Well-posedness and asymptotic behaviour for linear magneto-thermo-elasticity with second sound

Universität Konstanz

Well-posedness and asymptotic behaviour for linear magneto- thermo-elasticity with second sound

Martin Gubisch

Konstanzer Schriften in Mathematik Nr. 349, April 2011

ISSN 1430-3558

© Fachbereich Mathematik und Statistik Universität Konstanz

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-322691

Martin Gubisch

Universität Konstanz, FB Mathematik und Statistik Universitätsstraße 10, 78457 Konstanz, Germany

Abstract. We consider the Cauchy problem of magneto-thermo-elasticity with second sound inR³. After proving the existence of a unique solution, we use Fourier transform and multiplier methods to show polynomial decay rates for suitable initial data. We compare the qualitative and quantitative asymptotic behaviour of magneto-thermo-elasticity with second sound with that of the classical system.

1. Introduction. We study the reciprocal effects between temperature, elasticity and magnetism in some thermoconductive, elastic and magnetic 3D-medium.

Letθ =θ(t, x)∈Rdenote the temperature difference to some fixed reference temperature, u = u(t, x) ∈ R³ the displacement vector with respect to some reference configuration, and H =H(H₀, h)the (total) magnetic field, consisting of the primary fieldH₀ ∈R³ and the induced field h=h(t, x)∈R³.

The classical coupled system then reads as







M(x)u_tt−Eu + γ∇θ − α(∇ ×h)×H₀ = 0 µ₀(x)h_t− ∇^TΛ(x)∇h − β∇ ×(u_t×H₀) = 0 c(x)θ_t− ∇^TK(x)∇θ + γ∇^Tu_t = 0.

(1.1) The matrices M(x) (mass density), K(x) (thermal conductivity) and Λ(x) (magnetic coefficients) are assumed to be symmetric and positive definite uniformly in the variablex, c(x)≥c > 0denotes the specific heat capacity,µ₀(x)≥µ₀ >0the magnetic permeability and E the elasticity operator:

Eu:= 1 2

3

X

j,k,l=1

∂

∂x_js·jkl(x) ∂

∂x_kul+ ∂

∂x_luk

=D^TSDu,

depending on the matrix S(x) = (s_ijkl(x))_ijkl of the elastic coefficients which shall fullfill the symmetries s_ijkl = s_jikl = s_klij and be positive definite uniformly in x, too, and on the natural gradient D corresponding to E which is defined in (2.1)

The classical (parabolic) model for the temperature,

c(x)θt(t, x)− ∇^TK(x)∇θ(t, x) = 0, (1.2) implies the paradox of infinite propagation speed. Therefore, we split this heat equation in the transport equation for the heat flux q = q(t, x) ∈ R³ and Fourier’s law of heat conduction and introduce a small delay parameter τ > 0:

c(x)θ_t(t, x) +∇^Tq(t, x) = 0

q(t+τ, x) +K(x)∇θ(t, x) = 0. (1.3) Taylor expansion of the delay term q(t+τ)up to first order yields

c(x)θ_t+∇^Tq = 0

τ qt+q+K(x)∇θ = 0. (1.4)

Hereby, Fourier’s law is replaced by the so-called Cattaneo law of heat conduction which respects the finite propagation speed of the heat flux; we say that the second sound effect occurs in the model. Notice that any θ which solves this system with delay also is a solution of the damped wave equation

c(x)τ θ_tt+c(x)θ_t− ∇^TK(x)∇θ = 0, (1.5) i.e. the system (1.4) is of hyperbolic type. Our coupled system with relaxation now reads as











M(x)u_tt−Eu + γ∇θ − α(∇ ×h)×H₀ = 0 µ₀(x)h_t− ∇^TΛ(x)∇h − β∇ ×(u_t×H₀) = 0

c(x)θt+∇^Tq + γ∇^Tut = 0

τ q_t+q+K(x)∇θ = 0.

(1.6)

Additionally, we postulate initial conditions

(u, u_t, h, θ, q)(0) = (u₀, u₁, h₀, θ₀, q₀) (1.7) with some regularity which is determined in section 2.

A detailed derivation of the system and an overview of the results for classical magneto- elasticity and magneto-thermo-elasticity in the sixties was made by Paria [18]. The poly- nomial stablity of the magneto-elastic model, neglecting the thermal effects in (1.1) by choosing γ = 0 and θ ≡0, has been analyzed by Andreou & Dassios [01] using spectral analysis and techniques provided from thermo-elasticity. The equations of pure thermo- elasticity, i.e. neglecting the influence of the magnetic field in (1.1) resp. (1.6), choosing α = 0, β = 0 and h ≡ 0, have recently been discussed in details by Jiang & Racke [11], [22], [23], Irmscher [09], [10], Weinmann [26] and others, using different space dimensions and both bounded domains in R^d and the full space (d= 1,2,3).

