Very Regular Solutions for the Landau-Lifschitz Equation with Electric Current

Gilles CARBOU Rida JIZZINI

Abstract The authors consider a model of ferromagnetic material subject to an electric current,and prove the local in time existence of very regular solutions for this model in the scale of Hkspaces.In particular,they describe in detail the compatibility conditions at the boundary for the initial data.

Keywords Ferromagnetic materials,Compatibility conditions,Existence result

1 Introduction

Ferromagnetic materials are used in several applications:radar furtivity,electromagnets,storage of digital data.They are characterized by a spontaneous magnetization represented by the magnetic moment m defined on[0,T]×?,where ? is the ferromagnetic domain in which the material is confined.At low temperature(under the Curie temperature),the material is said to be saturated,that is,the norm of the magnetic moment is constant,so after renormalization,we have

The magnetic moment links the magnetic field H and the magnetic induction B by the constitutive relation

The most promising application of the ferromagnetic materials concerns the nano-electronics,and in particular the storage of digital informations with quick access.The ferromagnetic devices used for these applications are either nano wires,thin plates or really 3d components.For nano-electronic applications,one goal is to store informations by magnetic domains representing the bits,and to transmit it very rapidly along the magnetic structures via domain walls motion.A standard way to induce this motion is by applying a magnetic field on the sample.The effects of the applied magnetic field are measured by the Zeeman energy(see[5,11,15]).The problem of this solution is that the information is modified since the domains can be removed by the application of an applied field.As explained in[13–14],the good way to transport the information(to a reading head for example)without change is obtained by using of an applied electric current.This increases the access velocity to the memory without mechanical wear of the components,and preserving the data.

In this paper,we study the following Landau-Lifschitz model describing a ferromagnetic material submitted to an applied current:

where

(1) ? is an open bounded set with smooth boundary and ν is the outside unit normal on??;

(2)Heis the effective magnetic field including the demagnetizing field and the exchange field;

(3)v is the current speed.For physical reasons,the relevant boundary condition is that v satisfies v·ν=0 on the boundary(the electric current remains in the domain),but we do not use this condition in our analysis;

(4)?m is the exchange field;

(5)H(m)is the demagnetizing field.It is deduced from m by the static Maxwell equations and the law of Faraday: ?

Remark 1.1The applied current modeling by the transport term(v·?)m+m×(v·?)m is explained in[13](see also the references therein).

We denote by Hk(?;S2)the set

In[1,6,16],the existence of global weak solutions for the Landau-Lifschitz equation without electric current is obtained.With the same method as[1]or[6],Bonithon proved in[3]the global existence of weak solutions for(1.2).In the case of the Landau-Lifchitz equations without electric current when the effective field is reduced to?m,the weak solutions are non unique(see[1]).This non uniqueness is expected in more general cases but is totally open.In addition,the Landau-Lifschitz equation does not have regularizing effects.So the existence of strong solutions is not obvious.

Existence of strong solutions for the Landau-Lifschitz equation without applied current is obtained in[7–8].The solutions constructed in these papers are inand are local in time.

In the present paper,following the method of[7],we construct local in time strong solutions for(1.2)infor T0.In addition,for more regular initial data,if compatibility conditions are satisfied,we prove the existence of strong solutions with high regularity on the same maximal existence interval[0,T?[.In particular,we prove that an eventual blow up for the regular solution occurs for the H2-norm.To our knowledge,the construction of very regular solutions for the Landau-Lifschitz equation does not exist in the literature,even for models without applied current.

Our first result is the following theorem.

Theorem 1.1Letsatisfy the compatibility condition:

(3)m satisfies(1.2).

Without additional compatibility condition,we can construct solutions in H3(?).

Theorem 1.2Letsuch that

The key point for the proof of Theorems 1.1 and 1.2 is that the Landau-Lifschitz equation(1.2)is equivalent,for solutions satisfying the saturation constraint(1.1),to the following problem:

In order to obtain more regularity for the solutions of(1.4),we must assume compatibility conditions on the initial data.Since(1.2)is equivalent to(1.4),and since our strategy consists in working with this last equation,we write the compatibility conditions derived from(1.4).First,we define formally the initial value ofat t=0.

