global existence of weak solutions for landau-lifshitz

DISCRETE AND CONTINUOUS Website: http://aimSciences.orgDYNAMICAL SYSTEMSVolume 17, Number 4, April 2007 pp. 867–890

GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR

LANDAU-LIFSHITZ-MAXWELL EQUATIONS

Shijin Ding

School of Mathematical Sciences, South China Normal University,Guangzhou, Guangdong 510631, China

Boling Guo

Center for Nonlinear StudiesInstitute of Applied Physics and Computational Mathematics

P.O.Box 8009, Beijing 100088, China

Junyu Lin

School of Mathematical Sciences, South China Normal University,Guangzhou, Guangdong 510631, China

Ming Zeng

College of Applied Sciences, Beijing University of Technology,PingLeYuan100, Chaoyang District,

Beijing 100022 P. R. China.

Abstract. In this paper we study the model that the usual Maxwell’s equa-tions are supplemented with a constitution relation in which the electric dis-placement equals a constant time the electric field plus an internal polarizationvariable and the magnetic displacement equals a constant time the magneticfield plus the microscopic magnetization. Using the Galerkin method and vis-cosity vanishing approach, we obtain the existence of the global weak solutionfor the Landau-Lifshitz-Maxwell equations. The main difficulties in this studyare due to the loss of compactness in the system.

1. Introduction. In this paper, we study the three dimensional Landau-Lifshitz-Maxwell equations as follows

~Zt = α1~Z ×

(~Z + ~H

)− α2

~Z ×(

~Z ×(~Z + ~H

))(1)

∇× ~H =∂( ~E + ~P )

∂t+ σ ~E (2)

∇× ~E = −∂ ~H

∂t− β

∂ ~Z

∂t(3)

∂2 ~P

∂t2+ λ2curl2 ~P + µ

∂ ~P

∂t= ν

(~E − 2 ~PΦ′(|~P |2)

)(4)

where ~Z(x, t) = (Z1(x, t), Z2(x, t), Z3(x, t)) denotes the magnetization field, ~H(x, t)

= (H1(x, t), H2(x, t), H3(x, t)) the magnetic field, ~E(x, t) = (E1(x, t), E2(x, t),

E3(x, t)) the electric field, ~P (x, t) = (P1(x, t), P2(x, t), P3(x, t)) the electric polariza-

tion, ~He = ~Z+ ~H the effective magnetic field, E(~P ) = 2 ~PΦ′(|~P |2) the equilibrium

2000 Mathematics Subject Classification. Primary: 35D10 ; Secondary: 35Q80.Key words and phrases. Landau-Lifshitz-Maxwell equations, global weak solution, existence .

867

868 SHIJIN DING, BOLING GUO, JUNYU LIN AND MING ZENG

electric field, curl2 ~P = curl(curl ~P ) = ∇× (∇× ~P ). α1, α2, β, σ, λ, µ, ν are con-stants, where α2 ≥ 0 is the Gilbert damping coefficient; λ > 0 denotes the speedof light for the internal fields; σ ≥ 0 denotes the constant conductivity, Constant β

can be viewed as the magnetic permeability of free space. The physical meaningsof parameters µ, ν can be found in [17].

We assume that Φ : R+ → R is a C2 convex function such that

|Φ′(r)| ≤ C0, rΦ′′(r) ≤ C1 (5)

for all r ≥ 0. We also assume that function Φ(r2) has unique minimum at somepoint r2

0 . These assumptions guarantee that rΦ′(r2) ≤ C2 for all r ≥ 0, whereC2 = C0 + 2C1. Therefore we have

∣∣∣ ~XΦ′(| ~X|2) − ~Y Φ′(|~Y |2)∣∣∣ ≤ C2

∣∣∣ ~X − ~Y∣∣∣ for all ~X, ~Y ∈ R3 (6)

Much more about the equilibrium relation of Φ may be found in Landau and Lifshitz[27],P84-91.

System (1)-(4) models the dynamics of magnetization, magnetic field, electricfield and electric polarization for the ferromagnetic-ferroelectric materials which,compared with the classical Landau-Lifshitz-Maxwell system in [20], includes a new

equation for polarization ~P . As we know that, some ferromagnetic substances, suchas ferrites, are not only ferromagnetic materials, but also ferroelectric ones (such asLiFePO4 ), we call them the ferromagnetic-ferroelectrics [34].

If an electric field is applied to a medium (such as a dielectric one) made upof a large number of atoms or molecules, the charges bound in each molecule willrespond to the applied field and will execute perturbed motions: the molecularcharge density will be distorted. The multipole moments of each molecule will bedifferent from what they were in the absence of the field. In simple substances,when there is no applied field the multipole moments are all zero, at least when theaveraged over many molecules. The dominant molecular multipole with the appliedfields is the dipole. There is thus produced in the medium an electric polarization~P (the dipole moment per unit volume). A dielectric in which ~P differs from zero is

said to be polarized. The vector ~P determines not only the volume charge densitybut also the density of the charge on the surface of the polarized dielectric[22]. Onecan learn more details about polarization in [6],[12],[16],[27].

(1)-(3) (without ~P ) is a well-known classical Landau-Lifshitz-Maxwell system forferromagnets [20]. The coupling of this classical Landau-Lifshitz-Maxwell system

with ~P and the equation (4) for ~P can be derived from the full Maxwell system asfollows

∂ ~B

∂t= −curl ~E and

∂ ~D

∂t+ σ ~E = curl ~H (7)

here ~E and ~H are the electric and magnetic fields, σ ≥ 0 is the conductivity, ~D and~B the electric and magnetic displacements defined by

~D = ǫ0 ~E + ~P , ~B = µ0( ~H + ~Z)

where ǫ0 is the permittivity of free space, µ0 is the the magnetic permeability of

free space, ~Z is the magnetization and ~P is the electric polarization. Substituting

these definition into (7), one may couple ~Z, ~E, ~H and ~P by systems like (1)-(3).For the derivation of (4), we refer to [17].

LANDAU-LIFSHITZ-MAXWELL EQUATIONS 869

In (1)-(4), if ~H = 0, ~E = 0, ~P = 0, β = 0, we obtain the classical Landau-Lifshitzequation:

~Zt = α1~Z ×~Z − α2

~Z × (~Z ×~Z) (8)

This equation has been studied extensively in recent years. For the global ex-istence of weak solutions of (8), we refer to [10],[18],[19] and [34]. Guo and Hong[18] studied the links between the solution and the harmonic map on the compactRiemannian manifold. For more recent results on the regularity of the solutions, werefer to [13],[14],[28],[30] and [34].

If, in addition, α2 = 0, system (8) becomes

~Zt = α1~Z ×~Z (9)

As pointed out in [41], system (9) is a strongly coupled and strongly degenerateparabolic system. In [35]-[40] (and references therein), the authors investigatedextensively the global existence of classical and generalized solutions to system (9).

If ignoring the polarization, one gets the classical Landau-Lifshitz-Maxwell’s sys-

tem (1)-(3) (without ~P ) which was proposed by Landau and Lifshitz in [26]. Guoand Su in [20] obtained the global weak solutions for this system subject to theperiodic initial data by Galerkin method. Similar discussions can be found in [8],[9] by Carbou et. al.

If only considering the dynamics of electric filed ~E, megnetic field ~H and po-

larization ~P which models the dynamics of ferroelectric materials, one has the so

called nonlinear Maxwell systems (2)-(4) (without ~Z) which was first consideredby Greenberg et. al [17] for a simple case. And in [4], Habib Ammari and KamelHamdache generated the discussions of [17] to general cases and obtained the globalexistence, uniqueness and regularity of weak solutions by the theory of semigroupsand the a priori estimates. There are also several other papers studied Maxwell’sequations with polarization effect (see [1],[2],[7],[11],[15],[21]-[25],[29] and [32]).

In this paper, we are concerned with the full Landau-Lifshitz-Maxwell systemwhich includes both Landau-Lifshitz equation and nonlinear Maxwell equations.We shall prove the global existence of periodic weak solutions with periodic initialconditions.

We call a function f(x) is 2D-periodic if f(x + 2Dei) = f(x), (i = 1, 2, 3), where(e1, e2, e3) forms the unit orthogonal basis of R3, D > 0 is a constant.

For the system (1)-(4), we impose the following initial conditions

~Z(x, 0) = ~Z0(x), ~H(x, 0) = ~H0(x), ~E(x, 0) = ~E0(x),

~P (x, 0) = ~P0(x), ~Pt(x, 0) = ~P1(x) (10)

Throughout this paper, we always assume that ~Z0(x), ~H0(x), ~E0(x), ~P0(x), ~P1(x)are 2D− periodic. We denote by Ω ⊂ R3 the three dimensional cube with width2D along each direction, i.e. Ω = x = (x1, x2, x3)| |xi| < D; (i = 1, 2, 3) andQT = (x, t)| x ∈ Ω, 0 < t ≤ T .

Since equation (1) is strongly coupled, it is not easy to obtain weak solutions byuse of the theory of semigroups as Habib Ammari in [4]. What we are going to dois to use Galerkin method.

