ANISOTROPIC THERMO-ELASTICITY IN 2D
PART I: A UNIFIED TREATMENT
MICHAEL REISSIG AND JENS WIRTH
Abstract. In this note we develop tools and techniques for the treatment of anisotropic
thermo-elasticity in two space dimensions. We use a diagonalisation technique to obtain
properties of the characteristic roots of the full symbol of the system in order to prove Lp–Lq
decay rates for its solutions.
Keywords: thermo-elasticity, a-priori estimates, anisotropic media
1. The problem under consideration
Systems of thermo-elasticity are hyperbolic-parabolic or hyperbolic-hyperbolic coupled sys-
tems (type-1, type-2 or type-3 models) describing the elastic and thermal behaviour of elastic,
heat-conducting media. The classical type-1 model of thermo-elasticity is based on Fourier’s law,
which means, that the heat flux is proportional to the gradient of the temperature. The present
paper is devoted to the study of type-1 systems for homogeneous but anisotropic media in R2.
There are different results in the literature for certain anisotropic media (cubic in [3]; rhombic
in [4]). Our goal is to present an approach, which allows to consider (an)isotropic models in R2
from a unified point of view.
We consider the type-1 system of thermo-elasticity
Utt +A(D)U + γ∇θ = 0, (1.1a)
θt − κ∆θ + γ∇TUt = 0. (1.1b)
Here A(D) denotes the elastic operator, which is assumed to be a homogeneous second order
2×2 matrix of (pseudo) differential operators and models the elastic properties of the underlying
medium. Furthermore, κ describes the conduction of heat and γ the thermo-elastic coupling of
the system. We assume κ > 0 and γ 6= 0. We solve the Cauchy problem for system (1.1) with
initial data
U(0, ·) = U1, Ut(0, ·) = U2, θ(0, ·) = θ0, (1.2)
for simplicity we assume U1, U2 ∈ S(R2,R2) and θ0 ∈ S(R2). We denote by A(ξ) the symbol of
the elastic operator and we set η = ξ/|ξ| ∈ S1. Then some basic examples for our approach are
given as follows. The material constants are always specified in such a way that the matrix A(η)
becomes positive.
Example 1.1. Cubic media in 2D are modelled by
A(η) =
(τ − µ)η21 + µ (λ+ µ)η1η2
(λ+ µ)η1η2 (τ − µ)η22 + µ
(1.3)
with constants τ, µ > 0, −2µ−τ < λ < τ . This case was treated e.g. in [3]. For the corresponding
elastic system see [12].
Example 1.2. Rhombic media in 2D are modelled by
A(η) =
(τ1 − µ)η21 + µ (λ + µ)η1η2
(λ+ µ)η1η2 (τ2 − µ)η22 + µ
(1.4)
http://arxiv.org/abs/0704.0125v3
2 MICHAEL REISSIG AND JENS WIRTH
with constants τ1, τ2, µ > 0 and −2µ−
τ1τ2 < λ <
τ1τ2. For this case we refer also to [4].
Example 1.3. Although it is not the main point of this note, we can consider isotropic media,
where
A(η) = µI + (λ + µ)η ⊗ η
(λ+ µ)η21 + µ (λ+ µ)η1η2
(λ+ µ)η1η2 (λ+ µ)η
2 + µ
(1.5)
with Lamé constants µ > 0 and λ+ µ > 0.
We will present a unified treatment of these cases of (in general) anisotropic thermo-elasticity.
For this we assume that the homogeneous symbol A = A(ξ) = |ξ|2A(η), η = ξ/|ξ|, is given as a
function
A : S1 → C2×2 (1.6)
subject to the conditions
(A1): A is real-analytic in η ∈ S1,
(A2): A(η) is self-adjoint and positive for all η ∈ S1.
For some results we require that
(A3): A(η) has two distinct eigenvalues, # specA(η) = 2.
Under assumption (A3) the direction η ∈ S1 is called (elastically) non-degenerate. In this case we
know that the elasticity equation Utt+A(D)U = 0 is strictly hyperbolic and can be diagonalised
smoothly using a corresponding system of normalised eigenvectors rj(η) to the eigenvalues κj(η)
of A(η).
If (A3) is violated we will call the corresponding directions η ∈ S1 degenerate. For these
directions we can use the one-dimensionality of S1 in connection with the analytic perturbation
theory of self-adjoint matrices (cf. [6]). So we can always find locally smooth eigenvalues κj(η)
and corresponding locally smooth normalised eigenvectors rj(η) of A(η). For the following we
assume for simplicity that these functions extend to global smooth functions on S1.
