QUARTERLY OF APPLIED MATHEMATICSVOLUME LIX, NUMBER 1MARCH 2001. PAGES 67-84

LINEAR STABILITY FOR A THERMOELECTROMAGNETICMATERIAL WITH MEMORY

By

GIOVAMBATTISTA AMENDOLA

Dipartimento di Matematica Applicata "U. Dini", Facolta di Ingegneria, University of Pisa, viaDiotisalvi 2, 56126-Pisa, Italy

Abstract. In this paper we study the behaviour of a three-dimensional linear ther-moelectromagnetic material, which has constitutive equations with memory effects forboth the heat flux and the electric current density. We develop a linearized theory ofthermodynamics, in which context we are able to introduce a maximal free energy definedin the frequency domain. Using this free energy, a domain of dependence is obtained.Moreover, we prove a theorem of uniqueness, existence, and asymptotic stability.

1. Introduction. Thermal effects in electromagnetism have been considered by Cole-man and Dill [5], [6]. They derived the thermodynamic restrictions on the constitutiveequations. The dissipative effects of heat conduction on the asymptotic decay of thesolution have been studied in [10] for a dielectric nondissipative medium, but only forone-dimensional bodies.

In this paper we consider the linear theory of the thermodynamics of a homogeneousconductor. The material is characterized by two linearized constitutive equations for theheat flux and the electric current density. Both these constitutive functionals exhibitlong-term memory.

In Sec. 2 we introduce the constitutive relations and the field equations, the latterconsisting of Maxwell's equations and the energy equation. Then in Sec. 3 we consider thethermodynamics of simple materials [1], [2], which imposes restrictions on the constitutiveequations. Moreover, we define a maximal free energy which allows us to derive aninequality, that is used in Sec. 4 to show a domain-of-dependence inequality. In thelast section, theorems of existence, uniqueness, and asymptotic stability of solutions areproved for the three-dimensional model.

2. Constitutive equations. Let $7 be a region in the three-dimensional Euclideanspace R3, that an electromagnetic solid B occupies at time t. We suppose that Q is

Received January 5, 1999.2000 Mathematics Subject Classification. Primary 35Q60, 78A25, 83C50.Work performed under the support of C.N.R. and M.U.R.S.T..

©2001 Brown University67

68 GIOVAMBATTISTA AMENDOLA

a, bounded and regular domain, that is simply-connected and has a smooth boundary,whose unit outward normal is n.

We regard B as a homogeneous conductor with memory effects both for the electriccurrent density J and for the heat flux q within the linear theory of thermoelectromag-netism. Thus, the following constitutive equations

D(x, t) = fE(x, t) + i?(x, t)a, B(x, t) = /xH(x, t), (2.1)r+oc /*+oc

J(x, t) = / ct(s)E'(x, s) ds, q(x,i) = — / k(s)g'(x,s) ds, (2.2)Jo J o

h(x,t) = c&{x,t) + 0o[£_1Ax • D(x, t) + m_1A2 ■ B(x, <)] (2.3)

are assumed for the electric displacement D, the magnetic induction B. the rate at whichheat is absorbed per unit volume h, for J and q. In these, E and H denote the electricand magnetic fields, respectively, is the temperature relative to the absolute referencetemperature 0o, uniform in 0, g = W is the temperature gradient,

Ef(s) = E(£ — s), g'(s) = g(t — s) Vs e [0, +oo) (2.4)

are the histories of E and g up to time t. Moreover, the positive coefficients e, /x, and c are,respectively, the dielectric constant, the permeability, and the specific heat, while a, Aj,and A2 are three constant vectors. Finally, both the electric conductivity <r: R+ —> Rand the thermal conductivity k: R+ —> R are functions belonging to L1 (R+) H H1 (R+).

It is useful to introduce the integrated histories of E and g. that is, the functionsg' (x, •): R+ —> R3 and Ef (x, ■): R+ R'! defined by

E'(x,s)= f E(x, t) dr, g'(x,s)= f g(x,r)dr, (2.5)Jt—s Jt—s

with which (2.2) can be rewritten [4] as

f+oo r+ocJ(x, f) = — / ct'(s)E (x, s) ds, q(x,i)= / k'(s)g( (x, s) ds. (2.6)

Jo JoThe field equations that we must consider are Maxwell's equations, whose local form

V x H(x, t) = D(x, t) + J(x, t) + Jf(x, t), V • B(x, t) = 0, (2.7)

VxE(x,t) = -B(x,t), V • D(x, t) = p, (2.8)

where J f is a known function of (x,t), p is the free charge density, and the energyequation

h{x.,t) = HV • q(x,t) + r(x,t), (2.9)

r being the heat sources.We observe that the free charge density p is determined by means of Eq. (2.8)2.Sometimes the dependence on x will be understood.

LINEAR THERMOELECTROMAGNETIC MATERIAL WITH MEMORY 69

3. Thermodynamics of electromagnetic materials: its restrictions and apseudo-free energy. The constitutive equations (2.1)—(2.3), that we have assumed,allow us to consider our electromagnetic solid as a simple material [1], [2], for which thelaws of thermodynamics state that ([5], [6]) the following relations

^[h(t) + E(t) • D(t) + H(t) • B(t) + E(t) ■ J(i)] dt = 0, (3.1)

j){[0O + ■d{t)]^lh(t) + [0O + tf(£)]~2q(i) • g(<)} dt <0 (3.2)

must hold for any cyclic process, the equality sign in (3.2) referring only to reversibleprocesses. Here the mass density is supposed unitary.