We set V0=m0.We denote

Using Equation(1.4),if m is regular enough,we obtain the value ofby taking(1.4)at t=0 and by writing that.So,we define V1by

We remark that,if m0∈ Hk(?)and if v is sufficiently regular,then V1∈ Hk?2(?)for k ≥ 2.

In order to write the compatibility conditions,we recall the following formulas for the derivation of a product:

and

with

(1)AK={α =(α1,α2,α3)∈ {0,···,K}3, α1+ α2+ α3=K},

By differentiating(1.4)k times with respect to t,and by taking the obtained result at t=0,we define recursively

Definition 1.1Let k∈ N,We say that m0satisfies the compatibility condition at order k,if

Then,for all T

and

The present paper is organized as follows.

In Section 2,we recall useful lemmas.In particular,we study the regularity properties for the operator H.

Section 3 is devoted to the proof of Theorem 1.1.Basically,we follow the method of[7].As said above,the key point is that Equation(1.2)is equivalent to(1.4)for solutions satisfying the saturation constraint(1.1).In this new formulation,the dissipation due to the exchange field?m appears completely though it only appears m×?m in(1.2).We construct by the Galerkin method a solution of(1.4)inand we prove that this solution satisfies in addition the saturation constraint,so that it is a solution for(1.2).

Construction of more regular solutions for(1.2)is totally new and entails several difficulties.In Section 4,in order to prove the H3regularity for the solution of(1.4),a direct energy estimate for the third order space derivatives is not possible because of the non-local term H(m)which does not satisfy the homogeneous Neumann boundary condition.Our method consists in differentiating the Galerkin approximation with respect to time and proving by this way thatWe recover the H3-regularity for m by a bootstrap argument using Equation(1.4).This operation is complicated by the non-linear term

Section 5 is devoted to the proof of Theorem 1.3.As already said for the proof of the H3-regularity,variational estimates for high order space derivatives are not possible because of the non-local term H(m)which does not satisfy the homogeneous Neumann boundary condition.In addition,we are unable to perform H2estimates for the time derivative of mn,solution for the Galerkin approximation of(1.4).Indeed,we need for that compatibility conditions for all n which are not satisfied because of the non-local term H(mn).

Let us explain briefly our strategy in the simplest case:k=4.We already know from Theorem 1.2 that the solution m is inWe derivate(1.4)with respect to time.By a Galerkin process,using compatibility conditions on?tm(t=0),we construct a solution for the obtained problem inNow,?tm is a solution of this problem and a uniqueness argument ensures then thatFinally,we improve the regularity of m by elliptic regularity theorems writing?m in function of?tm.

In the general case,we proceed by induction with,roughly speaking,the same method.

2 Preliminary Results

2.1 Equivalent norms

We use the following notations:

and

We recall the following result established in[7].

Lemma 2.1Let ? be a bounded regular open set.There exist c1and c2such that for all u ∈ H2(?)such thaton ??,

and for all u ∈ H3(?)such thaton ??,

From Lemma 2.1 and using the standard interpolation inequality,we rewrite Sobolev and Gagliardo-Nirenberg inequalities on the following form.

Lemma 2.2Let ? be a regular bounded domain of R3.There exists a constant C such that for all u ∈ H2(?)such thaton ??,

and for all u ∈ H3(?)such thaton ??,

The following lemma will be useful to estimate products of functions.

Lemma 2.3If u ∈ H1(?)and v∈ H2(?),then uv∈ H1(?)and

ProofSince ? is a smooth bounded domain of R3,by Sobolev embedding,so that uv∈ L2(?).

2.2 Comparison lemma

We recall without proof the following standard comparison lemma.

Lemma 2.4Letand locally lipschtz with respect to its second variable.Letbe the maximal solution of the Cauchy problem:

Let y:R+?→R,C1such that

Then

2.3 Galerkin basis

Let us denote by Vnthe finite space spanned on the n first eigenfunctions of the operatorwithand Pn,the orthogonal projection from L2(?)onto Vn.