This classification of directions is not sufficient for a precise study of Lp–Lq decay estimates
for solutions to the thermo-elastic system. It turns out that different microlocal directions
η = ξ/|ξ| ∈ S1 from the phase space have different influence on decay estimates. But how to
distinguish these directions and how to understand their influence? In general, this can be done
by a refined diagonalisation procedure applied to a corresponding first order system (first order
with respect to time). Applying a partial Fourier transform and chosing a suitable energy (of
minimal dimension) this system reads as DtV = B(ξ)V . The properties of the matrix B(ξ) are
essential for our understanding:
• the notions of hyperbolic and parabolic directions depend on the behaviour of the eigen-
values of B(ξ) (see (2.3), (2.5), Definition 1);
• the matrix B(ξ) contains spectral data of A(ξ) together with certain coupling functions
(see (2.6)) between different components of the energy. The behaviour of these coupling
functions close to hyperbolic directions has an essential influence on decay rates (see
Theorems 3.1, 3.5 and 3.6).
It turns out that we have to exclude some exceptional values of the coupling constant γ by
assuming that (see Definition 1 for the notion of hyperbolic directions)
(A4): γ2 6= 2κj0(η̄)− trA(η̄) for all hyperbolic directions η̄ with respect to κj0 .
Basically this implies the non-degeneracy of the 1-homogeneous part of B(ξ). In the following
we will call a hyperbolic direction violating (A4) a γ-degenerate direction. Assumption (A4) is
used for the treatment of small hyperbolic frequencies and plays there a similar rôle like (A3)
for large frequencies.
ANISOTROPIC THERMO-ELASTICITY IN 2D 3
In Section 2 we will give the transformation of the thermo-elastic system (1.1) to a system
of first order and the diagonalisation procedure in detail. The proposed procedure generalises
those from [14], [15], [9], [17]. The obtained results are used to represent solutions of the original
system as Fourier integrals with complex phases. Based on these representations we give micro-
localised decay estimates for solutions in Section 3. They and their method of proof depend
• the classification of directions (to be hyperbolic or parabolic);
• the order of contact between Fresnel curves (coming from the elastic part) and their
tangents for hyperbolic directions;
• the vanishing order of the coupling functions in hyperbolic directions.
Let us formulate some of the results. The first one follows from Theorem 3.1 and Corollary 3.2.
Result. Under assumptions (A1) to (A4) and if the coupling functions vanish to first order in
hyperbolic directions the solutions U(t, x) and θ(t, x) to (1.1) satisfy the Lp–Lq estimate
‖DtU(t, ·)‖q + ‖
A(D)U(t, ·)‖q + ‖θ(t, ·)‖q
. (1 + t)−
) (‖U1‖p,r+1 + ‖U2‖p,r + ‖θ0‖p,r) (1.7)
for dual indices p ∈ (1, 2], pq = p+ q, and Sobolev regularity r > 2(1/p− 1/q).
If the coupling functions vanish to higher order we have to relate their vanishing order ℓ to
the order of contact γ̄ between the Fresnel curve and its tangent in the corresponding direction
and for the corresponding sheet. In Theorems 3.5 and 3.6 we show that in this case the 1/2 in
the exponent is changed to 1/min(2ℓ, γ̄).
Our main motivation to write this paper is to provide a unified way to treat anisotropic
models of thermo-elasiticity. New analytical tools presented in these notes generalise to higher
dimensions and allow to treat especially models in 3D (outside degenerate directions). The two-
dimensional results from [3] for cubic media and [4] for rhombic media are contained / extended;
general anisotropic media can be treated. This is discussed in some detail in the second part [16]
of this note.
Acknowledgements. The first author thanks Prof. Wang Ya-Guang (Shanghai Jiao Tong
University) for the discussion about some basic ideas of the approach presented in this paper
during his stay at the TU Bergakademie Freiberg in August 2004. The stay was supported by
the German-Chinese research project 446 CHV 113/170/0-2.
2. General treatment of the thermo-elastic system
We use a partial Fourier transform with respect to the spatial variables to reduce the Cauchy
problem for (1.1) to the system of ordinary differential equations
Ûtt + |ξ|2A(η)Û + iγξθ̂ = 0, (2.1a)
θ̂t + κ|ξ|2θ̂ + iγξ · Ût = 0, (2.1b)
Û(0, ·) = Û1, Ût(0, ·) = Û2, θ̂(0, ·) = θ̂0 (2.1c)
parameterised by the frequency variable ξ.