However, it is easy to see that equation (4) lacks compactness which we need to

get the estimate of H1-norm for ~P . In fact, from this equation, we only be able to ob-

tain the L∞(0, T ; L2(Ω)) estimates for curl ~P , but do not have the L∞(0, T ; L2(Ω))


estimates for div ~P . To overcome this difficulty, we firstly apply the viscosity van-ishing argument to get the weak solution for the viscosity problem and then, wewant to send the viscosity constant to zero. But the lack of compactness reappearsin the limit procedure. Therefore we secondly consider more regular (than energyones) class weak solution and then obtain the additional a priori estimate for thediv-component of Maxwell fields(cf.Lemma 9). Finaly we obtain the desired weaksolution to the original problem.

Replacing (4) by

∂2 ~P

∂t2+ λ2curl2 ~P + µ

∂ ~P

∂t− ǫ~P = ν( ~E − 2 ~PΦ′(|~P |2)) (11)

We get a viscosity system (1)-(3) and (11) with the 2D-periodic initial conditions(10). By Galerkin method we shall obtain the global 2D-periodic weak solution

~Zǫ, ~Hǫ, ~Eǫ, ~P ǫ to the viscosity problem (1)-(3), (10) and (11). By establishingthe estimates uniform in ǫ and sending ǫ to 0, we may get the global weak solutionto problem (1)-(4) and (10).

Definition 1. ([5]) The space HP (curl, Ω) is defined by

HP (curl, Ω) = ~V ∈ L2(Ω); ~V is 2D − Periodic and curl~V ∈ L2(Ω),

and is provided with the norm

‖~V ‖HP (curl,Ω) = ‖~V ‖2L2(Ω) + ‖curl~V ‖2

L2(Ω)1/2.

The space HP (div, Ω) is defined by

HP (div, Ω) = ~V ∈ L2(Ω); ~V is 2D − Periodic and div~V ∈ L2(Ω),

and is provided with the norm

‖~V ‖HP (div,Ω) = ‖~V ‖2L2(Ω) + ‖div~V ‖2

L2(Ω)1/2.

Finally, we set

XP (Ω) = HP (curl, Ω) ∩ HP (div, Ω)

with the norm

‖~V ‖XP (Ω) = ‖~V ‖2L2(Ω) + ‖curl~V ‖2

L2(Ω) + ‖div~V ‖2L2(Ω)

1/2.

Definition 2. A 2D-periodic vector (~Z(x, t), ~E(x, t), ~H(x, t), ~P (x, t)) ∈ (L∞(0, T ;H1(Ω)), L∞(0, T ; L2(Ω)), L∞(0, T ; L2(Ω)), W 1,∞(0, T ; L2(Ω))

⋂L∞(0, T ; H1(Ω))) is

called a weak solution to problem (1)-(4) and (10), if for any 2D-periodic vector-

valued test function ~Ψ(x, t) ∈ C1(QT ) such that ~Ψ(x, T ) = 0, the following equalitieshold

∫ ∫

QT

~Z · ~Ψt + α1

∫ ∫

QT

(~Z ×∇~Z) · ∇~Ψ − α1

∫ ∫

QT

(~Z × ~H) · ~Ψ +

∫

Ω

~Z0 · ~Ψ(x, 0)

+α2

∫ ∫

QT

(~Z ×∇~Z

)· ∇(

~Z × ~Ψ)− α2

∫ ∫

QT

(~Z × ~H

)· ∇(

~Z × ~Ψ)

= 0 (12)


∫ ∫

QT

(~E + ~P

)· ~Ψte

σt + σ

∫ ∫

QT

eσt ~P · ~Ψ

+

∫ ∫

QT

eσt∇× ~Ψ · ~H +

∫

Ω

(~E0 + ~P0

)· Ψ(x, 0) = 0 (13)

∫ ∫

QT

(~H + β ~Z

)· ~Ψt −

∫ ∫

QT

(∇× ~Ψ

)· ~E +

∫

Ω

(~H0 + β ~Z0

)· ~Ψ(x, 0) = 0 (14)

∫ ∫

QT

~Pt · ~Ψt − λ2

∫ ∫

QT

curl ~P · cur~Ψ − µ

∫ ∫

QT

~Pt · ~Ψ + ν

∫ ∫

QT

~E · ~Ψ

− 2ν

∫ ∫

QT

Φ′(|~P |2)~P · ~Ψ +

∫

Ω

~P1 · ~Ψ(x, 0) = 0 (15)

Lemma 1. ([31]) Assume X ⊂ E ⊂ Y are Banach spaces and X →→ E. Thenthe following imbedding are compact:

(i)Lq(0, T ; X)⋂

ϕ :∂ϕ

∂t∈ L1(0, T ; Y ) →→ Lq(0, T ; E), if1 ≤ q ≤ ∞ (16)

(ii)L∞(0, T ; X)⋂

ϕ :∂ϕ

∂t∈ Lr(0, T ; Y ) →→ C([0, T ]; E), if1 < r ≤ ∞ (17)

2. Solutions to the Viscosity Problem. In this section, we will use Galerkinmethod to establish the global existence of weak solutions to the viscocity problem(1)-(3), (11) and (10).

Let ωn(x), (n = 1, 2, 3, · · · ) be the unit eigenfunctions satisfying the equation−ωn = λnωn, with periodicity ωn(x−Dei) = ωn(x+Dei) and λn, (n = 1, 2, 3, · · · )the corresponding eigenvalues different from each other. Denote the approximate so-

lutions of the problem (1)-(3), (11) and (10) by ~ZǫN (x, t), ~Hǫ

N (x, t), ~EǫN (x, t), ~P ǫ

N (x, t)in the following form:

~ZǫN (x, t) =

N∑

s=1

~αǫsN (t)ωs(x), ~Hǫ

N (x, t) =

N∑

s=1

~βǫsN (t)ωs(x)

~EǫN (x, t) =

N∑

s=1

~γǫsN (t)ωs(x), ~P ǫ

N (x, t) =

N∑

s=1

~δǫsN (t)ωs(x)

where ~αǫsN (t), ~βǫ

sN (t), ~γǫsN (t), ~δǫ

sN (t), (t ∈ R+), (s = 1, 2, · · · , N ; N = 1, 2, · · · ) arethree dimensional vector-valued functions satisfying the following system of ordinarydifferential equations:

∫

Ω

~ZǫNtωs = α1

∫

Ω

~ZǫN ×

(~Zǫ

N + ~HǫN

)ωs

−α2

∫

Ω

~ZǫN ×

(~Zǫ

N ×(~Zǫ

N + ~HǫN

))ωs (18)

∫

Ω

(~Hǫ

Nt + β ~ZǫNt

)ωs = −

∫

Ω

(∇× ~Eǫ

N

)ωs(x) (19)


∫

Ω

(~Eǫ

Nt + ~P ǫNt

)ωs(x) + σ

∫

Ω

(~Eǫ

N + ~P ǫN

)ωs(x)

=

∫

Ω

(∇× ~Hǫ

N

)ωs(x) + σ

∫

Ω

~P ǫNωs(x) (20)

∫

Ω

~P ǫNttωs + λ2

∫

Ω

curl2 ~P ǫNωs(x) + µ

∫

Ω

~P ǫNtωs(x) − ǫ

∫

Ω

~P ǫNωs(x)

= ν

∫

Ω

~EǫNωs(x) − 2ν

∫

Ω

~P ǫNΦ′(|~P ǫ

N |2)ωs(x) (21)

with initial conditions∫

Ω

~ZǫN (x, 0)ωs(x) =

∫

Ω

~Z0(x)ωs(x),

∫

Ω

~HǫN (x, 0)ωs(x) =

∫

Ω

~H0(x)ωs(x) (22)

∫

Ω

~EǫN (x, 0)ωs(x) =

∫

Ω

~E0(x)ωs(x),

∫

Ω

~P ǫN (x, 0)ωs(x)dx =

∫

Ω

~P0(x)ωs(x)dx (23)

∫

Ω

~P ǫNt(x, 0)ωs(x)dx =

∫

Ω

~P1(x)ωs(x)dx (24)

It follows from the standard theory on nonlinear ordinary differential equationsthat (18)-(24) admits unique local solution. The following a priori estimates makeus be able to take the limit N → ∞ in (18)-(24) to obtain the global solution tothe viscosity problem.

For the sake of simplicity, we denote ‖ · ‖Lp(Ω) = ‖ · ‖p, p ≥ 2.

Lemma 2. Assume(

~Z0(x), ~H0(x), ~E0(x), ~P0(x), ~P1(x))∈ (H1(Ω), L2(Ω), L2(Ω),

H1(Ω), L2(Ω)) Then for the solution of the initial value problem (18)-(24), we havethe following estimates:

sup0≤t≤T

[‖~ZǫN (·, t)‖2

H1(Ω) + ‖ ~EǫN (·, t)‖2

2 + ‖ ~HǫN (·, t)‖2

2 + ‖ ~P ǫN(·, t)‖2

2

+‖curl ~P ǫN(·, t)‖2

2 + ǫ‖∇~P ǫN(·, t)‖2

2 + ‖ ~P ǫNt(·, t)‖

22] ≤ C3 (25)

sup0≤t≤T

‖~ZǫN(·, t)‖2

6 ≤ C4, sup0≤t≤T

‖~ZǫN(·, t) ×∇~Zǫ

N (·, t)‖L

3

2 (Ω)≤ C5, (26)

where the constant C3, C4, C5 are independent of N, α2, and D. When α2 > 0, thereis

‖~ZǫN × (~Zǫ

N + ~HǫN )‖L2(0,T ;L2(Ω)) ≤ C6 (27)

where the constant C6 is independent of N, and D.