Under the additional assumption that the rotation of the initial data V₀ = (SDu₀, u₁, θ₀) is constantly zero and the condition

Z

R³

1

C₃(ξ)^m|FV(0, ξ)| dξ <∞, C₃(ξ) := |ξ|²

1 +|ξ|² (1.8)

holds, the solution V = (SDu, u_t, θ) to the classical system of thermo-elasticity in R³ decays polynomially with rate m in the Lebesgue spaceL² (a proof using methods of the Fourier analysis can be found in [24], for example). For the system with delay, qualitatively equal results are shown. However, this result does not hold for all thermo-elastic models:

Fernández & Racke [06] showed that solutions to classical damped Timoshenko systems are exponentially stable and that the second sound effect destroys this stability. Since Muñoz Rivera and Racke [16] proved polynomial stability for the classical equations of magneto- thermo-elasticity if a condition corresponding to (1.8) holds, it is therefore natural to investigate if this behaviour inherits on the equations with delay parameter.

The paper is structured as follows: In the first section, the well-posedness of the Cauchy problem is shown by using standard methods of semigroup theory. In the second section, a corrected version of the decay rates for the classical system is presented which

can be received by slightly modifying the original proof of Muñoz Rivera and Racke given in [16]. On this basis, decay rates for the second sound system are derived afterwards. The third section observes the asymptotic behaviour of solutions for vanishing delay parameter τ₀ and the short-time behaviour of solutions. In the forth section, usual simplifying assumptions on the magnetic field are dropped; in this case, the coupling between elastic and magnetic effects becomes stronger. The techniques used in section three are modified and decay rates are given under additional assumptions on the initial data. Finally, in the last section, those initial conditions which refer to the Fourier-transformed data are retranslated into regularity assumptions on the original data.

2. Well-posedness. We apply methods presented in [11] for the well-posedness of the initial value problem in thermo-elasticity. Reduction of (1.6) to a first order systems yields

V_t+AV = 0

V(0) = V₀ (2.1)

where

V :=











 SDu

ut

h θ q













, V₀ :=











 SDu₀

u1

h₀ θ₀ q0













, Γ :=















 γ γ γ 0 0 0

















, D:=

















∂₁ 0 0 0 ∂₂ 0 0 0 ∂₃ 0 ∂₃ ∂₂

∂₃ 0 ∂₁

∂₂ ∂₁ 0

















and the differential operator in xis given as

A:=













S 0 0 0 0

0 M⁻¹ 0 0 0

0 0 _µ¹

0

β

αId 0 0

0 0 0 ¹_c 0

0 0 0 0 ¹_τK













| {z }

=:Q













0 −D 0 0 0

−D^T 0 −α(∇ × ·)×H₀ D^TΓ 0 0 −α∇ ×(· ×H₀) −^α_β∇^TΛ∇ 0 0

0 Γ^TD 0 0 ∇^T

0 0 0 ∇ K⁻¹











 .

Notice that the canonical domain of A,

D(A) ={V ∈ H | ∃W ∈ H :∀Φ∈ C₀^∞:hV, A^TΦi_L² =−hW,Φi}_L²

is dense in the Hilbert space H :=L² and A : D(A)→ H defines a closed operator. We provide H with the modified L²-scalar product

h·,·iH :=h·, Q⁻¹·i_L²

which is equivalent to h·,·iL², thenA is dissipative with respect to h·,·iH: Let V ∈ D(A). SinceΛ and K are positive definite, we get

RehAV, ViH = α

βRehΛ∇V³,∇V³iL² +RehK⁻¹V⁵, V⁵iL²

≥ α

βλ₀||∇V³||²_L2 +k₀||K⁻¹V⁵||²_L2 ≥ 0. (2.2)

Therefore, for any λ∈(−∞,0) and V ∈ D(A) the following inequality holds:

||(A−λ)V||H||V||H ≥ Reh(A−λ)V, ViH

≥ λ₀α

β||∇V³||²_L2 +k₀||K⁻¹V⁵||²_L2 −λ||V||²_H

and we get ||(A−λ)V||H ≥ −λ||V||H for any V 6= 0. Especially, A−λ is injective and (A−λ)⁻¹ :im(A−λ)→ H is continous.

Finally, since the adjoint operator A^∗ of A is defined on D(A) and has the form (A^∗)_ij = (−1)^1+δijA_ij (i, j = 1, ...,5)

where δij is the Kronecker Delta,A^∗−λ is injective, too, and the partition H=im(A−λ)⊕ker(A^∗−λ) =im(A−λ)⊕ {0}

implies that the image of A−λ is dense in H. Therefore, any λ <0is an element of the resolvent set %(A) and the corresponding resolvent fullfills the estimate

||(A−λ)⁻¹|| ≤ −1 λ.

