{e} α β \int_{D_{T}} \nabla (| m |^{2}) \otimes m \cdot \nabla ϕ d x d t . \end{aligned}$

The rest of the paper is organized as follows. Global existence of weak solution to the model (6) is proved in Section 2 by essentially using Faedo–Galerkin method. In Section 3, we prove the existence of a regular solution for (6) while Section 4 reveals the relationships between the LLBar equation and both LLB and Heisenberg equations. Some concluding remarks are presented in the last section.
2. Weak solutions to LLBar equation

This portion is dedicated to prove a global existence of weak solutions result.

Theorem 1.
Let $T > 0$ be fixed and $m_{0} \in L_{per}^{2} (Ω)$ . Then there exists a global weak solution of the problem ( 6 ) in the sense of Definition 1 .

To prove the existence of weak solutions of the problem (6) we follow [1,3,5,15,19,21,25,28]. We use Faedo–Galerkin method, we approximate $m$ by $m^{K}$ . We set $\begin{matrix} m^{K} (t, x) = \sum_{j = 1}^{K} a_{j} (t) ϕ_{j} (x) \end{matrix}$ where the $ϕ_{j}$ are the orthonormalized eigenfunctions of the eigenvalue problems $\begin{matrix} \{\begin{array}{l} - △ ϕ_{j} = μ_{j} ϕ_{j}, in Ω, j = 1, 2, \dots, \\ ϕ_{j} (x - D e_{i}) = ϕ_{j} (x + D e_{i}) . \end{array} \end{matrix}$ Here $a_{j} (t)$ are $R^{3}$ -valued vectors. Let $S_{K} = span {ϕ_{1}, \dots, ϕ_{K}}$ and $P_{K} : L^{2} (Ω) \to S_{K}$ is the orthogonal projection. Recall that each of the $ϕ_{i}$ also satisfies $\begin{matrix} \{\begin{array}{l} △^{2} ϕ_{j} = μ_{j}^{2} ϕ_{j}, in Ω, j = 1, 2, \dots, \\ ϕ_{j} (x - D e_{i}) = ϕ_{j} (x + D e_{i}) . \end{array} \end{matrix}$

We obtain the following approximated problem $\begin{matrix} (9) & \{\begin{array}{l} \partial_{t} m^{K} = σ △ m^{K} - λ_{e} △^{2} m^{K} \\ + P_{K} [- γ m^{K} \times △ m^{K} - λ_{r} α (1 + β | m^{K} |^{2}) m^{K} + λ_{e} α β △ (| m^{K} |^{2} m^{K})], \\ m^{K} (0, \cdot) = P_{K} (m_{0}) . \end{array} \end{matrix}$ The vector function $m^{K} (t, x) = \sum_{j = 1}^{K} a_{j} (t) ϕ_{j} (x)$ is a solution of (9) if the components of the vector $a (t) = (a_{1} (t), a_{2} (t), \dots, a_{K} (t))$ is a solution of the following ordinary differential system $\begin{matrix} (10) & \{\begin{array}{l} \partial_{t} a_{j} = {(F_{K} (a))}_{j}, j = 1, \dots, K, \\ a_{j} (0) = {(a_{0})}_{j}, \end{array} \end{matrix}$ where $a_{0}$ is the projection of $m_{0}$ on $S_{K}$ , and the components of the vector $F_{K} (a)$ is defined by $\begin{aligned} {(F_{K} (a))}_{j} & = ⟨ - γ m^{K} \times △ m^{K} + σ △ m^{K} - λ_{r} α (1 + β {| m^{K} |}^{2}) m^{K} - λ_{e} △^{2} m^{K} \\ + λ_{e} α β △ ({| m^{K} |}^{2} m^{K}), ϕ_{j} ⟩, j = 1, \dots, K . \end{aligned}$ Here the inner product in $L^{2} (Ω)$ is denoted by $⟨ \cdot, \cdot ⟩$ . Standard results on nonlinear ordinary differential equations allows to get local existence of a solution to (10).

In order to ensure that all the functions are defined on interval $[0, T]$ for all $T > 0$ , and hence a global solution to the problem (6) by taking $K \to \infty$ , the following estimates need to be hold.
2.1. Some a priori estimates

Lemma 1.
Let $m_{0} \in L_{per}^{2} (Ω)$ . Then for any $0 < T < \infty$ , for the solution $m^{K}$ to the approximated problem, the following estimates hold $\begin{matrix} (11) & \begin{aligned} \frac{1}{2} \int_{Ω} {| m^{K} |}^{2} d x + σ \int_{0}^{t} \int_{Ω} {| \nabla m^{K} |}^{2} d x d t + λ_{e} \int_{0}^{t} \int_{Ω} {| △ m^{K} |}^{2} d x d t \\ + λ_{r} α \int_{0}^{t} \int_{Ω} {| m^{K} |}^{2} (1 + β {| m^{K} |}^{2}) d x d t + α λ_{e} β \int_{0}^{t} \int_{Ω} {| \nabla m^{K} |}^{2} {| m^{K} |}^{2} d x d t \\ + \frac{α λ_{e} β}{2} \int_{0}^{t} \int_{Ω} {| \nabla ({| m^{K} |}^{2}) |}^{2} d x d t = \frac{1}{2} \int_{Ω} {| m^{K} (0, \cdot) |}^{2} d x, \end{aligned} \end{matrix}$ and $\begin{matrix} (12) & {‖ \partial_{t} m^{K} ‖}_{L_{T}^{\frac{4}{3}} (H^{- 2} (Ω))} ⩽ C, \end{matrix}$ for some constant C independent of K.
Proof.
The inner product of the two sides of (9) with $m^{K}$ , and integrating by parts, allow to obtain $\begin{matrix} (13) & \begin{aligned} \frac{1}{2} \frac{d}{d t} \int_{Ω} {| m^{K} |}^{2} d x + σ \int_{Ω} {| \nabla m^{K} |}^{2} d x + λ_{e} \int_{Ω} {| △ m^{K} |}^{2} d x \\ + λ_{r} α \int_{Ω} (1 + β {| m^{K} |}^{2}) {| m^{K} |}^{2} d x + α λ_{e} β \int_{Ω} {| m^{K} |}^{2} {| \nabla m^{K} |}^{2} d x \\ + \frac{α λ_{e} β}{2} \int_{Ω} {| \nabla ({| m^{K} |}^{2}) |}^{2} d x = 0 . \end{aligned} \end{matrix}$ Integrating with respect to t, we get (11).

In order to prove (12), let $ψ \in L_{T}^{4} (H_{per}^{2} (Ω))$ and taking the inner product of the two sides of the first Eq. of (9) with $ψ$ , we obtain $\begin{aligned} \int_{D_{T}} \partial_{t} m^{K} \cdot ψ d x d t & = γ \int_{D_{T}} m^{K} \times \nabla m^{K} \cdot \nabla ψ d x d t \\ - σ \int_{D_{T}} \nabla m^{K} \cdot \nabla ψ d x d t - λ_{e} \int_{D_{T}} △ m^{K} \cdot △ ψ d x d t \\ - λ_{r} α \int_{D_{T}} (1 + β {| m^{K} |}^{2}) m^{K} \cdot ψ d x d t - λ_{e} α β \int_{D_{T}} {| m^{K} |}^{2} \nabla m^{K} \cdot \nabla ψ d x d t \\ (14) & - λ_{e} α β \int_{D_{T}} \nabla ({| m^{K} |}^{2}) \otimes m^{K} \cdot \nabla ψ d x d t . \end{aligned}$ By Young’s inequality, we get $\begin{aligned} | \int_{D_{T}} \partial_{t} m^{K} \cdot ψ d x d t | & ⩽ γ {‖ m^{K} ‖}_{L_{T}^{4} (L_{per}^{4} (Ω))} {‖ \nabla m^{K} ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} ‖ \nabla ψ ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} \\ + σ {‖ \nabla m^{K} ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} ‖ \nabla ψ ‖_{L_{T}^{2} (L_{per}^{2} (Ω))} \\ + λ_{e} {‖ △ m^{K} ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} ‖ △ ψ ‖_{L_{T}^{2} (L_{per}^{2} (Ω))} \\ + λ_{r} α {‖ (1 + β {| m^{K} |}^{2}) ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} {‖ m^{K} ‖}_{L_{T}^{4} (L_{per}^{4} (Ω))} ‖ ψ ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} \\ + λ_{e} α β {‖ | m^{K} | \nabla m^{K} ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} {‖ m^{K} ‖}_{L_{T}^{4} (L_{per}^{4} (Ω))} ‖ \nabla ψ ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} \\ + λ_{e} α β {‖ \nabla ({| m^{K} |}^{2}) ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} {‖ m^{K} ‖}_{L_{T}^{4} (L_{per}^{4} (Ω))} ‖ \nabla ψ ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} . \end{aligned}$ By estimates (11), and for some constant C independent of K, we have $\begin{matrix} (15) & | \int_{D_{T}} \partial_{t} m^{K} \cdot ψ d x d t | ⩽ C ‖ ψ ‖_{L_{T}^{4} (H_{per}^{2} (Ω))} . \end{matrix}$ This complete the proof of Lemma 1. The next step is to prove the Theorem 1. □
2.2. Convergence of the Galerkin approximation

This subsection is dedicated to the convergence of the approximate solutions by letting $K \to \infty$ in the above analysis. Let us first recall a compactness lemma in order to deal with nonlinear terms. More details can be found in Lions [23].

Lemma 2.
If $Y_{0}$ , $Y$ , $Y_{1}$ are three Banach spaces satisfying $\begin{matrix} Y_{0} ↪ Y ↪ Y_{1} \end{matrix}$ where the injections are continuous and $\begin{matrix} Y_{0}, Y_{1} are separable reflexive with Y_{0} ↪ Y is compact . \end{matrix}$ Let $\begin{matrix} W = {v / v \in L_{T}^{σ_{0}} (Y_{0}), and \frac{d v}{d t} \in L_{T}^{σ_{1}} (Y_{1})} \end{matrix}$ with T is finite and $1 < σ_{0}, σ_{1} < \infty$ . Then $W$ equipped with the norm $\begin{matrix} ‖ v ‖_{W} = ‖ v ‖_{L_{T}^{σ_{0}} (Y_{0})} + {‖ \frac{d v}{d t} ‖}_{L_{T}^{σ_{1}} (Y_{1})} \end{matrix}$ is a Banach space and the embedding $W ↪ L_{T}^{σ_{0}} (Y)$ is compact. When $σ_{0} = \infty$ , $1 < σ_{1} ⩽ \infty$ , the embedding $W ↪ C ([0, T], Y)$ is compact.