It is well known that from these laws it is possible to prove the existence of the internalenergy e and the entropy r] and to derive, for smooth processes, the following relations

[7]:

e{t) = h{t) + E(t) • D(t) + H(t) • B(t) + E(t) • J(t), (3.3)V(t) > [©o + fi(t)]-lhit) + [0O + d(t,)}-'2qit) • git). (3.4)

We must consider the approximate expressions of (3.2) and (3.4), that is,

©o 2 j> {M<)[0o - + q(0 • g(t)}dt < 0, (3.5)

fl(t) > ©o2{^W[0o - + q(0 ■ §(0}, (3.6)

from which, on account of (3.1) and (3.3), we get

©o 2 j>{h{t)ti(t) + 0o[E(t) • Dit) + Hit) • Bit) + E(t) ■ J(t)]

~ q(t) ■ g(t)j dt > 0, (3.7)

©077(t) > e(t) - [E(t) ■ t>it) + H(t) ■ B(t) + E(t) ■ J(t)]

- 0o 1 Wt) ' Sit) ~ h(t)0(t)]. (3.8)

If we introduce the approximate pseudo-free energy

i/>(x, t) = e(x, t) - 0ot?(x, t), (3.9)

from (3.8) we derive

TP(t) < E it) ■ D it) + Hit) ■ Bit) + E it) ■ J it) + ©o ̂ l{h(t)#it) - q it) ■ g(t)]. (3.10)

We remember that the potentials r\ and ip are not uniquely determined in the presenceof memory effects; we introduce the following definition.

Definition 3.1. The potential ipM, such that (3.10) becomes the equality

i>Mit) = E it) ■ Bit) + H it) ■ B it) + E(t) ■ J it) + ©o l[hit)dit) - q it) ■ g it)], (3.11)

is called maximal free energy.

70 GIOVAMBATTISTA AMENDOLA

Let us consider the inequality (3.7), that, upon an integration on cycles, can be writtenas

©o 2 j{h{t)d(t) - 0o[E(t) • D(i) + H(i) • B(t) - Eft) ■ J(t)] - qft) ■ gft)}dt > 0.(3.12)

This expression, substituting

h(x.,t) = (c + 0o£-1A! ■ a)i?(x,i) + 0o[Ai • E(x,£) + A2 • H(x,t)], (3.13)

derived by (2.3) taking account of (2.1)^2, becomes

©o 2 j) 10o ^[(A]_ - a) • E(x, t) + A2 • H(x, t)]i9(x, t)r+oo \ r+oo

+ J cr(s)Ef(x, s)ds ■ E(x, t)\ + J^ fc(s)gf(x, 5) ds ■ g(x, t)j dt > 0, (3.14)

since it holds for any cycle.Taking account of the arbitrariness of i?,E, and g, from (3.14) we first obtain

j) [(Ai — a) • E(x, t) + A2 • H(x, £)]^(x, t) dt > 0, (3.15)

whence, since 19 is independent of E and H, it follows that

Ai = a, A2 = 0; (3.16)

then, we also haver+oc

j(s)Ef(x, s) ds ■ E(x, t) > 0,

(3.17)<r(s)Ef(x, s) ds ■ E(x, t) > 0,

j) J fc(s)g4(x, s) ds • g(x, t) dt > 0.

Thus, on account of (3.16), (3.13) becomes [10]

/i(x, t) = (c + 0o£_1a2)i?(x, t) + ©0a ■ E(x, t). (3.18)

In (3.17) the equality signs correspond to constant histories; therefore, if we considerperiodic processes of g(t) and of E(i) with the same frequency, we find that

r+00 r+00

/ k(s) cos(o;s) ds > 0, / cr(s) cos(o;s) ds > 0 Vu; ̂ 0. (3.19)Jo Jo

If we introduce the Fourier transform of the function /: R+ —> R" , identified with itscausal extension on R,

f(uj) = f f(s) exp[—iws] ds = fc(u) - if8(w), (3.20)J — OO

where we have considered the half-range Fourier cosine and sine transforms/• + OG r + oc

fc{ u) = / f(s)cos(ujs)ds, fs{u)= / f(s)sin(ujs)ds, (3.21)Jo Jo

LINEAR THERMOELECTROMAGNETIC MATERIAL WITH MEMORY 71

relations (3.19) assume the following form:

kc{uf) > 0, crc(oj) > 0 Voj ̂ 0. (3.22)

We observe that, if / and /' G L2(R+), then

1 r+°° .f'{w) = iuf(u) - /(0), /(0) = - / f(co)dco. (3.23)

^ J — oo

Moreover, we have

/» = -u>fc{w), /» = wfB(w) - /(0) (3.24)

and their consequences

2 Z"1-00uk's{io) < 0, fc(0) = — / w_1fc;(w) dw > 0, (3.25)

n Jo2 r+oc

^<7s(a') — 0' CT(0) = — / w~1(Js(a;) ^ > 0- (3.26)^ io

If k" e L2(R+), |fc'(0)l < +°° and o" € £2(R+)> |f'(0)l < +°°> follows that

sup |wfcg(u;)| < +oo, lim ujks(uj) = — lim ui2kc(u>) = k'(0) < 0, (3.27)ugR uj-+oc w—oo

sup \u}<t's(u>)\ < +oo, lim wcrs(u>) = — lim w2erc(u;) = <j'(0) < 0. (3.28)

In the following we shall assume: kc(0) > 0, k'(0) < 0 and erc(0) > 0, cr'(0) < 0.Finally, if g4 € L2(R+) and e' g L2(R+), we get

gc(x,w) = -wgts(x,w), il(x,w) =w_1g(x,t) +wg^(x,w), (3.29)

Ec(x,w) = -wE'(x,w), Es(x,u>) = w_1E(x, £) + wE'(x, uj). (3.30)

We can now prove the following theorem.

