## Abstract

Using the differential–difference method and viscosity vanishing approach, we obtain the existence and uniqueness of the global smooth solution to the periodic initial-value problem of the inhomogeneous, non-automorphic Landau–Lifshitz equation without Gilbert damping terms in one dimension. To establish the uniform estimates, we use some identities resulting from the fact and the fact that the vectors form an orthogonal base of the space .

## 1. Introduction

In this paper, we are concerned with the existence and uniqueness of the smooth solution to the one-dimensional periodic initial-value problem of the inhomogeneous non-automorphic Landau–Lifshitz equation,(1.1)(1.2)where *D*>0 is a constant, and are smooth functions and for some constant , . We assume that and are periodic functions with period .

Equation (1.1) is related to the generalized model of inhomogeneous ferromagnetisms and the simplified compressible ferromagnetisms.

As we know, the equation for the spin wave of inhomogeneous ferromagnetic medium was raised in the condensed matter physics given by Balakrishnan (1982). It can be derived from the inhomogeneous, isotropic Heisenberg exchange Hamiltonianfor a magnetic chain with a *site-dependent* nearest-neighbour interaction. The equation of motion obtained for iswhere *J* is the exchange constant. This equation can be shown to be valid in both the quantum and the classical cases. A continuum description in which , is suitable when , vary slowly over one lattice separation *a*. Inserting Taylor expansions of and in the above equation, we get in continuum limit(1.3)where . Then, equation (1.1) is generalized from (1.3) by replacing by , that is, the inhomogeneous function depends on time. Then, we call (1.1) the non-automorphic equation.

Another background of equation (1.1) is the model of compressible ferromagnetisms. Fievez (1982) revisited the one-dimensional classical compressible Heisenberg chain described by the Hamiltonianconsidered earlier by Cieplak & Turski (1980*a*,*b*), where is the displacement of the magnetic ion from equilibrium, without spin–phonon coupling, is the spring constant and . In the continuum limit, which corresponds to long-wavelength excitations, the equations of motion deduced by Fievez read as(1.4)(1.5)where and the substitution , has been made, a dot denotes derivative with respect to *t*, a prime with respect to *x*.

Fievez tried the solution of the form , , where , with (the lattice and spin wave are assumed to travel at the same velocity *c*). Equation (1.4) now becomeswith boundary conditions .

By integration, one has and .

Hence, Fievez derived the following Heisenberg chain equation:(1.6)in which with , , where .

The solitons of (1.6) were given by Magyari (1982). Equation (1.6) is called the compressible Heisenberg chain equation (or compressible Landau–Lifshitz equation). Equation (1.6) with *B*=0, corresponds to the inhomogeneous Heisenberg chain equation derived by Balakrishnan (1982), and (1.6) with *B*=0, is just equation (1.1).

To our knowledge, there are some discussions on the periodic initial-value problems of inhomogeneous and compressible equation. Ding *et al*. (1999*a*) obtained the existence of measure-valued solution to the compressible equation (1.6). In the same year, these authors obtained the existence and uniqueness of the smooth solution to the inhomogeneous equation (1.3) by the viscosity vanishing method.

The novelties of this paper are in the following aspects. First of all, we find some new energy laws, which are different from those in the article given by Ding *et al*. (1999*b*) to establish the *a priori* estimates. Secondly, the most interesting one is to apply the property that the family of vectors forms an orthogonal base of since . This fact was first introduced by Zhou *et al*. (1991) for the case . Some other identities resulting from the fact that are also used in the estimates. For the inhomogeneous model given by Ding *et al*. (1999*b*), they did not successfully use these ideas. Finally, since depends on time *t*, we confront some special difficulties in the estimates. However, we have given sufficient conditions to get the existence of the smooth solutions to problems (1.1) and (1.2), in the case where the inhomogeneous function depends on time.

In order to prove the existence of smooth solutions to problems (1.1) and (1.2), we use the viscosity vanishing method. First, by the difference method, we establish the existence and uniqueness of the smooth solution to the following problems:(1.7)(1.8)Then we prove some *a priori* estimates for the solutions of (1.7) and (1.8) uniform in by constructing some new energy laws, we then send to zero to obtain the existence of smooth solutions to problems (1.1) and (1.2). In this procedure, the usage of the orthogonal base may simplify our proof and make the idea clear compared with the proof in the paper given by Ding *et al*. (1999*b*). Some other orthogonal bases are also used in the proof. Again, since , we have used many other useful identities as we can see in §3.

As pointed out by Zhou *et al*. (1991) and Ding *et al*. (1999*b*), in the classical sense, equation (1.7) is equivalent to(1.9)since and .