We denote by κ1,κ2 ∈ C∞(S1) the eigenvalues of A(η) and by r1, r2 ∈ C∞(S1, S1) corre-
sponding normalised eigenvectors. Both depend in a real-analytic way on η ∈ S1. In a first step
we reduce (2.1) to a first order system. For this we use the diagonaliser of the elastic operator,
i.e. the matrix M(η) = (r1(η)|r2(η)) build up from the normalised eigenvectors, and define
U (0)(t, ξ) =MT (η)Û (t, ξ). (2.2)
4 MICHAEL REISSIG AND JENS WIRTH
Then we define by the aid of
D1/2(η) = diag(ω1(η), ω2(η)), ωj(η) =
κj(η) ∈ C∞(S1), (2.3)
the vector-valued function
V (0)(t, ξ) =
(Dt +D1/2(ξ))U (0)(t, ξ)
(Dt −D1/2(ξ))U (0)(t, ξ)
θ̂(t, ξ)
 , (2.4)
where as usual Dt = −i∂t. It satisfies a first order system with apparently simple structure. A
short calculation yields DtV
(0)(t, ξ) = B(ξ)V (0)(t, ξ) with
B(ξ) =
ω1(ξ) iγa1(ξ)
ω2(ξ) iγa2(ξ)
−ω1(ξ) iγa1(ξ)
−ω2(ξ) iγa2(ξ)
a1(ξ) − iγ2 a2(ξ) −
a1(ξ) − iγ2 a2(ξ) iκ|ξ|
, (2.5)
where we used the coupling functions
aj(ξ) = rj(η) · ξ. (2.6)
For later use we introduce the notation B1(ξ) and B2(ξ) for the homogeneous components of
B(ξ) of order 1 and 2, respectively. The coupling functions aj(η) can be understood as the
co-ordinates of η with respect to the orthonormal eigenvector basis {r1(η), r2(η)}. Therefore, it
holds a21(η) + a
2(η) = 1. Furthermore, they are well-defined and real-analytic functions on S
In the following proposition we collect some information on the characteristic polynomial of
the matrix B(ξ).
Proposition 2.1. (1) trB(ξ) = iκ|ξ|2 and detB(ξ) = iκ|ξ|6 detA(η).
(2) The characteristic polynomial of B(ξ) is given by
det(νI −B(ξ)) =(ν − iκ|ξ|2)(ν2 − κ1(ξ))(ν2 − κ2(ξ))
− νγ2|ξ|2a21(η)(ν2 − κ2(ξ))− νγ2|ξ|2a22(η)(ν2 − κ1(ξ)). (2.7)
(3) An eigenvalue ν ∈ specB(ξ), ξ 6= 0, is real if and only if ν2 = κj0 (ξ) for an index
j0 = 1, 2. If the direction is non-degenerate this is equivalent to aj0(η) = 0.
(4) If aj(η) 6= 0, j = 1, 2 the eigenvalues ν ∈ specB(ξ) satisfy
iκ|ξ|2
a21(ξ)
ν2 − κ1(ξ)
a22(ξ)
ν2 − κ2(ξ)
. (2.8)
It turns out that the property of B(ξ) to have real eigenvalues depends only on the direction
η = ξ/|ξ| ∈ S1. We will introduce a notation.
Definition 1. We call a direction η ∈ S1 hyperbolic if B(ξ) has a real eigenvalue and parabolic
if all eigenvalues of B(ξ) have non-zero imaginary part.
In hyperbolic directions we always have a pair of real eigenvalues. If η ∈ S1 is hyperbolic
with ±ωj0(ξ) ∈ specB(ξ) for ξ = |ξ|η, we call η hyperbolic with respect to the index j0 (or
with respect to the eigenvalue κj0(η) of A(η)) and ν±(ξ) = ±ωj0(ξ) the corresponding pair of
hyperbolic eigenvalues of B(ξ).
A non-degenerate direction is parabolic if and only if aj(η) 6= 0, j = 1, 2, while for non-
degenerate hyperbolic directions one of the coupling functions aj0(η) vanishes. Degenerate di-
rections are always hyperbolic (in 2D), see (2.7).
ANISOTROPIC THERMO-ELASTICITY IN 2D 5
Example 2.1. If the medium is isotropic, A(η) = µI+(λ+µ)η⊗η, the eigenvalues of A are µ and
λ+µ with corresponding eigenvectors η and η⊥. Thus all directions are hyperbolic (with respect
to the second eigenvalue). In this case the matrix B(ξ) decomposes into a diagonal hyperbolic
2 × 2-block and a parabolic 3 × 3-block. This decomposition coincides with the Helmholtz
decomposition as used in the standard treatment of isotropic thermo-elasticity.