Using Lemma 1 and the fact that $m^{K} (0, \cdot) \to m_{0}$ in $L^{2} (Ω)$ we conclude that the right-hand side of (11) is uniformly bounded with respect to K. Then we get the uniform boundedness of ${(m^{K})}_{K}$ in $L_{T}^{2} (H_{per}^{2} (Ω))$ , and of ${(\partial_{t} m^{K})}_{K}$ in $L_{T}^{\frac{4}{3}} (H^{- 2} (Ω))$ .

Applying the compactness Lemma 2 with $Y_{0} = H_{per}^{2} (Ω)$ , $Y = H_{per}^{1} (Ω)$ and $Y_{1} = H^{- 2} (Ω)$ , for a subsequence still labeled by $m^{K}$ ), the following convergences hold $\begin{array}{c} (16) & m^{K} ⇀ m weakly in L_{T}^{2} (H_{per}^{2} (Ω)), \\ (17) & \partial_{t} m^{K} ⇀ \partial_{t} m weakly in L_{T}^{\frac{4}{3}} (H^{- 2} (Ω)), \\ m^{K} \to m strongly in L_{T}^{2} (H_{per}^{1} (Ω)) . \end{array}$ Consequently $\begin{matrix} (18) & (m^{K}, \nabla m^{K}) \to (m, \nabla m) strongly in {(L_{T}^{2} (L_{per}^{2} (Ω)))}^{2} and a.e in D_{T} . \end{matrix}$ Let $ϕ \in C^{\infty} (D_{T})$ which is 2D-periodic. Multiplying the first Eq. of (9) by $ϕ$ and integrating by parts on Ω, we obtain $\begin{matrix} (19) & \begin{aligned} \int_{D_{T}} \partial_{t} m^{K} \cdot ϕ d x d t & = γ \int_{D_{T}} m^{K} \times \nabla m^{K} \cdot \nabla ϕ d x d t \\ - σ \int_{D_{T}} \nabla m^{K} \cdot \nabla ϕ d x d t - λ_{e} \int_{D_{T}} △ m^{K} \cdot △ ϕ d x d t \\ - λ_{r} α \int_{D_{T}} (1 + β {| m^{K} |}^{2}) m^{K} \cdot ϕ d x d t \\ - λ_{e} α β \int_{D_{T}} {| m^{K} |}^{2} \nabla m^{K} \cdot \nabla ϕ d x d t \\ - λ_{e} α β \int_{D_{T}} \nabla ({| m^{K} |}^{2}) \otimes m^{K} \cdot \nabla ϕ d x d t . \end{aligned} \end{matrix}$ Convergence of the first, the third and the fourth term is obvious since they are linear, $\begin{array}{c} \int_{D_{T}} \partial_{t} m^{K} \cdot ϕ d x d t \to \int_{D_{T}} \partial_{t} m \cdot ϕ d x d t, \\ \int_{D_{T}} \nabla m^{K} \cdot \nabla ϕ d x d t \to \int_{D_{T}} \nabla m \cdot \nabla ϕ d x d t, \\ \int_{D_{T}} △ m^{K} \cdot △ ϕ d x d t \to \int_{D_{T}} △ m \cdot △ ϕ d x d t . \end{array}$ The boundedness of Ω and of $(m^{K})$ in $L_{T}^{4} (L_{per}^{4} (Ω))$ allows to deduce the boundedness of ${(1 + β | m^{K} |^{2})}_{K}$ in $L_{T}^{2} (L^{2} (Ω))$ hence $1 + β | m^{K} |^{2} ⇀ ζ$ in $L^{2} (D_{T})$ . From[[23], Lemma 1.3, Chap. 1], one can easily show that $ζ = 1 + β | m |^{2}$ . The weak convergence of $m^{K}$ allows to get $\begin{matrix} \int_{D_{T}} (1 + β {| m^{K} |}^{2}) m^{K} \cdot ϕ d x d t \to \int_{D_{T}} (1 + β | m |^{2}) m \cdot ϕ d x d t . \end{matrix}$ For the second term, the weak convergence of ${(\nabla m^{K})}_{K}$ and the strong convergence of ${(m^{K})}_{K}$ in $L_{T}^{2} (L_{per}^{2} (Ω))$ , enable to obtain $\begin{matrix} \int_{D_{T}} m^{K} \times \nabla m^{K} \cdot \nabla ϕ d x d t \to \int_{D_{T}} m \times \nabla m \cdot \nabla ϕ d x d t . \end{matrix}$ Similarly $\begin{aligned} | \int_{D_{T}} {| m^{K} |}^{2} \nabla m^{K} \cdot \nabla ϕ d x d t - \int_{D_{T}} | m |^{2} \nabla m \cdot \nabla ϕ d x d t | \\ ⩽ {‖ m^{K} ‖}_{L_{T}^{4} (L_{per}^{4} (Ω))}^{2} {‖ \nabla m^{K} - \nabla m ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} ‖ \nabla ϕ ‖_{L_{T}^{\infty} (L_{per}^{\infty} (Ω))} \\ + | \int_{D_{T}} ({| m^{K} |}^{2} - | m |^{2}) \nabla m \cdot \nabla ϕ d x d t |, \end{aligned}$ which tend to 0 when K tend to infinity, thanks to the strong convergence of ${(\nabla m^{K})}_{K}$ in $L_{T}^{2} (L^{2} (Ω))$ and the weak convergence of ${(| m^{K} |^{2})}_{K}$ in $L_{T}^{2} (L_{per}^{2} (Ω))$ with $\nabla m \cdot \nabla ϕ \in L_{T}^{2} (L_{per}^{2} (Ω))$ since $\nabla ϕ \in L_{T}^{\infty} (L_{per}^{\infty} (Ω))$ .

To pass to the limit in the last term of (19), we need the convergence almost everywhere of ${(\nabla m^{K})}_{K}$ in $D_{T}$ due to the presence of the term $\nabla (| m^{K} |^{2})$ , hence $\begin{aligned} | \int_{D_{T}} \nabla ({| m^{K} |}^{2}) \otimes m^{K} \cdot \nabla ϕ d x d t - \int_{D_{T}} \nabla (| m |^{2}) \otimes m \cdot \nabla ϕ d x d t | \\ ⩽ {‖ \nabla ({| m^{K} |}^{2}) ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} {‖ m^{K} - m ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} ‖ \nabla ϕ ‖_{L_{T}^{\infty} (L_{per}^{\infty} (Ω))} \\ + | \int_{D_{T}} (\nabla ({| m^{K} |}^{2}) - \nabla (| m |^{2})) \otimes m \cdot \nabla ϕ d x d t |, \end{aligned}$ and the last term in this inequality tends to zero when K tends to infinity because $\nabla (| m^{K} |^{2}) ⇀ \nabla (| m |^{2})$ weakly in $L_{T}^{2} (L_{per}^{2} (Ω))$ as a consequence of (18) and $\nabla ϕ \in L_{T}^{\infty} (L_{per}^{\infty} (Ω))$ .

The limit as $K \to \infty$ in (19), permit to get $\begin{matrix} (20) & \begin{aligned} \int_{D_{T}} \partial_{t} m \cdot ϕ d x d t & = γ \int_{D_{T}} m \times \nabla m \cdot \nabla ϕ d x d t \\ - σ \int_{D_{T}} \nabla m \cdot \nabla ϕ d x d t - λ_{e} \int_{D_{T}} △ m \cdot △ ϕ d x d t \\ - λ_{r} α \int_{D_{T}} (1 + β | m |^{2}) m \cdot ϕ d x d t - λ_{e} α β \int_{D_{T}} | m |^{2} \nabla m \cdot \nabla ϕ d x d t \\ - λ_{e} α β \int_{D_{T}} \nabla (| m |^{2}) \otimes m \cdot \nabla ϕ d x d t . \end{aligned} \end{matrix}$ Remark 1.
The a priori estimates holds true even for the solution $m$ to problem (6). Accordingly $m$ satisfies the estimates of Lemma 1.

3. Regular solution to LLBar equation

Here we shall investigate a local well-posedness result for the problem (6). More precisely, we have

Theorem 2.
Assume $m_{0} \in H_{per}^{2} (Ω)$ . Then there exists $T^{} > 0$ and a unique strong solution to (* 6 ) such that for all $T < T^{}$ , $\begin{matrix} m \in C ([0, T], H_{per}^{2} (Ω)) \cap L_{T}^{2} (H_{per}^{4} (Ω)), \partial_{t} m \in L_{T}^{2} (L_{per}^{2} (Ω)), \end{matrix}$ and for all* $ϕ \in L_{T}^{2} (H_{per}^{2} (Ω))$ , there holds $\begin{aligned} \int_{D_{T}} \partial_{t} m \cdot ϕ d x d t & = γ \int_{D_{T}} m \times \nabla m \cdot \nabla ϕ d x d t - σ \int_{D_{T}} \nabla m \cdot \nabla ϕ d x d t \\ - λ_{e} \int_{D_{T}} △ m \cdot △ ϕ d x d t - λ_{r} α \int_{D_{T}} (1 + β | m |^{2}) m \cdot ϕ d x d t \\ - λ_{e} α β \int_{D_{T}} | m |^{2} \nabla m \cdot \nabla ϕ d x d t - λ_{e} α β \int_{D_{T}} \nabla (| m |^{2}) \otimes m \cdot \nabla ϕ d x d t . \end{aligned}$

Let us recall some preliminary results. Lemma 3.
Let $g : R^{+} \times R \to R$ continuous and locally lipschitz with respect to its second variable. Let $Θ : [0, T^{} [\to R$ be the maximal solution of the Cauchy problem:* $\begin{matrix} \{\begin{array}{l} Θ^{'} = g (t, Θ), \\ Θ (0) = Θ_{0} . \end{array} \end{matrix}$ Let $Φ : R^{+} \to R$ , $C^{1}$ such that $\begin{matrix} \{\begin{array}{l} Φ^{'} (t) ⩽ g (t, Φ (t)), \\ Φ (0) ⩽ Φ_{0} . \end{array} \end{matrix}$ Then $\begin{matrix} \forall t \in [0, T^{} [, Φ (t) ⩽ Θ (t) . \end{matrix}$
Proposition 1.
One can find a constant C such that for all K, the orthogonal projection* $P_{K}$ satisfies
$\forall u \in H^{1} (Ω)$ , $‖ P_{K} (u) ‖_{1, 2} ⩽ ‖ u ‖_{1, 2}$ .