The Hille & Yosida Theorem implies that Ais the generator of the C₀-semigroup (e^tA)t≥0, i.e. the Cauchy Problem (2.1) is well-posed:

Theorem 2.1 (existence and uniqueness)For any initial valueV₀ ∈ H, the initial value problem (2.1) has a unique solution V ∈ C⁰([0,∞),H) which depends continously on the data.

Additionally, if V₀ ∈ D(A), we get V ∈ C⁰([0,∞),D(A))∩ C¹([0,∞),H).

3. Asymptotic behaviour. From now on, we assume that our medium is homogenous and isotropic, i.e. we take M =Id, K =κId,Λ =Id, c= 1, µ₀ = 1 and

E =µ∆ + (µ+λ)∇∇^T

for some constants µ, λ > 0 which are called the “Lamé moduli”. Furthermore, we take H₀ := (0,0, H)^T, H > 0 (which is not assumed without loss of generality as we will see in section 5).

Let F =(·)c denote the Fourier transform on L² and v := ˆu, w := ˆh, ϑ:= ˆθ, r := ˆq, then the transformed system (1.6) reads as











v_tt+µ|ξ|²v+ (µ+λ)(ξ·v)ξ−iγϑξ−iαHw³ξ+iαHξ₃w = 0 w_t+|ξ|²w+iβHξ₃v_t−iβH(0,0, ξ·v_t)^T = 0 ϑ_t−i(ξ·r)−iγ(ξ·v_t) = 0 τ r_t+r−iκϑξ = 0.

(3.1)

According to the Plancherel Theorem, the energy associated to (3.1), E^τ(t) := 1

2 Z

R³

(|vt|²+µ|ξ|²|v|²+ (µ+λ)(ξ·v)²+ α

β|w|²+|ϑ|²+ τ κ|r|²

(t, ξ) dξ,

is equal to the energy of the original system (1.6):

E^τ(t) = 1 2

Z

R³

|u_t|²+µ|∇u|²+ (µ+λ)|∇^Tu|²+α

β|h|²+|θ|²+ τ κ|q|²

(t, x)dx.

We show that the integrand Eˆ^τ(t, ξ) := 1

2

|v_t|²+µ|ξ|²|v|²+ (µ+λ)(ξ·v)²+α

β|w|²+|ϑ|²+ τ κ|r|²

(t, ξ) decays exponentially in t for any fixed ξ ∈ R³. For suitable initial data, this yields to polynomial decay rates for E^τ where the decay rate depends on the data. To construct a Lyapunov functional L^τ =L^τ(t, ξ)which is equivalent to Eˆ^τ and decays exponentially in t, we modify the methods used in [16] for the classical system (1.1).

In the following, we present a corrected version of the proof for polynomial decay rates in classical magneto-thermo-elasticity. Let (v, w, ϑ) the Fourier-transformed solution to (1.1). With multiplier methods, the following estimate can be shown, replacing Lemma 2.1, 2.2 & 2.3 in [16]:

Define

Φ₁(t) := Re i ξ₃

v_t¹w¹+v_t²w²

!

(t); Φ₄(t) := Re v²_ttv_t³

(t);

Φ₂(t) := Re i ξ₃

v_tt¹w¹_t +v²_ttw_t²

!

(t); Φ₅(t) := Re(v_tv) (t);

Φ3(t) := Re

v¹_ttv_t³

(t); Φ6(t) := Re

v¹_tv¹ +v²_tv²

(t)

and let a:=|ξ|² 1 ξ₁²+ 1

ξ₃²

!

, then the function

Φ(t) := (1 +a)|ξ|²Φ₁(t) +aΦ₂(t) +δ ξ₁

ξ₃ +ξ₃ ξ₁

Φ₃(t) +δξ₂

ξ₃Φ₄(t) + δ

2|ξ|²Φ₅(t) + 2µ µ+λ

δ 8

|ξ|⁴ ξ₃² Φ₆(t) fullfills the inequality

d

dtΦ(t) ≤ c

δa²|ξ|²(1 +|ξ|²)|w|²+ c

δa²(1 +|ξ|²)|w_t|²+ca|ξ|²|ϑ|² +cδa|ϑ_t|²− βH

4 a(|v_tt¹|²+|v_tt²|²)− δ

16|ξ|²Eˆ⁰(t) (3.2) for some c > 0 which depends only on the coefficients of the differential equation and some δ =δ(c)>0.