Example 2.2. For cubic media (where we assume in addition µ 6= τ and µ + λ 6= 0) there exist
eight hyperbolic directions determined by η1η2 = 0 or η
1 = η
2 . The functions aj(η) have simple
zeros at these directions.
Example 2.3. Weakly coupled cubic media with λ+µ = 0, µ 6= τ , have the degenerate directions
η21 = η
2 , media with µ = τ , λ + µ 6= 0, for η1η2 = 0. In both cases the coupling functions aj(η)
do not vanish in these directions.
If µ = τ = −λ, the elastic system decouples directly into two wave equations with propagation
speed
µ. In this case all directions are degenerate.
Example 2.4. For rhombic media we have to distinguish between three cases.
Case 1. If the material constants satisfy (λ+2µ− τ1)(λ+2µ− τ2) > 0, we are close to the cubic
case and there exist eight hyperbolic directions given by η1η2 = 0 and
η21(λ+ 2µ− τ1) = η22(λ+ 2µ− τ2).
Case 2. If we assume on the contrary that (λ+2µ−τ1)(λ+2µ−τ2) < 0, only the four hyperbolic
directions η1η2 = 0 exist. In the Cases 1 and 2 in each hyperbolic direction one of the coupling
functions aj(η) vanishes to first order.
Case 3. In the borderline case τ1 = λ+2µ or τ2 = λ+2µ, but τ1 6= τ2, three hyperbolic directions
collapse to one. We have the four hyperbolic directions η1η2 = 0, at two of them (ηj = ±1 if
τj = λ+ 2µ) the vanishing order of the coupling function is three.
Rhombic media are degenerate if a) µ = τ1 (or µ = τ2) with degenerate direction (0, 1)
T (or
(1, 0)T ) or b) λ+µ = 0 (weakly coupled case) and (µ−τ1)(µ−τ2) > 0 with degenerate directions
determined by η21(µ− τ1) = η22(µ− τ2) or c) τi = µ = −λ (exceptional case) where all directions
are degenerate.
Proposition 2.2. Let the direction η̄ ∈ S1 be non-degenerate and hyperbolic with respect to the
index j0. Then the corresponding eigenvalues ν±(ξ) satisfy
a2j0(ξ)
ν2±(ξ) − κj0(ξ)
= qη̄(|ξ|) = Cη̄ ∓ iDη̄|ξ| (2.9)
for all non-tangential limits with real constants Cη̄, Dη̄ ∈ R, Dη̄ > 0. Furthermore, the imaginary
part of the hyperbolic eigenvalue satisfies
Im ν±(ξ)
a2j0(η)
D2η̄|ξ|2
C2η̄ + |ξ|2D2η̄
> 0, (2.10)
and thus vanishes like Im ν(|ξ|η) ∼ a2j0(η) as η → η̄ for all ξ 6= 0.
6 MICHAEL REISSIG AND JENS WIRTH
Proof. Let for simplicity j0 = 1. We use the characteristic polynomial of B(ξ) to deduce
γ2|ξ|2a21(η)
ν2± − |ξ|2κ1(η)
iκ|ξ|2
γ2|ξ|2a22(η)
ν2± − |ξ|2κ2(η)
→ 1∓ iκ|ξ|
ω1(η̄)
κ1(η̄)− κ2(η̄)
(2.11)
= 1− γ
κ1(η̄)− κ2(η̄)
︸ ︷︷ ︸
γ2Cη̄
ω1(η̄)
︸ ︷︷ ︸
γ2Dη̄
|ξ| = γ2qη̄(|ξ|). (2.12)
The existence of the limit is implied by ν± 6= 0 and ν2± 6= κ2(ξ) as consequence of ν±(ξ) →
±|ξ|ω1(η) as η → η̄ by Proposition 2.1.
Obviously, Im qη̄(|ξ|) = ∓κ|ξ|/γ2ω1(η̄) = ∓Dη̄|ξ| is non-zero for ξ 6= 0 and considering the
imaginary part of the first limit expression
Im qη̄(|ξ|) = lim
a21(ξ)
ν2±(ξ)− κ1(ξ)
= lim
−2Re ν±(ξ) Im ν±(ξ) a21(ξ)
|ν2±(ξ)− κ1(ξ)|2
= ∓2ω1(|ξ|η̄)|qη̄(|ξ|)|2 lim
Im ν±(|ξ|η)
|ξ|2a21(η)
proves the second statement, limη→η̄
Im ν±(ξ)
a21(η)
Dη̄ |ξ|
2ω1(η̄)(C
η̄+|ξ|
2D2η̄)
In the case of isolated degenerate directions (others are not of interest, because then the
system is decoupled) we can find a replacement for Proposition 2.1.