Proof: 1. Multiplying (18) by ~αǫsN (t), summing up the products for s = 1, 2,

· · · , N , we get

d

dt

∫

Ω

|~ZǫN(·, t)|2dx = 0

Then we have

‖~ZǫN(·, t)‖2

2 = ‖~ZǫN(·, 0)‖2

2 ≤ ‖~Z0(x)‖22, ∀t ≥ 0 (28)

2. Making the scalar product of(−λs~α

ǫsN (t) + ~βǫ

sN (t))

with (18), summing up

the resulting product for s = 1, 2, · · · , N and then integrating by parts, we have

1

2

d

dt

∫

Ω

∣∣∣∇~ZǫN (·, t)

∣∣∣2

dx + α2

∥∥∥~ZǫN ×

(~Zǫ

N + ~HǫN

)∥∥∥2

2−

∫

Ω

~ZǫNt ·

~HǫNdx = 0 (29)


Multiplying (19) by ~βǫsN (t) and (20) by ~γǫ

sN (t), summing up and integrating byparts, we obtain

1

2

d

dt

∫

Ω

(∣∣∣ ~HǫN (·, t)

∣∣∣2

+∣∣∣ ~Eǫ

N (·, t)∣∣∣2)

+ σ

∫

Ω

∣∣∣~EǫN

∣∣∣2

+

∫

Ω

~P ǫNt · ~Eǫ

N + β

∫

Ω

~ZǫNt · ~Hǫ

N = 0 (30)

Multiplying (20) by (~γǫsN (t) + ~δǫ

sN (t)), summing up and integrating by parts, weget

1

2

d

dt

∫

Ω

∣∣∣~EǫN (·, t) + ~P ǫ

N (·, t)∣∣∣2

dx + σ

∫

Ω

∣∣∣ ~EǫN + ~P ǫ

N

∣∣∣2

dx

= σ

∫

Ω

~P ǫN ·(

~EǫN + ~P ǫ

N

)dx +

∫

Ω

(∇× ~Hǫ

N

)·(

~EǫN + ~P ǫ

N

)dx (31)

Putting these equalities together, we have

1

2

d

dt

∫

Ω

[∣∣∣~EǫN (·, t) + ~P ǫ

N (·, t)∣∣∣2

+ 2∣∣∣ ~Hǫ

N (·, t)∣∣∣+∣∣∣~Eǫ

N (·, t)∣∣∣]

dx

+ σ

∫

Ω

∣∣∣~EǫN + ~P ǫ

N

∣∣∣2

dx + σ

∫

Ω

∣∣∣ ~EǫN

∣∣∣2

dx + 2β

∫

Ω

~ZǫNt · ~Hǫ

Ndx +

∫

Ω

~P ǫNt · ~Eǫ

Ndx

= σ

∫

Ω

~P ǫN ·(

~EǫN + ~P ǫ

N

)dx +

∫

Ω

(∇× ~Hǫ

N

)· ~P ǫ

Ndx (32)

Multiplying (21) by ~δǫsN

′(t), summing up the product for s = 1, 2, · · · , N and

integrating by parts, one has, by noticing that ~P ǫN is periodic

1

2

d

dt

∫

Ω

∣∣∣~P ǫNt(·, t)

∣∣∣2

dx +λ2

2

d

dt

∫

Ω

∣∣∣∇× ~P ǫN (·, t)

∣∣∣2

dx +ǫ

2

d

dt

∫

Ω

∣∣∣∇~P ǫN (·, t)

∣∣∣2

dx

+µ

∫

Ω

∣∣∣~P ǫNt

∣∣∣2

dx = ν

∫

Ω

~EǫN · ~P ǫ

Ntdx − 2ν

∫

Ω

Φ′(|~P ǫN |2)~P ǫ

N · ~P ǫNtdx (33)

From (29) and (32) (multiplying (32) by δ0, chosen later), it follows that

1

2

d

dt

∫

Ω

[∣∣∣∇~Zǫ

N (·, t)∣∣∣2

+ δ0

∣∣∣~EǫN (·, t) + ~P ǫ

N (·, t)∣∣∣2

+ 2δ0

∣∣∣ ~HǫN (·, t)

∣∣∣

+δ0

∣∣∣~EǫN (·, t)

∣∣∣] + α2

∥∥∥~ZǫN ×

(~Zǫ

N + ~HǫN

)∥∥∥2

2+ (2βδ0 − 1)

∫

Ω

~ZǫNt ·

~HǫNdx

+δ0σ

∫

Ω

∣∣∣~EǫN + ~P ǫ

N

∣∣∣2

+ δ0σ

∫

Ω

∣∣∣~EǫN

∣∣∣2

+ δ0

∫

Ω

~P ǫNt ·

~EǫNdx

= δ0σ

∫

Ω

~P ǫN ·(

~EǫN + ~P ǫ

N

)+ δ0

∫

Ω

(∇× ~Hǫ

N

)· ~P ǫ

N (34)

In order to deal with the term∫Ω

~ZǫNt ·

~HǫNdx, we multiplying (19) by (2βδ0

−1)~αǫsN and sum up the product for s = 1, 2, · · · , N to obtain that

(2βδ0 − 1)

∫

Ω

~HǫNt ·

~ZǫN + (2βδ0 − 1)

∫

Ω

(∇× ~Eǫ

N

)· ~Zǫ

N = 0 (35)


Adding (34) and (35), one gets

1

2

d

dt

∫

Ω

[∣∣∣∇~Zǫ

N (·, t)∣∣∣2

+ δ0

∣∣∣~EǫN (·, t) + ~P ǫ

N (·, t)∣∣∣2

+ 2δ0

∣∣∣ ~HǫN (·, t)

∣∣∣

+δ0

∣∣∣~EǫN (·, t)

∣∣∣]dx + α2

∥∥∥~ZǫN ×

(~Zǫ

N + ~HǫN

)∥∥∥2

2

+(2βδ0 − 1)d

dt

∫

Ω

~ZǫN · ~Hǫ

Ndx

= −δ0σ

∫

Ω

∣∣∣ ~EǫN + ~P ǫ

N

∣∣∣2

dx − δ0σ

∫

Ω

∣∣∣~EǫN

∣∣∣2

dx − δ0

∫

Ω

~P ǫNt · ~Eǫ

Ndx

+δ0σ

∫

Ω

~P ǫN ·(

~EǫN + ~P ǫ

N

)dx − (2βδ0 − 1)

∫

Ω

(∇× ~Eǫ

N

)· ~Zǫ

Ndx

+δ0

∫

Ω

(∇× ~Hǫ

N

)· ~P ǫ

Ndx (36)

Putting (33) and (36) together, we have

1

2

d

dt

∫

Ω

[∣∣∣∇~Zǫ

N (·, t)∣∣∣2

+ δ0

∣∣∣ ~EǫN (·, t) + ~P ǫ

N (·, t)∣∣∣2

+ 2δ0

∣∣∣ ~HǫN (·, t)

∣∣∣

+δ0

∣∣∣~EǫN (·, t)

∣∣∣+∣∣∣~P ǫ

Nt(·, t)∣∣∣2

+ λ2∣∣∣∇× ~P ǫ

N (·, t)∣∣∣2

+ ǫ∣∣∣∇~P ǫ

N (·, t)∣∣∣2

]dx

+(2βδ0 − 1)d

dt

∫

Ω

~ZǫN · ~Hǫ

Ndx + α2

∥∥∥~ZǫN ×

(~Zǫ

N + ~HǫN

)∥∥∥2

2

≤

(µ + 2 +

(δ0 − ν)2

4

)∥∥∥~P ǫNt

∥∥∥2

2+

(8ν2C2

0 +σ2δ2

0

4+ δ0σ

)∥∥∥~P ǫN

∥∥∥2

2

+|3 − σδ0|∥∥∥ ~Eǫ

N

∥∥∥2

2+ δ0σ

∥∥∥ ~EǫN + ~P ǫ

N

∥∥∥2

2+∥∥∥ ~Hǫ

N

∥∥∥2

2+

(1 − 2βδ0)2

4

∥∥∥∇~ZǫN

∥∥∥2

2

+δ20

4

∥∥∥curl ~P ǫN

∥∥∥2

2

where C0 is given by (5).Integrating the above inequality with respect to t, we obtain

1

2

∥∥∥∇~ZǫN (·, t)

∥∥∥2

2+

δ0

2

∥∥∥ ~EǫN (·, t) + ~P ǫ

N (·, t)∥∥∥

2

2+ δ0

∥∥∥ ~HǫN (·, t)

∥∥∥2

2

+δ0

2

∥∥∥ ~EǫN (·, t)

∥∥∥2

2+

1

2

∥∥∥ ~P ǫNt(·, t)

∥∥∥2

2+

λ2

2

∥∥∥curl ~P ǫN (·, t)

∥∥∥2

2+

ǫ

2

∥∥∥∇~P ǫN (·, t)

∥∥∥2

2

+(2βδ0 − 1)

