Global Existence of Weak Solutions for Landau-Lifshitz-Maxwell Equations

DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 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 Studies Institute 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 equations are supplemented with a constitution relation in which the electric displacement equals a constant time the electric field plus an internal polarization variable and the magnetic displacement equals a constant time the magnetic field plus the microscopic magnetization. Using the Galerkin method and viscosity vanishing approach, we obtain the existence of the global weak solution for the Landau-Lifshitz-Maxwell equations. The main difficulties in this study are 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 = α1Z × △Z + H − α2Z × Z × △Z + H (1) ∂(E~ + P~ ) ∇× H~ = + σE~ (2) ∂t ∂H~ ∂Z~ ∇× E~ = − − β (3) ∂t ∂t ∂2P~ ∂P~ + λ2curl2P~ + µ = ν E~ − 2P~Φ′(|P~ |2) (4) ∂t2 ∂t ~ ~ 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 polarization, H~ e = △Z~ +H~ the effective magnetic field, E(P~ )=2P~Φ′(|P~ |2) the equilibrium

2000 Mathematics Subject Classiﬁcation. Primary: 35D10b ; 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

2 ~ ~ ~ electric field, curl P = curl(curlP )= ∇× (∇× P ). α1, α2, β, σ, λ, µ, ν are constants, where α2 ≥ 0 is the Gilbert damping coefficient; λ > 0 denotes the speed of 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 meanings of 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 some 2 ′ 2 point r0. These assumptions guarantee that rΦ (r ) ≤ C2 for all r ≥ 0, where C2 = C0 +2C1. Therefore we have ~ ′ ~ 2 ~ ′ ~ 2 ~ ~ ~ ~ 3 XΦ (|X| ) − Y Φ (|Y | ) ≤ C2 X − Y for all X, Y ∈ R (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, electric field 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, such as ferrites, are not only ferromagnetic materials, but also ferroelectric ones (such as LiFePO4 ), we call them the ferromagnetic-ferroelectrics [34]. If an electric field is applied to a medium (such as a dielectric one) made up of a large number of atoms or molecules, the charges bound in each molecule will respond to the applied field and will execute perturbed motions: the molecular charge density will be distorted. The multipole moments of each molecule will be different 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 the averaged over many molecules. The dominant molecular multipole with the applied fields 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 density but also the density of the charge on the surface of the polarized dielectric[22]. One can learn more details about polarization in [6],[12],[16],[27]. (1)-(3) (without P~ ) is a well-known classical Landau-Lifshitz-Maxwell system for ferromagnets [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 as follows ∂B~ ∂D~ = −curlE~ and + σE~ = curlH~ (7) ∂t ∂t 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 = ǫ0E + 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-Lifshitz equation: ~ ~ ~ ~ ~ ~ Zt = α1Z ×△Z − α2Z × (Z ×△Z) (8) This equation has been studied extensively in recent years. For the global existence 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 compact Riemannian manifold. For more recent results on the regularity of the solutions, we refer to [13],[14],[28],[30] and [34]. If, in addition, α2 = 0, system (8) becomes ~ ~ ~ Zt = α1Z ×△Z (9) As pointed out in [41], system (9) is a strongly coupled and strongly degenerate parabolic system. In [35]-[40] (and references therein), the authors investigated extensively the global existence of classical and generalized solutions to system (9). If ignoring the polarization, one gets the classical Landau-Lifshitz-Maxwell’s system (1)-(3) (without P~ ) which was proposed by Landau and Lifshitz in [26]. Guo and Su in [20] obtained the global weak solutions for this system subject to the periodic 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 polarization P~ which models the dynamics of ferroelectric materials, one has the so called nonlinear Maxwell systems (2)-(4) (without Z~) which was first considered by Greenberg et. al [17] for a simple case. And in [4], Habib Ammari and Kamel Hamdache generated the discussions of [17] to general cases and obtained the global existence, uniqueness and regularity of weak solutions by the theory of semigroups and the a priori estimates. There are also several other papers studied Maxwell’s equations 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 system which includes both Landau-Lifshitz equation and nonlinear Maxwell equations. We shall prove the global existence of periodic weak solutions with periodic initial conditions. We call a function f(x)is 2D-periodic if f(x +2Dei)= f(x), (i =1, 2, 3), where 3 (e1,e2,e3) forms the unit orthogonal basis of R , 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 width 2D along each direction, i.e. Ω = {x = (x1, x2, x3)| |xi| < D; (i = 1, 2, 3)} and QT = {(x, t)| x ∈ Ω, 0

∂2P~ ∂P~ + λ2curl2P~ + µ − ǫ△P~ = ν(E~ − 2P~Φ′(|P~ |2)) (11) ∂t2 ∂t 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 {Z~ ǫ, H~ ǫ, E~ ǫ, P~ ǫ} to the viscosity problem (1)-(3), (10) and (11). By establishing the estimates uniform in ǫ and sending ǫ to 0, we may get the global weak solution to problem (1)-(4) and (10).

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

~ 2 ~ ~ 2 HP (curl, Ω) = {V ∈ L (Ω); V is 2D − Periodic and curlV ∈ L (Ω)}, and is provided with the norm

2 2 1/2 ~ ~ 2 ~ 2 kV kHP (curl,Ω) = {kV kL (Ω) + kcurlV kL (Ω)} .

The space HP (div, Ω) is deﬁned by

~ 2 ~ ~ 2 HP (div, Ω) = {V ∈ L (Ω); V is 2D − Periodic and divV ∈ L (Ω)}, and is provided with the norm

2 2 1/2 ~ ~ 2 ~ 2 kV kHP (div,Ω) = {kV kL (Ω) + kdivV kL (Ω)} . Finally, we set

XP (Ω) = HP (curl, Ω) ∩ HP (div, Ω) with the norm

2 2 2 1/2 ~ ~ 2 ~ 2 ~ 2 kV kXP (Ω) = {kV kL (Ω) + kcurlV kL (Ω) + kdivV kL (Ω)} .

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- ~ 1 ~ T valued test function Ψ(x, t) ∈ C (QT ) such that Ψ(x, T ) = 0, the following equalities hold

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Z · Ψt + α1 (Z × ∇Z) · ∇Ψ − α1 (Z × H) · Ψ+ Z0 · Ψ(x, 0) Ω ZQ ZT ZQ ZT ZQ ZT Z ~ ~ ~ ~ ~ ~ ~ ~ +α2 Z × ∇Z · ∇ Z × Ψ − α2 Z × H · ∇ Z × Ψ = 0 (12)

ZQ ZT ZQ ZT LANDAU-LIFSHITZ-MAXWELL EQUATIONS 871

~ ~ ~ σt σt ~ ~ E + P · Ψte + σ e P · Ψ

ZQ ZT ZQ ZT σt ~ ~ ~ ~ + e ∇× Ψ · H + E0 + P0 · Ψ(x, 0)=0 (13) Ω ZQ ZT Z

~ ~ ~ ~ ~ ~ ~ ~ H + βZ · Ψt − ∇× Ψ · E + H0 + βZ0 · Ψ(x, 0)=0 (14) Ω ZQ ZT ZQ ZT Z

~ ~ 2 ~ ~ ~ ~ ~ ~ Pt · Ψt − λ curlP · curΨ − µ Pt · Ψ+ ν E · Ψ

ZQ ZT ZQ ZT ZQ ZT ZQ ZT ′ ~ 2 ~ ~ ~ ~ − 2ν Φ (|P | )P · Ψ+ P1 · Ψ(x, 0)=0 (15) Ω ZQ ZT Z Lemma 1. ([31]) Assume X ⊂ E ⊂ Y are Banach spaces and X ֒→֒→ E. Then the following imbedding are compact: ∂ϕ (i)Lq(0,T ; X) {ϕ : ∈ L1(0,T ; Y )} ֒→֒→ Lq(0,T ; E),if1 ≤ q ≤ ∞ (16) ∂t \ ∂ϕ (ii)L∞(0,T ; X) {ϕ : ∈ Lr(0,T ; Y )} ֒→֒→ C([0,T ]; E),if1 < r ≤ ∞ (17) ∂t \ 2. Solutions to the Viscosity Problem. In this section, we will use Galerkin method 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- ~ ǫ ~ ǫ ~ ǫ ~ ǫ lutions of the problem (1)-(3), (11) and (10) by ZN (x, t), HN (x, t), EN (x, t), PN (x, t) in the following form:

N N ~ ǫ ǫ ~ ǫ ~ǫ ZN (x, t)= ~αsN (t)ωs(x), HN (x, t)= βsN (t)ωs(x) s=1 s=1 X X N N ~ ǫ ǫ ~ ǫ ~ǫ EN (x, t)= ~γsN (t)ωs(x), PN (x, t)= δsN (t)ωs(x) s=1 s=1 X X ǫ ~ǫ ǫ ~ǫ + where ~αsN (t), βsN (t),~γsN (t), δsN (t), (t ∈ R ), (s = 1, 2, · · · ,N; N = 1, 2, · · · ) are three dimensional vector-valued functions satisfying the following system of ordinary diﬀerential equations:

~ ǫ ~ ǫ ~ ǫ ~ ǫ ZNtωs = α1 ZN × △ZN + HN ωs Ω Ω Z Z ~ ǫ ~ ǫ ~ ǫ ~ ǫ −α2 ZN × ZN × △ZN + HN ωs (18) Ω Z ~ ǫ ~ ǫ ~ ǫ HNt + βZNt ωs = − ∇× EN ωs(x) (19) Ω Ω Z Z 872 SHIJIN DING, BOLING GUO, JUNYU LIN AND MING ZENG

~ ǫ ~ ǫ ~ ǫ ~ ǫ ENt + PNt ωs(x)+ σ EN + PN ωs(x) Ω Ω Z Z ~ ǫ ~ ǫ = ∇× HN ωs(x)+ σ PN ωs(x) (20) Ω Ω Z Z ~ ǫ 2 2 ~ ǫ ~ ǫ ~ ǫ PNttωs + λ curl PN ωs(x)+ µ PNtωs(x) − ǫ △PN ωs(x) ZΩ ZΩ ZΩ ZΩ ~ ǫ ~ ǫ ′ ~ ǫ 2 = ν EN ωs(x) − 2ν PN Φ (|PN | )ωs(x) (21) ZΩ ZΩ with initial conditions ~ ǫ ~ ~ ǫ ~ ZN (x, 0)ωs(x)= Z0(x)ωs(x), HN (x, 0)ωs(x)= H0(x)ωs(x) (22) ZΩ ZΩ ZΩ ZΩ ~ ǫ ~ ~ ǫ ~ EN (x, 0)ωs(x)= E0(x)ωs(x), PN (x, 0)ωs(x)dx = P0(x)ωs(x)dx (23) ZΩ ZΩ ZΩ ZΩ ~ ǫ ~ PNt(x, 0)ωs(x)dx = P1(x)ωs(x)dx (24) ZΩ ZΩ It follows from the standard theory on nonlinear ordinary diﬀerential equations that (18)-(24) admits unique local solution. The following a priori estimates make us be able to take the limit N → ∞ in (18)-(24) to obtain the global solution to the viscosity problem. For the sake of simplicity, we denote k·kLp(Ω) = k·kp, p ≥ 2.

~ ~ ~ ~ ~ 1 2 2 Lemma 2. Assume Z0(x), H0(x), E0(x), P0(x), P1(x) ∈ (H (Ω),L (Ω),L (Ω), H1(Ω),L2(Ω)) Then for the solution of the initial value problem (18)-(24), we have the following estimates: ~ ǫ 2 ~ ǫ 2 ~ ǫ 2 ~ ǫ 2 sup [kZN (·,t)kH1(Ω) + kEN (·,t)k2 + kHN (·,t)k2 + kPN (·,t)k2 0≤t≤T ~ ǫ 2 ~ ǫ 2 ~ ǫ 2 +kcurlPN (·,t)k2 + ǫk∇PN (·,t)k2 + kPNt(·,t)k2] ≤ C3 (25)

~ ǫ 2 ~ ǫ ~ ǫ 3 sup kZN (·,t)k6 ≤ C4, sup kZN (·,t) × ∇ZN (·,t)k 2 ≤ C5, (26) 0≤t≤T 0≤t≤T L (Ω) where the constant C3, C4, C5 are independent of N, α2, and D. When α2 > 0, there is ~ ǫ ~ ǫ ~ ǫ kZN × (△ZN + HN )kL2(0,T ;L2(Ω)) ≤ C6 (27) where the constant C6 is independent of N, and D. ǫ Proof: 1. Multiplying (18) by ~αsN (t), summing up the products for s = 1, 2, · · · ,N, we get d |Z~ ǫ (·,t)|2dx =0 dt N ZΩ Then we have ~ ǫ 2 ~ ǫ 2 ~ 2 kZN (·,t)k2 = kZN (·, 0)k2 ≤kZ0(x)k2, ∀t ≥ 0 (28) ǫ ~ǫ 2. Making the scalar product of −λs~αsN (t)+ βsN (t) with (18), summing up the resulting product for s =1, 2, · · · ,N and then integrating by parts, we have 2 2 1 d ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ∇ZN (·,t) dx + α2 ZN × △ZN + HN − ZNt · HN dx = 0 (29) 2 dt Ω 2 Ω Z Z

LANDAU-LIFSHITZ-MAXWELL EQUATIONS 873

~ǫ ǫ Multiplying (19) by βsN (t) and (20) by ~γsN (t), summing up and integrating by parts, we obtain

2 2 2 1 d ~ ǫ ~ ǫ ~ ǫ HN (·,t) + EN (·,t) + σ EN 2 dt Ω Ω Z Z ~ ǫ ~ ǫ ~ ǫ ~ ǫ + PNt · EN + β Z Nt · HN = 0 (30) ZΩ ZΩ ǫ ~ǫ Multiplying (20) by (~γsN (t)+ δsN (t)), summing up and integrating by parts, we get

2 2 1 d ~ ǫ ~ ǫ ~ ǫ ~ ǫ EN (·,t)+ PN (·,t) dx + σ EN + PN dx 2 dt Ω Ω Z Z ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ = σ PN · EN + PN dx + ∇× HN · EN + PN dx (31) Ω Ω Z Z Putting these equalities together, we have

1 d 2 E~ ǫ (·,t)+ P~ ǫ (·,t) +2 H~ ǫ (·,t) + E~ ǫ (·,t) dx 2 dt N N N N ZΩ 2 2 ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ + σ EN + PN dx + σ EN dx +2β Z Nt · HN dx + PNt · EN dx Ω Ω Ω Ω Z Z Z Z ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ = σ P N · EN + PN dx + ∇× HN · PN dx (32) Ω Ω Z Z ~ǫ ′ Multiplying (21) by δsN (t), summing up the product for s = 1, 2, · · · ,N and ~ ǫ integrating by parts, one has, by noticing that PN is periodic