Proposition 2.1There exists a constant C such that for all n,the orthogonal projection Pnsatisfies the following properties:

ProofWe write u on the formwhere Qn(u)belongs toSincewe obtain

Using integration by parts and the fact that?Pn(u)belongs to Vnso thaton ??,we obtain

So

In the same way,for the H2and the H3estimates,we remark thaton the boundary.For the H2estimate,we have

For the H3estimate,

Therefore,using Lemma 2.1,

2.4 Demagnetizing field

The operator H takes its values in the space L2(R3).We can observe thatis the orthogonal projection ofon the vector fields of gradients in L2(R3).Let us consider the restriction of H to ?.Classicaly,we have

In the proposition below,following Ladyshenskaya[10,p.196],we can derive the following regularity result which describe the continuity of the operator H on the spaces Wk,p(?)for p∈]1,+∞[and k∈N.

Proposition 2.2LetThen for k∈N,if u belongs to Wk,p(?),the restriction of H(u)to ? belongs to Wk,p(?)and there exists a constant Ck,psuch that

ProofSee[8].

3 Proof of the H2Regularity

In[7],Carbou and Fabrie proved the existence of local strong solutions infor the Landau-Lifschitz equation without electric current.Our proof for Theorem 1.1 is basically the same.For the convenience of the reader,we give the complete proof in the present paper,emphasizing the changes due to the electric current.

3.1 Equivalent system

A dissipative term of the formappears if we take the inner product in L2of(1.2)with ?m.This dissipation is not sufficient to obtain energy estimate in the space H2(?).

We observe that,for m regular enough and|m|=1 in ?,then

So,if m is regular enough and satisfies the saturation constraint,then m is solution for the Landau-Lifschitz equation(1.2)if and only if m satisfies the following system

This equation has the advantage to highlight the dissipative term and is more convenient to build regular solutions for(1.2).

3.2 Galerkin approximation for the modified Landau-Lifschitz equation

We recall that we denote by Vnthe finite space built on the n first eigenfunctions of the operatorwithand by Pnthe orthogonal projection from L2(?)on Vn.We aim to find a solution mntaking its values in Vnfor the following Galerkin approximation of(1.4):

Cauchy-Lipschitz theorem ensures the existence of a unique solution of(3.2)defined on[0,Tn[.

3.2.1 L2estimate for(3.1)

Taking the inner product in L2(?)of(3.2)by mn,we obtain

3.2.2 H2estimate for(3.1)

Now,we take the inner product in L2(?)of(3.2)with ?2mn,and integrate by parts to get

where

We bound separately each term.

The terms I1,I2and I3are estimated in[7].For the convenience of the reader,we rewrite these estimates.Using Lemma 2.2,we have

(1)Estimate on I1:

(2)Estimate on I2:We remark thatso

(3)Estimate on I3:

(4)Estimate on I4:

By addition of all these estimates and using Young inequality,we obtain the following inequality

3.2.3 Uniform estimate on the Galerkin approximation

Summing inequalities(3.3)and(3.4),we obtain that there exists a constant C such that

where C does not depend on n.We denote

From the assumptions about v,c∈C0(R+).

In addition,mn(0)=Pn(m0),and since m0satisfies the compatibility condition(1.3),by Proposition 2.1,we obtain that for all n,

We consider z the maximal solution of the O.D.E.:

and we denote by T?the maximal existence time of z.By the comparison lemma,we have

Therefore,for all T

and by integration of(3.5)on the interval[0,T],we obtain

By Equation(3.2)we conclude that

and

3.3 Limit when n tends to+∞

These uniform estimates insure the existence of a subsequence(mnk)and a function m such that

Thanks to Aubin-Simon compactness lemma(see[2,12]),we can conclude that

and

because H is a continuous map on Sobolev spaces Hs(?)for s=0,1,2.

Taking the limit in(3.2),we obtain that m satisfies

In addition,we remark that from[4,Theorem II.5.14],

3.4 Conservation of the ponctual norm

We prove now that the solution m constructed in the previous part satisfies the physical constraint|m|=1,so that m satisfies the Landau-Lifschitz equation.Contrary to[7],a transport term due to electric current appears in Equation(1.4).

Using the scalar product in R3of(1.4)by m,we get

For d≤3 and for all u∈L∞(0,T;H2(?)),we have

Taking the inner product of this equation by a,we obtain

Under the assumption|m|=1,the equations(1.2)and(1.4)are equivalent.Hence,we have proven the existence of a strong solution for(1.2)in the spacefor T

3.5 Uniqueness for the strong solution of(1.2)

Let m1and m2insatisfying(1.2).Since(1.2)preserves the saturation constraint(1.1)for the strong solutions,they satisfy(1.4).We denote w=m1?m2.Thenand is a solution to

where

We take the inner product of(3.9)with w.Since

So by using the Young inequality,we obtain

and since m1and m2are in L2(0,T;H3(?)),we conclude by the comparison lemma that for all t≥0,

and since w(0)=0,we obtain that w=0 for all t,which concludes the proof of Theorem 1.1.