Since the dissipation of the classical system reads as d

dt

Eˆ⁰(t) = −α

β|ξ|²|w|²−κ|ξ|²|ϑ|², (3.3)

it is easy to see (compare Theorem 3.3 in this work) that for sufficiently large constants N₁, N₂ >0,

L⁰(t) := Φ(t) +N1a²(1 +|ξ|²) ˆE₁⁰(t) +N2a²1 +|ξ|²

|ξ|² Eˆ₂⁰(t) (3.4) is a Lyapunov funcional with the properties described above where the energy terms of first, second and third order (which we need in section 4) are defined as

Eˆ_i^τ(t, ξ) := 1 2

|∂_tⁱv|²+µ|ξ|²|∂_tⁱ⁻¹v|²+ (µ+λ)(ξ·∂_tⁱ⁻¹v)² +α

β|∂_tⁱ⁻¹w|²+|∂_tⁱ⁻¹ϑ|²+ τ

κ|∂_tⁱ⁻¹r|²

(t, ξ) (i= 1,2,3; τ ≥0).

The proof for the main decay result of Muñoz Rivera and Racke in [16] then reads as follows:

Theorem 3.1 (decay rates for classical magneto-thermo-elasticity) Let τ0 = 0 and let m ∈ N0 arbitrary. If the Fourier-transformed initial data (ˆu₀,uˆ₁,ˆh,θ)ˆ of classical magneto-thermo-elasticity fullfill

Z

R³

(1 +|ξ|²)

ξ₁⁴ξ₃⁴

(ξ₁²+ξ₃²)|ξ|²(1 +|ξ|²)^{2 −m}

Eˆ⁰(0, ξ) dξ <∞

where

Eˆ⁰(0, ξ) := 1 2

|ˆu₁|²+µ|ξ|²|ˆu₀|²+ (µ+λ)(ξ·uˆ₀)²+ α

β|ˆh₀|²+γ δ|θˆ₀|²

(ξ), then the energy of the solution (u, h, θ),

E⁰(t) = 1 2

Z

R³

|u_t|²+µ|∇u|²+ (µ+λ)|divu|² +α

β|h|²+ γ δ|θ|²

(t, x) dx, decays polynomially with rate m: E⁰(t) =O(t^−m).

Now let (v, w, ϑ, r) the Fourier-transformed solution of the second sound system. To transfer the result above, we just have to estimate the terms|ϑ|²and|ϑ_t|² in (3.2) towards

|r|², |rt|², |w|²and|wt|²since the dissipation of the system with second sound is, according to (2.2),

d dt

Eˆ^τ(t) = d dt

1

2||V(t)||²_H =−RehAV, ViH=−α

β|ξ|²|w|²− 1 κ

|r|². (3.5) Notice that (3.2) also holds for the second sound system since only the first two differential equations of (1.1) or (1.6), respectively, are needed for the proof. Therefore, we get

d

dtΦ(t) ≤ c

δa²|ξ|²(1 +|ξ|²)|w|²+c

δa²(1 +|ξ|²)|w_t|²+ca|ξ|²|ϑ|²+cδa|ϑ_t|²

−βH

4 a(|v¹_tt|²+|v_tt²|²) + δ 32

τ

κ|ξ|²|r|²− δ

16|ξ|²Eˆ^τ(t). (3.6) Using the last differential equation of (3.1), we can estimate |ϑ|² directly:

|ϑ|² ≤ c

|ξ|²(|r_t|²+|r|²). (3.7)

Lemma 3.2 For any 0< <1 the functional Φ7(t) := aRe

i ξ₃r_t³ϑt

(t) fullfills the estimate

d

dtΦ7 ≤ −κ

2τa|ϑt|²+ c a³

1 + 1

|ξ|²

|rt|² +βH

8 a(|v¹_tt|²+|v_tt²|²) +

2 µ|ξ|⁴|v|²+|ξ|²|ϑ|² .

Proof. Multiplication of the fourth resp. third equation of (3.1) with ϑ_t resp. r³_t yields d

dtΦ₇ = aRe i

ξ3

r_tt³ϑ_t+r³_tϑ_tt

= −1 τ

1

ξ₃aRe ir_t³ϑt

− κ

τaRe ϑtϑt

+1

ξ₃aRe r³_t(ξr_t) +γ 1

ξ₃aRe r_t³(ξv_tt)

≤ 1

2a|ϑ_t|² + c 2₁a|ξ|²

ξ₃²

|rt|²

|ξ|² − κ

τa|ϑ_t|²+ca|ξ|

|ξ₃||r_t|² +₂

2a(|v¹_tt|²+|v²_tt|²) + c 2₂a|ξ|²

ξ²₃ |r_t|² +₃

2|v_tt³|²+ c

2₃a²|ξ|² ξ₃² |r_t|²

≤ −κ

2τa|ϑ_t|²+c a³

1 + 1

|ξ|²

|r_t|²+βH

8 (|v_tt¹|²+|v²_tt|²) +

2 µ|ξ|⁴|v|²+|ξ|²|ϑ|² with ₁ = κ

τ, ₂ = βH

4 and some ∈(0,1) which will be determined in the next proof.