∫

Ω

~ZǫN · ~Hǫ

Ndx + α2

∫ t

0

∥∥∥~ZǫN ×

(~Zǫ

N + ~HǫN

)∥∥∥2

2dt

≤1

2

∥∥∥∇~Z0

∥∥∥2

2+

δ0

2

∥∥∥ ~E0 + ~P0

∥∥∥2

2+ δ0

∥∥∥ ~H0

∥∥∥2

2+

δ0

2

∥∥∥ ~E0

∥∥∥2

2+

1

2

∥∥∥~P1

∥∥∥2

2

+|2βδ0 − 1|∥∥∥ ~H0

∥∥∥2

∥∥∥~Z0

∥∥∥2

+ δ0σ

∫ t

0

∥∥∥ ~EǫN + ~P ǫ

N

∥∥∥2

2dt

+λ2

2

∥∥∥curl ~P0

∥∥∥2

2+

ǫ

2

∥∥∥∇~P0

∥∥∥2

2+

(µ + 2 +

(δ0 − ν)2

4

)∫ t

0

∥∥∥~P ǫNt

∥∥∥2

2dt

+

(8ν2C2

0 +σ2δ2

0

4+ δ0σ

)∫ t

0

∥∥∥ ~P ǫN

∥∥∥2

2dt + |3 − σδ0|

∫ t

0

∥∥∥ ~EǫN

∥∥∥2

2dt

+

∫ t

0

∥∥∥ ~HǫN

∥∥∥2

2dt +

(1 − 2βδ0)2

4

∫ t

0

∥∥∥∇~ZǫN

∥∥∥2

2dt +

δ20

4

∫ t

0


∥∥∥2

2dt


Therefore we have∥∥∥∇~Zǫ

N (·, t)∥∥∥

2

2+ δ0

∥∥∥ ~EǫN (·, t) + ~P ǫ

N (·, t)∥∥∥

2

2+ 2δ0

∥∥∥ ~HǫN (·, t)

∥∥∥2

2

+δ0

∥∥∥ ~EǫN (·, t)

∥∥∥2

2+∥∥∥ ~P ǫ

Nt(·, t)∥∥∥

2

2+ λ2


∥∥∥2

2+ ǫ∥∥∥∇~P ǫ

N (·, t)∥∥∥

2

2

+2α2

∫ t

0

∥∥∥~ZǫN ×

(~Zǫ

N + ~HǫN

)∥∥∥2

2dt

≤ C7 + 2

(µ + 2 +

(δ0 − ν)2

4

)∫ t

0

∥∥∥~P ǫNt

∥∥∥2

2dt

+

(16ν2C2

0 +σ2δ2

0

2+ 8|δ0σ|

)∫ t

0

∥∥∥~P ǫN

∥∥∥2

2dt

+(|6 − 2σδ0| + 4|δ0σ|)

∫ t

0

∥∥∥ ~EǫN

∥∥∥2

2dt + 2

∫ t

0

∥∥∥ ~HǫN

∥∥∥2

2dt

+(1 − 2βδ0)

2

2

∫ t

0

∥∥∥∇~ZǫN

∥∥∥2

2dt +

δ20

2

∫ t

0


∥∥∥2

2dt (37)

where

C7 =∥∥∥∇~Z0

∥∥∥2

2+ δ0

∥∥∥ ~E0 + ~P0

∥∥∥2

2+ 2δ0

∥∥∥ ~H0

∥∥∥2

2+ δ0

∥∥∥ ~E0

∥∥∥2

2+∥∥∥~P1

∥∥∥2

2

+ 2|2βδ0 − 1|∥∥∥ ~H0

∥∥∥2

∥∥∥~Z0

∥∥∥2+ λ2

∥∥∥curl ~P0

∥∥∥2

2+ ǫ∥∥∥∇~P0

∥∥∥2

2dx

+(1 − 2βδ0)

2

δ0

∥∥∥~Z0

∥∥∥2

2

On the other hand, we have∥∥∥~P ǫ

N (·, t)∥∥∥

2

2− 2

∥∥∥ ~EǫN (·, t)

∥∥∥2

2≤ 2

∥∥∥~P ǫN (·, t) + ~Eǫ

N (·, t)∥∥∥

2

2

≤ 3∥∥∥~P ǫ

N (·, t) + ~EǫN (·, t)

∥∥∥2

2(38)

Taking δ0 = 3, we get from (37) and (38) that∥∥∥∇~Zǫ

N (·, t)∥∥∥

2

2+∥∥∥ ~P ǫ

N (·, t)∥∥∥

2

2+ 6

∥∥∥ ~HǫN (·, t)

∥∥∥2

2+∥∥∥ ~Eǫ

N (·, t)∥∥∥

2

2+ ǫ

∥∥∥∇~P ǫN (·, t)

∥∥∥2

2

+∥∥∥~P ǫ

Nt(·, t)∥∥∥

2

2+ λ2


∥∥∥2

2+ 2α2

∫ t

0

∥∥∥~ZǫN ×

(~Zǫ

N + ~HǫN

)∥∥∥2

2dt

≤ C7 + C8

∫ t

0

[∥∥∥∇~Zǫ

N

∥∥∥2

2+∥∥∥~P ǫ

N

∥∥∥2

2+∥∥∥ ~Eǫ

N

∥∥∥2

2+∥∥∥ ~Hǫ

N

∥∥∥2

2

+∥∥∥~P ǫ

Nt

∥∥∥2

2+∥∥∥curl ~P ǫ

N

∥∥∥2

2]dt (39)

where

C8 = max(1 − 6β)2

2, (16ν2C2

0 +9σ2

2+ 24σ), (|6 − 6σ| + 12σ), 2,

2[µ + 2 + (3 − ν)2],9

2

For λ2 > 0 (in fact, λ > 0 denotes the speed of light for the internal field), (39)combined with Gronwall’s inequality yields (25).


Step 3. By Sobolev imbedding theorem and Holder inequality, we have (26).Combining (36) and (25), we obtain (27) if α2 > 0. Lemma 2 is proved.

Lemma 3. Under the condition of Lemma 2, for the solution (~ZǫN , ~Hǫ

N , ~EǫN , ~P ǫ

N ) ofthe problem (18)-(24), there exist C9 > 0 and C10 > 0, both independent of N, D,

and ǫ, such that(i) when α2 = 0,

sup0≤t≤T

[∥∥∥~ZǫNt

∥∥∥H−2(Ω)

+∥∥∥ ~Hǫ

Nt

∥∥∥H−2(Ω)

+∥∥∥ ~Eǫ

Nt

∥∥∥H−2(Ω)

+∥∥∥ ~P ǫ

Ntt

∥∥∥H−2(Ω)

]≤ C9

(40)(ii) when α2 > 0,

∥∥∥~ZǫNt

∥∥∥L

3

2 (QT )+

∥∥∥ ~HǫNt

∥∥∥L2(0,T ;H−1(Ω))

+∥∥∥ ~Eǫ

Nt

∥∥∥L2(0,T ;H−1(Ω))

+∥∥∥~P ǫ

Ntt

∥∥∥L2(0,T ;H−2(Ω))

≤ C10 (41)

Remark 1. This lemma shows that if α2 > 0, then we may get better estimatelike above lemma.

Proof: (i) When α2 = 0, for any periodic function ϕ ∈ H20 (Ω), ϕ can be represented

by

ϕ = ϕN + ϕN , ϕN =

N∑

s=1

ηsωs(x), ϕN =

∞∑

s=N+1

ηsωs(x) (42)

For s ≥ N + 1, we have∫

Ω

~ZǫNtωs(x)dx = 0

Then by Lemma 2, there holds∣∣∣∣∫

Ω

~ZǫNtϕ(x)dx

∣∣∣∣ =

∫

Ω

~ZǫNtϕN (x)dx

= α1

∫

Ω

~ZǫN ×

(~Zǫ

N + ~HǫN

)ϕN (x)dx

≤ |α1|(‖∇~Zǫ

N‖2‖~ZǫN‖6 + ‖~Zǫ

N‖2‖ ~HǫN‖2

)(‖∇ϕN‖3 + ‖ϕN‖∞)

≤ C11‖ϕ‖H2(Ω)

where we have used Gagliardo-Nirenberg inequalities

‖ϕN‖∞ ≤ C‖ϕN‖1

4

2 ‖ϕN‖3

4

2 , ‖∇ϕN‖3 ≤ C‖∇ϕN‖1

2

2 ‖ϕN‖1

2

2

In the similar manner, we have∣∣∣∣∫

Ω

~HǫNtϕ(x)dx

∣∣∣∣ ≤ C12‖ϕ‖H2(Ω),

∣∣∣∣∫

Ω

~EǫNtϕ(x)dx

∣∣∣∣ ≤ C13‖ϕ‖H2(Ω),

∣∣∣∣∫

Ω

~P ǫNttϕdx

∣∣∣∣ ≤ C14‖ϕ‖H2(Ω)

where C11, C12, C13, C14 are independent of N, D and ǫ. (40) follows.