2 2 2 2 1 d ~ ǫ λ d ~ ǫ ǫ d ~ ǫ PNt(·,t) dx + ∇× PN (·,t) dx + ∇PN (·,t) dx 2 dt Ω 2 dt Ω 2 dt Ω Z 2 Z Z ~ ǫ ~ ǫ ~ ǫ ′ ~ ǫ 2 ~ ǫ ~ ǫ +µ P Nt dx = ν EN · PNtdx − 2ν Φ (|PN | )PN · PNt dx (33) Ω Ω Ω Z Z Z

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

2 2 1 d ~ ǫ ~ ǫ ~ ǫ ~ ǫ [ ∇ZN (·,t) + δ0 EN (·,t)+ PN (·,t) +2δ0 HN (·,t) 2 dt Ω Z 2 ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ +δ0 EN (·,t) ]+ α 2 ZN × △ZN + HN + (2βδ0 − 1) Z Nt · HN dx 2 Ω 2 2 Z ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ +δ0σ EN + PN + δ0σ EN + δ0 PNt · EN dx Ω Ω Ω Z Z Z ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ = δ0σ PN · EN + PN + δ0 ∇× HN · PN (34) Ω Ω Z Z ~ ǫ ~ ǫ In order to deal with the term Ω ZNt · HN dx, we multiplying (19) by (2βδ0 −1)~αǫ and sum up the product for s =1, 2, · · · ,N to obtain that sN R ~ ǫ ~ ǫ ~ ǫ ~ ǫ (2βδ0 − 1) HNt · ZN + (2βδ0 − 1) ∇× EN · ZN = 0 (35) Ω Ω Z Z 874 SHIJIN DING, BOLING GUO, JUNYU LIN AND MING ZENG

Adding (34) and (35), one gets 2 2 1 d ~ ǫ ~ ǫ ~ ǫ ~ ǫ [ ∇ZN (·,t) + δ0 EN (·,t)+ PN (·,t) +2δ0 HN (·,t) 2 dt Ω Z 2 ~ ǫ ~ ǫ ~ ǫ ~ ǫ +δ0 EN (·,t) ]dx + α2 ZN × △ZN + HN 2 d ~ ǫ ~ ǫ +(2βδ 0 − 1) ZN · HN dx dt Ω Z 2 2 ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ = −δ0σ EN + PN dx − δ0σ EN dx − δ0 PNt · EN dx Ω Ω Ω Z Z Z ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ +δ0σ PN · EN + P N dx − (2βδ 0 − 1) ∇× EN · ZN dx Ω Ω Z Z ~ ǫ ~ ǫ +δ0 ∇× HN · PN dx (36) Ω Z Putting (33) and (36) together, we have 2 2 1 d ~ ǫ ~ ǫ ~ ǫ ~ ǫ [ ∇ZN (·,t) + δ0 EN (·,t)+ PN (·,t) +2δ0 HN (·,t) 2 dt Ω Z 2 2 2 ~ ǫ ~ ǫ 2 ~ ǫ ~ ǫ +δ0 EN (·,t) + P Nt(·,t) + λ ∇× PN (·,t ) + ǫ ∇ PN (·,t) ]dx 2 d ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ +(2βδ0 − 1) ZN · HN dx + α2 ZN × △ZN + HN dt 2 ZΩ 2 2 2 2 2 (δ0 − ν) ~ ǫ 2 2 σ δ0 ~ ǫ ≤ µ +2+ PNt + 8ν C0 + + δ0σ PN 4 2 4 2 2 2 2 2 2 ~ ǫ ~ ǫ ~ ǫ ~ ǫ (1 − 2βδ 0) ~ ǫ +|3 − σδ0| EN + δ0σ EN + PN + HN + ∇ZN 2 2 2 4 2 2 2 δ0 ~ ǫ + curlP N 4 2 where C0 is given by (5). Integrating the above inequality with respect to t, we obtain 2 2 2 1 ~ ǫ δ0 ~ ǫ ~ ǫ ~ ǫ ∇ZN (·,t) + EN (·,t)+ PN (·,t) + δ0 HN (·,t) 2 2 2 2 2 2 2 2 2 2 δ 0 ~ ǫ 1 ~ ǫ λ ~ ǫ ǫ ~ ǫ + EN (·,t ) + PNt(·,t) + curl PN ( ·,t) + ∇PN (·,t) 2 2 2 2 2 2 2 2 t 2 ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ +(2βδ 0 − 1) ZN · H N dx + α 2 ZN × △ZN + HN dt 2 ZΩ Z0 2 2 2 2 2 1 ~ δ0 ~ ~ ~ δ0 ~ 1 ~ ≤ ∇Z0 + E0 + P0 + δ0 H0 + E0 + P1 2 2 2 2 2 2 2 2 2 t 2 ~ ~ ~ ǫ ~ ǫ +|2βδ 0 − 1 | H0 Z0 + δ0σ EN + PN dt 2 2 2 Z0 2 2 2 2 t 2 λ ~ ǫ ~ (δ0 − ν) ~ ǫ + curlP0 + ∇ P0 + µ +2+ PNt dt 2 2 2 2 4 0 2 Z 2 2 t 2 t 2 2 2 σ δ0 ~ ǫ ~ ǫ + 8ν C0 + + δ0σ PN dt + |3 − σδ0| E N dt 4 0 2 0 2 Z Z t 2 2 t 2 2 t 2 ~ ǫ (1 − 2βδ0) ~ ǫ δ0 ~ ǫ + HN dt + ∇ZN dt + curl PN dt 0 2 4 0 2 4 0 2 Z Z Z

LANDAU-LIFSHITZ-MAXWELL EQUATIONS 875

Therefore we have 2 2 2 ~ ǫ ~ ǫ ~ ǫ ~ ǫ ∇ZN (·,t) + δ0 EN (·,t)+ PN (·,t) +2δ0 HN (·,t) 2 2 2 2 2 2 2 ~ ǫ ~ ǫ 2 ~ ǫ ~ ǫ +δ0 EN (·,t) + PNt(·,t) + λ curlPN (·,t) + ǫ ∇PN (·,t) 2 2 2 2 t 2 ~ ǫ ~ ǫ ~ ǫ +2α 2 ZN × △ ZN + H N dt 2 Z0 2 t 2 (δ0 − ν) ~ ǫ ≤ C7 +2 µ +2+ PNt dt 4 0 2 Z 2 2 t 2 2 2 σ δ0 ~ ǫ + 16ν C0 + +8|δ0σ| PN dt 2 0 2 Z t 2 t 2 ~ ǫ ~ ǫ +(|6 − 2σδ0| +4|δ0σ|) EN dt +2 HN dt 2 2 Z0 Z0 2 t 2 2 t 2 (1 − 2βδ0) ~ ǫ δ0 ~ ǫ + ∇ZN dt + curl PN dt (37) 2 0 2 2 0 2 Z Z where 2 2 2 2 2 ~ ~ ~ ~ ~ ~ C7 = ∇Z0 + δ0 E0 + P0 +2δ0 H0 + δ0 E0 + P1 2 2 2 2 2 2 2 ~ ~ 2 ~ ~ + 2|2βδ0 − 1| H0 Z0 + λ curlP0 + ǫ ∇P0 dx 2 2 2 2 2 2 (1 − 2βδ0) ~ + Z0 δ0 2