Here, and in the following, when discussing the periodic problem, we assume that and are periodic functions. When discussing the initial-value problem, we first replace them by and , where and in and periodic in and then let . Denote .

## 2. : local smooth solution

To get the existence of local smooth solutions of (1.7) and (1.8), we apply the difference method. The proof is generally similar to that of the paper given by Ding *et al*. (1999*b*), since we only make a difference in the space direction. However, for completion, we give the proof in this section. For simplicity, we let .

We need the following well-known lemmas.

## Lemma 2.1 Zhou et al. (1991)

*Let q, r be real numbers and j, m be integers such that* *,* . *If* *, then**where* *,* *,* *and*

## Lemma 2.2 Zhou (1984)

*Let p be a real number and j, m be integers such that* *,* . *Then**where* *,* *,* *,* *,*

## Lemma 2.3 Zhou (1984)

*Let* *,* *such that* *,* . *We have*

,

,

,

*where* *,* *denote the forward and backward difference, respectively*.

We use the differential–difference method to prove the local existence of smooth solutions of (1.7) and (1.8). We establish the following difference–differential equations:(2.1)(2.2)(2.3)where , , *J*>0.

It is clear that the initial-value problem for ordinary differential equations (2.1)–(2.3) admits a local smooth solution. For such solution, we shall give some estimates uniformly in *h* and then get a local smooth solution to problems (1.7) and (1.8). In this section, we always denote a smooth solution of (2.1)–(2.3) by .

*If* *,* *then there are constants* *independent of h such that*(2.4)

(2.5)

Multiplying (2.1) by and summing from *j*=1 to *J*, we have

It follows from lemma 2.2 that .

Therefore, we have(2.6)

Moreover, multiplying (2.1) by and summing from *j*=1 to *J*, we get

Therefore, one gets

Applying lemma 2.2, we have

Then, it follows from Young's inequality that(2.7)

Hence, putting (2.6) and (2.7) together, we have

This inequality, combined with Gronwall's inequality, implies that there are constants , independent of *h* such that

Lemma 2.4 is proved. ▪

*Under the conditions in* *lemma 2.4* *and* *, we have, for some constant C independent of h,*(2.8)

*If* *,* *,* *, then there are constants* *independent of h such that*(2.9)

(2.10)

It follows from (2.1) that

Multiplying this equality by , summing it from *j*=1 to *J* and noting thatthe others can be given in a similar way. By the following inequalitieswe can get

Lemma 2.6 follows from Gronwall's inequality. ▪

*Under the conditions in* *lemma 2.6**, we have, for some C independent of h,*(2.11)

(2.12)

We will give the following lemma.

*If* *,* *,* *, then there are constants* *independent of h such that*(2.13)

(2.14)

From (2.1) we have

From lemma 2.2, we have .

By lemmas 2.3, 2.4 and 2.6 and a similar method to lemma 2.6, we can have

Lemma 2.8 follows from Gronwall's inequality. ▪

By the induction method we have the following lemma.

*If* *,* *,**, then there are constants* *independent of h such that*(2.15)

(2.16)

(2.17)

From lemma 2.9, we obtain the *a priori* estimates for solutions to the differential–difference equations (2.1)–(2.3). Using the same method as in Zhou *et al*. (1991), we conclude that there exists a constant such that problems (1.7) and (1.8) admit a smooth solution in . This result is stated as follows.

*Let ϵ>0,* *,* *,* *,* . *Then* *(1.7) and (1.8)* *admit a local smooth solution* *in* *with* *depending on k:*

## 3. : global solution and uniform estimates

In §2, we have obtained a local smooth solutions for (1.7) and (1.8) when is fixed. In this section, we intend to prove the global existence of smooth solutions to problems (1.7) and (1.8) for fixed by deriving the global (in time) estimates. To meet the need to send to zero, we want these estimates to be both global in time and uniform in . The difficulty in the uniform estimate is overcome by constructing some new energy laws and by using the fact that is a base of , seeing and estimates below.

In the following, we always suppose is a global smooth solution of problems (1.7) and (1.8) for . We intend to derive the following global and uniform estimates.

*If* and *is a global smooth solution of problems* *(1.7) and (1.8)**, then we have*(3.1)

Multiplying (1.9) by , we have . This implies the conclusion of lemma 3.1. ▪

*Under the same conditions of* *lemma 3.1* *and* *, we have, for any given T>0, there exists C>0 independent of* *and D,*(3.2)

From (1.9), we have

By , we get

Therefore, one gets

It is easy to see that

Then lemma 3.2 follows. ▪

*Let* *,* *,* *,* *, then for any given T>0, there are constants C>0 independent of* *, such that*(3.3)

(3.4)

In the proof, we shall use the following identities which follow from the fact :(3.5)

It follows from (1.7) that(3.6)

From (3.6) we get(3.7)

Note that, if , then the vectors form an orthogonal basis of . Letand it is easy to see that

Therefore, we have(3.8)where we have used the fact that .