Theorem 3.1. The potential ipM defined by

ipM{x,t) = ^ -^-i92(x,t) + -D2(x, t) + —B2(x, t)00 £ M

1 /»-1-00

/ a>CTg(w)[E^(x,u;) ■ E^(x,w) + e'(x,w) • E*(x,w)]dw (3.31)^ Jo

1 f+°° _ _ _ _/ w^s(a;)[gs(x!w)' gs(x, w) + g'(x,w)."'g£(x,u;)] duj

Jo

is the maximal free energy; moreover, for any unit vector u,

-[0(71i9(x,^q(x,t) + E(x,t) x H(x,t)] • u < xVjw(x,i), (3-32)

where

X = 2{[fc(0)/c]1/2 + l/(e/i)[e1/2 + |a|(0o/c)1/2]/i1/2}. (3.33)

72 GIOVAMBATTISTA AMENDOLA

Proof. On applying Plancherel's theorem to (2.6) and using (3.29) and (3.30), we get1 i /•-fOC

J(x, t) • E(x, t) = - — - / wo'a (u>) [E,(x, u>) ■ Es(x, uj)at it J()+ S'(x,w)-E *(x,w)]dw, (3.34)

q(x, t) ■ g(x. t) = f uk's{u>)[g's(x,u;)-g^x.w)at it Jq

+ g'(x,w)-g £(x,w)]du;, (3.35)

the latter of which has been derived in [10].Then, the derivative of (3.31) with respect to time, taking into account (3.34), (3.35),

(2.1)i.2 and (3.18), yields (3.11).To establish the second assertion, we consider (3.31), from which, on account of (3.25)i

and (3.26) 1, it follows that

|0| < (20oc"Vw)1/2, |D| < (2^A/)1/2, |B| < (2/xVa/)1/2, (3.36)

whence (2.1)i,2 gives

|E| < 2i/2£-1[£1/2 + laKOoc"1)1/2]^2, |H| < n~\2(3.37)

moreover, (2.6)2, as in [10], gives

|q| < [2fc(O)0oVA/]1/2; (3.38)

thus, (3.32) follows. □

4. Field equations and domain-of-dependence inequality. The system of fieldequations for the linear theory of thermoelectromagnetism consists of (2.7)-(2.9). Takinginto account (2.1), (2.2), and (3.18), it becomes

V x H(x, t) = eE(x, t) + t?(x,i)a+ f o-(s)E4(x,s) ds + Ja(x, t) + J/(x, t), (4.1)Jo

V x E(x,t) = -/*H(x,t), V • H(x, t) = 0, (4.2)

(c+ 0oe_1a2)?9(x,t) + ©()a ■ E(x, t) = V • f k(s)g'(x, s) ds - V • qo(x, t) + r(x, t),Jo

(4.3)where

p + OC /» + oc

Ja(x, £) = / cr(s)E'(x, 5) ds = / a(t + r)E°(x, r) dr, (4.4)Jt J o

/ + OC /- + 00k(s)g'(x,s)ds = - k(t + t)g°(x, r) dr. (4.5)

We must add to (4.1)-(4.3) the initial conditions

H(x, 0) = H(j(x), E(x,0) = E„(x), i?(x, 0) = i?0(x), (4.6)

which are assumed consistent with the following boundary conditions:

E(x,f)xn = 0, i9(x,i) = 0 V(x,f)€dfix R+. (4.7)

LINEAR THERMOELECTROMAGNETIC MATERIAL WITH MEMORY 73

Moreover, we suppose Ho(x) is such that V • H0(x) = 0 and Hq(x) ■ n = 0 on Weobserve that these conditions, by use of Maxwell's equation (2.8)i and (2.1)2, imply thatV ■ H(x, t) = 0 in fl and H(x, t) ■ n = 0 on dCl are satisfied for all t.

We can now prove that there exists a domain of dependence by introducing the totalenergy

E(D.t)=f V>A/(E(x,i),H(x,f),i?(x,f),gf(x))dx VDcft. (4.8)Jd

Theorem 4.1. Let (E, H, 1?) be a solution of (4.1)-(4.3) and (4.6)-(4.7). Then

E{B(x0, p),T) < E(B{x0, P + \T), 0)

+ [ f [0q 1T'(x, i)$(x, t) — Jf(x,t) • E(x, t)]dxdt (4.9)

for any fixed (x0,T) G fi x R+, where \ is given by (3.33) and B(xo,p) is the ball ofcenter x<j and radius p.