(ii) Now we assume α2 > 0. For any periodic function ϕ ∈ L3(QT ), we have∣∣∣∣∣∣

∫ ∫

QT

~ZǫNϕdxdt

∣∣∣∣∣∣≤ |α1|

∣∣∣∣∣∣

∫ ∫

QT

~ZǫN ×

(~Zǫ

N + ~HǫN

)ϕdxdt

∣∣∣∣∣∣

+ α2

∣∣∣∣∣∣

∫ ∫

QT

~ZǫN ×

(~Zǫ

N ×(~Zǫ

N + ~HǫN

))ϕdxdt

∣∣∣∣∣∣

≤ |α1|‖~ZǫN ×

(~Zǫ

N + ~HǫN

)‖L2(QT )‖ϕ‖L2(QT )

+ α2‖~ZǫN‖L6(QT )‖~Zǫ

N ×(~Zǫ

N + ~HǫN

)‖L2(QT )‖ϕ‖L3(QT )

≤ C15‖ϕ‖L3(QT )

Similarly, for any periodic function ϕ ∈ L2(0, T ; H10(Ω)), using (42) and Lemma

2, we get ∣∣∣∣∫

Ω

~ZǫNtϕdx

∣∣∣∣ ≤ C16‖ϕ‖H1(Ω)

∣∣∣∣∣∣

∫ ∫

QT

~HǫNtϕdxdt

∣∣∣∣∣∣=

∣∣∣∣∣∣−

∫ ∫

QT

(∇× ~Eǫ

N

)ϕdxdt − β

∫ ∫

QT

~ZǫNtϕNdxdt

∣∣∣∣∣∣

≤ C17‖ϕ‖L2(0,T ;H1(Ω))

∣∣∣∣∣∣

∫ ∫

QT

~EǫNtϕdxdt

∣∣∣∣∣∣

=

∣∣∣∣∣∣

∫ ∫

QT

(∇× ~Hǫ

N

)ϕdxdt −

∫ ∫

QT

~PNtϕtdxdt − σ

∫ ∫

QT

~EǫNϕNdxdt

∣∣∣∣∣∣

≤ C18‖ϕ‖L2(0,T ;H1(Ω))

For any periodic function ϕ ∈ L2(0, T ; H2(Ω)), using (42) again, we obtain

|

∫ ∫

QT

~P ǫNttϕdxdt| = |ν

∫ ∫

QT

~EǫNϕNdxdt + ǫ

∫ ∫

QT

~P ǫNϕNdxdt

−2ν

∫ ∫

QT

Φ′(|~P ǫN |2)~P ǫ

NϕNdxdt − µ

∫ ∫

QT

~P ǫNtϕNdxdt

−λ2

∫ ∫

QT

curl2 ~P ǫNϕNdxdt|

≤ C19‖ϕN‖L2(0,T ;H1(Ω)) + ǫ

∫ T

0

‖ ~P ǫN‖2‖ϕN‖2dt

+λ2

∫ T

0

‖curl ~P ǫN‖2‖∇ϕN‖2dt

≤ C20‖ϕ‖L2(0,T ;H2(Ω))

where C15−C20 are independent of N, D and ǫ. The proof of Lemma 3 is completed.


Lemma 4. Under the condition of Lemma 2, for the solution (~ZǫN , ~Hǫ

N , ~EǫN , ~P ǫ

N )of the problem (18)-(24), there exist constants C21 > 0, C22 > 0, C23 > 0, C24 > 0and C25 > 0, independent of N, D, and ǫ, such that(i) When α2 = 0,

‖~ZǫN(·, t1) − ~Zǫ

N (·, t2)‖2 ≤ C21|t1 − t2|1

2 (43)

~HǫN , ~Eǫ

N , ~P ǫN , ~P ǫ

Nt ∈ C([0, T ]; H−1(Ω)) (44)

(ii) When α2 > 0,

‖~ZǫN(·, t1) − ~Zǫ

N (·, t2)‖3 ≤ C22|t1 − t2|2

3 (45)

‖ ~HǫN (·, t1)− ~Hǫ

N (·, t2)‖H−1(Ω) + ‖ ~EǫN (·, t1)− ~Eǫ

N (·, t2)‖H−1(Ω) ≤ C23|t1 − t2|1

2 (46)

‖ ~P ǫNt(·, t1) −

~P ǫNt(·, t2)‖H−2(Ω) ≤ C24|t1 − t2|

1

2 (47)

‖ ~P ǫN (·, t1) − ~P ǫ

N (·, t2)‖2 ≤ C25|t1 − t2|1

2 (48)

Proof: (i) When α2 = 0, by the Sobolev interpolation of negative order, thereholds

‖~ZǫN(·, t1) − ~Zǫ

N (·, t2)‖2

≤ C26‖~ZǫN(·, t1) − ~Zǫ

N (·, t2)‖1

3

H−2‖~ZǫN(·, t1) − ~Zǫ

N(·, t2)‖2

3

H1

≤ C27

∥∥∥∥∥

∫ t2

t1

∂ ~ZǫN

∂tdt

∥∥∥∥∥

1

3

H−2

≤ C22|t1 − t2|1

3

On the other hand, it follows from Lemma 1 and

L2(Ω) →→ H−1(Ω) → H−2(Ω);

~HǫN ∈ L∞(0, T ; L2(Ω))

⋂Ψ :

∂Ψ

∂t∈ L∞(0, T ; H−2(Ω))

that~Hǫ

N ∈ C([0, T ]; H−1(Ω))

Similarly, we also have

~EǫN , ~P ǫ

N , ~P ǫNt ∈ C([0, T ]; H−1(Ω))

(ii) When α2 > 0, we have

∥∥∥~ZǫN (·, t1) − ~Zǫ

N (·, t2)∥∥∥

3=

∥∥∥∥∥

∫ t2

t1

∂ ~ZǫN

∂tdt

∥∥∥∥∥3

≤ |t1 − t2|2

3

∫ ∫

QT

∣∣∣∣∣∂ ~Zǫ

N

∂t

∣∣∣∣∣ dxdt

1

3

≤ C22|t1 − t2|2

3

∥∥∥ ~HǫN (·, t1) − ~Hǫ

N (·, t2)∥∥∥

H−1(Ω)=

∥∥∥∥∥

∫ t2

t1

∂ ~HǫN

∂tdt

∥∥∥∥∥H−1

≤ |t1 − t2|1

2

∫ T

0

∥∥∥∥∥∂ ~Hǫ

N

∂t

∥∥∥∥∥

2

H−1(Ω)

dt

1

2

≤ C23|t1 − t2|1

2


For | ~EǫN (·, t1)− ~Eǫ

N (·, t2)|, a similar inequality holds. At the same time, we have

∥∥∥~P ǫNt(·, t1) −

~P ǫNt(·, t2)

∥∥∥H−2(Ω)

=

∥∥∥∥∥

∫ t2

t1

∂2 ~P ǫN

∂t2dt

∥∥∥∥∥H−2(Ω)

≤ |t1 − t2|1

2

∫ T

0

∥∥∥∥∥∂2 ~P ǫ

N

∂t2

∥∥∥∥∥

2

H−2(Ω)

dt

1

2

≤ C24|t1 − t2|1

2

∥∥∥ ~P ǫN (·, t1) − ~P ǫ

N (·, t2)∥∥∥

2=

∥∥∥∥∥

∫ t2

t1

∂ ~P ǫN

∂tdt

∥∥∥∥∥2

≤ |t1 − t2|1

2

∫ T

0

∥∥∥∥∥∂ ~P ǫ

N

∂t

∥∥∥∥∥

2

2

dt

1

2

≤ C25|t1 − t2|1

2

Lemma 4 follows.

In fact, it follows from (25)-(26) that the solution of ODE (18)-(24) does notblow-up at any finite time. Hence from the theory of ODE theory , Lemma 2,Lemma 3 and Lemma 4, we have the following lemma:

Lemma 5. Under the conditions of Lemma 2, the initial value problem for the sys-tem of the ordinary differential equation (18)-(24) admits at least one continuouslydifferentiable global solution

~αǫsN (t), ~βǫ

sN (t), ~γǫsN (t), ~δǫ

sN (t), (s = 1, 2, · · · , N ; t ∈ [0, T ])

3. Existence of Weak Solution for the Viscosity Problem. First of all, sim-ilarly to Definition 2, we may define the weak solution for the viscosity problem(1)-(3), (11), (10). In the proof of the following theorem, we must use the followinglemma which is well-known to all.

Lemma 6. If un → u strongly in L2(QT ) and vn → v weakly in L2(QT ), thenunvn → uv weakly in L1(QT ) and in the sense of distribution.