Combining (3.7) with (3.8), we get(3.9)

On the other hand, from (1.7) we get(3.10)

Putting (3.9) and (3.10) together and using (1.5), we have(3.11)

By lemmas 2.1 and 3.1, we have

Letting , we get(3.12)where we have used the fact that .

Therefore, one gets(3.13)

Integrating (3.13), by lemma 3.2 and , we have

The conclusion of lemma 3.3 follows from this inequality and Gronwall's inequality. ▪

Similarly, we have the following lemma:

*Let* *,* *,* *,* . *For any given T>0, there are C>0 independent of* *such that*(3.14)

(3.15)

It follows from (3.6) that(3.16)

Then, if , let

It is easy to see that

Then(3.17)where we have used the fact .

Putting (3.16) and (3.17) together, we have

Employing Gronwall's inequality, we obtain (3.14). Then, one can obtain (3.15) easily. ▪

By induction, we have the following lemma.

*Let* *,* *,* *,* . *For any given T>0, there is C>0 independent of* *such that*(3.18)

Combining the local existence obtained in §2 and the global in time estimates in lemmas 3.1–3.5, we can get the global existence of smooth solutions to problems (1.7) and (1.8) for fixed in the following sense.

*Let* *,* *,* *,* . *Then, for any given T>0, problems* *(1.7) and (1.8)* *admit at least one smooth solution* *in* *:*(3.19)

We should note that the *a priori* estimates in lemmas 3.1–3.5 are uniform in .

## 4. : global solution and uniqueness

In §3, we have obtained a global smooth solution for (1.7) and (1.8) for fixed , and the estimates are all uniform in . These uniform estimates enable us to pass to the limit in equation (1.7), and then to get the global smooth solutions of problems (1.1) and (1.2). Therefore, we have the following theorem.

*Let* *,* *,* *,* *,* . *Then* *(1.1) and (1.2)* *admit a global smooth solution* *:*

On the other hand, all the estimates in §3 are independent of *D*. Thus, by sending *D* to , we get the global existence of smooth solution to the Cauchy problem of (1.1).

*Let* *,* *,* *,* *,* . *Then, the Cauchy problem of* *(1.1)* *admits a global smooth solution* *:*

For any given , it is not difficult to see that the problems (1.7) and (1.8) (or the Cauchy problem of (1.7)) admit at least one global smooth solution.

Now, we turn to prove that the global smooth solution of (1.1) and (1.2) obtained in theorem 4.1 is unique. That is, we prove the following theorem.

*The global smooth solutions of* *(1.1) and (1.2)* *obtained in* *theorem 4.1* *is unique*.

First of all, we prove that the periodic problems (1.7) and (1.8) admit a unique global smooth solution. Let and be smooth solutions of (1.7) and (1.8). Let .

Then, we have(4.1)with homogeneous initial boundary conditions. Multiplying (4.1) by , using Hölder's inequality and noting that are smooth, we have(4.2)

On the other hand, multiplying (4.1) by , we have(4.3)

Combining (4.2) and (4.3), using Gronwall's inequality and noting that , we can get the uniqueness when .

In the second step, we let in (4.1) to get(4.4)

Therefore, we have(4.5)

On the other hand, from (4.4) we have

Therefore, using Hölder's inequality and noting that are smooth, we have(4.6)

Putting (4.5) and (4.6) together, we get

Using Gronwall's inequality and noting that , we can obtain the conclusion to theorem 4.1. ▪

## 5. Conclusion

In this paper, we obtain existence and uniqueness of global smooth solutions to the periodic initial problem of the one-dimensional inhomogeneous non-automorphic Landau–Lifshitz equation, using the differential–difference method and viscosity vanishing approach. First, we establish the existence and uniqueness of the smooth solution to (1.7) and (1.8) by differential–difference method. Secondly, we give some *a priori* estimates for the solution of the viscosity problem uniform in by constructing some new energy laws. Finally, we send to zero to obtain the existence of the smooth solution to problems (1.1) and (1.2). In this procedure, the usage of the orthogonal base simplifies our proof.

## Acknowledgements

The authors are grateful to the referees for their helpful comments and suggestions. The authors are supported by the National Natural Science Foundation of China (grant nos. 19971030 and 10471050) and by Guangdong Provincial Natural Science Foundation (grant nos. 000671 and 031495).

## Footnotes

- Received May 10, 2005.
- Accepted February 6, 2006.

- © 2006 The Royal Society