$\forall u \in H^{k} (Ω)$ with $\partial_{ν} u = 0$ on $\partial Ω$ , $‖ P_{K} (u) ‖_{k, 2} ⩽ C ‖ u ‖_{k, 2}$ for $k = 2, 3, 4, 5$ .

Proof.
The proof of $H^{k}$ estimates for $k = 1, 2, 3$ can be found in [10]. We prove in the same way the $H^{4}$ and $H^{5}$ estimates. First, we write $u = P_{K} (u) + R_{K} (u)$ , where $R_{K} (u) \in S_{K}^{⊥}$ . For $k = 4$ , since $\partial_{ν} △ P_{K} (u) = \partial_{ν} △^{2} P_{K} (u) = 0$ on $\partial Ω$ , we have $\begin{aligned} {‖ △^{2} P_{K} (u) ‖}_{2}^{2} & = - \int_{Ω} \nabla △ P_{K} (u) \cdot \nabla △^{2} P_{K} (u) d x \\ = \int_{Ω} △ P_{K} (u) \cdot △^{3} P_{K} (u) d x \\ = \int_{Ω} u \cdot △^{4} P_{K} (u) d x \\ = - \int_{Ω} \nabla u \cdot \nabla △^{3} P_{K} (u) d x \\ = \int_{Ω} △ u \cdot △^{3} P_{K} (u) d x \\ = \int_{Ω} △^{2} u \cdot △^{2} P_{K} (u) d x \\ ⩽ {‖ △^{2} u ‖}_{2} {‖ △^{2} P_{K} (u) ‖}_{2}, \end{aligned}$ hence $‖ △^{2} P_{K} (u) ‖_{2} ⩽ ‖ △^{2} u ‖_{2}$ . Then $\begin{matrix} {‖ P_{K} (u) ‖}_{4, 2} ⩽ C ‖ u ‖_{4, 2} . \end{matrix}$ In the same manner, we prove that $‖ \nabla △^{2} P_{K} (u) ‖_{2} ⩽ ‖ \nabla △^{2} u ‖_{2}$ , then we deduce that the property is true for $k = 5$ . This complete the proof of the Proposition 1. □
Lemma 4 ([11,17]).

Let Ω be a regular bounded set of $R^{3}$ . There exists a constant C such that for all $u \in H^{2} (Ω)$ with $\partial_{ν} u = 0$ on $\partial Ω$ , $\begin{array}{c} (21) & ‖ u ‖_{2, 2} ⩽ C {(‖ u ‖_{2}^{2} + ‖ △ u ‖_{2}^{2})}^{\frac{1}{2}}, \\ (22) & ‖ \nabla u ‖_{1, 2} ⩽ C {(‖ \nabla u ‖_{2}^{2} + ‖ △ u ‖_{2}^{2})}^{\frac{1}{2}}, \end{array}$ and $\forall u \in H^{3} (Ω)$ with $\partial_{ν} u = 0$ on $\partial Ω$ , $\begin{matrix} (23) & ‖ \nabla u ‖_{2, 2} ⩽ C {(‖ \nabla u ‖_{2}^{2} + ‖ △ u ‖_{2}^{2} + ‖ \nabla △ u ‖_{2}^{2})}^{\frac{1}{2}} . \end{matrix}$

Lemma 5 ([11,17]).

Let Ω be a regular bounded domain of $R^{3}$ . There exists a constant C such that $\forall u \in H^{2} (Ω)$ with $\partial_{ν} u = 0$ on $\partial Ω$ , $\begin{array}{c} (24) & ‖ u ‖_{\infty} ⩽ C {(‖ u ‖_{2}^{2} + ‖ △ u ‖_{2}^{2})}^{\frac{1}{2}}, \\ (25) & ‖ \nabla u ‖_{6} ⩽ C {(‖ u ‖_{2}^{2} + ‖ △ u ‖_{2}^{2})}^{\frac{1}{2}}, \\ (26) & ‖ \nabla u ‖_{4} ⩽ C ‖ u ‖_{\infty} {(‖ u ‖_{2}^{2} + ‖ △ u ‖_{2}^{2})}^{\frac{1}{2}}, \end{array}$ and $\forall u \in H^{3} (Ω)$ with $\partial_{ν} u = 0$ on $\partial Ω$ , $\begin{matrix} (27) & {‖ D^{2} u ‖}_{3} ⩽ C ({(‖ u ‖_{2}^{2} + ‖ △ u ‖_{2}^{2})}^{\frac{1}{2}} + {(‖ u ‖_{2}^{2} + ‖ △ u ‖_{2}^{2})}^{\frac{1}{4}} ‖ \nabla △ u ‖_{2}^{\frac{1}{2}}) . \end{matrix}$

3.1. Proof of Theorem 2

Now we prove result of theorem 2. We first provide some uniform estimates. We take the inner product of the two sides of (9) with $m^{K}$ and $△^{2} m^{K}$ respectively, integrating by parts, we obtain $\begin{matrix} (28) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ m^{K} ‖}_{2}^{2} + σ {‖ \nabla m^{K} ‖}_{2}^{2} + λ_{e} {‖ △ m^{K} ‖}_{2}^{2} + α λ_{e} β \int_{Ω} {| \nabla m^{K} |}^{2} {| m^{K} |}^{2} d x \\ + λ_{r} α \int_{Ω} {| m^{K} |}^{2} (1 + β {| m^{K} |}^{2}) d x + \frac{α λ_{e} β}{2} {‖ \nabla ({| m^{K} |}^{2}) ‖}_{2}^{2} = 0, \end{aligned} \end{matrix}$ and $\begin{matrix} (29) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ △ m^{K} ‖}_{2}^{2} + σ {‖ \nabla △ m^{K} ‖}_{2}^{2} + λ_{e} {‖ △^{2} m^{K} ‖}_{2}^{2} + λ_{r} α β {‖ △ m^{K} ‖}_{2}^{2} \\ ⩽ γ {‖ △ m^{K} ‖}_{2} {‖ m^{K} ‖}_{\infty} {‖ △^{2} m^{K} ‖}_{2} + λ_{r} α β {‖ m^{K} ‖}_{\infty}^{3} {‖ △^{2} m^{K} ‖}_{2} \\ + λ_{e} α β \int_{Ω} | △^{2} m^{K} | | △ ({| m^{K} |}^{2} m^{K}) | d x, \end{aligned} \end{matrix}$ where $\begin{matrix} \int_{Ω} | △^{2} m^{K} | | △ ({| m^{K} |}^{2} m^{K}) | d x ⩽ C ({‖ △ m^{K} ‖}_{2} {‖ m^{K} ‖}_{\infty}^{2} + {‖ \nabla m^{K} ‖}_{4}^{2} {‖ m^{K} ‖}_{\infty}) {‖ △^{2} m^{K} ‖}_{2} . \end{matrix}$ Then $\begin{matrix} (30) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ △ m^{K} ‖}_{2}^{2} + σ {‖ \nabla △ m^{K} ‖}_{2}^{2} + λ_{e} {‖ △^{2} m^{K} ‖}_{2}^{2} + λ_{r} α β {‖ △ m^{K} ‖}_{2}^{2} \\ ⩽ C ({‖ m^{K} ‖}_{\infty}^{3} + {‖ △ m^{K} ‖}_{2} {‖ m^{K} ‖}_{\infty} + {‖ △ m^{K} ‖}_{2} {‖ m^{K} ‖}_{\infty}^{2} \\ + {‖ \nabla m^{K} ‖}_{4}^{2} {‖ m^{K} ‖}_{\infty}) {‖ △^{2} m^{K} ‖}_{2} \\ ⩽ C (({‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2}) + {({‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{\frac{3}{2}}) {‖ △^{2} m^{K} ‖}_{2} . \end{aligned} \end{matrix}$ Summing (28) and (30) and absorbing $‖ △^{2} m^{K} ‖_{2}$ , we obtain in particular that there exists a constant C such that $\begin{matrix} (31) & \begin{aligned} \frac{d}{d t} ({‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2}) + {‖ △^{2} m^{K} ‖}_{2}^{2} \\ ⩽ C (({‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2}) + {({‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{\frac{3}{2}}) . \end{aligned} \end{matrix}$ We set $y_{K} (t) = ‖ △ m^{K} ‖_{2}^{2} + ‖ m^{K} ‖_{2}^{2}$ . We have proven that for all K $\begin{matrix} \{\begin{array}{l} \frac{d}{d t} y_{K} (t) ⩽ C (y_{K}^{2} (t) + y_{K}^{3} (t)), \\ y_{K} (0) ⩽ C (‖ △ m_{0} ‖_{2}^{2} + ‖ m_{0} ‖_{2}^{2}) . \end{array} \end{matrix}$ Consider Θ the maximal solution to the following ordinary differential equation $\begin{matrix} \{\begin{array}{l} Θ^{'} (t) = C (Θ^{2} (t) + Θ^{3} (t)), \\ Θ (0) = C (‖ △ m_{0} ‖_{2}^{2} + ‖ m_{0} ‖_{2}^{2}) . \end{array} \end{matrix}$ and let $T^{*}$ the maximal existence time of θ. From the comparison lemma, we get $\begin{matrix} \forall t \in [0, T^{*} [, y_{K} (t) ⩽ Θ (t) . \end{matrix}$ Then there exists a constant C wich depend only on the size of $m_{0}$ in $H_{per}^{2} (Ω)$ such that for all $T < T^{*}$ we have $\begin{matrix} (32) & sup_{t \in [0, T]} {‖ m^{K} ‖}_{2, 2}^{2} ⩽ C . \end{matrix}$ Using again (30), we deduce that $\begin{matrix} \int_{0}^{T} ({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △^{2} m^{K} ‖}_{2}^{2}) d t ⩽ C . \end{matrix}$ Since $m^{K} \in L_{T}^{\infty} (H_{per}^{2} (Ω))$ and $Ω \subset R^{3}$ , we can write the term $△ (| m^{K} |^{2} m^{K})$ in the form $\begin{matrix} (33) & \begin{aligned} △ ({| m^{K} |}^{2} m^{K}) & = 2 (m^{K} \cdot △ m^{K}) m^{K} + 2 {| \nabla m^{K} |}^{2} m^{K} \\ + 4 (m^{K} \cdot \nabla m^{K}) \nabla m^{K} + {| m^{K} |}^{2} △ m^{K} . \end{aligned} \end{matrix}$ Consequently, by using the continuous embedding $L_{T}^{\infty} (H_{per}^{2} (Ω)) ↪ L_{T}^{\infty} (L_{per}^{\infty} (Ω))$ , we deduce from (32) and (33) that the right hand side of (9) is uniformly bounded in $L_{T}^{2} (L_{per}^{2} (Ω))$ with respect K. Then $\begin{matrix} (34) & \int_{0}^{T} {‖ \partial_{t} m^{K} ‖}_{2}^{2} d t ⩽ C . \end{matrix}$ Now, we pass to the limit as K goes to infinity. Previous estimates implies that there exists a subsequence of ${(m^{K})}_{K}$ (not relabeled) such that $\begin{array}{c} (35) & m^{K} ⇀ m weakly in L_{T}^{2} (H_{per}^{4} (Ω)), \\ (36) & m^{K} ⇀ m weakly- ⋆ in L_{T}^{\infty} (H_{per}^{2} (Ω)), \\ (37) & \partial_{t} m^{K} ⇀ \partial_{t} m weakly in L_{T}^{2} (L_{per}^{2} (Ω)) . \end{array}$ Thanks to Aubin–Lions compactness lemma 2, we obtain $\begin{matrix} (38) & m^{K} \to m strongly in L_{T}^{2} (H_{per}^{s} (Ω)), with 0 ⩽ s < 4, \end{matrix}$ and $\begin{matrix} (39) & m^{K} \to m strongly in C ([0, T], H_{per}^{s^{'}} (Ω)) for 0 ⩽ s^{'} < 2 . \end{matrix}$ Let $ϕ \in L_{T}^{2} (H_{per}^{2} (Ω))$ be a test function. From (9) we get $\begin{matrix} (40) & \begin{aligned} \int_{D_{T}} \partial_{t} m^{K} \cdot ϕ d x d t & = γ \int_{D_{T}} m^{K} \times \nabla m^{K} \cdot \nabla ϕ d x d t - σ \int_{D_{T}} \nabla m^{K} \cdot \nabla ϕ d x d t \\ - λ_{e} \int_{D_{T}} △ m^{K} \cdot △ ϕ d x d t - λ_{r} α \int_{D_{T}} (1 + β {| m^{K} |}^{2}) m^{K} \cdot ϕ d x d t \\ - λ_{e} α β \int_{D_{T}} {| m^{K} |}^{2} \nabla m^{K} \cdot \nabla ϕ d x d t \\ - λ_{e} α β \int_{D_{T}} \nabla ({| m^{K} |}^{2}) \otimes m^{K} \cdot \nabla ϕ d x d t . \end{aligned} \end{matrix}$ We take to the limit in each term of (40) as $K \to \infty$ by using the principle of weak-strong convergence as a consequence of (35), (36), (37), (38) and (39).