Theorem 1. Assume the 2D-periodic initial data (~Z0(x), ~H0(x), ~E0(x), ~P0(x),~P1(x)) ∈ (H1(Ω), L2(Ω), L2(Ω), H1(Ω), L2(Ω)). Then the periodic initial value

problem (1)-(3), (11), (10) admits at least one global weak solution ~Zǫ(x, t), ~Hǫ(x, t),~Eǫ(x, t), ~P ǫ(x, t) such that

(i) When α2 = 0, there hold

~Zǫ(x, t) ∈ L∞(0, T ; H1(Ω))⋂

C(0, 1

3)(0, T ; L2(Ω)); (49)

~Hǫ(x, t) ∈ L∞(0, T ; L2(Ω))⋂

C(0, T ; H−1(Ω)); (50)

~Eǫ(x, t) ∈ L∞(0, T ; L2(Ω))⋂

C(0, T ; H−1(Ω)); (51)

~P ǫ(x, t) ∈ L∞(0, T ; H1(Ω))⋂

C(0, T ; H−1(Ω)); (52)

~P ǫt (x, t) ∈ L∞(0, T ; L2(Ω))

⋂C(0, T ; H−1(Ω)). (53)


(ii) When α2 > 0, we have

~Zǫ(x, t) ∈ L∞(0, T ; H1(Ω))⋂

C(0, 2

3)(0, T ; L3(Ω)); (54)

~Hǫ(x, t) ∈ L∞(0, T ; L2(Ω))⋂

C(0, 1

2)(0, T ; H−1(Ω)); (55)

~Eǫ(x, t) ∈ L∞(0, T ; L2(Ω))⋂

C(0, 1

2)(0, T ; H−1(Ω)); (56)

~P ǫ(x, t) ∈ L∞(0, T ; H1(Ω))⋂

C(0, 1

2)(0, T ; L2(Ω)); (57)

~P ǫt (x, t) ∈ L∞(0, T ; L2(Ω))

⋂C(0, 1

2)(0, T ; H−2(Ω)). (58)

Proof: The uniform estimates for the approximate solution ~ZǫN (x, t), ~Hǫ

N (x, t),~Eǫ

N (x, t), ~P ǫN (x, t) in section 2 yield that there is a subsequence of ~Zǫ

N (x, t), ~HǫN (x, t),

~EǫN (x, t), ~P ǫ

N (x, t), still denoted by ~ZǫN (x, t), ~Hǫ

N (x, t), ~EǫN (x, t), ~P ǫ

N (x, t), and~Zǫ(x, t), ~Hǫ(x, t), ~Eǫ(x, t), ~P ǫ(x, t), such that

~ZǫN (x, t) → ~Zǫ(x, t), weakly in L6(QT ); (59)

~ZǫN (x, t) → ~Zǫ(x, t), strongly in L6−(QT ), ( > 0); (60)

~ZǫN (x, t) → ~Zǫ(x, t), weakly start in L∞(0, T ; H1(Ω)); (61)

~HǫN (x, t) → ~Hǫ(x, t), weakly start in L∞(0, T ; L2(Ω)); (62)

~EǫN (x, t) → ~Eǫ(x, t), weakly start in L∞(0, T ; L2(Ω)); (63)

~P ǫN (x, t) → ~P ǫ(x, t), weakly start in L∞(0, T ; H1(Ω)); (64)

~P ǫNt(x, t) → ~P ǫ

t (x, t), weakly start in L∞(0, T ; L2(Ω)); (65)

curl ~P ǫN (x, t) → curl ~P ǫ(x, t), weakly start in L∞(0, T ; L2(Ω)); (66)

~ZǫNt(x, t) → ~Zǫ

t (x, t), weakly in L3

2 (QT ), (α2 > 0); (67)

~ZǫNt(x, t) → ~Zǫ

t (x, t), weakly star in L∞(0, T ; H−2(Ω)), (α2 = 0). (68)

From Lemma 1 (ii), we get that

L∞(0, T ; H1(Ω))⋂

ϕ :∂ϕ

∂t∈ L∞(0, T ; L2(Ω))

→→ C([0, T ]; L2(Ω)) ⊂ L2(0, T ; L2(Ω))

Since ~P ǫN is bounded uniformly in L∞(0, T ; H1(Ω)) and ∂t

~P ǫN is bounded uni-

formly in L∞(0, T ; L2(Ω)), we deduce that there exists a subsequence of ~P ǫN, still

denoted by ~P ǫN, such that as N → ∞

~P ǫN (x, t) → ~P ǫ(x, t), strongly in L∞(0, T ; L2(Ω)) (69)

For any vector-valued periodic test function ~Ψ(x, t) ∈ C1(QT ), ~Ψ(x, T ) = 0, wedefine an approximate sequence

~ΨN(x, t) =

N∑

s=1

~ηs(t)ωs(x)

where ~ηs(t) =∫Ω

~Ψ(x, t)ωs(x)dx, then

~ΨN (x, t) → ~Ψ(x, t) in C1(QT ) and in Lp(QT ), ∀p > 1 (70)


Making the scalar product of ~ηs(t) with (18), (19) and eσt~ηs(t) with (20), ~ηs(t)with (21), summing up the products for s = 1, 2, · · · , N , we get from integration byparts

∫ ∫

QT

~ZǫN · ~ΨNtdxdt + α1

∫ ∫

QT

(~ZǫN ×∇~Zǫ

N ) · ∇~ΨNdxdt

−α1

∫ ∫

QT

(~ZǫN × ~Hǫ

N ) · ~ΨNdxdt −

∫

Ω

~ZǫN(x, 0) · ~Ψ(x, 0)dx

+α2

∫ ∫

QT

(~Zǫ

N ×∇~ZǫN

)· ∇(

~ZǫN × ~ΨN

)dxdt

−α2

∫ ∫

QT

(~Zǫ

N × ~HǫN

)·(

~ZǫN × ~ΨN

)dxdt = 0 (71)

∫ ∫

QT

(~Hǫ

N + β ~ZǫN

)· ~ΨNtdxdt −

∫ ∫

QT

(∇× ~ΨN

)· ~Eǫ

Ndxdt

+

∫

Ω

(~Hǫ

N (x, 0) + β ~ZǫN (x, 0)

)· ~ΨN (x, 0)dx = 0 (72)

∫ ∫

QT

( ~EǫN + ~P ǫ

N ) · (eσt~ΨNt)dxdt +

∫ ∫

QT

eσt∇× ΨN · ~HǫNdxdt

+σ

∫ ∫

QT

eσt ~P ǫN · ~ΨNdxdt +

∫

Ω

( ~EǫN (·, 0) + ~P ǫ

N (·, 0)) · ~ΨN (·, 0)dx = 0 (73)

∫ ∫

QT

~P ǫNt · ~ΨNtdxdt − λ2

∫ ∫

QT

curl ~P ǫN · curl~ΨNdxdt

+ν

∫ ∫

QT

~EǫN · ~ΨNdxdt − ǫ

∫ ∫

QT

∇~P ǫN · ∇~ΨNdxdt − µ

∫ ∫

QT

~P ǫNt ·

~ΨNdxdt

−2ν

∫ ∫

QT

Φ′(|~P ǫN |2)~P ǫ

N · ~ΨNdxdt +

∫

Ω

~P ǫNt(·, 0) · ~ΨN (·, 0)dx = 0 (74)

Now we are in the position to prove that (~Zǫ(x, t), ~Hǫ(x, t), ~Eǫ(x, t), ~P ǫ(x, t)) isa weak solution of (1)-(3), (11) and (10). To this aim, one should send N to ∞ in(71)-(74). From (59)-(70) and Lemma 6, it suffices to deal with the nonlinear termsin (71)-(74).

First of all, we are able to prove

∫ ∫

QT

Φ′(|~P ǫN |2)~P ǫ

N · ~ΨNdxdt →

∫ ∫

QT

Φ′(|~P ǫ|2)~P ǫ · ~Ψdxdt (75)


In fact, using the Lipschitz condition (6), we get∣∣∣∣∣∣

∫ ∫

QT

Φ′(|~P ǫN |2)~P ǫ

N · ~ΨNdxdt −

∫ ∫

QT

Φ′(|~P ǫ|2)~P ǫ · ~Ψdxdt

∣∣∣∣∣∣

≤ |

∫ ∫

QT

[Φ′(|~P ǫ

N |2)~P ǫN − Φ′(|~P ǫ|2)~P ǫ

]· ~ΨNdxdt

+

∫ ∫

QT

Φ′(|~P ǫ|2)~P ǫ ·(~ΨN − ~Ψ

)dxdt|

≤ C∗

∫ ∫

QT

|~P ǫN − ~P ǫ||~ΨN |dxdt

+C‖ ~P ǫ‖L∞(0,T ;L2(Ω))

∫ T

0

‖~ΨN − ~Ψ‖L2(Ω)dt

≤ C∗‖~ΨN‖L2(0,T ;L2(Ω))‖ ~P ǫN − ~P ǫ‖L2(0,T ;L2(Ω))

+C‖ ~P ǫ‖L∞(0,T ;L2(Ω))

∫ T

0

‖~ΨN − ~Ψ‖L2(Ω)dt → 0 (as N → +∞)

where we have used (69).

Secondly, we claim that there exist subsequences of ~ZǫN , still denoted by ~Zǫ

N ,such that, as N → +∞,

(1). ~ZǫN ×

∂ ~ZǫN

∂xi→ ~Zǫ ×

∂ ~Zǫ

∂xiweakly star in L∞(0, T ; L

3

2 (Ω)), i = 1, 2, 3; (76)

(2). (~ZǫN ×

∂ ~ZǫN

∂xi)xi

→ (~Zǫ ×∂ ~Zǫ

∂xi)xi

weakly in L2(QT ), i = 1, 2, 3; (α2 > 0) (77)

In fact, for any periodic test function ~Ψ(x, t) ∈ C1(QT ), we obtain

∫ ∫

QT

(~Zǫ

N ×∂ ~Zǫ

N

∂xi− ~Zǫ ×

∂ ~Zǫ

∂xi

)· ~Ψdxdt

=

∫ ∫

QT

[(~Zǫ

N − ~Zǫ) ×∂ ~Zǫ

N

∂xi

]· ~Ψdxdt +

∫ ∫

QT

[~Zǫ ×

(∂ ~Zǫ

N

∂xi−

∂ ~Zǫ

∂xi

)]· ~Ψdxdt

≤ ‖~Ψ‖L∞(QT )‖∂ ~Zǫ

N

∂xi‖L2(QT )‖~Zǫ

N − ~Zǫ‖L2(QT )

+

∫ ∫

QT

[~Zǫ ×

(∂ ~Zǫ

N

∂xi−

∂ ~Zǫ

∂xi

)]· ~Ψdxdt → 0, (as N → +∞)

where we have used (59) and (61). Therefore (76) is proved.