4 Study of the Case s=3

In order to obtain more regular solutions,it would be standard to multiply the Galerkin approximation(3.2)by?3mnto obtain a H3-estimate.Unfortunately,the non-local term H(mn)does not satisfy the homogeneous Neumann boundary condition,so that the necessary double integration by parts is not possible.Therefore,the H3regularity is obtained by derivation of(3.2)with respect to t in order to obtain a H1estimate on?tm.We conclude the proof by a bootstrap argument to obtain that?m and?tm have the same regularity.

4.1 H1regularity for?tm

We differentiate the Galerkin approximation(3.2)with respect to t.Denoting by w1,n=?tmnthe time derivative of mn,we obtain

with

We multiply(4.1)by??w1,nand we integrate on ?.Sincewe obtain

Using the continuity of the operator H on Hk(?)for k=0,1,2,we estimate each term:

Since v is sufficiently regular,by absorbing,we obtain

where

and

Let us now estimate the initial value of?tmn.By Equation(3.2)taken at t=0,we have

Here we recall that v0(x)=v(0,x).

Using Proposition 2.1,we can estimate the H1norm of w1,n(0)without compatibility condition on the following way:

using the compatibility condition(1.3)and Proposition 2.1.

Let ξnbe the solution of

We have

From Estimates(3.6)–(3.7),we obtain that for all T

so we infer that

Therefore,by the comparison Lemma 2.4,we obtain that for every T

and by integration of(4.2)on the interval[0,T],we obtain

From these inequalities,we can conclude that there exists a subsequence(w1,nk)such that

Therefore we obtain that?tm is bounded in the spacefor all T

4.2 Regularity of?m

In this paragraph,we prove

so that

By the equation,we have

Let us consider gmthe map defined byThis map is linear bijective which inverseis given by

where W=m×(m×H(m))?m×(v·?)m.

Lemma 4.1For all

ProofWe recall that

In the previous section,we have proven thatandSo by Lemma 2.3,

Now,H2(?)is an algebra.Since m ∈ L∞(0,T;H2(?))andwe obtain

Lemma 4.2For all

ProofWe know that m ∈ L∞(0,T;H2(?)).From Proposition 2.1,the same holds for H(m).Since L∞(0,T;H2(?))is an algebra,

So a fortiori this term belongs to

Concerning the other term,m and v are bounded in L∞(0;T;H2(?))and?m is bounded in L∞(0,T;H1(?)).So by Lemma 2.3,

On the other hand,m and v are bounded in L∞(0;T;H2(?))and?m is bounded in L2(0,T;H2(?)),thus,since H2(?)is an algebra,

We aim to deduce from(4.4)more regularity for m.We have

so

Using Lemmas 4.1–4.2,we obtain from(4.4)that

By using Lemmas 4.1–4.2,we obtain

and so

By[4,Theorem II.5.14],since ?tm ∈ L2(0,T;H2(?)),we obtain

This concludes the proof of Theorem 1.2.

5 Very Regular Solutions

It is not possible to obtain H2estimates on w1,n=?tmnusing Equation(4.1).Indeed,we would need a uniform estimate on the initial value w1,n(0)in the H2norm,and using Proposition 2.1,it would be necessary to check a compatibility condition of the form:

Because of the non-local term H(mn(0)),this condition can not be satisfied for all n.

We assume that m0∈ H4(?;S2)and satisfies the compatibility condition at order one(see Definition 1.1).In order to obtain more regular solutions,our strategy is the following:

(1)We compute the equation(5.2)(see below)satisfied by the time derivative?tm.

(2)We construct a solution w1for this equation inAt this step we need a compatibility condition at the boundary for?tm(t=0).

(4)Writing?m in function of?tm,we proveso that

If m0∈ H5(?;S2)satisfies the compatibility condition at order one,we obtain the desired regularity by differentiating the Galerkin approximation of(5.2)with respect to time(in the same spirit we obtained the H3regularity for m in the previous part).