Theorem 3.3 We define the functional

L^τ(t) := Φ(t) + Φ7(t) +N1a²(1 +|ξ|²) ˆE₁^τ(t) +N2a³

1 + 1

|ξ|²

Eˆ₂^τ(t). (3.8) If N₁, N₂ >0 are large enough, then L^τ is equivalent to Eˆ^τ and decays exponentially, i.e.

for any ξ ∈ R³ there are constants C₁(ξ), C₂(ξ), C₃(ξ) > 0 such that for any t ≥ 0 the following holds:

C₁(ξ) ˆE^τ(t, ξ)≤ L^τ(t, ξ)≤C₂(ξ) ˆE^τ(t, ξ), L(t, ξ)≤ L(0, ξ)e^−C3(ξ)t. Proof. Putting (3.6), (3.7) and Lemma 3.2 together, we get

d

dtL^τ(t) ≤

2− δ 16

|ξ|²Eˆ^τ(t) +

cδ− κ 2τ

|ϑ_t|²+βHa 1

8 − 1 4

(|v¹_tt|²+|v_tt²|²) +

c

δ −N₁α β

a²|ξ|²(1 +|ξ|²)|w|² + c

δ −N₂α β

a³(1 +|ξ|²)|w_t|² +

c− N₁ κ

a²(1 +|ξ|²)|r|² + c

δ − N₂ κ

a³

1 + 1

|ξ|²

|r_t|².

Choosing , δ small enough and N₁, N₂ large enough, this implies d

dtL^τ(t) ≤ − δ

32|ξ|²Eˆ^τ(t). (3.9)

Using the differential equations (3.1), we estimate

|Φ| ≤ C

a(1 +|ξ|²) ˆE₁^τ+a²1 +|ξ|²

|ξ|² Eˆ₂^τ

; (3.10)

Eˆ₂^τ ≤ C(1 +|ξ|²)²Eˆ₁ (3.11)

which yields

L^τ ≤ |Φ|+N₁a²(1 +|ξ|²) ˆE₁^τ +N₂a³

1 + 1

|ξ|²

Eˆ₂^τ

≤

(C+N₁)a²(1 +|ξ|²) + (C+N₂)a³

1 + 1

|ξ|²

C(1 +|ξ|²)²

Eˆ₁^τ

≤ d₁a³

|ξ|⁴+ 1

|ξ|²

Eˆ₁^τ,

L^τ ≥ −|Φ|+N₁a²(1 +|ξ|²) ˆE₁^τ +N₂a³

1 + 1

|ξ|²

Eˆ₂^τ

≥ (N₁−C)a²(1 +|ξ|²) ˆE₁^τ + (N₂−C)a³

1 + 1

|ξ|²

Eˆ₂^τ

≥ d₂a²(1 +|ξ|²) ˆE₁^τ

with d₁, d₂ >0 if N₁, N₂ are large enough. We choose

C₁(ξ) := d₂a²(1 +|ξ|²), (3.12) C₂(ξ) := d₁a³

|ξ|⁴+ 1

|ξ|²

(3.13) and it remains to show the existence of C₃(ξ). (3.9) implies

d

dtL^τ(t) ≤ − δ 32

|ξ|⁴

d₁a³(1 +|ξ|⁶)L^τ(t) =: −C₃(ξ)L^τ(t) (3.14) and, usings the Lemma of Gronwall, we get the exponential decay of L^τ:

L^τ(t) ≤ L^τ(0)e^−C3(ξ)t.

To deduce the polynomial decay of E^τ(t)from the exponential decay ofEˆ^τ(t, ξ)pointwise in ξ, we need the following lemma:

Lemma 3.4 For all m ∈ N0 there exists some c(m) > 0 such that, for any t ≥ 0, the following holds:

Z t 0

s^me^−C3(ξ)s ds ≤ c(m) C₃^m+1(ξ)

Proof. Via induction. Let m = 0, then

t

Z

0

e^−C3(ξ)s ds = 1

C3(ξ)(1−e^−C3(ξ)t) ≤

1 C3(ξ)

1

.

Now assume that the formula holds for m, then we receive by partial integration:

t

Z

0

s^m+1e^−C3(ξ)s ds = − 1

C₃(ξ) t^m+1e^−C3(ξ)t

+ m+ 1 C₃(ξ)

t

Z

0

s^me^−C3(ξ)s ds

≤ c(m)(m+ 1) 1

C₃(ξ) m+2

.

Theorem 3.5 (decay rates for magneto-thermo-elasticity with second sound) For any m ∈N0, let the initial energy

Eˆ^τ(0, ξ) := 1 2

|v₁|²+µ|ξ|²|v₀|²+ (µ+λ)(ξ·v₀)²+ α

β|w₀|²+|ϑ₀|² + τ κ

|r₀|²

(t, ξ) fullfill

Z

R³

C₂(ξ)

C₁(ξ)C₃^m(ξ)Eˆ₁^τ(0) dξ <∞. (3.15) Then the energy associated to (u, h, θ, q) decays polynomially with order m: E^τ(t) = O(t^−m).