Proof. Let (E,H,i?) be a solution of our problem and <p(x,t) E C£°(R3,R+) be anonnegative function. We define

£^(fi,£) = [ V»M(x,t)0(x,f)dx, (4.10)Jn

whose derivative with respect to time, taking into account (3.11), (2.7)-(2.9), and (4.7),gives

E${n,t)= f {[0q 1r{x,t)'0(x,t) - J/(x,f) • E(x,£)]</>(x,£)Jn

+ [0o 1t?(x,f)q(x, t) +E(x,t) x H(x, £)] ■ V<A(x,t) + ^/(x,t)<j)(x,t)} dx. (4.11)

If we put

f 1 Vw < —6</>(x,f) =i^(x,() = <fo(x-x0 \-p-x{T~t)) =Mv) = w r > (4-12)[0 \/y > 6

where 6 > 0, p > 0 and x is given by (3.33), for any fixed point x() £ Q and T E R+,t E (0,T), <j>s E C°°(R) and <p's{y) < 0, then we have

= <j>'s(y)x, V<A«(x,f) = <^(y)V|x-x0| = ^(y)u(x). (4.13)Substituting (4.13) into (4.11), the condition <p's(y) < 0 and (3.32) allow us to derive

the inequality

E<t>(tt,t) < f [Q^1r(x,t)i)(x,t) - J/(x, t) -E(x,f)]<fo(x,t)dx, (4.14)Jn

which, with an integration over (0,T), yields

E^,,(fl,t) - £^(Q,0) < [ [ [0O 1r(x,<)^(x,f) - Jf(x,t) ■ E(x,t)]0g(x,t) dxdt,Jo Jn

(4.15)whence, the limit as 5 —■> 0, since <f>s(x,t) tends to the characteristic function of the ballof center Xq and radius p + \(T — t), gives (4.9). □

74 GIOVAMBATTISTA AMENDOLA

5. Existence, uniqueness, and asymptotic stability. The linear system of equa-tions for the thermoelectromagnetic body is given by (4.1)-(4.3) together with the con-ditions (4.6) and (4.7). For simplicity, we can transform this problem into an equivalentone with zero initial data by putting in (4.1)-(4.3)

E(x, t) = E(x, t) + w(x, t),

H(x, t) = H(x, t) + v(x, t),

i9(x,t) = i9(x,t) + a(x,t),

with (w, v, a) and (E, H, ■$), that belong to the same function spaces of (E, H, fl), suchthat eV • w = — Va • a, V • v = 0, and

v(x, 0) = H0(x), w0(x,t) = E0(x,r), a0(x, t) = i?0(x> r) V(x, r) G fl x (-oo, 0],

but also changing the sources Jra + J/ in (4.1) with

r+oo

F(x, t) = J/(x, t) — V x v(x, t) + ew(x, t) + d(x, t)a + / cr(s)w<(x, s) ds,Jo

the sources r — V • qo in (4.3) with

r+oc

R(x, t) = r(x, t) + V ■ / ^(s)Vq'(x, s) ds — (c + 0()e_1a2)Q(x, t) — 0oa • w(x, t)Jo

and introducing a new source in (4.2)j given by

G(x, t) = —V x w(x, t) — /xv(x, t).

Thus, denoting again by (E, H, i?) the (E, H, §), we can consider the following equiv-alent problem:

V x H (x, t) - eE(x, t) — i?(x, t)a — f cr(s)E'(x, s) ds = F(x, t), (5.1)Jo

V x E(x, t) + |iH(x, t) — G(x, t), (5.2)

(c + 0oe_1a2)i?(x, t) + 0ya • E(x, t) — V • f fc(s)W'(x, s) ds = /?(x, t) (5.3)Jo

for any (x, t) 6 Q. x R+, with

H(x, 0) = 0, E(x, 0) = 0, i9(x, 0) = 0, (5.4)

E(x,()xn = 0, i9(x, t) = 0 V(x, t) £ dfl x R+. (5.5)

LINEAR THERMOELECTROMAGNETIC MATERIAL WITH MEMORY 75

We introduce the following function spaces:

Z2(ft) = {E(x) G L2(ft): f E-Vpdx = 0 \/<p G (fi)},Jn

H\n) = {H(x) G H\Q): V • H = 0},tf^(fi) = {E(x) G H\n): E(x) x n|an = 0},Hl0(ft) = {tf(x) G 0(x)|an = 0},

H{Q, R+) = L2(R+;ii^(ft)) x L^R+jH1^)) x L2(R+; H& (fi)),W(ft,R+) = H1(R+; L2(0)) x H^R+j L2(f2))

x{i/1(R+;L2(n))nL2(R+;H01(fl))}I

V(fi, R+) = L2(R+; L2(Q)) x L2(R+; Z2(fi)) x L2(R+; L2(0)).

They are Hilbert spaces with the usual scalar products.We suppose that the new sources (F, G, R) G V(fi, R+) are such that

p(FlGlfi)GV(ll)R+),dn7T-(F,G,R)dtn

= 0 (n = 0,1,2,3), (5.6)t=o

the last of which can be obtained by choosing the corresponding derivatives of w, v, anda opportunely.

We denote by i/(fi,R), W(fl. R), and V(f2, R) the spaces of the Fourier transformswith respect to time of the functions of H(Sl,R+), VF(fi,R+), and y(fi,R+). Each ofthese spaces is isomorphic to the corresponding one by virtue of the Plancherel theoremthat allows us to define in a natural way the scalar products by use of those introducedfor H(i2, R+), W(ft, R+), and V(fl,R+).