Now we turn to prove (77). By Lemma 2, when α2 > 0, (~ZǫN × (~Zǫ

N + ~HǫN ))

is bounded in L2(QT ) uniformly respect to N. Then there exist a subsequence of

(~ZǫN × (~Zǫ

N + ~HǫN )), still denoted by (~Zǫ

N × (~ZǫN + ~Hǫ

N)), and a vector ~U ǫ(x, t) ∈

L2(QT ), such that for any test function ~Ψ(x, t) ∈ C1(QT ), there holds that as


N → +∞∫ ∫

QT

~ZǫN ×

(~Zǫ

N + ~HǫN

)· ~Ψdxdt →

∫ ∫

QT

~U ǫ · ~Ψdxdt

On the other hand, as N → +∞,∫ ∫

QT

~ZǫN ×

(~Zǫ

N + ~HǫN

)· ~Ψdxdt

= −

∫ ∫

QT

(~Zǫ

N ×∇~ZǫN

)· ∇~Ψdxdt +

∫ ∫

QT

(~Zǫ

N × ~HǫN

)· ~Ψdxdt

→ −

∫ ∫

QT

(~Zǫ ×∇~Zǫ

)· ∇~Ψdxdt +

∫ ∫

QT

(~Zǫ × ~Hǫ

)· ~Ψdxdt

where we have used (76) and the fact that, as N → ∞∣∣∣∣∣∣

∫ ∫

QT

(~Zǫ

N × ~HǫN − ~Zǫ × ~Hǫ

)· ~Ψdxdt

∣∣∣∣∣∣

≤

∣∣∣∣∣∣

∫ ∫

QT

(~Zǫ

N − ~Zǫ)× ~Hǫ

N · ~Ψdxdt

∣∣∣∣∣∣+

∣∣∣∣∣∣

∫ ∫

QT

~Zǫ ×(

~HǫN − ~Hǫ

)· ~Ψdxdt

∣∣∣∣∣∣

≤

∫ T

0

‖~ZǫN − ~Zǫ‖5‖ ~Hǫ

N‖2‖~Ψ‖ 10

3

dt +

∣∣∣∣∣∣

∫ ∫

QT

~Zǫ × ( ~HǫN − ~Hǫ) · ~Ψdxdt

∣∣∣∣∣∣

→ 0 (78)

Then we have∫ ∫

QT

~U ǫ · ~Ψdxdt −

∫ ∫

QT

(~Zǫ × ~Hǫ

)· ~Ψdxdt = −

∫ ∫

QT

(~Zǫ ×∇~Zǫ

)· ∇~Ψdxdt

Therefore one gets in the sense of distribution

~Zǫ ×~Zǫ = (~U ǫ − (~Zǫ × ~Hǫ)) ∈ L2(QT ).

So (77) is proved.It remains to prove that

∫ ∫

QT

(~Zǫ

N ×∇~ZǫN

)· ∇(

~ZǫN × ~ΨN

)dxdt

−

∫ ∫

QT

(~Zǫ

N × ~HǫN

)·(

~ZǫN × ~ΨN

)dxdt

→

∫ ∫

QT

(~Zǫ ×∇~Zǫ

)· ∇(

~Zǫ × ~Ψ)

dxdt

−

∫ ∫

QT

(~Zǫ × ~Hǫ

)·(

~Zǫ × ~Ψ)

dxdt


In fact, we have∫ ∫

QT

(~Zǫ

N ×∇~ZǫN

)· ∇(

~ZǫN × ~ΨN

)dxdt

−

∫ ∫

QT

(~Zǫ

N × ~HǫN

)·(

~ZǫN × ~ΨN

)dxdt

−

∫ ∫

QT

(~Zǫ ×∇~Zǫ

)· ∇(

~Zǫ × ~Ψ)

dxdt

+

∫ ∫

QT

(~Zǫ × ~Hǫ

)·(

~Zǫ × ~Ψ)

dxdt

=

∫ ∫

QT

~ZǫN ×

(~Zǫ

N + ~HǫN

)·(

~ZǫN × ~ΨN

)dxdt

−

∫ ∫

QT

~Zǫ ×(~Zǫ + ~Hǫ

)·(

~Zǫ × ~Ψ)

dxdt

=

∫ ∫

QT

[(~Zǫ

N ×~ZǫN

)−(

~Zǫ ×~Zǫ)]

·(

~ZǫN × ~Ψ

)dxdt

+

∫ ∫

QT

(~Zǫ

N ×~ZǫN

)·[(

~ZǫN × ~ΨN

)−(

~Zǫ × ~Ψ)]

dxdt

+

∫ ∫

QT

[(~Zǫ

N × ~HǫN

)−(

~Zǫ × ~Hǫ)]

·(

~ZǫN × ~Ψ

)dxdt

+

∫ ∫

QT

(~Zǫ

N × ~HǫN

)·[(

~ZǫN × ~ΨN

)−(

~Zǫ × ~Ψ)]

dxdt

.= Iǫ

N + JǫN + Kǫ

N + LǫN

From (77), we get IǫN → 0 as N → +∞. At the same time, as N → +∞, we

have

|JǫN | ≤ ‖~Zǫ

N ×~ZǫN‖L2(QT )

∫ ∫

QT

|~ZǫN × ~ΨN − ~Zǫ × ~Ψ|2dxdt

1

2

≤ C

∫ ∫

QT

|~ZǫN × (~ΨN − ~Ψ) +

(~Zǫ

N − ~Z)× ~Ψ|2dxdt

1

2

→ 0

Similarly, one gets that KǫN → 0, as N → +∞ and

|LǫN | ≤ ‖ ~Hǫ

N‖L2(QT )‖~ZǫN‖L4(QT )‖~Zǫ

N × ~ΨN − ~Zǫ × ~Ψ‖L4(QT ) → 0.

Finally, from above arguments, one may take N → +∞ in (71)-(74) to obtain

that (~Zǫ(x, t), ~Hǫ(x, t), ~Eǫ(x, t), ~P ǫ(x, t)) is a global weak solution of the viscosityproblem (1)-(3), (11) and (10). This completes the proof.


Note that the a priori estimates in section 2 are independent of D. By using thediagonal method and letting D → +∞, we can obtain the global existence of weaksolution to the Cauchy problem of system (1)-(3) and (11). For simplicity, we donot state the theorem here.

4. A Priori Estimates Uniform in ε. In section 3, we have obtained a globalweak solution for viscosity problem (1)-(3), (11) and (10) for fixed ε > 0. In thissection we will derive the a priori estimates uniform in ε for solutions to viscosityproblem. These uniform estimates make us be able to pass to the limit ε → 0 andthen get the global weak solution to the problem (1)-(4) and (10).

We need the following lemmas

Lemma 7. Assume Ω = x = (x1, x2, x3); |xi| < D, i = 1, 2, 3, ~Q ∈ Xp(Ω). Then~Q ∈ H1(Ω) and there holds

‖ ~Q‖2H1(Ω) = ‖ ~Q‖2

Xp(Ω)

Proof: It follows from the relation

∆ ~Q = ∇(∇ · ~Q) −∇× (∇× ~Q)

that ∫

Ω

~Q∆ ~Q =

∫

Ω

~Q∇(∇ · ~Q) −

∫

Ω

~Q∇× (∇× ~Q)

The periodicity of ~Q implies∫

Ω

|∇ ~Q|2 =

∫

Ω

|∇ · ~Q|2 +

∫

Ω

|∇ × ~Q|2

and therefore we obtain the conclusion of the lemma.From the estimates in section 2 and the convergence in section 3, one easily gets

Lemma 8. Assume (~Z0(x), ~H0(x), ~E0(x), ~P0(x), ~P1(x)) ∈ (H1(Ω), L2(Ω), L2(Ω),H1(Ω), L2(Ω)). Then for the solution of the initial value problem (1)-(3), (11) and(10), there hold following estimates:

sup0≤t≤T

[‖~Zǫ(·, t)‖2H1(Ω) + ‖ ~Eǫ(·, t)‖2

2 + ‖ ~Hǫ(·, t)‖22

+‖ ~P ǫ(·, t)‖22 + ‖curl ~P ǫ(·, t)‖2

2 + ‖ ~P ǫt (·, t)‖2

2] ≤ M1 (79)

sup0≤t≤T

‖~Zǫ(·, t)‖26 ≤ M1, sup

0≤t≤T‖~Zǫ(·, t) ×∇~Zǫ(·, t)‖

L3

2 (Ω)≤ M1, (80)

where the constant M1 is independent of α2, D, and ǫ. When α2 > 0, there is

‖~Zǫ × (~Zǫ + ~Hǫ)‖L2(0,T ;L2(Ω)) ≤ M2 (81)

where the constant M2 is independent of ǫ, D.