Let us now discuss the uniqueness of the solution to problem (6). To deal with, let $m_{1}$ and $m_{2}$ be two solutions of (6) and $T^{*} = min (T_{1}^{*}, T_{2}^{*})$ . We introduce the difference $u = m_{1} - m_{2}$ . Then $u$ verifies in the sense of Theorem 2 the following equation $\begin{matrix} (41) & \begin{aligned} \partial_{t} u & = - γ m_{1} \times △ u - γ u \times △ m_{2} + σ △ u - λ_{r} α (1 + β | m_{1} |^{2}) u - λ_{e} △^{2} u \\ - λ_{r} α β [u \cdot (m_{1} + m_{2})] m_{2} + λ_{e} α β △ (| m_{1} |^{2} m_{1}) - λ_{e} α β △ (| m_{2} |^{2} m_{2}), \end{aligned} \end{matrix}$ in $D_{T}$ , and $0 < T < T^{*}$ subject to initial conditions $u (0, \cdot) = 0$ in Ω.

Since $u \in L_{T}^{2} (H_{per}^{4} (Ω))$ then $△ u \in L_{T}^{2} (H_{per}^{2} (Ω))$ . Therefore we can take $u$ and $△ u$ as a test functions. Consequently, multiplying equation (41) respectively by $u$ and $△ u$ and integrating over Ω, we get $\begin{matrix} (42) & \begin{aligned} \frac{1}{2} \frac{d}{d t} \int_{Ω} | u |^{2} d x + σ \int_{Ω} | \nabla u |^{2} d x + λ_{e} \int_{Ω} | △ u |^{2} d x + λ_{r} α \int_{Ω} (1 + β | m_{1} |^{2}) | u |^{2} d x \\ = γ \int_{Ω} \nabla u \cdot u \times \nabla m_{1} d x - λ_{r} α β \int_{Ω} [u \cdot (m_{1} + m_{2})] m_{2} \cdot u d x \\ - λ_{e} α β \int_{Ω} \nabla (| m_{1} |^{2} m_{1}) \cdot \nabla u d x + λ_{e} α β \int_{Ω} \nabla (| m_{2} |^{2} m_{2}) \cdot \nabla u d x, \end{aligned} \end{matrix}$ and $\begin{array}{c} \frac{1}{2} \frac{d}{d t} \int_{Ω} | \nabla u |^{2} d x + σ \int_{Ω} | △ u |^{2} d x + λ_{e} \int_{Ω} | \nabla △ u |^{2} d x + λ_{r} α \int_{Ω} | \nabla u |^{2} d x \\ = γ \int_{Ω} \nabla u \times \nabla m_{2} \cdot \nabla △ u d x - λ_{r} α β \int_{Ω} [(m_{1} + m_{2}) \cdot \nabla u] \nabla u \cdot m_{2} d x \\ - λ_{r} α β \int_{Ω} [(\nabla m_{1} + \nabla m_{2}) \cdot u] m_{2} \cdot \nabla u d x - λ_{r} α β \int_{Ω} [(m_{1} + m_{2}) \cdot u] \nabla m_{2} \cdot \nabla u d x \\ - λ_{r} α β \int_{Ω} \nabla (| m_{1} |^{2}) u \cdot \nabla u d x + λ_{e} α β \int_{Ω} \nabla (| m_{1} |^{2} m_{1}) \cdot \nabla △ u d x \\ (43) & - λ_{e} α β \int_{Ω} \nabla (| m_{2} |^{2} m_{2}) \cdot \nabla △ u d x . \end{array}$ Summing up (42) and (43), we obtain $\begin{matrix} (44) & \begin{aligned} \frac{1}{2} \frac{d}{d t} ‖ u ‖_{1, 2}^{2} + (σ + λ_{r} α) \int_{Ω} | \nabla u |^{2} d x \\ + (σ + λ_{e}) \int_{Ω} | △ u |^{2} d x + λ_{e} \int_{Ω} | \nabla △ u |^{2} d x + λ_{r} α \int_{Ω} (1 + β | m_{1} |^{2}) | u |^{2} d x \\ : = I_{1} + I_{2} + I_{3} \end{aligned} \end{matrix}$ where $\begin{aligned} I_{1} & = γ \int_{Ω} \nabla u \cdot u \times \nabla m_{1} d x - λ_{r} α β \int_{Ω} [u \cdot (m_{1} + m_{2})] m_{2} \cdot u d x \\ - λ_{r} α β \int_{Ω} [(m_{1} + m_{2}) \cdot \nabla u] m_{2} \cdot \nabla u d x - λ_{r} α β \int_{Ω} [(\nabla m_{1} + \nabla m_{2}) \cdot u] m_{2} \cdot \nabla u d x \\ - λ_{r} α β \int_{Ω} [u \cdot (m_{1} + m_{2})] \nabla u \cdot \nabla m_{2} d x - λ_{r} α β \int_{Ω} \nabla u \cdot \nabla (| m_{1} |^{2}) u d x, \\ I_{2} & = γ \int_{Ω} \nabla u \times \nabla m_{2} \cdot \nabla △ u d x + λ_{e} α β \int_{Ω} \nabla (| m_{1} |^{2} m_{1}) \cdot \nabla △ u d x \\ - λ_{e} α β \int_{Ω} \nabla (| m_{2} |^{2} m_{2}) \cdot \nabla △ u d x, \end{aligned}$ and $\begin{matrix} I_{3} = - λ_{e} α β \int_{Ω} \nabla (| m_{1} |^{2} m_{1}) \cdot \nabla u d x + λ_{e} α β \int_{Ω} \nabla (| m_{2} |^{2} m_{2}) \cdot \nabla u d x . \end{matrix}$

Let us estimate $I_{1}$ . By interpolation inequality, and the continuous embedding $H_{per}^{2} (Ω) ↪ L_{per}^{\infty} (Ω)$ (for $Ω \subset R^{3}$ ) and $H_{per}^{1} (Ω) ↪ L_{per}^{4} (Ω)$ , we obtain $\begin{aligned} | I_{1} | & ⩽ γ ‖ \nabla u ‖_{2} ‖ u ‖_{4} ‖ \nabla m_{1} ‖_{4} + λ_{r} α β ‖ u ‖_{2}^{2} ‖ m_{1} + m_{2} ‖_{\infty} ‖ m_{2} ‖_{\infty} \\ + λ_{r} α β ‖ \nabla u ‖_{2}^{2} ‖ m_{1} + m_{2} ‖_{\infty} ‖ m_{2} ‖_{\infty} + λ_{r} α β ‖ u ‖_{4} ‖ \nabla u ‖_{2} ‖ \nabla m_{1} + \nabla m_{2} ‖_{4} ‖ m_{2} ‖_{\infty} \\ + λ_{r} α β ‖ u ‖_{4} ‖ \nabla u ‖_{2} ‖ m_{1} + m_{2} ‖_{\infty} ‖ \nabla m_{2} ‖_{4} + 2 λ_{r} α β ‖ \nabla m_{1} ‖_{4} ‖ m_{1} ‖_{\infty} ‖ u ‖_{4} ‖ \nabla u ‖_{2} \\ ⩽ F_{1} (t) ‖ u ‖_{1, 2}^{2}, \end{aligned}$ where $F_{1}$ depends on the data and linearly on $‖ \nabla m_{i} ‖_{4}$ for $i = 1, 2$ .