Thus, for example, we have the norm

||(E(x,w),H(x,u;),i?(x,w))||^= f°° [ (|V x E|2 + |V x H|2H 2tt J_x Jq (5.7)

+ |E|2 + |H|2 + |V$|2 + |i?|2) dxdui.

We shall use * to denote the complex conjugate.Definition 5.1. A triplet (E, H,$) G iJ(S7,R+) is called a weak solution of (5.1)—

(5.5), with sources (F, G, R) G V(f2,R+) that satisfy (5.6), if

f + OO P/ / {—[eE(x,t) + i?(x, £)a] • e(x,f) + /zH(x,f) • h(x,t)

Jo Jn+ [(c + 0o£_1a2)'i9(x, t) + 0oa • E(x, t)\/3(x, ()-Vx H(x, t) ■ e(x, t)

- V x E(x, t) ■ h(x, t) + J a(s)Et(x, s)ds ■ e{x,t) ^

— f fc(s)V$4(x, s) ds ■ V/3(x,t)} dxdtJo

i-+oo r

+ / [F(x, t) ■ e(x, t) + G(x, t) ■ h(x,i) + R(x,t)/3(x,t)\ dxdt = 0Jo Jn

for any (e, h, /3) G W(0, R+) such that e(x, 0) = 0, h(x, 0) = 0, /3(x, 0) = 0.

76 GIOVAMBATTISTA AMENDOLA

Plancherel's theorem allows us to transform our problem. Equation (5.8), on accountof (3.23) i with zero initial data, becomes

— f [ ({-w[eE(x,oj) + i?(x,w)aj + V x H(x,u) - ij(ai)E(x,u)}27r J-OG JQ

■ e*(x,w) + [iuj/j,H(x,u;) + V x E(x,u;)] • h*(x,w)

+ ico[(c + 0oe~'a2)i9(x,uj) + 0oa ■ E(x, w)]/3*(x,w) (5.9)

+ £;(a;)W(x,uj) ■ V/3*(x,u;)) dxduj = — [ [ [F(x,w) ■ e*(x,w)27r J-oc J£2

+ G(x, u) • h*(x,u;) + R(x,uj)(3*(x,lo)] dxdco

for any (e, h,/3) £ W(fi, R).Denoting by /1[(E, H, •$), (e,h,/3)] the expression in the left-hand side of (5.9) and by

((F, G, R), (e, h, (3)) the last integral of the same (5.9), we show the following theorem.

Theorem 5.1. There exists one and only one solution (E,H,?9) e H(fl,H) for any(F, G, R) e V(Ct, R) such that

i[(iEtf),(e,h,/j)] = i((F,G,fi),(e,M)) V(e,h,/3) € VT(fi,R). (5.10)

Proof. We first observe that, if (E, H, i9) 6 H(fl, R+) is a weak solution of our problem,its transform (E,H,$) G ) is a weak solution of the following problem:

—«j[eE(x,uj) + $(x,w)a] — <r(o;)E(x,w) + V x H(x,o>) = F(x,w), (5.11)

-Hw/iH(x,o>) + V x E(x,uj) = G(x,w), (5.12)

iuj[(c + 0(|£_1a2)i3(x,a;) + 0oa ■ E(x,u>)] — V • [fc(w) W(x, w)] = R(x,uj), (5.13)

for almost all uj 6 R, with

E(x,w) x n = 0, ^(x,w) = 0 V(x,w) € dfl x R. (5-14)

(Uniqueness). We must show because of linearity that the system with zero data hasonly the solution E*(x,o;) = 0, H* (x,o>) = 0, and ^*(x,w) = 0 for almost all uj G R.

Upon taking the inner products of the modified (5.11)—(5.13) with E*,H*, and $*,respectively, we obtain

JJ Q{ — [icue + cx(u;)]|E(x,u;)|2 — iw&(x,u))a • E*(x.o>)

(5.15)+ V x H(x,w) • E*(x,a;)} dx = 0,

/1Jtt

fJq

[^^^(Xjw)!2 + V x E(x,o;) • H*(x,w)] dx = 0, (5.16)

{iuj[{c + 0oe_1a2)|i9(x,u;)|2 + 0oa • E(x, w)i?*(x, w)l(5.17)

+ fc(w)|Vi?(x, uj)\2} d,x = 0.

J dx

|2 dx,(5.19)

LINEAR THERMOELECTROMAGNETIC MATERIAL WITH MEMORY 77

Hence, the real part of (5.16) gives

Re f V x E(x, lu) • H*(x,u>) dx = Re f E*(x,u)-VxH(x,w)(ix = 0, (5.18)J n Jn

on account of (5.14)i too.Moreover, the real parts of (5.15) and (5.17), by use of (5.18)2, yield

/ crc(u>)|E(x,a;)|2 dx = — Im / wa-E(x,w)^*(x,w)iJn Jn

=-©(71 [ fcc(w)|Vi?(x,w)|2Jn

that is,

f [Oocrc(w)|E(x, w)|2 + fcc(w)|V^(x, w)|2] dx — 0. (5.20)Jn

Thus, the conditions (3.22) allow us to write

[ |E(x,w)|2dx = 0, [ |W(x,w)|2dx = 0, (5.21)J n J n

where i9(x, uj) G Hq(CI). Therefore, it follows that

I |tf(x,w)|2dx = 0. (5.22)Jn

Hence a similar relation for H(x,uj) can be derived from (5.15) and (5.16), or directlyfrom the modified (5.12), which reduces to

iwH(x,ta;) = 0. (5.23)

(Existence). The proof is based on the following lemmas.