This strategy will be used at any order to obtain very regular solutions for the Landau-Lifschitz equation.

We detail this process at any order by proving by induction the following property P(k):

This property is proven for k=1 in Sections 3–4.

Let k≥1.Let us assume that P(k)is true,and let us establish P(k+1).

Let m0∈ H2(k+1)(?;S2)satisfy the compatibility condition at order k.By the property P(k),we already know that

where

with

and

We aim to construct a more regular solution for(5.2).We establish the following estimates.

Lemma 5.1For all j ∈ {1,···,k},and for all T

ProofWe fix j ≤ k.From the property P(k),for all i∈ {0,···,j?1},

and a fortiori this quantity is in

In the same way,

5.1 Construction of more regular solution for(5.3)

We consider the following Cauchy problem:

We aim to construct a solution for(5.3)in

We recall that,from the compatibility condition at order k,Vk∈ H2(?)andon??.

We consider the following Galerkin approximation for(5.3):

Since the coefficients of this ordinary differential equation are continuous on[0,T?[,since the equation is linear,the maximal existence time Tnequals T?.Let us obtain uniform estimates on wn.

Bound for the initial dataUsing the compatibility condition at order k,we know that Vk∈ H2(?)and thaton ??,so we can apply Proposition 2.1 and obtain

L2estimateWe multiply(5.4)by wnand obtain

where

H2estimateWe multiply(5.4)byand obtain

where

By adding up the previous L2and H2estimates,after absorbingwe obtain that

where

We already know that for allsofor all T

By Equation(5.4),we obtain in addition the following estimate on the time derivatives

By standard arguments(see Subsection 3.3),we obtain by extracting subsequences that there exists w satisfying(5.3)such that

5.2 Uniqueness for(5.3)

Let w1and w2be two solutions of(5.3)inWe denote W=w1?w2,and we remark that w satisfies the following problem:

We multiply Equation(5.9)by??W and integrate on[0,T]×? for all T

We have

and

We integrate the L2estimate from 0 to T.Adding up with the H1estimate and using Young formula,we obtain

where

By properties of m and v,g ∈ L1(0,T)for all T

5.3 Regularity for m

From the property P(k),we know that

In addition,we proved in the previous subsection that

Let us establish by induction the following claim.

Claim 5.1For all T

ProofThis property is true for j=0 by(5.10).

Let j ∈ {1,···,k ?1}.Let us assume that the property is true at the rank j?1.Then,from Equation(5.1),replacing j by k?j,we have

where ψmis defined on page 904.We have

On the one hand,we recall that

Since 2j≤ 2k,?m,?wk?j,v and m are in L∞(0,T;H2j(?)),which is an algebra(since j≥1),we have

Since 2j+1≤ 2k,?m,v and m are inand since

we have

On the other hand,we recall that

From Proposition 2.2,H(wk?j)and wk?jhave the same regularity,i.e.,they are in L∞(0,T;H2j+1(?)).We remark that j≤k?1 so that 2j+1≤2k?1.So?m,H(m),m and?m are in.The same holds for v.Since this space is an algebra,we obtain

From Lemma 5.1,

From the property at rank j?1,

and by elliptic regularity results,

This concludes the proof of the claim.

In particular,we have proven that

Now,we have

5.4 Regularity for in the case

We assume now in addition thatand satisfies the compatibility condition at order k.From the previous case,we already know that

We derivate(5.4)with respect to time.Denoting by αn:= ?twn,we obtain

where

We multiply(5.11)by ?αn.After integration on ? and using Young formula,we obtain

We have

for all T

for all T

In addition,

by(5.7).

So,

for all t≤ T

Using Proposition 2.2,(5.7)and the known bounds on?tm,we obtain

Now from Proposition 2.1,we have

From the compatibility condition at orderon ??,so from Proposition 2.1,

for all n.Therefore,there exists a constant C such that for all n,

Estimates(5.12)and(5.13)coupled with Lemma 2.4 yield

5.5 Regularity for m in the case m0∈ H2k+3(?)

We know from Subsection 5.3 that

So coupling this estimate with(5.14),we gain one rank for the regularity of each wj,and exactly with the same method as in Subsection 5.3,we prove

so that the proof of the property P(k+1)is complete.