In following we will prove that ∇~P ǫ is uniformly bounded in L∞(0, T ; L2(Ω)).We shall consider the compatibility conditions associated with the viscosity problemgiven by the following set of equations that hold in the sense of distributions

∂(eǫ + pǫ)

∂t+ σeǫ = 0 (82)

∂(hǫ + β∇ · ~Zǫ)

∂t= 0 (83)

∂2pǫ

∂t2+ µ

∂pǫ

∂t− ǫpǫ − νeǫ + 2νΦ′(|~P ǫ|2)pǫ = −4νΦ(2)(|~P ǫ|2)P ǫ

i P ǫj

∂P ǫj

∂xi(84)


wherehǫ = div ~Hǫ, eǫ = div ~Eǫ, pǫ = div ~P ǫ

and P ǫi is the i − th component of ~P ǫ and the relation:

div(~P ǫΦ′(|~P ǫ|2)) = Φ′(|~P ǫ|2)pǫ + 2Φ(2)(|~P ǫ|2)P ǫi P ǫ

j

∂P ǫj

∂xi

In order to obtain the L2(Ω) estimate of ∇~P ǫ(·, t), we shall assume that

div ~H0, div ~E0, div ~P0, div ~P1 ∈ L2(Ω). (85)

We have the following lemma:

Lemma 9. Under the conditions of Lemma 8 and assuming that the hypotheses(85) hold. Then for the solutions of the viscocity problem, we have

sup0≤t≤T

‖∇~P ǫ(·, t)‖L2(Ω) ≤ M3 (86)

where M3 is independent of D and ǫ.

Proof: For simplicity we present the proof for the case that ∇(div ~P0) ∈ L2(Ω),

since the general case of div ~P0 ∈ L2(Ω) can be handled by the modifying technique

or the proper approximation of the div ~P0.Multiplying (82) by 3eǫ and 2(eǫ + pǫ), we have

3

2

d

dt

∫

Ω

|eǫ|2dx + 3

∫

Ω

eǫ ∂pǫ

∂tdx + 3σ

∫

Ω

|eǫ|2dx = 0 (87)

d

dt

∫

Ω

|eǫ + pǫ|2dx + 2σ

∫

Ω

|eǫ + pǫ|2dx − 2σ

∫

Ω

(eǫ + pǫ)pǫdx = 0 (88)

Multiplying (84) by ∂pǫ

∂t , one gets

1

2

d

dt

∫

Ω

|∂pǫ

∂t|2dx + µ

∫

Ω

|∂pǫ

∂t|2dx +

ǫ

2

d

dt

∫

Ω

|∇pǫ|2dx − ν

∫

Ω

eǫ ∂pǫ

∂tdx

+ 2ν

∫

Ω

Φ′(|~P ǫ|2)pǫ ∂pǫ

∂tdx + 4ν

∫

Ω

Φ(2)(|~P ǫ|2)P ǫi P ǫ

j

∂P ǫj

∂xi

∂pǫ

∂tdx = 0 (89)

Combining (87)-(89), we obtain

1

2

d

dt

∫

Ω

[2|eǫ + pǫ|2 + 3|eǫ|2 + |

∂pǫ

∂t|2 + ǫ|∇pǫ|2

]dx

+ 2σ

∫

Ω

|eǫ + pǫ|2dx + µ

∫

Ω

|∂pǫ

∂t|2dx

= −3σ

∫

Ω

|eǫ|2dx + 2σ

∫

Ω

(pǫ + eǫ)pǫdx + (ν − 3)

∫

Ω

eǫ ∂pǫ

∂tdx

− 2ν

∫

Ω

Φ′(|~P ǫ|2)pǫ ∂pǫ

∂tdx − 4ν

∫

Ω

Φ(2)(|~P ǫ|2)P ǫi P ǫ

j

∂P ǫj

∂xi

∂pǫ

∂tdx

≤ M3

∫

Ω

[|eǫ|2 + |pǫ|2 + |

∂pǫ

∂t|2]

dx + M4 + M5

∫

Ω

|∇~P ǫ|2dx

Therefore we get that

1

2

d

dt

∫

Ω

[2|eǫ + pǫ|2 + 3|eǫ|2 + |

∂pǫ

∂t|2 + ǫ|∇pǫ|2

]dx

≤ M3

∫

Ω

[|eǫ|2 + |pǫ|2 + |

∂pǫ

∂t|2]

dx + M4 + M5

∫

Ω

|∇~P ǫ|2dx


Integrating with respect to t, we have

2‖(eǫ + pǫ)(·, t)‖22 + 3‖eǫ(·, t)‖2

2 + |∂pǫ

∂t(·, t)‖2

2 + ǫ‖∇pǫ(·, t)‖22

≤ M6 + 2M5

∫ t

0

∫

Ω

|∇~P ǫ|2dxdt + 2M3

∫ t

0

∫

Ω

[|eǫ|2 + |pǫ|2 + |

∂pǫ

∂t|2]

dxdt

where

M6 = 2‖div ~E0 + div ~P0‖22 + 3‖div ~E0‖

22 + ‖div ~P1‖

22 + ǫ‖∇(div ~P0)‖

22 + 2M4

is a constant from hypotheses (85).On the other hand,we get

‖pǫ(·, t)‖22 = ‖pǫ(·, t) + eǫ(·, t) − eǫ(·, t)‖2

2

≤ 2‖pǫ(·, t) + eǫ(·, t)‖22 + 2‖eǫ(·, t)‖2

2

We obtain that

‖pǫ(·, t)‖22 + ‖eǫ(·, t)‖2

2 + |∂pǫ

∂t(·, t)‖2

2 + ǫ‖∇pǫ(·, t)‖22

≤ M6 + 2M5

∫ t

0

∫

Ω

|∇~P ǫ|2dxdt + 2M3

∫ t

0

∫

Ω

[|eǫ|2 + |pǫ|2 + |

∂pǫ

∂t|2]

dxdt

By Gronwall inequality we get

‖pǫ(·, t)‖22 + ‖eǫ(·, t)‖2

2 + |∂pǫ

∂t(·, t)‖2

2

≤

(M6 + M5

∫ t

0

∫

Ω

|∇~P ǫ|2dxdt

)(1 + M3te

M3t)

≤ M7 + M8

∫ t

0

∫

Ω

|∇~P ǫ|2dxdt

Therefore we obtain that

‖pǫ(·, t)‖22 ≤ M7 + M8

∫ t

0

∫

Ω

|∇~P ǫ|2dxdt (90)

Using Lemma 7 for ~P ǫ(x, t), we get

‖∇~P ǫ(·, t)‖22 ≤ M9(‖cur ~P ǫ(·, t)‖2

2 + ‖div ~P ǫ(·, t)‖22 + ‖ ~P ǫ(·, t)‖2

2)

≤ M10 + M11

∫ t

0

∫

Ω

|∇~P ǫ|2dxdt

By Gronwall’s inequality one gets

‖∇~P ǫ(·, t)‖22 ≤ M12

where M12 is independent of ǫ. Lemma 9 is proved.

Remark 2. Lemma 8 and Lemma 9 show that ~P ǫ is bounded in L∞(0, T ; H1(Ω)).

5. Global Existence of Weak Solutions. By a priori estimates uniform in ε

obtained in section 4 for the viscosity problem and passing to the limit ε → 0 inequation (1)-(3), (11), we will get the global weak solution of problem (1)-(4) and(10).


Theorem 2. Assume the 2D-periodic functions (~Z0(x), ~H0(x), ~E0(x), ~P0(x), ~P1(x))∈ (H1(Ω), L2(Ω), L2(Ω), H1(Ω), L2(Ω)) and satisfying (85). Then the periodicinitial value problem (1)-(4) and (10) admits at least one global 2D-periodic weak

solution ~Z(x, t), ~H(x, t), ~E(x, t), ~P (x, t) such that(i) When α2 = 0,

~Z(x, t) ∈ L∞(0, T ; H1(Ω))⋂

C(0, 1

3)(0, T ; L2(Ω));

~H(x, t) ∈ L∞(0, T ; L2(Ω))⋂

C(0, T ; H−1(Ω));

~E(x, t) ∈ L∞(0, T ; L2(Ω))⋂

C(0, T ; H−1(Ω));

~P (x, t) ∈ L∞(0, T ; H1(Ω))⋂

C(0, T ; H−1(Ω));

~Pt(x, t) ∈ L∞(0, T ; L2(Ω))⋂

C(0, T ; H−1(Ω)). (91)

(ii) When α2 > 0,

~Z(x, t) ∈ L∞(0, T ; H1(Ω))⋂

C(0, 2

3)(0, T ; L3(Ω));

~H(x, t) ∈ L∞(0, T ; L2(Ω))⋂

C(0, 1

2)(0, T ; H−1(Ω));

~E(x, t) ∈ L∞(0, T ; L2(Ω))⋂

C(0, 1

2)(0, T ; H−1(Ω));

~P (x, t) ∈ L∞(0, T ; H1(Ω))⋂

C(0, 1

2)(0, T ; L2(Ω));

~Pt(x, t) ∈ L∞(0, T ; L2(Ω))⋂

C(0, 1

2)(0, T ; H−2(Ω)). (92)

Proof: The proof of this theorem is similar to that of Theorem 1. We omit it.

Acknowledgements. The first author is supported by the National Natural Sci-ence Foundation of China (Grant No.19971030, No.10471050), National 973 Project(No.2006CB805902) and Guangdong Provincial Natural Science Foundation (GrantNo.000671, No.031495).We would like to thank the referees very much for their valu-able comments and suggestions.

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Received December 2005; 1st revision May 2006; 2nd revision November 2006.

E-mail address: [email protected]




global existence of weak solutions for landau-lifshitz

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