Proposition 2.3. Let η̄ ∈ S1 be an isolated degenerate direction, κ1(η̄) = κ2(η̄). Then the
corresponding hyperbolic eigenvalues ν±(ξ) satisfy
ω1(ξ)− ν2±(ξ)
ω1(ξ)− ω2(ξ)
= a21(η̄) > 0, (2.13)
and, therefore,
Im ν±(ξ)
ω1(ξ)− ω2(ξ)
= 0, lim
ω1(ξ)− Re ν±(ξ)
ω1(ξ)− ω2(ξ)
= a21(η̄). (2.14)
Thus, if a1(η̄) 6= 0 then the eigenvalues ν±(ξ) approach ±ω1(ξ) at the contact order between
ω1(ξ) and ω2(ξ) (while they approach ±ω1(ξ) with a higher order if a1(ξ) = 0).
2.1. Asymptotic expansion of the eigenvalues as |ξ| → 0. If |ξ| is small the first order part
B1(ξ) dominates B2(ξ), so the properties of the eigenvalues are governed by spectral properties
of B1(ξ).
Proposition 2.4. (1) trB1(ξ) = 0 and detB1(ξ) = 0.
(2) If the direction η ∈ S1 is parabolic the nonzero eigenvalues ν̃ of B1(η) satisfy
γ−2 =
a21(η)
ν̃2 − κ1(η)
a22(η)
ν̃2 − κ2(η)
(2.15)
and are thus real and related to κj(η) by
0 < κ1(η) < ν̃
1(η) < κ2(η) < ν̃
2(η) (2.16)
(if κ1(η) < κ2(η)).
ANISOTROPIC THERMO-ELASTICITY IN 2D 7
(3) If η is non-degenerate and hyperbolic with respect to the index 1 we have κ1(η) = ν̃
1(η),
while for hyperbolic directions with respect to the index 2 three cases occur depending on
the size of the coupling constant γ:
γ2 < κ2(η̄)− κ1(η̄) : κ2(η) = ν̃22 (η),
γ2 = κ2(η̄)− κ1(η̄) : ν̃21(η) = κ2(η) = ν̃22 (η),
γ2 > κ2(η̄)− κ1(η̄) : ν̃21(η) = κ2(η).
(2.17)
(4) If the direction is degenerate, κ1(η) = κ2(η), we have the eigenvalues ±
κ1(η) and
κ1(η) + γ2.
The existence of five distinct eigenvalues of the homogeneous principal part B1(η) for all
parabolic and most hyperbolic directions allows us to calculate the full asymptotic expansion of
the eigenvalues ν(ξ) of B(ξ) as |ξ| → 0. We will give only the first terms in detail, but provide
the whole diagonalisation procedure. Assumption (A4) guarantees the non-degeneracy of B1(ξ).
Note that, even if (A4) is violated the matrix B1(ξ) is diagonalisable (as consequence of its block
structure).
Proposition 2.5. As |ξ| → 0 the eigenvalues of the matrix B(ξ) behave as
ν0(ξ) = iκ|ξ|2b0(η) +O(|ξ|3), (2.18a)
ν±j (ξ) = ±|ξ|ν̃j(η) + iκ|ξ|
2bj(η) +O(|ξ|3) (2.18b)
for all non-γ-degenerate directions, where the functions bj ∈ C∞(S1) are given by
b0(η) =
γ2a21(η)
κ1(η)
γ2a22(η)
κ2(η)
> 0 (2.19)
bj(η) =
1 + γ2a21(η)
ν̃2j + κ1(η)
(ν̃2j − κ1(η))2
+ γ2a22(η)
ν̃2j + κ2(η)
(ν̃2j − κ2(η))2
≥ 0. (2.20)
Furthermore, bj(η) > 0 if η is parabolic and bj0(η) = 0 if η is hyperbolic with respect to the index
Proof. We apply a diagonalisation scheme in order to extract the spectral information for B(ξ).
We assume that the eigenvalues are denoted such that κ1(η) ≤ κ2(η).