Proof. Let m≥1. Using the dissipation (3.5) and Lemma 3.3 & 3.4, we get t^mE^τ(t) =

Z

R³ t

Z

0

d

dss^mEˆ₁^τ(s, ξ)ds dξ

≤ m

Z

R³ t

Z

0

s^m−1Eˆ₁^τ(s, ξ)ds dξ

≤ m

Z

R³

C₂(ξ)

C₁(ξ)Eˆ₁^τ(0, ξ)

t

Z

0

s^m−1e^−C3(ξ)s ds dξ

≤ m·c(m−1) Z

R³

C₂(ξ)

C₁(ξ)C₃(ξ)^mEˆ₁^τ(0, ξ) dξ

< ∞.

4. The behaviour of the energy for vanishing delay parameter τ → 0. Let (u⁰, h⁰, θ⁰, q⁰) resp. (u^τ0, h^τ0, θ^τ0, q^τ0) the solution to (1.6) for τ = 0 resp. τ = τ₀. Our objective is to estimate the difference of the corresponding energies against a function depending on τ₀.

Let(u^d, h^d, θ^d, q^d) := (u^τ0−u⁰, h^τ0−h⁰, θ^τ0−θ⁰, q^τ0−q⁰)the difference of the solutions. If the compability condition q₀⁰ =−κ∇θ₀⁰ holds for the classical system, then(u^d, h^d, θ^d, q^d) solves











u^d_tt−µ∆u^d−(µ+λ)∇∇^Tu^d + γ∇θ^d − α(∇ ×h^d)×H₀ = 0 h^d_t −∆h^d − β∇ ×(u^d_t ×H₀) = 0 θ_t^d+∇^Tq^d + γ∇^Tu^d_t = 0

τ q^d_t +q^d+κ∇θ^d = −τ₀q⁰_t

(4.1)

to the initial condition

(u^d, u^d_t, h^d, θ^d, q^d)(0) = (0,0,0,0,0). (4.2) The energy term corresponding to the Fourier transform (v^d, w^d, ϑ^d, r^d) of (u^d, h^d, θ^d, q^d) fullfills

d dt

Eˆ^d(t, ξ) = −α

β|ξ|²|w^d|²− 1

κ|r^d|²−τ₀Re(iϑ⁰_t(ξ·r^d)) ≤ τ₀²κ

2|ξ|²|ϑ⁰_t|². (4.3) The dissipation of the classical system yields for the energy term of second order:

d dt

Eˆ₂⁰(t) = −α

β|ξ|²|w⁰_t|²−κ|ξ|²|ϑ⁰_t|². (4.4) By integration over [0,∞) and overR³ we get

E₂⁰(0) ≥ E₂⁰(0)− lim

t→∞E₂⁰(t) = α β

∞

Z

0

||∇h⁰_t||²_L2 ds+κ

∞

Z

0

||∇θ⁰_t||²_L2 (4.5)

and with (4.3) and the initial condition (4.2) it follows, again by intergration over[0,∞), that

E^d(t) = Z

R³

∞

Z

0

d ds

Eˆ^d(s, ξ) ds dξ≤τ₀²κ 2

∞

Z

0

||∇θ_t⁰||²_L2 dt≤ τ₀²

2E₂⁰(0). (4.6)

Theorem 4.1 (comparison of energies)Letτ₀ →0, then the energyE^τ0 corresponding to the system with second sound converges quadratically towards the energyE⁰ of the classical system.

Sometimes it is useful to get sharper results for the short-time behaviour of energies: In physical applications like the laser cleaning [14] or the pulsed laser heating [25], very small time periodes are considered. In the following we apply methods introduced in [10]:

Lemma 4.2 (Irmscher, 06) The second order energy of a dissipative system satisfies

E₂⁰(0)− E₂⁰(t)≤E₂⁰(t) :=





 2tp

E₂⁰(0)E₃⁰(0)−t²E₃⁰(t) if 0≤t≤

sE₂⁰(0) E₃⁰(0); E₂⁰(0) else

Inequality (4.6) then reads as E^d(t) ≤ τ₀²κ

2

t

Z

0

Z

R³

|ξ|²|ϑ⁰_t|² dξ ds ≤ −τ₀² 2

Z

R³ t

Z

0

d ds

Eˆ₂⁰(s, ξ)ds dξ

= τ₀²

2 (E₂⁰(0)− E₂⁰(t)) ≤ τ₀²

2E₂⁰(t). (4.7)

5. A generalized model for the stationary magnetic field H₀. At the beginning of section 3, we mentioned that technical difficulties arise if we do not assume H1, H2 = 0 for the stationary field H₀ = (H₁, H₂, H₃)– this assumption is made in most works about magneto-elastic and magneto-thermo-elastic models. Notice that the following results can be applied for classical magneto-thermo-elasticity (taking τ = 0) and magneto-elasticity (taking γ = 0 and θ≡0), too.