In order to estimate $I_{2}$ , we rewrite $I_{2}$ in the following way $\begin{aligned} I_{2} & = γ \int_{Ω} [\nabla u \times \nabla m_{2} + λ_{e} α β | m_{1} |^{2} \nabla u + 2 λ_{e} α β (\nabla m_{1} \cdot m_{1}) u \\ + λ_{e} α β [(m_{1} + m_{2}) \cdot \nabla u] m_{2} + λ_{e} α β [(\nabla m_{1} + \nabla m_{2}) \cdot u] m_{2} \\ + λ_{e} α β [u \cdot (m_{1} + m_{2})] \nabla m_{2}] \cdot \nabla △ u d x . \end{aligned}$ Noting that $m_{i} \in L_{T}^{2} (H_{per}^{4} (Ω))$ for $i = 1, 2$ , then $\nabla m_{i} \in L_{T}^{2} (L_{per}^{\infty} (Ω))$ . Therefore $\begin{aligned} | I_{2} | & ⩽ \frac{C}{2 d} [‖ \nabla u ‖_{2}^{2} ‖ \nabla m_{2} ‖_{\infty}^{2} + ‖ m_{1} ‖_{\infty}^{4} ‖ \nabla u ‖_{2}^{2} + ‖ m_{1} ‖_{\infty}^{2} ‖ \nabla m_{1} ‖_{\infty}^{2} ‖ u ‖_{2}^{2} \\ + ‖ m_{2} ‖_{\infty}^{2} ‖ m_{1} + m_{2} ‖_{\infty}^{2} ‖ \nabla u ‖_{2}^{2} + ‖ m_{2} ‖_{\infty}^{2} ‖ \nabla m_{1} + \nabla m_{2} ‖_{\infty}^{2} ‖ u ‖_{2}^{2} \\ + ‖ m_{1} + m_{2} ‖_{\infty}^{2} ‖ \nabla m_{2} ‖_{\infty}^{2} ‖ u ‖_{2}^{2}] + \frac{d}{2} ‖ \nabla △ u ‖_{2}^{2} \\ ⩽ \frac{C}{2 d} F_{2} (t) ‖ u ‖_{1, 2}^{2} + \frac{d}{2} ‖ \nabla △ u ‖_{2}^{2}, \end{aligned}$ where $F_{2}$ is given by $\begin{aligned} F_{2} (t) & = ‖ \nabla m_{2} ‖_{\infty}^{2} + ‖ m_{1} ‖_{\infty}^{4} + ‖ \nabla m_{1} ‖_{\infty}^{2} ‖ m_{1} ‖_{\infty}^{2} + ‖ m_{2} ‖_{\infty}^{2} ‖ m_{1} + m_{2} ‖_{\infty}^{2} \\ + ‖ m_{2} ‖_{\infty}^{2} ‖ \nabla m_{1} + \nabla m_{2} ‖_{\infty}^{2} + ‖ \nabla m_{2} ‖_{\infty}^{2} ‖ m_{1} + m_{2} ‖_{\infty}^{2}, \end{aligned}$ and for all $d > 0$ .

For $I_{3}$ , we have $\begin{aligned} I_{3} & = - λ_{e} α β [\int_{Ω} | m_{1} |^{2} | \nabla u |^{2} d x + 2 \int_{Ω} (u \cdot \nabla u) (m_{1} \cdot \nabla m_{1}) d x \\ + \int_{Ω} [(m_{1} + m_{2}) \cdot \nabla u] m_{2} \cdot m_{2} d x + \int_{Ω} [u \cdot (\nabla m_{1} + \nabla m_{2})] m_{2} \cdot \nabla u d x \\ + \int_{Ω} [u \cdot (m_{1} + m_{2})] \nabla m_{2} \cdot \nabla u d x] . \end{aligned}$ Then $\begin{aligned} | I_{3} | & ⩽ λ_{e} α β [‖ m_{1} ‖_{\infty}^{2} ‖ \nabla u ‖_{2}^{2} + 2 ‖ m_{1} ‖_{\infty} ‖ \nabla m_{1} ‖_{4} ‖ u ‖_{4} ‖ \nabla u ‖_{2} \\ + ‖ m_{1} + m_{2} ‖_{\infty} ‖ m_{2} ‖_{\infty} ‖ \nabla u ‖_{2}^{2} + ‖ \nabla m_{1} + \nabla m_{2} ‖_{4} ‖ m_{2} ‖_{\infty} ‖ u ‖_{4} ‖ \nabla u ‖_{2} \\ + ‖ m_{1} + m_{2} ‖_{\infty} ‖ \nabla m_{2} ‖_{4} ‖ u ‖_{4} ‖ \nabla u ‖_{2}] \\ ⩽ F_{3} (t) ‖ u ‖_{1, 2}^{2} . \end{aligned}$ Equation (44) allows to get $\begin{aligned} \frac{1}{2} \frac{d}{d t} ‖ u ‖_{1, 2}^{2} + (σ + λ_{r} α) \int_{Ω} | \nabla u |^{2} d x \\ + (σ + λ_{e}) \int_{Ω} | △ u |^{2} d x + λ_{e} \int_{Ω} | \nabla △ u |^{2} d x + λ_{r} α \int_{Ω} (1 + β | m_{1} |^{2}) | u |^{2} d x \\ ⩽ (F_{1} (t) + \frac{C}{2 d} F_{2} (t) + F_{3} (t)) ‖ u ‖_{1, 2}^{2} + \frac{d}{2} ‖ \nabla △ u ‖_{2}^{2} . \end{aligned}$ Choosing d such that $λ_{e} > \frac{d}{2}$ , say $d = λ_{e}$ , we get $\begin{aligned} \frac{1}{2} \frac{d}{d t} ‖ u ‖_{1, 2}^{2} + (σ + λ_{r} α) \int_{Ω} | \nabla u |^{2} d x \\ + (σ + λ_{e}) \int_{Ω} | △ u |^{2} d x + \frac{λ_{e}}{2} \int_{Ω} | \nabla △ u |^{2} d x \\ + λ_{r} α \int_{Ω} (1 + β | m_{1} |^{2}) | u |^{2} d x ⩽ F (t) ‖ u ‖_{1, 2}^{2} \end{aligned}$ where $\begin{matrix} F (t) : = F_{1} (t) + \frac{C}{2 λ_{e}} F_{2} (t) + F_{3} (t) . \end{matrix}$ Note that $F \in L^{1} (0, T)$ . Since $m_{i} \in L_{T}^{\infty} (H_{per}^{2} (Ω)) \cap L_{T}^{2} (H_{per}^{4} (Ω))$ by the continuous embeddings $H_{per}^{2} (Ω) ↪ L_{per}^{\infty} (Ω)$ , we have $m_{i} \in L_{T}^{\infty} (L_{per}^{\infty} (Ω))$ and $\nabla m_{i} \in L_{T}^{2} (L_{per}^{\infty} (Ω))$ for $i = 1, 2$ . Accordingly, Gronwall’s inequality yields $\begin{aligned} \frac{1}{2} ‖ u ‖_{1, 2}^{2} + (σ + λ_{r} α) \int_{0}^{t} \int_{Ω} | \nabla u |^{2} d x d s \\ + (σ + λ_{e}) \int_{0}^{t} \int_{Ω} | △ u |^{2} d x d s + \frac{λ_{e}}{2} \int_{0}^{t} \int_{Ω} | \nabla △ u |^{2} d x d s \\ + λ_{r} α \int_{0}^{t} \int_{Ω} (1 + β | m_{1} |^{2}) | u |^{2} d x d s ⩽ exp (2 \int_{0}^{t} F F (s) d s) ‖ u_{0} ‖_{1, 2}^{2} = 0 . \end{aligned}$ This completes the proof.

Further regular solution to the problem (6) can be obtained. More precisely we construct more regular solution to the problem (6) in the class $L_{T}^{\infty} (H_{per}^{3} (Ω)) \cap L_{T}^{2} (H_{per}^{5} (Ω))$ with $\partial_{t} m \in L_{T}^{\infty} (H_{per}^{1} (Ω)) \cap L_{T}^{2} (H_{per}^{3} (Ω))$ . We have

Theorem 3.
Assume $m_{0} \in H_{per}^{5} (Ω)$ . Let $T^{} > 0$ and* $m \in C ([0, T], H_{per}^{2} (Ω)) \cap L_{T}^{2} (H_{per}^{4} (Ω))$ for $T < T^{}$ given by Theorem* 2 . Then for all $T < T^{}$ , $\begin{array}{c} m \in C ([0, T], H_{per}^{3} (Ω)) \cap L_{T}^{2} (H_{per}^{5} (Ω)), \\ \partial_{t} m \in L_{T}^{\infty} (H_{per}^{1} (Ω)) \cap L_{T}^{2} (H_{per}^{3} (Ω)) . \end{array}$
Proof.
Taking the inner product of (9) by $△^{3} m^{K}$ and integrating by parts, we get $\begin{matrix} (45) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ \nabla △ m^{K} ‖}_{2}^{2} + σ {‖ △^{2} m^{K} ‖}_{2}^{2} + λ_{r} α β {‖ \nabla △^{2} m^{K} ‖}_{2}^{2} \\ = - γ \int_{Ω} m^{K} \times △ m^{K} \cdot △^{3} m^{K} d x - λ_{r} α β \int_{Ω} △^{3} m^{K} \cdot ({| m^{K} |}^{2} m^{K}) d x \\ + λ_{e} α β \int_{Ω} △ ({| m^{K} |}^{2} m^{K}) \cdot △^{3} m^{K} d x \\ : = A_{1} + A_{2} + A_{3} . \end{aligned} \end{matrix}$ In the following, we bound each term separately.