Step 1. By Proposition 2.4 we know that the homogeneous first order part B1(η) has the
distinct eigenvalues ν̃0 = 0 and ν̃
j (η) = ±ν̃j(η), which are ordered as κ1(η) ≤ ν̃1(η) ≤ κ2(η) ≤
ν̃2(η) (where equality holds only under the exceptions stated in Proposition 2.4). We denote
corresponding normalised and bi-orthogonal left and right eigenvectors of the matrix B1(η) by
±(η) and e±j (η). If we collect them in the matrices
L(η) = (0e(η)|1e+(η)|1e−(η)|2e+(η)|2e−(η)), (2.21a)
R(η) = (e0(η)|e+1 (η)|e
1 (η)|e
2 (η)|e
2 (η)), (2.21b)
we have L∗(η)R(η) = I and
L∗(η)B1(η)R(η)) = D1(η) = diag(0, ν̃1(η),−ν̃1(η), ν̃2(η),−ν̃2(η)). (2.22)
Further we get
L∗(η)B2(η)R(η) = iκb∗(η)⊗ ∗b(η), (2.23)
where b∗(η) and ∗b(η) are vectors collecting the last entries b0(η), b
j (η) and 0b(η), jb
±(η) of the
eigenvectors e0(η), e
j (η) and 0e(η), je
±(η), respectively.
8 MICHAEL REISSIG AND JENS WIRTH
The matrix
B(0)(ξ) = L∗(η)B(ξ)R(η) = |ξ|D1(η) + |ξ|2iκb∗(η)⊗ ∗b(η) (2.24)
is diagonalised modulo O(|ξ|2) as |ξ| → 0 and has a main part with distinct entries. We denote
R(2)(ξ) = |ξ|2iκb∗ ⊗ ∗b.
Step 2. We construct a diagonaliser of B(0)(ξ) as |ξ| → 0 of the form
Nk(ξ) = I +
|ξ|jN (j)(η). (2.25)
For this we denote the k-homogeneous part of R(k)(ξ) by R̃(k)(ξ) and its entries by R̃
ij (η).
Then we set for k = 1, 2, . . .
Dk+1(η) = diag R̃(k+1)(η), (2.26)
N (k)(η) =
(k+1)
12 (η)
d1(η)−d2(η)
· · · R̃
(k+1)
15 (η)
d1(η)−d5(η)
(k+1)
21 (η)
d2(η)−d1(η)
0 · · · R̃
(k+1)
25 (η)
d2(η)−d5(η)
. . .
(k+1)
51 (η)
d5(η)−d1(η)
(k+1)
52 (η)
d5(η)−d2(η)
· · · 0
, (2.27)
where dj(η) are the entries of D1(η). By construction we have the commutator relation
[D1(η), N (k)(η)] = Dk+1(η)− R̃(k+1)(η), (2.28)
such that
R(k+2)(ξ) = B(0)(ξ)Nk(ξ)−Nk(ξ)
|ξ|jDj(η)
= R(k+1)(ξ)
+ |ξ|kB(0)(ξ)N (k)(η)− |ξ|kN (k)(η)
|ξ|jDj(η) −Nk(ξ)|ξ|k+1Dk+1(η)
= R(2)(ξ)|ξ|kN (k)(η)− |ξ|kN (k)(η)
|ξ|jDj(η)− (Nk(ξ)− I)|ξ|k+1Dk+1(η)
= O(|ξ|k+2).
Using Nk(ξ) − I = O(|ξ|) we see that for |ξ| ≤ ck, ck sufficiently small, the matrix Nk(ξ) is
invertible and
N−1k (ξ)B
(0)(ξ)Nk(ξ) =
|ξ|jDj(η) +O(|ξ|k+2). (2.29)
Thus, the entries of Dj(η) contain the asymptotic expansion of the eigenvalues, while the rows
of Nk(ξ)R(η) (and L(η)N
k (ξ)) give asymptotic expansions of the right (and left) eigenvectors
of B(ξ).
Furthermore, the construction implies that all occurring matrices are smooth functions of
η ∈ S1.
Step 3. We calculate the first terms explicitly. For this we need the diagonal entries of the
matrix b∗ ⊗ ∗b. Therefore, we determine the left and right eigenvectors of B1(η). If we assume
ANISOTROPIC THERMO-ELASTICITY IN 2D 9
that the direction η is non-degenerate we get for e0(η) = (r
1 , r
1 , r
2 , r
2 , r0)
T and 0e(η) =
(ℓ+1 , ℓ
1 , ℓ
2 , ℓ
2 , ℓ0)
T the equations
±r±j ωj + iγajr0 = 0, ±ℓ
j ωj −
ajℓ0 = 0, (2.30a)
1 + a2r
2 + a1r
1 + a2r
2 = 0, a1ℓ
1 + a2ℓ
2 + a1ℓ
1 + a2ℓ
2 = 0, (2.30b)
together with the normalisation condition
r+1 ℓ
1 + · · · r
2 + r0ℓ0 = 1. (2.31)
The first equations imply the representation
±r±j (η) = −
iγaj(η)
ωj(η)
r0(η), ±ℓ±j (η) =
iγaj(η)
2ωj(η)
ℓ0(η), (2.32)
the second line of equations follows from the first, while the normalisation condition yields
b0(η) = r0(η) ℓ0(η) =
γ2a21(η)
κ1(η)
γ2a22(η)
κ2(η)
6= 0. (2.33)
To calculate the eigenvectors we can further require r0(η) = ℓ0(η) =
b0(η) > 0.