On the other hand, we have 2 2 2 ~ ǫ ~ ǫ ~ ǫ ~ ǫ PN (·,t) − 2 EN (·,t) ≤ 2 PN (·,t)+ EN (·,t) 2 2 2 2 ~ ǫ ~ ǫ ≤ 3 PN (·,t)+ EN (·,t) (38) 2

Taking δ = 3, we get from (37) and (38) that 0 2 2 2 2 2 ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ∇ZN (·,t) + PN (·,t) +6 HN (·,t) + EN (·,t) + ǫ ∇PN (·,t) 2 2 2 2 2 2 2 t 2 ~ ǫ 2 ~ ǫ ~ ǫ ~ ǫ ~ ǫ + PNt(·,t) + λ curl PN (·,t ) +2α2 ZN × △ ZN + HN dt 2 2 2 Z0 t 2 2 2 2 ~ ǫ ~ ǫ ~ ǫ ~ ǫ ≤ C7 + C8 [ ∇ZN + PN + EN + HN 0 2 2 2 2 2 Z 2 ~ ǫ ~ ǫ + PNt + curl PN ]dt (39) 2 2 where

(1 − 6β)2 9σ2 C = max{ , (16ν2C2 + + 24σ), (|6 − 6σ| + 12σ), 2, 8 2 0 2 9 2[µ +2+(3 − ν)2], } 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). 876 SHIJIN DING, BOLING GUO, JUNYU LIN AND MING ZENG

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

~ ǫ ~ ǫ ~ ǫ ~ ǫ Lemma 3. Under the condition of Lemma 2, for the solution (ZN , HN , EN , PN ) of the problem (18)-(24), there exist C9 > 0 and C10 > 0, both independent of N,D, and ǫ, such that (i) when α2 =0,

sup Z~ ǫ + H~ ǫ + E~ ǫ + P~ ǫ ≤ C Nt −2 Nt −2 Nt −2 Ntt −2 9 0≤t≤T H (Ω) H (Ω) H (Ω) H (Ω) (40)

(ii) when α2 > 0,

ǫ ǫ ǫ Z~ 3 + H~ + E~ Nt 2 Nt 2 −1 Nt 2 −1 L (QT ) L (0,T ;H (Ω)) L (0,T ;H (Ω))

~ ǫ + PNtt ≤ C10 (41) L2(0,T ;H−2(Ω))

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

2 Proof: (i) When α2 = 0, for any periodic function ϕ ∈ H0 (Ω), ϕ can be represented by N ∞

ϕ = ϕN + ϕN , ϕN = ηsωs(x), ϕN = ηsωs(x) (42) s=1 X s=XN+1 For s ≥ N + 1, we have ~ ǫ ZNtωs(x)dx =0 ZΩ Then by Lemma 2, there holds

~ ǫ ~ ǫ ZNtϕ(x)dx = ZNtϕN (x)dx ZΩ ZΩ

~ ǫ ~ ǫ ~ ǫ = α 1 ZN × △ ZN + HN ϕN (x)dx Ω Z ~ ǫ ~ ǫ ~ ǫ ~ ǫ ≤ |α1| k∇ZN k2kZN k6 + kZN k2kHN k2 (k∇ϕN k3 + kϕN k∞)

≤ C11kϕkH2(Ω) where we have used Gagliardo-Nirenberg inequalities

1 3 1 1 4 4 2 2 kϕN k∞ ≤ CkϕN k2 k△ϕN k2 , k∇ϕN k3 ≤ Ck∇ϕN k2 k△ϕN k2 In the similar manner, we have

~ ǫ ~ ǫ HNtϕ(x)dx ≤ C12kϕkH2(Ω), ENtϕ(x)dx ≤ C13kϕkH2(Ω), ZΩ ZΩ

~ ǫ PNttϕdx ≤ C14kϕkH2(Ω) ZΩ

where C11, C12, C13, C14 are independent of N,D and ǫ. (40) follows. LANDAU-LIFSHITZ-MAXWELL EQUATIONS 877

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

Z~ ǫ ϕdxdt ≤ |α | Z~ ǫ × △Z~ ǫ + H~ ǫ ϕdxdt N 1 N N N Z Z Z Z QT QT

+ α Z~ ǫ × Z~ ǫ × △Z~ ǫ + H~ ǫ ϕdxdt 2 N N N N Z Z QT ǫ ǫ ǫ α Z~ Z~ H~ 2 ϕ 2 ≤ | 1 |k N × △ N + N kL (QT )k kL (QT ) ~ ǫ 6 ~ ǫ ~ ǫ ~ ǫ 2 3 + α2kZN kL (QT )kZN × △ZN + HN kL (QT )kϕkL (QT )

3 ≤ C15kϕkL (QT ) 2 1 Similarly, for any periodic function ϕ ∈ L (0,T ; H0 (Ω)), using (42) and Lemma 2, we get ~ ǫ ZNtϕdx ≤ C16kϕkH1(Ω) ZΩ

H~ ǫ ϕdxdt = − ∇× E~ ǫ ϕdxdt − β Z~ ǫ ϕ dxdt Nt N Nt N Z Z Z Z Z Z QT QT QT

≤ C 17kϕkL2(0,T ;H1(Ω))

E~ ǫ ϕdxdt Nt Z Z QT

= ∇× H~ ǫ ϕdxdt − P~ ϕ dxdt − σ E~ ǫ ϕ dxdt N Nt t N N Z Z Z Z Z Z QT QT QT

≤ C 18kϕkL2(0,T ;H1(Ω))

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

~ ǫ ~ ǫ ~ ǫ | PNttϕdxdt| = |ν EN ϕN dxdt + ǫ △PN ϕN dxdt

ZQ ZT ZQ ZT ZQ ZT ′ ~ ǫ 2 ~ ǫ ~ ǫ −2ν Φ (|PN | )PN ϕN dxdt − µ PNtϕN dxdt

ZQ ZT ZQ ZT 2 2 ~ ǫ −λ curl PN ϕN dxdt|

ZQ ZT T ~ ǫ ≤ C19kϕN kL2(0,T ;H1(Ω)) + ǫ kPN k2k△ϕN k2dt Z0 T 2 ~ ǫ +λ kcurlPN k2k∇ϕN k2dt Z0 ≤ C20kϕkL2(0,T ;H2(Ω)) where C15−C20 are independent of N,D and ǫ. The proof of Lemma 3 is completed. 878 SHIJIN DING, BOLING GUO, JUNYU LIN AND MING ZENG

~ ǫ ~ ǫ ~ ǫ ~ ǫ Lemma 4. Under the condition of Lemma 2, for the solution (ZN , HN , EN , PN ) of the problem (18)-(24), there exist constants C21 > 0, C22 > 0, C23 > 0, C24 > 0 and C25 > 0, independent of N,D, and ǫ, such that (i) When α2 =0, 1 ~ ǫ ~ ǫ 2 kZN (·,t1) − ZN (·,t2)k2 ≤ C21|t1 − t2| (43) ~ ǫ ~ ǫ ~ ǫ ~ ǫ −1 HN , EN , PN , PNt ∈ C([0,T ]; H (Ω)) (44) (ii) When α2 > 0, 2 ~ ǫ ~ ǫ 3 kZN (·,t1) − ZN (·,t2)k3 ≤ C22|t1 − t2| (45) 1 ~ ǫ ~ ǫ ~ ǫ ~ ǫ 2 kHN (·,t1) − HN (·,t2)kH−1(Ω) + kEN (·,t1) − EN (·,t2)kH−1(Ω) ≤ C23|t1 − t2| (46) 1 ~ ǫ ~ ǫ 2 kPNt(·,t1) − PNt(·,t2)kH−2(Ω) ≤ C24|t1 − t2| (47) 1 ~ ǫ ~ ǫ 2 kPN (·,t1) − PN (·,t2)k2 ≤ C25|t1 − t2| (48)