Initially, we take H1, H2 and H3 ≥ 0 arbitrary. Then the first two Fourier-transformed equations of (1.6) in components have the form

v_tt¹ +µ|ξ|²v¹+ (µ+λ)(ξ·v)ξ₁−iγϑξ₁

−iαH₃(ξ₁w³ −ξ₃w¹) +iαH₂(ξ₂w¹−ξ₁w²) = 0; (5.1) v_tt² +µ|ξ|²v²+ (µ+λ)(ξ·v)ξ₂−iγϑξ₂

−iαH₁(ξ2w¹ −ξ1w²) +iαH3(ξ3w²−ξ2w³) = 0; (5.2) v_tt³ +µ|ξ|²v³+ (µ+λ)(ξ·v)ξ3−iγϑξ3

−iαH₂(ξ₃w² −ξ₂w³) +iαH₁(ξ₁w³−ξ₃w¹) = 0; (5.3) w_t¹+|ξ|²w¹−iβH₁(ξ₂v²_t +ξ₃v_t³) +iβH₂ξ₂v_t¹+iβH₃ξ₃v_t¹ = 0 (5.4) w_t²+|ξ|²w²+iβH₁ξ₁v_t²−iβH₂(ξ₁v¹_t +ξ₃v_t³) +iβH₃ξ₃v_t² = 0 (5.5) w_t²+|ξ|²w²+iβH₁ξ₁v_t³+iβH₂ξ₂v³_t −iβH₃(ξ₁v_t¹+ξ₂v²_t) = 0 (5.6) In the situation of section 3, i.e. if H₁ = 0 and H₂ = 0, the derivative of Φ₁(t) can be estimated towards

d

dtΦ₁(t) = Re i

ξ₃

v¹_ttw¹+v_t¹w_t¹+v_tt²w²+v_t²w_t²

≤ −βH

2 (|v_t¹|²+|v²_t|²) + c ε

|ξ|²

ξ₃² 1 +|ξ|²

|w|²+ε(|ξ|²|v|²+|ϑ|²) (5.7) by multiplication of (5.1), (5.2) with w¹, w² and (5.4),(5.5) with v_t¹, v_t² where ε > 0 can be chosen as small as desired, compare [16]. Therefore, we can deal with the following mixed terms:

Re v_t¹v_t³

+Re v_t²v³_t

≤ 1

2ε(|v_t¹|²+|v²_t|²) +ε|v³_t|² (5.8) arising in the derivatives of Φ₂(t) and Φ₃(t).

On the other hand, if we choose H₂ >0, the derivative of the counterpartΦ˜₁(t) toΦ₁(t), Φ˜₁(t) := − 1

H₂ξ₂+H₃ξ₃

| {z }

=:ζ

Re

iv¹_tw¹

, (5.9)

just creates the negative term −|v_t¹|²: d

dt

Φ˜₁(t) ≤ −β

2|v_t¹|²+ c

ε(1 +|ξ|⁴)(1 +|ζ|²)|w|²+ε(|ξ|²|v|²+|ϑ|²) (5.10) which does not suffice to estimate Re

v²_tv_t³ . Theorem 5.1 Define the functionals

Φ˜₁(t) := −Re

iζv¹_tw¹

(t); Φ˜₄(t) := Re H₂ H₃

v_tt¹v_t² ξ₁ξ₃ +H₃

H₂ v_tt¹v_t²

ξ₁ξ₂

! (t);

Φ˜₂(t) := −Re

iζv¹_ttw_t¹

(t); Φ˜₅(t) := Re(v_tv) (t);

Φ˜₃(t) := −Re iv_t²w²

H₃ξ₃ +iv³_tw³ H₂ξ₂

!

(t); Φ˜₆(t) := τ

κRe i ξ₁r¹_tϑ_t

! (t)

and, for ˜c sufficiently large, ξ1 6= 0, ξ2ξ3 >0 and a :=|ξ|² 1 ξ₁²+ 1

ξ₃²

! ,

Φ(t) :=˜ 2 β

7 4

64

β²˜c²a+7 4

64

(µ+λ)²˜c²|ξ|² ξ₁² a+ 8

Φ˜₁(t) + 8 βΦ˜₃(t) +2

β

8(µ²+ (µ+λ)²) µ

64 (µ+λ)²

H₂ H₃ + H₃

H₂ 2

a

ξ₁²Φ˜₂(t) + 8

µ+λΦ˜₄(t) +4 ˜Φ5(t) + 7

4 128 (µ+λ)²

H₂ H₃ +H₃

H₂ 2

γ² a ξ₁² + 1

! Φ˜6(t).