By using integration by parts, we rewrite $A_{1}$ in the form $\begin{array}{r} A_{1} = γ \int_{Ω} (\nabla m^{K} \times △ m^{K} + m^{K} \times \nabla △ m^{K}) \cdot \nabla △^{2} m^{K} d x . \end{array}$ Then by Lemma 5, we obtain that $\begin{matrix} (46) & \begin{aligned} | A_{1} | & ⩽ γ ({‖ \nabla m^{K} ‖}_{6} {‖ △ m^{K} ‖}_{3} + {‖ m^{K} ‖}_{\infty} {‖ \nabla △ m^{K} ‖}_{2}) {‖ \nabla △^{2} m^{K} ‖}_{2} \\ ⩽ C ({‖ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ \nabla △ m^{K} ‖}_{2}^{2}) {‖ \nabla △^{2} m^{K} ‖}_{2} . \end{aligned} \end{matrix}$ For $A_{2}$ , we have $\begin{matrix} (47) & \begin{aligned} | A_{2} | & = λ_{r} α β | \int_{Ω} ({| m^{K} |}^{2} \nabla m^{K} + 2 (m^{K} \cdot \nabla m^{K}) m^{K}) \cdot \nabla △^{2} m^{K} d x | \\ ⩽ 3 λ_{r} α β {‖ m^{K} ‖}_{\infty}^{2} {‖ \nabla m^{K} ‖}_{2} {‖ \nabla △^{2} m^{K} ‖}_{2} \\ ⩽ C {({‖ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ \nabla △ m^{K} ‖}_{2}^{2})}^{\frac{3}{2}} {‖ \nabla △^{2} m^{K} ‖}_{2} . \end{aligned} \end{matrix}$ For $A_{3}$ we use integration by parts and Young’s inequality to obtain $\begin{matrix} (48) & \begin{aligned} | A_{3} | & ⩽ C ({‖ \nabla m^{K} ‖}_{6} {‖ △ m^{K} ‖}_{3} {‖ m^{K} ‖}_{\infty}^{2} + {‖ \nabla △ m^{K} ‖}_{2} {‖ m^{K} ‖}_{\infty}^{2} + {‖ \nabla m^{K} ‖}_{6}^{3} \\ + {‖ \nabla m^{K} ‖}_{6} {‖ m^{K} ‖}_{\infty}^{2} {‖ D^{2} m^{K} ‖}_{3}) {‖ \nabla △^{2} m^{K} ‖}_{2} . \end{aligned} \end{matrix}$ On the other hand thanks to the Lemma 5, for the first and last terms of the right-hand side of (48) one can get the following estimate $\begin{aligned} {‖ \nabla m^{K} ‖}_{6} {‖ m^{K} ‖}_{\infty}^{2} {‖ △ m^{K} ‖}_{3} + {‖ \nabla m^{K} ‖}_{6} {‖ m^{K} ‖}_{\infty}^{2} {‖ D^{2} m^{K} ‖}_{3} \\ ⩽ C {({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{\frac{3}{2}} [{({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{\frac{1}{2}} \\ + {({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{\frac{1}{4}} {‖ \nabla △ m^{K} ‖}_{2}^{\frac{1}{2}}] \\ ⩽ C {({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{2} . \end{aligned}$ For the second and the third terms of the right-hand side of (48), we have $\begin{array}{r} {‖ \nabla △ m^{K} ‖}_{2} {‖ m^{K} ‖}_{\infty}^{2} + {‖ \nabla m^{K} ‖}_{6}^{3} ⩽ C {({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{\frac{3}{2}} . \end{array}$ Therefore $\begin{matrix} (49) & \begin{aligned} | A_{3} | & ⩽ C [{({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{2} \\ + {({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{\frac{3}{2}}] {‖ \nabla △^{2} m^{K} ‖}_{2} . \end{aligned} \end{matrix}$ From (45) and the estimates (46), (47) and (49), we obtain $\begin{matrix} (50) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ \nabla △ m^{K} ‖}_{2}^{2} + σ {‖ △^{2} m^{K} ‖}_{2}^{2} + λ_{r} α β {‖ \nabla △^{2} m^{K} ‖}_{2}^{2} \\ ⩽ C [({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2}) + {({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{2} \\ + {({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{\frac{3}{2}}] {‖ \nabla △^{2} m^{K} ‖}_{2} . \end{aligned} \end{matrix}$ Absorbing $‖ \nabla △^{2} m^{K} ‖_{2}$ by using Young’s inequality, one gets $\begin{matrix} (51) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ \nabla △ m^{K} ‖}_{2}^{2} + σ {‖ △^{2} m^{K} ‖}_{2}^{2} + \frac{λ_{r} α β}{2} {‖ \nabla △^{2} m^{K} ‖}_{2}^{2} \\ ⩽ C [{({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{2} + {({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{4} \\ + {({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{3}] . \end{aligned} \end{matrix}$ Exploiting (31) and (51) leads to $\begin{matrix} (52) & \begin{aligned} \frac{d}{d t} ({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2}) \\ ⩽ C [{({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{2} \\ + {({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{4} + {({‖ \nabla △ m^{K} ‖}_{2}^{2} + {‖ △ m^{K} ‖}_{2}^{2} + {‖ m^{K} ‖}_{2}^{2})}^{3}] . \end{aligned} \end{matrix}$ Using the same reasoning as in the proof of the Theorem 2, we deduce that there exists $C > 0$ which depend on the size of $m_{0}$ and $T^{} > 0$ such that $\begin{matrix} sup_{t \in [0, T]} {‖ m^{K} ‖}_{3, 2} ⩽ C, \end{matrix}$ for all $T < T^{}$ , and (51) implies the uniform boundedness of ${(m^{K})}_{K}$ in $L_{T}^{2} (H_{per}^{5} (Ω))$ .

