SMOOTH SOLUTIONS OF THE LANDAU-LIFSHITZ-BLOCH EQUATION ∗

Landau-Lifshitz-Bloch equation is often used to model micromag-netic phenomenon under high temperature. This article proves the existence of smooth solutions of the equation in R 2 and R 3 , and a small initial value condition should be added in the latter case. These results can also be generalized to periodical boundary value case.


Introduction
In this paper, we consider the following initial value problem, known as Landaulifshitz-Bloch equation where the constants k, µ > 0.
In [5][6][7]10,12,13], it is pointed out that the dynamics of the magnetization of ferromagnets is a phase-changing process. When the electronic temperature is higher than θ c (i.e. the temperature θ ≥ θ c , where θ c is the critical Carie temperature), LLB equation has proved to describe the magnetization dynamics. When the temperature θ < θ c , it is the normally well-known Landau-Lifshitz equation. For the Landau-Lifshitz equation, many workers has been studied (see [1-4, 11, 15, 16] and the book of Guo-Ding [9]). In [5][6][7] the following LLB equation has been proposed Ngonle consider (1.3) in the case L 1 = L 2 The effective field H ef f is given by (1.4) where χ 11 is the longitudinal susceptibity and let L 1 = L 2 = k 1 , then the equation (1.3) can be writed as follows under the coefficients k 1 , k 2 , γ, µ > 0. He obtained the existence of global weak In this paper, we study the smooth solution of problem (1.1)-(1.2) and get the following theorem.
The rest of this paper is divided into three parts. In section 2, we prove the existence of smooth solution in Theorem 1.1; In section 3, we prove the existence of smooth solution in Theorem 1.2; In section 4, we prove the uniqueness of smooth solution in Theorem 1.1 and Theorem 1.2.
Proof. Taking the scalar product of function u and equation ( This inequality implies that where we have used the embedding theorem of Sobolev spaces. Note the constant C is independent of p and let p → ∞, estimate (2.4) is obtained. Similarly, taking the scalar product of ∆u and equation (1.1) and then integrating the result over R d for the space variable x and over [0, t] for temporal variable x, we have we can get the estimate (2.2).
Proof. By simple calcalation, we get Taking the scalar product of ∆u and the equation (2.12), and then integrating the result over R 2 for the space variable x, we have Integrating by parts, we get By Holder inequality, it follows that By Gagliardu-Nirenberg's inequality, we have Using (2.13) and Gronwall's inequality, we obtain (2.8).
Next we are going to prove Theorem 1.1 for 2-dimensional case, and it is sufficient to prove the following Theorem.
where C depends on T and ∥∇u 0 ∥ H k .
Proof. We will use induction arguments to prove this theorem, but first of all it should be necessary to obtain the boundedness of ∥∇u∥ L ∞ . In fact, applying Laplace operator to the both sides of equation (1.3) and taking the scalar product with △ 2 u , then integrating over R 2 , we get Then follows from(2.13) and Galiardo-Nirenberg inequality. Next we utilize induction arguments, the m=1 case has been proved by Lemma 2.2, so it will be supposed that if m = m case holds, then m = m + 1 also holds. Applying the differential operator D m+1 to the both sides of equation (1.4) and taking the scalar product with D m+1 u , then integrating over R 2 , we get Consequently, using Gronwall's inequality, we conclude the theorem.

Proof of existence in Theorem 1.2
The proof of Theorem 1.2 is in line with that of Theorem 1.1, and the difference is that a priori estimates are more difficult to derive as the dimension changes, we propose an additional condition to overcome this situation. Precisely, we have the following Lemma

4)
Proof. By using the same arguments as in the proof of Lemma 2.11, we get where the estimate (2.4) and hypothesis ∥u 0 ∥ H 2 ≪ 1 has been used.
Inserting (3.7) to (3.6), we obtain (3.1). Now taking the scalar product of ∆ 2 u and the equation (3.5), then integrating the result over R 3 for the space variable x, we have Integrating by parts, we get By using the induction's method, we can prove this lemma.

Proof of uniqueness of the solution
In this section we will deal with the uniqueness of the solution in problem (1.1), (1.2), in fact, we have the following generalized result: Proof. The proof is standard, set w = u − v, we'll prove w ≡ 0. Since u and v satisfies (1.1) respectively, w satisfies the following equation the cross product in the above equation can be rewritten as Thus (4.1) becomes w t = △w + w × △u + v × △w − kw − k(|u| 2 + |v| 2 + u · v)w, (4.2) taking the inner product with w in both sides of (4.2), then and k| R d Since u and v are smooth, the norm ∥∇v∥ 2 L ∞ , ∥u∥ 2 L ∞ and ∥v∥ 2 L ∞ can be replaced by a constant C, thus it can be concluded that Gronwall's inequality and the fact that w(x, 0) ≡ 0 lead to w ≡ 0.