Lemma 5.1. Let

I(w) = f (|V x H|2 + |V x E|2 + |H|2 + |E|2 + |Vi?|2 + |tf|2) dx. (5.24)JQ

Then

I(w) < ?72(w) [ (|F|2 + |G|2 + |i?|2) dx, (5.25)Jn

where t]{uj) is a positive function of uj, the material constants and Q.

Proof. Let us consider the system (5.11)—(5.13) and the relations

-[iuie + cr{u))\ I |Ej2 dx—iuia ■ f i?E*dxJn Jn (5.26)

+ / VxH-E*dx= / F-E*dx,Jn Jn

i• dxin

— [iuje + cr(u;)]a • f E$*dx — iuia2 j |i9|2Jn Jn

+ f V x H • ar dx = [ F • ad*dx,Jn Jn

(5.27)

78 GIOVAMBATTISTA AMENDOLA

— [iuie + <j(u>)\ / E • V x H*dx — iuia •Jn J n

+ [ |V x H|2 = [ F V x H*dx,Jn Jn

E • V x H*dx - iu>a • / tfV x H*dx(5.28)

derived by integrating (5.11) after taking the inner products with E*, i?*a, and V x H*.Moreover,

itufi I |H|2 dx + I VxE-H*dx = I G H*rfx, (5.29)> [ |H|2 dx + / V x E ■ H* dx = [Jn Jn Jn

i f H • V x E*dx + [ \V x E|2 dx = [Jn Jn Jn

iun / H-VxE *dx + / |VxE|^dx = / G • V x E*dx, (5.30)7<2 Jn Jn

which again follow by taking the inner products of (5.12) with H* and V x E*, andfinally

iu>(c + 0o£~1a2) I ]$|2 dx + iw0oa • f Ed*dx + k(ui) f |W|2dx = f Rfl* dx,Jn Jn Jn Jn

(5.31)

obtained by multiplying (5.13) by d*.We first observe that the real parts of (5.28) and (5.30) yield

[ |V x H|2 dx = Re f F • V x H* dx - wlma • [ tfV x H *dxJn Jn Jn

+ ac{u)) Re [ V x H ■ E*dx — [as(w) — £lo\ Im / V x H • E*dx, (5.32)Jn Jn

/ |V x E|2 dx = Re / G • V x E* dx + fJ-oj Im / V x H • E* dx, (5.33)Jn Jn Jn

while the real part of (5.31) gives

kc{co) f | W|2 dx = Re f Rti*dx - 0owlma ■ [ ^E* dx. (5.34)Jn Jn Jn

From (5.29), its real part, given by

Re [ V x H ■ E*dx = Re [ G H* dx (5.35)Jn Jn

allows us to derive from the real part of (5.26)

wlma- j i9E* dx = Re f F • E* dx — Re f G • H* dx + (jc{uj) j |E|2dx. (5.36)Jn Jn Jn Jn

Thus, from (5.34), using (5.36), we get

kc{u>) I |Vi?|2 dx + 0of7c(w) f |E|2dx = Re f Rfl* dxJn Jn Jnin

+ ©„ ( Re I G H *dx -Re I F E *dx) .• f G ■ H *dx - Re / :Jn Jn

(5.37)

LINEAR THERMOELECTROMAGNETIC MATERIAL WITH MEMORY 79

Let us consider the imaginary part of (5.31):

a; Re a- f dWdx = 0q 1 Im f R^dx + ks(w) f |W|2<ixJn [ Jn Jn

— (c + 0o£-1a2)w f \d\2 dx ,Jn

which, again from the same (5.34), allows us to get the imaginary part of (5.27) in theform

Ima • f i9V x H*dx = — Im f F • ai?*dxJn Jn

H—— |crc(w)Re f R-d*dx + \os(w) — ew] Im f i?$*c(xlI Jn Jn J

+ ©o 1 ( ~(ks(w)[as(w) - ew] - kc(w)crc(w)) f |Vi?|2Iw Jn

+[cew - (c + ©0£_1a2)(7s(a;)] |$|2 dxj .

dx

(5.39)

From the real part of (5.27), on account of (5.36) and (5.38), we have

w Re a • / i9V x H*dx = uRe / F • atTdxJn •/Jn

+ 0O 1 |<jc(o;) Im f Rd*dx — [<ts(w) — ew] Re f R$*dx{ Jn Jn

+ ©0 1 |(/cc(w)[o-s(w) - ew] + ks(w)oc(w)) J |V??|2dx

(c + 0q£ a )wac(w) > J l^x},(5.40)

which, together with (5.35), allows us to obtain from the imaginary part of (3.28) thelast term, which we had to derive for (5.32) and (5.33),

Im / V x H ■ E*dx = —I Im [ F • V x H*dx + w Re / F • adxif V x H • E*dx = —{Im f F • V x H*dx + «Re /Jn crc{w) [ Jq

— [<rs(w) — ew] Re J g ■ H*dx|

+ 0(71 {im f iM*dx [<rs(tc/) — ew] Re f Rfl* dxI Jn a c\u) Jn

v-i \ (kc(w) \ f j|2.+ 0° { ( [o-s(oj) -ew] + ks(w) ) / |W| dx

— (c + 0o£ 'a2)a; j |i9|2dx | .