In order to get the boundedness of $\partial_{t} m^{K}$ in $L_{T}^{\infty} (H_{per}^{1} (Ω)) \cap L_{T}^{2} (H_{per}^{3} (Ω))$ , we derive the Galerkin approximation (9) with respect the time and using the identity (33), we obtain $\begin{aligned} \partial_{t}^{2} m^{K} & = σ △ \partial_{t} m^{K} - λ_{e} △^{2} \partial_{t} m^{K} + P_{K} [- γ \partial_{t} m^{K} \times △ m^{K} - γ m^{K} \times △ \partial_{t} m^{K} \\ - λ_{r} α (1 + β {| m^{K} |}^{2}) \partial_{t} m^{K} - 2 λ_{r} α β (m^{K} \cdot \partial_{t} m^{K}) m^{K} + λ_{e} α β (2 (\partial_{t} m^{K} \cdot △ m^{K}) m^{K} \\ + 2 (m^{K} \cdot △ \partial_{t} m^{K}) m^{K} + 2 (m^{K} \cdot △ m^{K}) \partial_{t} m^{K} + 2 {| \nabla m^{K} |}^{2} \partial_{t} m^{K} \\ + 4 (\nabla \partial_{t} m^{K} \cdot \nabla m^{K}) m^{K} + 4 (\partial_{t} m^{K} \cdot \nabla m^{K}) \nabla m^{K} + 4 (m^{K} \cdot \nabla \partial_{t} m^{K}) \nabla m^{K} \\ + 4 (m^{K} \cdot \nabla m^{K}) \nabla \partial_{t} m^{K} + {| m^{K} |}^{2} △ \partial_{t} m^{K} + 2 (m^{K} \cdot \partial_{t} m^{K}) △ m^{K})] . \end{aligned}$ We multiply this equality by $\partial_{t} m^{K}$ and $△ \partial_{t} m^{K}$ , and integrate over Ω, we obtain $\begin{matrix} (53) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ \partial_{t} m^{K} ‖}_{2}^{2} + σ {‖ \nabla \partial_{t} m^{K} ‖}_{2}^{2} + λ_{e} {‖ △ \partial_{t} m^{K} ‖}_{2}^{2} + λ_{r} α \int_{Ω} (1 + β {| m^{K} |}^{2}) {| \partial_{t} m^{K} |}^{2} d x \\ + 2 λ_{r} α β \int_{Ω} {(m^{K} \cdot \partial_{t} m^{K})}^{2} d x \\ ⩽ C_{1} ({‖ m^{K} ‖}_{\infty} {‖ △ m^{K} ‖}_{\infty} + {‖ \nabla m^{K} ‖}_{4}^{2}) {‖ \partial_{t} m^{K} ‖}_{2}^{2} \\ + C_{2} {‖ m^{K} ‖}_{\infty}^{2} {‖ △ \partial_{t} m^{K} ‖}_{2} {‖ \partial_{t} m^{K} ‖}_{2} + C_{3} {‖ m^{K} ‖}_{\infty} {‖ \nabla m^{K} ‖}_{\infty} {‖ \nabla \partial_{t} m^{K} ‖}_{2} {‖ \partial_{t} m^{K} ‖}_{2} . \end{aligned} \end{matrix}$ and $\begin{matrix} (54) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ \nabla \partial_{t} m^{K} ‖}_{2}^{2} + σ {‖ △ \partial_{t} m^{K} ‖}_{2}^{2} + λ_{e} {‖ \nabla △ \partial_{t} m^{K} ‖}_{2}^{2} + 2 λ_{e} α β \int_{Ω} {(m^{K} \cdot △ \partial_{t} m^{K})}^{2} d x \\ + λ_{e} α β \int_{Ω} {| m^{K} |}^{2} {| △ \partial_{t} m^{K} |}^{2} d x \\ ⩽ C_{4} [1 + {‖ △ m^{K} ‖}_{\infty} + {‖ m^{K} ‖}_{\infty}^{2} + {‖ △ m^{K} ‖}_{\infty} {‖ m^{K} ‖}_{\infty} \\ + {‖ \nabla m^{K} ‖}_{\infty}^{2}] {‖ \partial_{t} m^{K} ‖}_{2} {‖ △ \partial_{t} m^{K} ‖}_{2} + C_{5} {‖ \nabla m^{K} ‖}_{\infty} {‖ m^{K} ‖}_{\infty} {‖ \nabla \partial_{t} m^{K} ‖}_{2} {‖ △ \partial_{t} m^{K} ‖}_{2} . \end{aligned} \end{matrix}$ Summing (53) and (54) and absorbing $‖ △ \partial_{t} m^{K} ‖_{2}$ , we get $\begin{matrix} (55) & \begin{aligned} \frac{1}{2} \frac{d}{d t} {‖ \partial_{t} m^{K} ‖}_{1, 2}^{2} + σ {‖ \nabla \partial_{t} m^{K} ‖}_{2}^{2} + \frac{σ + λ_{e}}{2} {‖ △ \partial_{t} m^{K} ‖}_{2}^{2} + λ_{e} {‖ \nabla △ \partial_{t} m^{K} ‖}_{2}^{2} \\ + λ_{r} α \int_{Ω} (1 + β {| m^{K} |}^{2}) {| \partial_{t} m^{K} |}^{2} d x + 2 λ_{r} α β \int_{Ω} {(m^{K} \cdot \partial_{t} m^{K})}^{2} d x \\ + 2 λ_{e} α β \int_{Ω} {(m^{K} \cdot △ \partial_{t} m^{K})}^{2} d x + λ_{e} α β \int_{Ω} {| m^{K} |}^{2} {| △ \partial_{t} m^{K} |}^{2} d x \\ ⩽ C_{6} (1 + {‖ △ m^{K} ‖}_{\infty} {‖ m^{K} ‖}_{\infty} + {‖ m^{K} ‖}_{\infty}^{4} + {‖ \nabla m^{K} ‖}_{4}^{2} + {‖ \nabla m^{K} ‖}_{\infty}^{2} {‖ m^{K} ‖}_{\infty}^{2} + {‖ △ m^{K} ‖}_{\infty}^{2} \\ + {‖ △ m^{K} ‖}_{\infty}^{2} {‖ m^{K} ‖}_{\infty}^{2} + {‖ \nabla m^{K} ‖}_{\infty}^{4}) {‖ \partial_{t} m^{K} ‖}_{2}^{2} + C_{7} (1 + {‖ m^{K} ‖}_{\infty}^{2} {‖ \nabla m^{K} ‖}_{\infty}^{2}) {‖ \nabla \partial_{t} m^{K} ‖}_{2}^{2} \\ ⩽ f_{K} (t) {‖ \partial_{t} m^{K} ‖}_{1, 2}^{2}, \end{aligned} \end{matrix}$ where $f_{K}$ is defined by $\begin{aligned} f_{K} (t) : = & C_{6} (1 + {‖ △ m^{K} ‖}_{\infty} {‖ m^{K} ‖}_{\infty} + {‖ m^{K} ‖}_{\infty}^{4} + {‖ \nabla m^{K} ‖}_{4}^{2} + {‖ \nabla m^{K} ‖}_{\infty}^{2} {‖ m^{K} ‖}_{\infty}^{2} \\ + {‖ △ m^{K} ‖}_{\infty}^{2} + {‖ △ m^{K} ‖}_{\infty}^{2} {‖ m^{K} ‖}_{\infty}^{2} + {‖ \nabla m^{K} ‖}_{\infty}^{4}) + C_{7} (1 + {‖ \nabla m^{K} ‖}_{\infty}^{2} {‖ m^{K} ‖}_{\infty}^{2}) . \end{aligned}$ We will now estimate the initial value of $\partial_{t} m^{K}$ . By Equation (9) taken at $t = 0$ , we have $\begin{aligned} \partial_{t} m^{K} (0) & = σ △ m^{K} (0) - λ_{e} △^{2} m^{K} (0) + P_{K} [- γ m^{K} (0) \times △ m^{K} (0) \\ - λ_{r} α (1 + β {| m^{K} (0) |}^{2}) m^{K} (0) + λ_{e} α β [2 (m^{K} (0) \cdot △ m^{K} (0)) m^{K} (0) \\ + 2 {| \nabla m^{K} (0) |}^{2} m^{K} (0) + 4 (m^{K} (0) \cdot \nabla m^{K} (0)) \nabla m^{K} (0) + {| m^{K} (0) |}^{2} △ m^{K} (0)]] . \end{aligned}$ Using Proposition 1, the $H^{1}$ estimate of $\partial_{t} m^{K} (0)$ can be obtained as follows $\begin{matrix} (56) & \begin{aligned} {‖ \partial_{t} m^{K} (0) ‖}_{2} & ⩽ σ {‖ △ m^{K} (0) ‖}_{2} + λ_{e} {‖ △^{2} m^{K} (0) ‖}_{2} + γ {‖ m^{K} (0) ‖}_{\infty} {‖ △ m^{K} (0) ‖}_{2} \\ + λ_{r} α (1 + β {‖ m^{K} (0) ‖}_{\infty}^{2}) {‖ m^{K} (0) ‖}_{2} + λ_{e} α β [3 {‖ m^{K} (0) ‖}_{\infty}^{2} {‖ △ m^{K} (0) ‖}_{2} \\ + 6 {‖ \nabla m^{K} (0) ‖}_{4}^{2} {‖ m^{K} (0) ‖}_{\infty}] \\ ⩽ C [{‖ m^{K} (0) ‖}_{2, 4} + {‖ m^{K} (0) ‖}_{2, 4}^{2} + {‖ m^{K} (0) ‖}_{2, 4}^{3}] \\ ⩽ C [‖ m_{0} ‖_{2, 4} + ‖ m_{0} ‖_{2, 4}^{2} + ‖ m_{0} ‖_{2, 4}^{3}] . \end{aligned} \end{matrix}$ and $\begin{matrix} (57) & \begin{aligned} {‖ \nabla \partial_{t} m^{K} (0) ‖}_{2} \\ ⩽ σ {‖ \nabla △ m^{K} (0) ‖}_{2} + λ_{e} {‖ \nabla △^{2} m^{K} (0) ‖}_{2} + γ {‖ \nabla m^{K} (0) ‖}_{\infty} {‖ △ m^{K} (0) ‖}_{2} \\ + γ {‖ m^{K} (0) ‖}_{\infty} {‖ \nabla △ m^{K} (0) ‖}_{2} + λ_{r} α {‖ \nabla m^{K} (0) ‖}_{2} + 3 λ_{r} α β {‖ m^{K} (0) ‖}_{\infty}^{2} {‖ \nabla m^{K} (0) ‖}_{2} \\ + 6 λ_{e} α β {‖ m^{K} (0) ‖}_{\infty} {‖ \nabla m^{K} (0) ‖}_{\infty} {‖ △ m^{K} (0) ‖}_{2} + 3 λ_{e} α β {‖ m^{K} (0) ‖}_{\infty}^{2} {‖ \nabla △ m^{K} (0) ‖}_{2} \\ + 6 λ_{e} α β {‖ \nabla m^{K} (0) ‖}_{\infty}^{2} {‖ \nabla m^{K} (0) ‖}_{2} + 12 λ_{e} α β {‖ m^{K} (0) ‖}_{\infty} {‖ \nabla m^{K} (0) ‖}_{\infty} {‖ D^{2} m^{K} (0) ‖}_{2} \\ ⩽ C [{‖ m^{K} (0) ‖}_{2, 5} + {‖ m^{K} (0) ‖}_{2, 5}^{2} + {‖ m^{K} (0) ‖}_{2, 5}^{3}] \\ ⩽ C [‖ m_{0} ‖_{2, 5} + ‖ m_{0} ‖_{2, 5}^{2} + ‖ m_{0} ‖_{2, 5}^{3}] . \end{aligned} \end{matrix}$ By using the Sobolev embedding of $H_{per}^{5} (Ω)$ into $H_{per}^{4} (Ω)$ , and the fact that $f_{K}$ is bounded in $L^{1} (0, T)$ , we deduce from (55), (56) and (57) by Gronwall’s lemma the boundedness of ${(\partial_{t} m^{K})}_{K}$ in $L_{T}^{\infty} (H_{per}^{1} (Ω)) \cap L_{T}^{2} (H_{per}^{3} (Ω))$ .

Now, we pass to the limit as $K \to \infty$ . From previous estimates, one can extract a subsequence of ${(m^{K})}_{K}$ satisfying $\begin{array}{c} m^{K} ⇀ m weakly-star in L_{T}^{\infty} (H_{per}^{3} (Ω)), \\ m^{K} ⇀ m weakly in L_{T}^{2} (H_{per}^{5} (Ω)), \\ \partial_{t} m^{K} ⇀ \partial_{t} m weakly in L_{T}^{2} (H_{per}^{3} (Ω)), \end{array}$ and $\begin{matrix} \partial_{t} m^{K} ⇀ \partial_{t} m weakly-star in L_{T}^{\infty} (H_{per}^{1} (Ω)) . \end{matrix}$ According to Aubin–Lions Lemma 2, for a not relabeled subsequence $\begin{matrix} m^{K} \to m strong in C ([0, T], H_{per}^{r} (Ω)) for 1 ⩽ r < 3, \end{matrix}$ and $\begin{matrix} m^{K} \to m strong in L_{T}^{2} (H_{per}^{s} (Ω)) for 2 ⩽ s < 5 . \end{matrix}$ So, we can take the limit in (9) to obtain that $m$ satisfies $\begin{matrix} \{\begin{array}{l} \partial_{t} m = σ △ m - λ_{e} △^{2} m - γ m \times △ m - λ_{r} α (1 + β | m |^{2}) m \\ + λ_{e} α β △ (| m |^{2} m) in [0, T^{}) \times Ω, \\ m (0, \cdot) = m_{0}, \end{array} \end{matrix}$ and the proof of Theorem 3 is complete. □

4. Asymptotic behaviour of weak solutions

We shall establish to relationships between the LLBar equation and both LLB and Harmonic map equations. We begin by the following result.