Similarly, we obtain for the eigenvectors e+k (η) = (r
1 , . . . , r
2 , r0)
T and ke
+(η) = (ℓ+1 , . . . , ℓ
2 , ℓ0)
the equations (we use the same notation as above in the hope that this will not lead to confusion
here)
±r±j ωj + iγajr0 = ±ν̃kr
j , ±ℓ
j ωj −
ajℓ0 = ±ν̃k(η)ℓ±j (2.34)
together with the normalisation condition. Thus for parabolic directions we get
±r±j (η) =
iγaj(η)
ν̃k(η)− ωj(η)
r0(η), ±ℓ±j (η) = −
iγaj(η)
2(ν̃k(η)− ωj(η))
ℓ0(η), (2.35)
and hence
b+k (η) = r0(η) ℓ0(η) =
j=1,2
γ2a2j(η)
2(ν̃k(η)− ωj(η))2
j=1,2
γ2a2j(η)
2(ν̃k(η) + ωj(η))2
1 + γ2a21(η)
ν̃2k(η) + κ1(η)
(ν̃2k(η) − κ1(η))2
+ γ2a22(η)
ν̃2k(η) + κ2(η)
(ν̃2k(η)− κ2(η))2
. (2.36)
For e−k (η) and ke
−(η) we have to replace ν̃k(η) by −ν̃k(η) and obtain b−k (η) = b
k (η) = bk(η).
If the direction η is non-degenerate and hyperbolic with respect to the index j0, the entries r
and ℓ±j0 are undetermined by (2.34), while the other entries of the vectors are zero. Together with
the normalisation condition this determines the eigenvectors and gives bj0(η) = 0. It remains to
consider degenerate directions. Then we have ν̃1 = ω1 and ν̃2 > ω1 such that for k = 1 we have
ℓ0 = r0 = 0, r
1 and ℓ
1 are non-zero while r
2 = ℓ
2 = 0, especially b1(η) = 0. For k = 2 we get
from the above expression for b2(η) = (2 + 2κ
2/γ2)−1 > 0. �
Remark. 1. Note that for all non-degenerate hyperbolic directions η̄ ∈ S1 with respect to the
index 1 the limit
a21(η)b
1 (η) =
κ1(η̄)
κ1(η̄)− κ2(η̄)
(2.37)
is taken and non-zero, while for hyperbolic directions with respect to the index 2 the corre-
sponding limit is non-zero only if γ2 6= κ2(η̄) − κ1(η̄), i.e. if the direction is not γ-degenerate.
Near γ-degenerate directions Step 1 of the previous proof is still valid. Similar to Step 2 we
10 MICHAEL REISSIG AND JENS WIRTH
can diagonalise to a (2, 2, 1) block structure. The eigenvalues of these blocks can be calculated
explicitely.
2. For degenerate directions we obtain similarly
b1(η)
(κ1(η)− κ2(η))2
a21(η̄)a
2(η̄)
2γ2κ1(η̄)
(2.38)
and b1(η) vanishes to the double contact order.
2.2. Asymptotic expansion of the eigenvalues as |ξ| → ∞. If we consider large frequencies
the second order part B2(ξ) dominates B1(ξ). This makes it necessary to apply a different two-
step diagonalisation scheme. We follow partly ideas from [9], [14], [15] adapted to our special
situation.
Proposition 2.6. As |ξ| → ∞ the eigenvalues of the matrix B(ξ) behave as
ν0(ξ) = iκ|ξ|2 −
+O(|ξ|−1), (2.39a)
ν±j (ξ) = ±|ξ|ωj(η) +
a2j (η) +O(|ξ|−1). (2.39b)
for all non-degenerate directions ξ/|ξ| ∈ S1.
Remark. 1. Note that, while in hyperbolic directions we always have ν±j0(ξ) = ±|ξ|ωj0(η) for
one index j0, in degenerate hyperbolic directions all aj(η) may be non-zero. Hence the statement
of the above theorem cannot be valid in such directions in general.
2. Degenerate directions play for large frequencies a similar rôle as γ-degenerate directions play
for small frequencies.
Proof. The proof will be decomposed into several steps. In a first step we use the main part
B2(ξ) = iκ|ξ|2 diag(0, 0, 0, 0, 1) to block-diagonalise B(ξ). In a second step we diagonalise the
upper 4× 4 block for all non-degenerate directions.