(5.41)

80 GIOVAMBATTISTA AMENDOLA

We next observe that the boundary conditions (5.14) assure that there exist twopositive constants, depending on f2, such that

f |i?|2dx. < f |Vi?|2dx,Jn Jn

(5.42)

/ |H|2dx < a,(fi) [ |V x H|2 dx,Jn Jn

the second of which follows from (5.14) j taking into account that H £ HJ(f2) and thatfl is simply connected [9]. Therefore, from (5.24) it follows that

< [ {[1 + a, (fi)]|V x H|2 + |V x E|2 + [1 + a„ (fl)](|V^|2 + |Ej2)} dx. (5.43)Jn

Then, the positiveness of kc{u>), crc(oj) Vu> G R and of 0q allows us to derive from(5.43) the following inequality:

l(co) < |q2|VxH|2 + |VxE|2

[M^)|W|2 + 0o<tcH|E|2]J dx,+Ol1 1

+kc(u>) 0qctc( ui)

(5.44)

where we have put

Qi — 1 + a, (0),

a-2 = 1 + a„ (ft)(5.45)

and introduced the expression in the left-hand side of (5.37).Substituting (5.37) and the relations derived from (5.32) and (5.33), on account of

(5.39), (5.35) and (5.41), (5.44) yields

LINEAR THERMOELECTROMAGNETIC MATERIAL WITH MEMORY 81

2"(w) < ( cii Re [ F ■ V x H*dx H [{ea\ + i±)u>\ Jtt CcV^J

- ai<7g(w)] Im / F • V x H*dx ) + ( Re / G ■ V x E*dxin /V Ju

+ ( I —y—{ai((T^(a;) + [ew - <ts(w)]2) + /iui[£uj - crs(w)]}V lo-cM

M,+ a2 0o 1 G H *dxn_kc(uj) ac(ui)_

+ ( —7 r [(s^i + - aicrs(w)]a;Re /\°"CM Jn

+ f0"1 I ~~~ ^(w)]2 - al(u>)) + fxuj[£u - as(u)]}

F • aiiTdx + aiui Im / F • ai?*dxn J n

+C*2e„ i

kc(oj) oc (ui)

+ 0q 1 {2«i[ew - crs(w)] +/uoj} Im j Rd*dx

M /foTdx

in

+ (~a2 + -4rl ^ / F • E*dx)\ _kc(oj) ac(uj)_ Jn J

x-l I I 1 / 2/ \ r ( \i2\ i o^s(a') [

+l" " ^[n""^ 4=(")|v,'|2<ixO

h {"' Ks+ t) 'e [i;+11"2c az( 0^ + ~J ujJi fcc(w)I^Nx-

(5.46)

Using (5.42) i and (5.37), the last two integrals of this inequality, denoting by £vtf(w)and £,>(w) the corresponding coefficients, can be transformed in the following manner:

fvtfM [ kc{io)\Vd\2dx +f kc(a;)|$|2dxJn Jn

<[|CvtfH|+a„(fi)|^(o;)|] f [ftc(u;)|Vi?|2 + 0octc(w)|E|2] dx

< [|£vtfMI + a„ (Q)|£,?(a>)>/•Jn

(5.47)Re / m*dx

+0O ( Re / G H*dx -Re F E*dxIn Jn

82 GIOVAMBATTISTA AMENDOLA

We now consider

= max{|7a(w)| + |7i2(w)| + |7i(w)| (z = 1,2,..., 6)}, (5.48)

where 7 ij(u>) (i = 1,2,..., 6 ;j = 1,2) denote the coefficients of the real (j = 1) andimaginary (j = 2) parts of the six ordered integrals in round brackets in (5.46), and

7i(u) =0 (i = 1,2,4), 73(w) = -76M = ©075(0;) = ©o[|£v»?MI + a„(n)|^(u;)|].(5.49)

Thus, the inequality (5.46), on account of (5.47), (5.48), and (5.49), becomes

\l/2/ r \l/2 / /■ xl/2 , n \ 1 /2

I(w)<^(cj) ( I |F|2<ix ) / |VxH|2dx + / |G|2dx / |V x E|2dx/n / \Jn / Vu

+ ( I |G|W) ( £ |H|2<ix^ 7 + |a|^ |F|2dx) ' Qf |^|2riXX '

+ (£ l^l2^ ' Qf |tf|2cbc) 1 +(£ |F|2dxj 7 Qf |Ei2dxX 7 1

(5.50)

Then, taking max{l, |a|}, it follows that there exists rj(ui) such that we have theinequality

T(u>) < r)(u>) [ (|F|2 + |G|2 + |£|2)dxJtt

1/2I1/2M, (5.51)

from which (5.25) follows. □

Lemma 5.2. If the sources (F.G,/?) 6 V(f2,R+) satisfy (5.6) and the hypotheses ona(s) and k(s) hold, then the inverse Fourier transforms of (E. H. € H(£l. R) exist andare L2-functions with zero initial conditions (5.4).