Proof: (i) When α2 = 0, by the Sobolev interpolation of negative order, there holds ~ ǫ ~ ǫ kZN (·,t1) − ZN (·,t2)k2 1 2 ~ ǫ ~ ǫ 3 ~ ǫ ~ ǫ 3 ≤ C26kZN (·,t1) − ZN (·,t2)kH−2 kZN (·,t1) − ZN (·,t2)kH1 1 3 t2 ǫ ∂Z~ 1 ≤ C N dt ≤ C |t − t | 3 27 ∂t 22 1 2 t1 −2 Z H

On the other hand, it follows from Lemma 1 and

;(L2(Ω) ֒→֒→ H−1(Ω) ֒→ H−2(Ω ∂Ψ H~ ǫ ∈ L∞(0,T ; L2(Ω)) {Ψ : ∈ L∞(0,T ; H−2(Ω))} N ∂t that \ ~ ǫ −1 HN ∈ C([0,T ]; H (Ω)) Similarly, we also have ~ ǫ ~ ǫ ~ ǫ −1 EN , PN , PNt ∈ C([0,T ]; H (Ω))

(ii) When α2 > 0, we have t2 ~ ǫ ~ ǫ ~ ǫ ∂ZN ZN (·,t1) − ZN (·,t2) = dt 3 t1 ∂t Z 3 1 3 ǫ 2 ∂Z~ 3 N ≤ |t1 − t2| dxdt  ∂t  ZQ ZT 2 3  ≤ C22|t1 − t2|

t2 ~ ǫ ~ ǫ ~ ǫ ∂HN HN (·,t1) − HN (·,t2) = dt H−1(Ω) t1 ∂t −1 Z H 1 2 2 T ǫ 1 ∂H~ 2 N ≤ |t1 − t2| dt  0 ∂t −1  Z H (Ω) 1 2  ≤ C23|t1 − t2| LANDAU-LIFSHITZ-MAXWELL EQUATIONS 879

t2 ~ ǫ ~ ǫ ~ ǫ ∂PN PN (·,t1) − PN (·,t2) = dt 2 t1 ∂t Z 2 1 2 2 T ǫ 1 ∂P~ 1 2 N 2 ≤ |t1 − t2| dt ≤ C25|t1 − t2|  0 ∂t  Z 2

  Lemma 4 follows. In fact, it follows from (25)-(26) that the solution of ODE (18)-(24) does not blow-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 system of the ordinary differential equation (18)-(24) admits at least one continuously differentiable global solution ǫ ~ǫ ǫ ~ǫ ~αsN (t), βsN (t),~γsN (t), δsN (t), (s =1, 2, · · · ,N; t ∈ [0,T ])

3. Existence of Weak Solution for the Viscosity Problem. First of all, similarly 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 following lemma which is well-known to all.

2 2 Lemma 6. If un → u strongly in L (QT ) and vn → v weakly in L (QT ), then 1 unvn → uv weakly in L (QT ) and in the sense of distribution. ~ ~ ~ ~ Theorem 1. Assume the 2D-periodic initial data (Z0(x), H0(x), E0(x), P0(x), ~ 1 2 2 1 2 P1(x)) ∈ (H (Ω), L (Ω),L (Ω), H (Ω), L (Ω)). Then the periodic initial value problem (1)-(3), (11), (10) admits at least one global weak solution Z~ ǫ(x, t), H~ ǫ(x, t), E~ ǫ(x, t), P~ ǫ(x, t) such that (i) When α2 =0, there hold 1 Z~ ǫ(x, t) ∈ L∞(0,T ; H1(Ω)) C(0, 3 )(0,T ; L2(Ω)); (49) \ H~ ǫ(x, t) ∈ L∞(0,T ; L2(Ω)) C(0,T ; H−1(Ω)); (50)

E~ ǫ(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)

~ ǫ ∞ 2 \ −1 Pt (x, t) ∈ L (0,T ; L (Ω)) C(0,T ; H (Ω)). (53) \ 880 SHIJIN DING, BOLING GUO, JUNYU LIN AND MING ZENG

(ii) When α2 > 0, we have 2 Z~ ǫ(x, t) ∈ L∞(0,T ; H1(Ω)) C(0, 3 )(0,T ; L3(Ω)); (54)

\ 1 H~ ǫ(x, t) ∈ L∞(0,T ; L2(Ω)) C(0, 2 )(0,T ; H−1(Ω)); (55)

1 E~ ǫ(x, t) ∈ L∞(0,T ; L2(Ω)) \C(0, 2 )(0,T ; H−1(Ω)); (56)

1 P~ ǫ(x, t) ∈ L∞(0,T ; H1(Ω))\ C(0, 2 )(0,T ; L2(Ω)); (57)

1 ~ ǫ ∞ 2 \ (0, 2 ) −2 Pt (x, t) ∈ L (0,T ; L (Ω)) C (0,T ; H (Ω)). (58)

\ ~ ǫ ~ ǫ Proof: The uniform estimates for the approximate solution ZN (x, t), HN (x, t), ~ ǫ ~ ǫ ~ ǫ ~ ǫ EN (x, t), PN (x, t) in section 2 yield that there is a subsequence of ZN (x, t), HN (x, t), ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ EN (x, t), PN (x, t), still denoted by ZN (x, t), HN (x, t), EN (x, t), PN (x, t), and Z~ ǫ(x, t), H~ ǫ(x, t), E~ ǫ(x, t), P~ ǫ(x, t), such that ~ ǫ ~ ǫ 6 ZN (x, t) → Z (x, t),weakly in L (QT ); (59) ~ ǫ ~ ǫ 6−̺ ZN (x, t) → Z (x, t),strongly in L (QT ), (̺> 0); (60) ~ ǫ ~ ǫ ∞ 1 ZN (x, t) → Z (x, t),weakly start in L (0,T ; H (Ω)); (61) ~ ǫ ~ ǫ ∞ 2 HN (x, t) → H (x, t),weakly start in L (0,T ; L (Ω)); (62) ~ ǫ ~ ǫ ∞ 2 EN (x, t) → E (x, t),weakly start in L (0,T ; L (Ω)); (63) ~ ǫ ~ ǫ ∞ 1 PN (x, t) → P (x, t),weakly start in L (0,T ; H (Ω)); (64) ~ ǫ ~ ǫ ∞ 2 PNt(x, t) → Pt (x, t),weakly start in L (0,T ; L (Ω)); (65) ~ ǫ ~ ǫ ∞ 2 curlPN (x, t) → curlP (x, t),weakly start in L (0,T ; L (Ω)); (66) 3 ~ ǫ ~ ǫ 2 ZNt(x, t) → Zt (x, t),weakly in L (QT ), (α2 > 0); (67) ~ ǫ ~ ǫ ∞ −2 ZNt(x, t) → Zt (x, t),weakly star in L (0,T ; H (Ω)), (α2 = 0). (68) From Lemma 1 (ii), we get that ∂ϕ L∞(0,T ; H1(Ω)) {ϕ : ∈ L∞(0,T ; L2(Ω))} ∂t \ ((C([0,T ]; L2(Ω)) ⊂ L2(0,T ; L2(Ω →֒→֒ ~ ǫ ∞ 1 ~ ǫ Since PN is bounded uniformly in L (0,T ; H (Ω)) and ∂tPN is bounded uni- ∞ 2 ~ ǫ formly in L (0,T ; L (Ω)), we deduce that there exists a subsequence of {PN }, still ~ ǫ denoted by {PN }, such that as N → ∞ ~ ǫ ~ ǫ ∞ 2 PN (x, t) → P (x, t), strongly in L (0,T ; L (Ω)) (69) ~ 1 ~ For any vector-valued periodic test function Ψ(x, t) ∈ C (QT ), Ψ(x, T )=0, we deﬁne an approximate sequence N ~ ΨN (x, t)= ~ηs(t)ωs(x) s=1 X ~ where ~ηs(t)= Ω Ψ(x, t)ωs(x)dx, then ~ ~ 1 p ΨR N (x, t) → Ψ(x, t) in C (QT ) and in L (QT ), ∀p> 1 (70) LANDAU-LIFSHITZ-MAXWELL EQUATIONS 881