Then there is some c > 0 such that the following estimate holds:

d dt

Φ(t)˜ ≤ ca⁴|ξ|²+|ξ|⁴+|ξ|⁶

ξ₁⁴ |w|²+ca⁴1 +|ξ|²+|ξ|⁴ ξ⁴₁ |w_t|² +c1 +|ξ|²

|ξ|² |r|²+ca²|ξ|²+|ξ|⁴ +|ξ|⁶+|ξ|⁸

ξ₁⁶ |r_t|²−Eˆ(t).

Proof. Analogously to (5.10), multiplying the derivatives of (5.1), (5.4) withw¹_t, v¹_tt, we get

d dt

Φ˜₂ ≤ c|ζ||ξ|²|v_t||w_t|+c|ζ||ξ||ϑ_t||w_t| +c|ζ||ξ||w_t|²+cζ²|ξ|⁴|w_t|²− β

2|v_tt¹|². (5.11)

Multiplication of (5.2), (5.3) with w², w³ and (5.5), (5.6) withv_t², v_t³ yields d

dt

Φ˜₃(t) = −Re i

H₃ξ₃v_tt²w²+ i

H₃ξ₃v_t²w²_t + i

H₂ξ₂v_tt³w³ + i

H₂ξ₂v³_tw³_t

≤ c|ξ|²|w|

|v²|

|ξ₃| + |v³|

|ξ₂|

+c |ξ₂|

|ξ₃|+ |ξ₃|

|ξ₂|

|ξ||w||v|+c |ξ₂|

|ξ₃| +|ξ₃|

|ξ₂|

|w||ϑ|

+c |ξ₂|

|ξ₃| +|ξ₃|

|ξ₂|

|w|²+c|ξ|²|w|

|v_t²|

|ξ₃| +|v³_t|

|ξ₂|

+c|ξ||v_t¹| |v²_t|

|ξ₃| + |v_t³|

|ξ₂|

−α(|w²|²+|w³|²)−β(|v_t²|²+|v_t³|²) +β H₂

H₃ +H₃ H₂

Re

v_t²v³_t

. (5.12) To annulate the mixed term Re(v_t²v_t³), we multiply the derivative of (5.1) with v_t² resp.

v³_t and estimate _dt^dΦ˜4(t) as follows:

d dt

Φ˜₄(t) = Re H₂ H₃

v¹_tttv²_t ξ₁ξ₃ + H₂

H₃ v_tt¹v²_tt

ξ₁ξ₃ + H₃ H₂

v¹_tttv³_t ξ₁ξ₂ + H₃

H₂ v_tt¹v³_tt

ξ₁ξ₂

!

≤ c |ξ|²

|ξ₁| +|ξ|

|v_t¹| |v_t²|

|ξ₃| +|v³_t|

|ξ₂|

−(µ+λ) H₂

H₃ ξ₂

ξ₃|v_t²|²+ H₃ H₂

ξ₃ ξ₂|v_t³|²

−(µ+λ) H₂

H₃ +H₃ H₂

Re

v²_tv_t³

+γ

H₂ H₃ + H₃

H₂

|ϑt| |v²_t|

|ξ₃| + |v_t²|

|ξ₂|

+c|ξ|

|ξ₁||w_t| |v_t²|

|ξ₃| +|v_t³|

|ξ₂|

+ H₂

H₃ + H₃ H₂

|v¹_tt|

|ξ₁| |v²_tt|

|ξ₃| + |v_tt³|

|ξ₂|

. (5.13) At this point, we need the assumption that ξ2 and ξ3 have the same sign as postulated in the assumptions of Theorem 5.1 to get rid of the second summand in (5.13). The two remaining functionals create the missing components of the negative energy in _dt^dΦ(t):˜ Multiplying (5.1), (5.2), (5.3) with v¹, v², v³ yields

d dt

Φ˜₅(t) = Re(v_ttv+v_tv_t)

≤ −µ|ξ|²|v|²−(µ+λ)(ξv)²+c|ϑ|² +µ

4|ξ|²|v|²+c|w|²+ µ

4|ξ|²|v|²+|vt|² (5.14) and multiplying the derivatives of the third and fourth equation of (3.1) withϑtresp. r_t¹:

d dt

Φ˜₆(t) = Re iτ₀

κξ1

r_tt¹ϑ_t+ iτ₀ κξ1

r¹_tϑ_tt

≤ −1

2|ϑ_t|²+ c

ξ₁²|r_t|²+c|ξ|

|ξ₁||r_t||v_t|+c|ξ|²

|ξ₁||r_t||vtt|

|ξ| . (5.15) Finally, using the differential equations (3.1) directly, we get

|vtt|² ≤ 8|ξ|² µ²|ξ|²|v|²+ (µ+λ)²(ξ·v)²+γ²|ϑ|²+ 2α²|H0|²|w|²

;

|ϑ|² ≤ c

|ξ|² |r|²+|r_t|²

(5.16)