Proof. The hypotheses assumed for (F,G,/?) with (5.6), and the properties of a(s)and k(s), stated in Sec. 3, in particular, those given by (3.27) and (3.28), assure thatt)(lo) is bounded for any u> £ R. Therefore, the right-hand side of (5.25) is integrableover R, that is,

f [ ?y2(u;)(|F|2 + |G|2 + |i?|2) dxdco < +00. (5.52)J-oc J SI

Comparison of (5.52) and (5.25) givesr+oc

[ [ (|V X Hj2 + |V X E|2 + |H|2 + |E|2 + |Vi^|2 + |i9|2) thtdujJ-oc Jq

< [+ [ 772(w)(|F|2 + |G|2 + |R\2)dxdLU.J—DC J S2

Therefore, application of Plancherel's theorem to (5.53) allows us to prove the existenceof the inverse Flourier transforms of (E.H,'!?). □

LINEAR THERMOELECTROMAGNETIC MATERIAL WITH MEMORY 83

Corollary 5.1. If we consider two source fields (F^,(j = 1,2) and thecorresponding solutions (j = 1,2), then

||(E(1) -E(2),H(1) -H(2),<!?(1) -tf(2))||~

< — f [ r?2(w)(|F(1)-F(2)|2 + |G(n-G(2)|2 + |^(1)-JR(2)|2)dxdw.2tt 7-oc -hi

(5.54)In order to complete the proof of existence we have to show the following lemma.

Lemma 5.3. The subset

S = {(F, G, R) G V{fl, R): 3(E, H, d) G H{Q, R)

solution of (5.10) V(e,h, /?) G W(fi,R)}

is dense and closed in V(fi,R).

Proof. We denote by S the closure of S in V(fi.R). Let us suppose that there exists(F°, G°,/?°) ^ 0 such that (F°,G°,i?°) G V(fi,R)\S'. Application of the Hahn-Banachtheorem assures the existence of (e°,h0,/?0) G W(fi,R) such that

((F°, G°, i?0), (e°, h°, /30)) ̂ 0, <(F, G, R), (e°, h°, /30)) = 0 V(F.G,R)sS (5.56)

hold. The same proof of the uniqueness theorem can be used for the condition

A[(E, H, ■&), (e°, h°,/30)] = 0, V(E, H, $) G H{fl, R), (5.57)

that, on account of (5.10). is equivalent to (5.56)2 and yields

(e°, h°, (3°) = 0, (5.58)

contrary to (5.56) i. Therefore, S is dense.To prove that S is closed, let (F'"', G'"', R^) G S, n = 1,2,..., be a sequence of

sources with lim„_>+0C(F(n\ fiW, R^) = (F, G,R) G V"(f2,R) and (EW.HW^W) gH(Q, R), n = 1,2,..., be the corresponding solutions.

Application of Corollary 5.1 yields

||(E(,l) - E(m),H(rt) - - i?(m))||!

< — / / ??2(u;)(|F(")-F(m)|2 + |G(n)-G(m)|2 + |JR(ri)-fi("!)|2)dxdti;.2?r J-OC J <2

(5.59)

Therefore, (E<n),H(n\i?<n> ), n = 1.2,..., is a Cauchy sequence and, since the space iscomplete, we have

lira (E(n\H('°,i?(n)) = (E,H,0) G H(fi,R). (5.60)

Substitution of the corresponding solutions (E("),HW,i?W) and (F<n), G(n\£<")),n = 1,2,..., gives a sequence of identities, whose limit as n —> +oo is an analogousidentity expressed in terms of the limits (E, H,i?) and (F.G.i?). Therefore it followsthat (F. G, R) G S.

84 GIOVAMBATTISTA AMENDOLA

Application of Plancherel's theorem leads to the existence of (E, H, i?) G //(fLR+),the solution of our problem; therefore, the proof of Theorem 5.1 is complete. □

References[1] W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal. 48, 1-50

(1972)[2] B. D. Coleman and D. R. Owen, A mathematical foundation of thermodynamics, Arch. Rational

Mech. Anal. 54, 1-104 (1974)[3] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch.

Rational Mech. Anal. 31, 113 126 (1968)[4] M. McCarthy, Constitutive equations for thermodynamical materials with memory, Internat. J.

Engrg. Sci. 8, 467-474 (1970)[5] B. D. Coleman and E. H. Dill, Thermodynamic restrictions on the constitutive equations of elec-

tromagnetic theory, ZAMP 22. 691-702 (1971)[6] B. D. Coleman and E. H. Dill, On the thermodynamics of electromagnetic fields in materials with

memory, Arch. Rational Mech. Anal. 41, 132-162 (1971)[7] M. Fabrizio and A. Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied

Mathematics, Philadelphia, 1992[8] M. Fabrizio and A. Morro, A boundary condition with memory in electromagnetism, Arch. Rational

Mech. Anal. 136, 359 381 (1996)[9] E. B. Bykhousekis and N. V. Smirnov, On the orthogonal decomposition of the space of vector

functions square summable in a given domain, Trudy Mat. Inst. Steklov 59, 6-36 (1960)[10] G. Amendola, On thermodynamic conditions for the stability of a thermoelectromagnetic system,

Math. Methods Appl. Sci. 23, 17-39 (2000)[11] R. E. Showalter, Hilbert Space Methods for Partial Differential Equations, Monographs and Studies

in Mathematics, Vol. 1, Pitman, London, 1977[12] F. Treves, Basic Linear Partial Differential Equations, Academic Press, New York, 1975[13] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford,

1960[14] C. Muller, Foundation of Mathematical Theory of Electromagnetic Waves, Springer, Berlin, 1969