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

~ ǫ ~ ~ ǫ ~ ǫ ~ ZN · ΨNtdxdt + α1 (ZN × ∇ZN ) · ∇ΨN dxdt

ZQ ZT ZQ ZT ~ ǫ ~ ǫ ~ ~ ǫ ~ −α1 (ZN × HN ) · ΨN dxdt − ZN (x, 0) · Ψ(x, 0)dx Ω ZQ ZT Z ~ ǫ ~ ǫ ~ ǫ ~ +α2 ZN × ∇ZN · ∇ ZN × ΨN dxdt

ZQ ZT ~ ǫ ~ ǫ ~ ǫ ~ −α2 ZN × HN · ZN × ΨN dxdt = 0 (71)

ZQ ZT

~ ǫ ~ ǫ ~ ~ ~ ǫ HN + βZN · ΨNtdxdt − ∇× ΨN · EN dxdt

ZQ ZT ZQ ZT ~ ǫ ~ ǫ ~ + HN (x, 0) + βZN (x, 0) · ΨN (x, 0)dx = 0 (72) Ω Z

~ ǫ ~ ǫ σt ~ σt ~ ǫ (EN + PN ) · (e ΨNt)dxdt + e ∇× ΨN · HN dxdt

ZQ ZT ZQ ZT σt ~ ǫ ~ ~ ǫ ~ ǫ ~ +σ e PN · ΨN dxdt + (EN (·, 0) + PN (·, 0)) · ΨN (·, 0)dx = 0 (73) Ω ZQ ZT Z

~ ǫ ~ 2 ~ ǫ ~ PNt · ΨNtdxdt − λ curlPN · curlΨN dxdt

ZQ ZT ZQ ZT ~ ǫ ~ ~ ǫ ~ ~ ǫ ~ +ν EN · ΨN dxdt − ǫ ∇PN · ∇ΨN dxdt − µ PNt · ΨN dxdt

ZQ ZT ZQ ZT ZQ ZT ′ ~ ǫ 2 ~ ǫ ~ ~ ǫ ~ −2ν Φ (|PN | )PN · ΨN dxdt + PNt(·, 0) · ΨN (·, 0)dx = 0 (74) Ω ZQ ZT Z

Now we are in the position to prove that (Z~ ǫ(x, t), H~ ǫ(x, t), E~ ǫ(x, t), P~ ǫ(x, t)) is a 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 terms in (71)-(74). First of all, we are able to prove

′ ~ ǫ 2 ~ ǫ ~ ′ ~ ǫ 2 ~ ǫ ~ Φ (|PN | )PN · ΨN dxdt → Φ (|P | )P · Ψdxdt (75)

ZQ ZT ZQ ZT 882 SHIJIN DING, BOLING GUO, JUNYU LIN AND MING ZENG

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

Φ′(|P~ ǫ |2)P~ ǫ · Ψ~ dxdt − Φ′(|P~ ǫ|2)P~ ǫ · Ψ~ dxdt N N N Z Z Z Z QT QT

′ ~ ǫ 2 ~ ǫ ′ ~ ǫ 2 ~ ǫ ~ ≤ | Φ (|PN | )PN − Φ (|P | )P · ΨN dxdt

ZQ ZT h i ′ ~ ǫ 2 ~ ǫ ~ ~ + Φ (|P | )P · ΨN − Ψ dxdt|

ZQ ZT ~ ǫ ~ ǫ ~ ≤ C∗ |PN − P ||ΨN |dxdt

ZQ ZT T ~ ǫ ~ ~ +CkP kL∞(0,T ;L2(Ω)) kΨN − ΨkL2(Ω)dt Z0 ~ ~ ǫ ~ ǫ ≤ C∗kΨN kL2(0,T ;L2(Ω))kPN − P kL2(0,T ;L2(Ω)) T ~ ǫ ~ ~ +CkP kL∞(0,T ;L2(Ω)) kΨN − ΨkL2(Ω)dt → 0 (as N → +∞) Z0 where we have used (69). ~ ǫ ~ ǫ Secondly, we claim that there exist subsequences of ZN , still denoted by ZN , such that, as N → +∞,

ǫ ǫ ∂Z~ ∂Z~ 3 ~ ǫ N ~ ǫ ∞ 2 (1). ZN × → Z × weakly star in L (0,T ; L (Ω)),i =1, 2, 3; (76) ∂xi ∂xi

~ ǫ ~ ǫ ~ ǫ ∂ZN ~ ǫ ∂Z 2 (2). (ZN × )xi → (Z × )xi weakly in L (QT ),i =1, 2, 3; (α2 > 0) (77) ∂xi ∂xi 1 In fact, for any periodic test function Ψ(~ x, t) ∈ C (QT ), we obtain

~ ǫ ~ ǫ ~ ǫ ∂ZN ~ ǫ ∂Z ~ ZN × − Z × · Ψdxdt ∂xi ∂xi ! ZQ ZT ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ∂ZN ~ ~ ǫ ∂ZN ∂Z ~ = (ZN − Z ) × · Ψdxdt + Z × − · Ψdxdt " ∂xi # " ∂xi ∂xi !# ZQ ZT ZQ ZT ∂Z~ ǫ ~ ∞ N 2 ~ ǫ ~ ǫ 2 ≤ kΨkL (QT )k kL (QT )kZN − Z kL (QT ) ∂xi ∂Z~ ǫ ∂Z~ ǫ + Z~ ǫ × N − · Ψ~ dxdt → 0, (as N → +∞) " ∂xi ∂xi !# ZQ ZT where we have used (59) and (61). Therefore (76) is proved. ~ ǫ ~ ǫ ~ ǫ Now we turn to prove (77). By Lemma 2, when α2 > 0, (ZN × (△ZN + HN )) 2 is bounded in L (QT ) uniformly respect to N. Then there exist a subsequence of ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ (ZN × (△ZN + HN )), still denoted by (ZN × (△ZN + HN )), and a vector U (x, t) ∈ 2 ~ 1 L (QT ), such that for any test function Ψ(x, t) ∈ C (QT ), there holds that as LANDAU-LIFSHITZ-MAXWELL EQUATIONS 883

