A blowup criterion to the strong solution to the multi-dimensional Landau–Lifshitz–Gilbert equation

Abstract

In this paper, we will research the regularity of the strong solution to the Landau–Lifshitz–Gilbert equation. We obtain a blowup criterion of the strong solution to the Landau–Lifshitz–Gilbert equation in a multi-dimensional bounded domain with Neumann boundary condition.

Introduction

It is well-known that the Landau–Lifshitz equations is a fundamental evolution equation for the ferromagnetic spin chain and was proposed on the phenomenological ground in studying the dispersive theory of magnetization of ferromagnets. Later, in 1955 Gilbert derived a new Landau–Lifshitz equation with a dissipative term (the so-called Landau–Lifshitz–Gilbert equation):

$${\partial }_{t}u+\alpha u\times \left(\Delta u\right)+\beta u\times (u\times \left(\Delta u\right))=0$$

Here $u$, which is a magnetization vector in physics, is a mapping from $\Omega$ into a unit sphere ${\mathbf{S}}^2\subset {\mathbf{R}}^3$. $\Omega$ is a smooth bounded domain in the Euclidean space ${\mathbf{R}}^3$ or a flat torus ${\mathbf{T}}^3$. The constant $\beta >0$ is called the Gilbert damping coefficient which is a gyroscopic ratio. This torque is most commonly termed non-adiabatic and $\alpha$ characterizes its strength.

In this paper, we need to consider the following initial boundary conditions

$$\frac{\partial u}{\partial \nu }=0,(x,t)\in \partial \Omega \times (0,T),u(x,0)=u_0,x\in \Omega ,\left|u_0\right|=1.$$

Where $\nu$ is the outwards facing unit normal vector on $\partial \Omega$.

Next, we will introduce some previous relative results about Landau–Lifshitz equation. The first important result is obtained by Visintin [1] in 1985 which is the existence of weak solutions to Landau–Lifshitz equation with magnetostrictive effects. In 1986, P. L. Sulem, C. Sulem and C. Bardos [2] used the difference method to prove the existence of global weak solution and the local smooth solution to the Landau–Lifshitz equation without dissipation term (that is Schrödinger flow for maps into ${\mathbb{S}}^2$) defined on ${\mathbf{R}}^n$. These results were improved later by Ding and Wang in [3], [4]. In 1992, a non-uniqueness result of weak solution to LLG equation with Neumann boundary condition for $\Omega$ a unit ball in ${\mathbf{R}}^3$ was obtained by Alouges and Soyeur [5]. In 1993, Guo and Hong [6] established the relations between two-dimensional Landau–Lifshitz equations and harmonic maps and applied the approaches studying harmonic maps to get the global existence and uniqueness of partially regular weak solution. In 1998, Y.D. Wang [7] proved the global existence of the weak solutions to the Cauchy problems of Schrödinger flow (Landau–Lifshitz equation) from a $n$-dimensional Euclidean domain $\Omega$ or a $n$-dimensional closed Riemannian manifold $M$ into a 2-dimensional unit sphere ${\mathbf{S}}^2$. In 2001, Carbou and Fabrie [8] investigated local existence, global existence with small data and uniqueness of regular solutions for Landau–Lifshitz equations in ${\mathbb{R}}^3$, and proved in [9] the local existence of regular solutions in the 2D and 3D cases for Landau–Lifshitz equation on a bounded domains in ${\mathbb{R}}^n(n\le 3)$. In 2011, I. Bejenaru, A. D. Ionescu, C. E. Kenig and D. Tataru [10] addressed the global well-posedness result for LL equation for small data in the critical Besov spaces in dimensions $n⩾2$. The weak–strong uniqueness for the Landau–Lifshitz–Gilbert equation in researched by Di Fratta and his coauthors [11].

J.S. Fang and his coauthors [12], [13], [14] have given some blowup criterion for 3D Landau–Lifshits–Gilbert equation. We will give a different blowup criterion for multi-dimensional ($n\ge 2$) case in this paper. The main results are as follows.

Theorem 1.1

Let $\Omega$ be a bounded smooth domain in ${\mathbf{R}}^n(n\ge 2)$. Assume $\nabla u_0\in H^1\left(\Omega \right)$, Let $T^{\ast }$ be the maximum value such that the Landau–Lifshtiz–Gilbert Eq. (1.1)–(1.2) has a strong solution $u$ satisfying, $\forall 0<T<T^{\ast }$,

$$\nabla u\in L^{\infty }([0,T],H^1\left(\Omega \right))\cap L^2([0,T],H^2\left(\Omega \right)).$$

If $T^{\ast }<+\infty$, then it yields that

(i) $2\le n\le 3,n<p<4$

$$\underset{T\to T^{\ast }}{lim}{\int }_0^T{\Vert \nabla u\Vert }_{L^p\left(\Omega \right)}^{\frac{p(6-n)}{p-n}}dt=+\infty ;$$

(ii) $2\le n\le 3,4\le p<+\infty$, or $n\ge 4,n<p<+\infty$

$$\underset{T\to T^{\ast }}{lim}{\int }_0^T{\Vert \nabla u\Vert }_{L^p\left(\Omega \right)}^{\frac{4(2n+4p-pn-4)}{(p-2)(4-n)}}dt=+\infty .$$

Remark 1

For $n=3$, $3<p<4$, the criterion will be ${\int }_0^T{\Vert \nabla u\Vert }_{L^p\left(\Omega \right)}^{\frac{3p}{p-3}}dt<+\infty$. For $n=2$, $2<p<4$, the criterion will be ${\int }_0^T{\Vert \nabla u\Vert }_{L^p\left(\Omega \right)}^{\frac{4p}{p-2}}dt<+\infty$. This criterion also satisfies the Serrin condition $\frac{n}{p}+\frac{2}{r}<1$. These results still hold for the torus domain or the whole space. The local existence of the strong solution can be proved by the method in [3], [4]. In this paper, we omit the details.

The organization of this paper is the following. In Section 2, we will show some lemmas which will be used in following sections. In Section 3, we will establish the blowup criterion for the strong solution to the Landau–Lifshitz–Gilbert Eq. (1.1)–(1.2).