Proposition 2.
Let $ε = λ_{e} \to 0$ . The weak solution $m^{ε}$ of the problem ( 6 ) weakly converges, up to a subsequence, to a weak solution of the LLB equation, i.e, $\begin{matrix} (58) & \begin{aligned} \int_{D_{T}} \partial_{t} m \cdot ϕ d x d t & = γ \int_{D_{T}} m \times \nabla m \cdot \nabla ϕ d x d t \\ - λ_{r} \int_{D_{T}} \nabla m \cdot \nabla ϕ d x d t - λ_{r} α \int_{D_{T}} (1 + β | m |^{2}) m \cdot ϕ d x d t \end{aligned} \end{matrix}$ for all 2D-periodic function test $ϕ \in C^{\infty} (D_{T})$ . Moreover, the following estimate holds $\begin{matrix} (59) & \begin{aligned} \frac{1}{2} \int_{Ω} | m |^{2} d x + λ_{r} \int_{0}^{t} \int_{Ω} | \nabla m |^{2} d x d t + λ_{r} α \int_{0}^{t} \int_{Ω} (1 + β | m |^{2}) | m |^{2} d x d t \\ ⩽ \frac{1}{2} \int_{Ω} {| m (0, \cdot) |}^{2} d x . \end{aligned} \end{matrix}$
Proof.
From the previous section, the following estimate holds $\begin{matrix} (60) & \begin{aligned} \frac{1}{2} \int_{Ω} {| m^{ε} |}^{2} d x + (λ_{r} + ε α) \int_{0}^{t} \int_{Ω} {| \nabla m^{ε} |}^{2} d x d t + ε \int_{0}^{t} \int_{Ω} {| △ m^{ε} |}^{2} d x d t \\ + λ_{r} α \int_{0}^{t} \int_{Ω} (1 + β {| m^{ε} |}^{2}) {| m^{ε} |}^{2} d x d t + α ε β \int_{0}^{t} \int_{Ω} {| m^{ε} |}^{2} {| \nabla m^{ε} |}^{2} d x d t \\ + \frac{α ε β}{2} \int_{0}^{t} \int_{Ω} {| \nabla ({| m^{ε} |}^{2}) |}^{2} d x d t ⩽ \frac{1}{2} \int_{Ω} {| m (0, \cdot) |}^{2} d x . \end{aligned} \end{matrix}$ Then we get the boundedness of ${(m^{ε})}_{ε}$ in $L_{T}^{\infty} (L_{per}^{2} (Ω)) \cap L_{T}^{2} (H_{per}^{1} (Ω))$ , and of ${(\sqrt{ε} m^{ε})}_{ε}$ in $L_{T}^{2} (H_{per}^{2} (Ω))$ . Since we are interested in the case where $ε \to 0$ , we can assume that $0 < ε < 1$ , then by a density argument for $ψ$ in $L_{T}^{4} (H_{per}^{2} (Ω))$ , we obtain $\begin{aligned} | \int_{D_{T}} \partial_{t} m^{ε} \cdot ψ d x d t | & ⩽ γ {‖ m^{ε} ‖}_{L_{T}^{4} (L_{per}^{4} (Ω))} {‖ \nabla m^{ε} ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} ‖ \nabla ψ ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} \\ + (λ_{r} + α) {‖ \nabla m^{ε} ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} ‖ \nabla ψ ‖_{L_{T}^{2} (L_{per}^{2} (Ω))} \\ + \sqrt{ε} {‖ \sqrt{ε} △ m^{ε} ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} ‖ △ ψ ‖_{L_{T}^{2} (L_{per}^{2} (Ω))} \\ + λ_{r} α {‖ (1 + β {| m^{ε} |}^{2}) ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} {‖ m^{K} ‖}_{L_{T}^{4} (L_{per}^{4} (Ω))} ‖ ψ ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} \\ + α β \sqrt{ε} {‖ \sqrt{ε} | m^{ε} | \nabla m^{ε} ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} {‖ m^{ε} ‖}_{L_{T}^{4} (L_{per}^{4} (Ω))} ‖ \nabla ψ ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} \\ + α β \sqrt{ε} {‖ \sqrt{ε} \nabla ({| m^{ε} |}^{2}) ‖}_{L_{T}^{2} (L_{per}^{2} (Ω))} {‖ m^{ε} ‖}_{L_{T}^{4} (L_{per}^{4} (Ω))} ‖ \nabla ψ ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} \\ ⩽ C ‖ ψ ‖_{L_{T}^{4} (H_{per}^{2} (Ω))}, \end{aligned}$ with C is a positive constant independent of ε. Then $\begin{matrix} {(\partial_{t} m^{ε})}_{ε} is uniformly bounded in L_{T}^{\frac{4}{3}} (H^{- 2} (Ω)) . \end{matrix}$ Therefore, for a subsequence, we have $\begin{array}{c} m^{ε} ⇀ m weakly in L_{T}^{2} (H_{per}^{1} (Ω)), \\ \partial_{t} m^{ε} ⇀ \partial_{t} m weakly in L_{T}^{\frac{4}{3}} (H^{- 2} (Ω)) . \end{array}$ Apply the compactness Lemma 2, one has $\begin{matrix} m^{ε} \to m strongly in L_{T}^{2} (L_{per}^{2} (Ω)) . \end{matrix}$ Now, we can pass to the limit in the weak formulation $\begin{matrix} (61) & \begin{aligned} \int_{D_{T}} \partial_{t} m^{ε} \cdot ϕ d x d t & = γ \int_{D_{T}} m^{ε} \times \nabla m^{ε} \cdot \nabla ϕ d x d t \\ - (λ_{r} + ε α) \int_{D_{T}} \nabla m^{ε} \cdot \nabla ϕ d x d t - \sqrt{ε} \int_{D_{T}} \sqrt{ε} △ m^{ε} \cdot △ ϕ d x d t \\ - λ_{r} α \int_{D_{T}} (1 + β {| m^{ε} |}^{2}) m^{ε} \cdot ϕ d x d t \\ - \sqrt{ε} α β \int_{D_{T}} \sqrt{ε} {| m^{ε} |}^{2} \nabla m^{ε} \cdot \nabla ϕ d x d t \\ - \sqrt{ε} α β \int_{D_{T}} \sqrt{ε} \nabla ({| m^{ε} |}^{2}) \otimes m^{ε} \cdot \nabla ϕ d x d t \end{aligned} \end{matrix}$ we obtain $\begin{matrix} (62) & \begin{aligned} \int_{D_{T}} \partial_{t} m \cdot ϕ d x d t & = γ \int_{D_{T}} m \times \nabla m \cdot \nabla ϕ d x d t \\ - λ_{r} \int_{D_{T}} \nabla m \cdot \nabla ϕ d x d t - λ_{r} α \int_{D_{T}} (1 + β | m |^{2}) m \cdot ϕ d x d t \end{aligned} \end{matrix}$ for all $ϕ \in C^{\infty} (D_{T})$ which is 2D-periodic.

Taking the lower semi-continuous limit ( $ε \to 0$ ) in (60), we get (59). Then Proposition 2 is proved. □

Our second result in this subsection is the following. Proposition 3.
Consider $λ_{e} = 0$ and $λ_{r} = η \to 0$ . Let $m^{η}$ be a weak solution of the LLB equation and set $u^{η} = η^{\frac{3}{4}} m^{η}$ , then ${(u^{η})}_{η}$ converge weakly up to a subsequence, to a weak solution of the harmonic map equation, i.e., $\begin{matrix} (63) & \int_{D_{T}} u \times \nabla u \cdot \nabla ϕ d x d t = 0, \end{matrix}$ for all 2D-periodic function test $ϕ \in C_{per}^{\infty} (D_{T})$ .
Proof.
Using the estimates (59), we deduce that $\begin{array}{c} {(\sqrt{η} m^{η})}_{η} is uniformly bounded in L_{T}^{2} (H_{per}^{1} (Ω)), \\ {(\sqrt[4]{η} m^{η})}_{η} is uniformly bounded in L_{T}^{4} (L_{per}^{4} (Ω)) . \end{array}$ Multiply both sides of (62) by $η^{\frac{3}{4}}$ leads to $\begin{aligned} | \int_{D_{T}} η^{\frac{3}{4}} \partial_{t} m^{η} \cdot ϕ d x d t | & ⩽ γ ‖ \sqrt[4]{η} m^{η} ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} ‖ \sqrt{η} \nabla m^{η} ‖_{L_{T}^{2} (L_{per}^{2} (Ω))} ‖ \nabla ϕ ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} \\ + η^{\frac{5}{4}} ‖ \sqrt{η} \nabla m^{η} ‖_{L_{T}^{2} (L_{per}^{2} (Ω))} ‖ \nabla ϕ ‖_{L_{T}^{2} (L_{per}^{2} (Ω))} \\ + η α ‖ \sqrt{η} (1 + β {| m^{η} |}^{2}) ‖_{L_{T}^{2} (L_{per}^{2} (Ω))} ‖ \sqrt[4]{η} m^{η} ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} ‖ ϕ ‖_{L_{T}^{4} (L_{per}^{4} (Ω))} \\ ⩽ C ‖ ϕ ‖_{L_{T}^{4} (W_{0}^{1, 4} (Ω))}, \end{aligned}$ for some constant C independent of η. For η small enough $\begin{matrix} {(\partial_{t} u^{η})}_{η} is uniformly bounded in L_{T}^{\frac{4}{3}} (W^{- 1, \frac{4}{3}} (Ω)), \end{matrix}$ and $\begin{matrix} {(u^{η})}_{η} is uniformly bounded in L_{T}^{2} (H_{per}^{1} (Ω)) . \end{matrix}$ So for a subsequence $\begin{array}{c} \partial_{t} u^{η} ⇀ \partial_{t} u weakly in L_{T}^{\frac{4}{3}} (W^{- 1, \frac{4}{3}} (Ω)), \\ u^{η} ⇀ u weakly in L_{T}^{2} (H_{per}^{1} (Ω)), \end{array}$ and $\begin{matrix} u^{η} \to u strongly in L_{T}^{2} (L_{per}^{2} (Ω)) . \end{matrix}$ Multiply now both sides of (62) by $η^{\frac{3}{2}}$ , we get $\begin{matrix} (64) & \begin{aligned} η^{\frac{3}{4}} \int_{D_{T}} \partial_{t} u^{η} \cdot ϕ d x d t & = γ \int_{D_{T}} u^{η} \times \nabla u^{η} \cdot \nabla ϕ d x d t - η^{\frac{7}{4}} \int_{D_{T}} \nabla u^{η} \cdot \nabla ϕ d x d t \\ - η^{\frac{7}{4}} α \int_{D_{T}} u^{η} \cdot ϕ d x d t - β η^{\frac{1}{4}} \int_{D_{T}} {| u^{η} |}^{2} u^{η} \cdot ϕ d x d t . \end{aligned} \end{matrix}$ We now can pass to the limit ( $η \to 0$ ) in each terms of (64) to find $\begin{matrix} γ \int_{D_{T}} u \times \nabla u \cdot \nabla ϕ d x d t = 0 \end{matrix}$ for all $ϕ \in C^{\infty} (D_{T})$ , that is $u$ is a periodic solution of the harmonic map equation $u \times △ u = 0$ in the distribution sense. This complete the proof of Proposition 3. □

5. Conclusion

In this manuscript, a LLBar equation is investigated and global existence result of weak solutions is obtained. A local well-posedness result is also obtained. The relationships between the LLBar equation and both LLB and harmonic map equations are revealed by discussing the limit of the obtained solutions when the relativistic and exchange relaxation damping parameters tends to zero. Recently, the field of numerical analysis of micromagnetic materials has witnessed significant advancements, as evidenced by a multitude of notable research contributions exploring novel simulation techniques, more accurate modeling algorithms, and enhanced computational methodologies, we refer for example to [12–14,24,26,27]. Based on these pioneering works, an interesting direction of future research is to explore numerical schemes for LLBar equation in order to reflect the true nature of the model considered in this paper. Finally, introducing noninteger/fractional order derivatives represents a challenge to find effective methods for the time-domain analysis of fractional-order LLBar equation.

6. Declarations

Ethical Approval: Not applicable.

Competing interests: The authors declare that they have no competing interests.

Authors’ contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Funding: Not applicable.

Availability of data and materials: No data were used to support this study.

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Existence results for the Landau–Lifshitz–Baryakhtar equation

Abstract

Keywords

1. Introduction

Lemma 5 ([11,17]).

3.1. Proof of Theorem 2

6. Declarations

References