N → +∞

~ ǫ ~ ǫ ~ ǫ ~ ~ ǫ ~ ZN × △ZN + HN · Ψdxdt → U · Ψdxdt

ZQ ZT ZQ ZT On the other hand, as N → +∞,

~ ǫ ~ ǫ ~ ǫ ~ ZN × △ZN + HN · Ψdxdt

ZQ ZT ~ ǫ ~ ǫ ~ ~ ǫ ~ ǫ ~ = − ZN × ∇ZN · ∇Ψdxdt + ZN × HN · Ψdxdt

ZQ ZT ZQ ZT → − Z~ ǫ × ∇Z~ ǫ · ∇Ψ~ dxdt + Z~ ǫ × H~ ǫ · Ψ~ dxdt

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

Z~ ǫ × H~ ǫ − Z~ ǫ × H~ ǫ · Ψ~ dxdt N N Z Z QT

≤ Z~ ǫ − Z~ ǫ × H~ ǫ · Ψ~ dxdt + Z~ ǫ × H~ ǫ − H~ ǫ · Ψ~ dxdt N N N Z Z Z Z QT QT

T ~ ǫ ~ ǫ ~ ǫ ~ ~ ǫ ~ ǫ ~ ǫ ~ ≤ kZ − Z k5kH k2kΨk 10 dt + Z × (H − H ) · Ψdxdt N N 3 N Z0 Z Z QT

→ 0 (78)

Then we have

U~ ǫ · Ψ~ dxdt − Z~ ǫ × H~ ǫ · Ψ~ dxdt = − Z~ ǫ × ∇Z~ ǫ · ∇Ψ~ dxdt

ZQ ZT ZQ ZT ZQ ZT Therefore one gets in the sense of distribution ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ 2 Z ×△Z = (U − (Z × H )) ∈ L (QT ). So (77) is proved. It remains to prove that

~ ǫ ~ ǫ ~ ǫ ~ ZN × ∇ZN · ∇ ZN × ΨN dxdt

ZQ ZT ~ ǫ ~ ǫ ~ ǫ ~ − ZN × HN · ZN × ΨN dxdt

ZQ ZT → Z~ ǫ × ∇Z~ ǫ · ∇ Z~ ǫ × Ψ~ dxdt

ZQ ZT − Z~ ǫ × H~ ǫ · Z~ ǫ × Ψ~ dxdt

ZQ ZT 884 SHIJIN DING, BOLING GUO, JUNYU LIN AND MING ZENG

In fact, we have

~ ǫ ~ ǫ ~ ǫ ~ ZN × ∇ZN · ∇ ZN × ΨN dxdt

ZQ ZT ~ ǫ ~ ǫ ~ ǫ ~ − ZN × HN · ZN × ΨN dxdt

ZQ ZT − Z~ ǫ × ∇Z~ ǫ · ∇ Z~ ǫ × Ψ~ dxdt

ZQ ZT + Z~ ǫ × H~ ǫ · Z~ ǫ × Ψ~ dxdt

ZQ ZT ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ = ZN × △ZN + HN · ZN × ΨN dxdt

ZQ ZT − Z~ ǫ × △Z~ ǫ + H~ ǫ · Z~ ǫ × Ψ~ dxdt

ZQ ZT ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ = ZN ×△ZN − Z ×△Z · ZN × Ψ dxdt

ZQ ZT h i ~ ǫ ~ ǫ ~ ǫ ~ ~ ǫ ~ + ZN ×△ZN · ZN × ΨN − Z × Ψ dxdt

ZQ ZT h i ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ ǫ ~ + ZN × HN − Z × H · ZN × Ψ dxdt

ZQ ZT h i ~ ǫ ~ ǫ ~ ǫ ~ ~ ǫ ~ + ZN × HN · ZN × ΨN − Z × Ψ dxdt

ZQ ZT h i . ǫ ǫ ǫ ǫ = IN + JN + KN + LN ǫ From (77), we get IN → 0 as N → +∞. At the same time, as N → +∞, we have 1 2 ǫ ǫ ǫ ǫ ǫ 2 |J | ≤ kZ~ ×△Z~ k 2 |Z~ × Ψ~ − Z~ × Ψ~ | dxdt N N N L (QT )  N N  ZQ ZT 1  2  ≤ C |Z~ ǫ × (Ψ~ − Ψ)+~ Z~ ǫ − Z~ × Ψ~ |2dxdt  N N N  ZQ ZT → 0   ǫ Similarly, one gets that KN → 0, as N → +∞ and

ǫ ~ ǫ 2 ~ ǫ 4 ~ ǫ ~ ~ ǫ ~ 4 |LN |≤kHN kL (QT )kZN kL (QT )kZN × ΨN − Z × ΨkL (QT ) → 0. Finally, from above arguments, one may take N → +∞ in (71)-(74) to obtain that (Z~ ǫ(x, t), H~ ǫ(x, t), E~ ǫ(x, t), P~ ǫ(x, t)) is a global weak solution of the viscosity problem (1)-(3), (11) and (10). This completes the proof. LANDAU-LIFSHITZ-MAXWELL EQUATIONS 885

Note that the a priori estimates in section 2 are independent of D. By using the diagonal method and letting D → +∞, we can obtain the global existence of weak solution to the Cauchy problem of system (1)-(3) and (11). For simplicity, we do not state the theorem here.

4. A Priori Estimates Uniform in ε. In section 3, we have obtained a global weak solution for viscosity problem (1)-(3), (11) and (10) for ﬁxed ε > 0. In this section we will derive the a priori estimates uniform in ε for solutions to viscosity problem. These uniform estimates make us be able to pass to the limit ε → 0 and then 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 2 2 ~ 1 ~ kQkH (Ω) = kQkXp(Ω) Proof: It follows from the relation ∆Q~ = ∇(∇ · Q~ ) −∇× (∇× Q~ ) that Q~ ∆Q~ = Q~ ∇(∇ · Q~ ) − Q~ ∇× (∇× Q~ ) ZΩ ZΩ ZΩ The periodicity of Q~ implies

|∇Q~ |2 = |∇ · Q~ |2 + |∇ × Q~ |2 ZΩ ZΩ ZΩ and therefore we obtain the conclusion of the lemma. From the estimates in section 2 and the convergence in section 3, one easily gets ~ ~ ~ ~ ~ 1 2 2 Lemma 8. Assume (Z0(x), H0(x), E0(x), P0(x), P1(x)) ∈ (H (Ω),L (Ω),L (Ω), H1(Ω),L2(Ω)). Then for the solution of the initial value problem (1)-(3), (11) and (10), there hold following estimates: ~ ǫ 2 ~ ǫ 2 ~ ǫ 2 sup [kZ (·,t)kH1 (Ω) + kE (·,t)k2 + kH (·,t)k2 0≤t≤T ~ ǫ 2 ~ ǫ 2 ~ ǫ 2 +kP (·,t)k2 + kcurlP (·,t)k2 + kPt (·,t)k2] ≤ M1 (79)

~ ǫ 2 ~ ǫ ~ ǫ 3 sup kZ (·,t)k6 ≤ M1, sup kZ (·,t) × ∇Z (·,t)k 2 ≤ M1, (80) 0≤t≤T 0≤t≤T L (Ω) where the constant M1 is independent of α2, D, and ǫ. When α2 > 0, there is ~ ǫ ~ ǫ ~ ǫ kZ × (△Z + H )kL2(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 problem given by the following set of equations that hold in the sense of distributions ∂(eǫ + pǫ) + σeǫ = 0 (82) ∂t ∂(hǫ + β∇ · Z~ ǫ) = 0 (83) ∂t ∂2pǫ ∂pǫ ∂P ǫ µ ǫ pǫ νeǫ ν ′ P~ ǫ 2 pǫ ν (2) P~ ǫ 2 P ǫP ǫ j 2 + − △ − +2 Φ (| | ) = −4 Φ (| | ) i j (84) ∂t ∂t ∂xi 886 SHIJIN DING, BOLING GUO, JUNYU LIN AND MING ZENG where hǫ = divH~ ǫ, eǫ = divE~ ǫ, pǫ = divP~ ǫ ǫ ~ ǫ and Pi is the i − th component of P and the relation: ∂P ǫ ~ ǫ ′ ~ ǫ 2 ′ ~ ǫ 2 ǫ (2) ~ ǫ 2 ǫ ǫ j div(P Φ (|P | ))=Φ (|P | )p + 2Φ (|P | )Pi Pj ∂xi In order to obtain the L2(Ω) estimate of ∇P~ ǫ(·,t), we shall assume that ~ ~ ~ ~ 2 divH0, divE0, divP0, divP1 ∈ L (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 ~ ǫ sup k∇P (·,t)kL2(Ω) ≤ M3 (86) 0≤t≤T where M3 is independent of D and ǫ. ~ 2 Proof: For simplicity we present the proof for the case that ∇(divP0) ∈ L (Ω), ~ 2 since the general case of divP0 ∈ L (Ω) can be handled by the modifying technique ~ or the proper approximation of the divP0. Multiplying (82) by 3eǫ and 2(eǫ + pǫ), we have 3 d ∂pǫ |eǫ|2dx +3 eǫ dx +3σ |eǫ|2dx = 0 (87) 2 dt ∂t ZΩ ZΩ ZΩ d |eǫ + pǫ|2dx +2σ |eǫ + pǫ|2dx − 2σ (eǫ + pǫ)pǫdx = 0 (88) dt Ω Ω Ω Z ∂pǫ Z Z Multiplying (84) by ∂t , one gets 1 d ∂pǫ ∂pǫ ǫ d ∂pǫ | |2dx + µ | |2dx + |∇pǫ|2dx − ν eǫ dx 2 dt ∂t ∂t 2 dt ∂t ZΩ ZΩ ZΩ ZΩ ∂pǫ ∂P ǫ ∂pǫ +2ν Φ′(|P~ ǫ|2)pǫ dx +4ν Φ(2)(|P~ ǫ|2)P ǫP ǫ j dx = 0 (89) ∂t i j ∂x ∂t ZΩ ZΩ i Combining (87)-(89), we obtain 1 d ∂pǫ 2|eǫ + pǫ|2 +3|eǫ|2 + | |2 + ǫ|∇pǫ|2 dx 2 dt ∂t ZΩ ∂pǫ + 2σ |eǫ + pǫ|2dx + µ | |2dx ∂t ZΩ ZΩ ∂pǫ = −3σ |eǫ|2dx +2σ (pǫ + eǫ)pǫdx + (ν − 3) eǫ dx ∂t ZΩ ZΩ ZΩ ∂pǫ ∂P ǫ ∂pǫ − 2ν Φ′(|P~ ǫ|2)pǫ dx − 4ν Φ(2)(|P~ ǫ|2)P ǫP ǫ j dx ∂t i j ∂x ∂t ZΩ ZΩ i ∂pǫ ≤ M |eǫ|2 + |pǫ|2 + | |2 dx + M + M |∇P~ ǫ|2dx 3 ∂t 4 5 ZΩ ZΩ Therefore we get that 1 d ∂pǫ 2|eǫ + pǫ|2 +3|eǫ|2 + | |2 + ǫ|∇pǫ|2 dx 2 dt ∂t ZΩ ∂pǫ ≤ M |eǫ|2 + |pǫ|2 + | |2 dx + M + M |∇P~ ǫ|2dx 3 ∂t 4 5 ZΩ ZΩ LANDAU-LIFSHITZ-MAXWELL EQUATIONS 887

Integrating with respect to t, we have ∂pǫ 2k(eǫ + pǫ)(·,t)k2 +3keǫ(·,t)k2 + | (·,t)k2 + ǫk∇pǫ(·,t)k2 2 2 ∂t 2 2 t t ∂pǫ ≤ M +2M |∇P~ ǫ|2dxdt +2M |eǫ|2 + |pǫ|2 + | |2 dxdt 6 5 3 ∂t Z0 ZΩ Z0 ZΩ where ~ ~ 2 ~ 2 ~ 2 ~ 2 M6 =2kdivE0 + divP0k2 +3kdivE0k2 + kdivP1k2 + ǫk∇(divP0)k2 +2M4 is a constant from hypotheses (85). On the other hand,we get ǫ 2 ǫ ǫ ǫ 2 kp (·,t)k2 = kp (·,t)+ e (·,t) − e (·,t)k2 ǫ ǫ 2 ǫ 2 ≤ 2kp (·,t)+ e (·,t)k2 +2ke (·,t)k2 We obtain that ∂pǫ kpǫ(·,t)k2 + keǫ(·,t)k2 + | (·,t)k2 + ǫk∇pǫ(·,t)k2 2 2 ∂t 2 2 t t ∂pǫ ≤ M +2M |∇P~ ǫ|2dxdt +2M |eǫ|2 + |pǫ|2 + | |2 dxdt 6 5 3 ∂t Z0 ZΩ Z0 ZΩ By Gronwall inequality we get ∂pǫ kpǫ(·,t)k2 + keǫ(·,t)k2 + | (·,t)k2 2 2 ∂t 2 t ~ ǫ 2 M3t ≤ M6 + M5 |∇P | dxdt (1 + M3te ) Z0 ZΩ t ~ ǫ 2 ≤ M7 + M8 |∇P | dxdt Z0 ZΩ Therefore we obtain that t ǫ 2 ~ ǫ 2 kp (·,t)k2 ≤ M7 + M8 |∇P | dxdt (90) Z0 ZΩ Using Lemma 7 for P~ ǫ(x, t), we get ~ ǫ 2 ~ ǫ 2 ~ ǫ 2 ~ ǫ 2 k∇P (·,t)k2 ≤ M9(kcurP (·,t)k2 + kdivP (·,t)k2 + kP (·,t)k2) t ~ ǫ 2 ≤ M10 + M11 |∇P | dxdt Z0 ZΩ By Gronwall’s inequality one gets ~ ǫ 2 k∇P (·,t)k2 ≤ 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 in equation (1)-(3), (11), we will get the global weak solution of problem (1)-(4) and (10). 888 SHIJIN DING, BOLING GUO, JUNYU LIN AND MING ZENG

~ ~ ~ ~ ~ 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 periodic initial 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, 1 Z~(x, t) ∈ L∞(0,T ; H1(Ω)) C(0, 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)); ~ ∞ 2 −1 Pt(x, t) ∈ L (0,T ; L (Ω)) \ C(0,T ; H (Ω)). (91)

(ii) When α2 > 0, \ 2 Z~(x, t) ∈ L∞(0,T ; H1(Ω)) C(0, 3 )(0,T ; L3(Ω));

1 H~ (x, t) ∈ L∞(0,T ; L2(Ω)) \C(0, 2 )(0,T ; H−1(Ω));

1 E~ (x, t) ∈ L∞(0,T ; L2(Ω)) \C(0, 2 )(0,T ; H−1(Ω));

1 P~ (x, t) ∈ L∞(0,T ; H1(Ω))\ C(0, 2 )(0,T ; L2(Ω));

1 ~ ∞ 2 (0, 2 ) −2 Pt(x, t) ∈ L (0,T ; L (Ω)) \C (0,T ; H (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 (Grant No.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] E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]