Step 1. For a matrix B ∈ C5×5 we denote by b-diag4,1B the block diagonal of B consisting
of the upper 4 × 4 block and the lower corner entry. We construct a diagonalisation scheme to
block-diagonalise B(ξ) as |ξ| → ∞.
We set R(−1)(ξ) = B(ξ) − B2(ξ) = B1(ξ) and B−2(ξ) = B2(ξ) and construct recursively a
diagonaliser modulo the upper 4× 4 block,
Mk(ξ) = I +
|ξ|−jM (j)(η). (2.40)
Again we denote by R̃(k)(η) the (−k)-homogeneous part of R(k)(ξ) (which exists because it exists
for R(−1)(ξ) and the existence is transfered by the construction). Then we introduce the recursive
scheme
Bk−2(η) = b-diag4,1 R̃(k−2)(η), (2.41)
M (k)(η) =
(k−2)
15 (η)
(k−2)
45 (η)
−R̃(k−2)51 (η) · · · −R̃
(k−2)
54 (η) 0
(2.42)
for k = 1, 2, . . ., such that the commutator relation
[B−2(η),M (k)(η)] = Bk−2(η)− R̃(k−2)(η), (2.43)
ANISOTROPIC THERMO-ELASTICITY IN 2D 11
holds. Thus it follows
R(k−1)(ξ) = B(ξ)Mk(ξ)−Mk(ξ)
|ξ|−jBj(η) = O(|ξ|1−k) (2.44)
and using that Mk(ξ) is invertible for |ξ| ≥ Ck, Ck sufficiently large, we obtain the block diago-
nalisation
M−1k (ξ)B(ξ)Mk(ξ) =
|ξ|−jBj(η) +O(|ξ|1−k), (2.45)
where Bj(η) = b-diag4,1 Bj(η) is (4, 1)-block diagonal.
Step 2. By Step 1 we constructed Mk(ξ) such that M
k (ξ)B(ξ)Mk(ξ) is (4, 1)-block diagonal
modulo O(|ξ|1−k). The upper 4×4 block has already diagonal main part |ξ|−1D−1(η) = B1(ξ). If
the direction η = ξ/|ξ| is non-degenerate, the diagonal entries ±ω1(η) and ±ω2(η) are mutually
distinct and thus we can apply the standard diagonalisation procedure (cf. proof of Proposi-
tion 2.5) in the corresponding subspace. This does not alter the lower corner entry and gives
only combinations of the entries of the last column and of the last row (without changing their
asymptotics).
Thus we can construct a matrix Nk−1(ξ) = I +
j=1 |ξ|−jN (j)(ξ), which is invertible for
|ξ| > C̃k−1, C̃k−1 sufficiently large, such that
N−1k−1(ξ)M
k (ξ)B(ξ)Mk(ξ)Nk−1(ξ) = |ξ|
2B−2(η) +
|ξ|−jDj(η) +O(|ξ|1−k) (2.46)
is diagonal modulo O(|ξ|1−k).
Step 3. We give the first matrices explicitly. Following Step 1 we get
M (1)(η) =
γa1(η)
γa2(η)
γa1(η)
γa2(η)
γa1(η)
γa2(η)
γa1(η)
γa2(η)
(2.47)
together with B−1(η) = diag(ω1(η), ω2(η),−ω1(η),−ω2(η), 0) and
R̃(0)(η) = B1(η)M
(1)(η)−M (1)(η)B−1(η)
γ2a21(η)
γ2a1(η)a2(η)
· · · γa1(η)ω1(η)
γ2a1(η)a2(η)
γ2a22(η)
· · · γa2(η)ω2(η)
. . .
γa1(η)ω1(η)
γa2(η)ω2(η)
· · · −iγ
, (2.48)
B0(η) = b-diag4,1 R̃(0)(η) (2.49)
in the first diagonalisation step. Applying a second step alters only the last row and column to
O(|ξ|−1). Following Step 2 we diagonalise the upper 4× 4 block to |ξ|B−1(η) +B0(η) +O(|ξ|−1)
modulo O(|ξ|−1) and the statement is proven. �
Remark. If the direction η is degenerate, i.e. ω1(η) = ω2(η), we can block-diagonalise in Step
2 to (2, 2, 1)-block form. To diagonalise further we have to know that the 0-homogeneous part
of these 2× 2-blocks has distinct eigenvalues.
12 MICHAEL REISSIG AND JENS WIRTH
One possible treatment of degenerate directions is given in the following proposition. Note
that a corresponding statement can be obtained for γ-degenerate directions as |ξ| → 0.
Proposition 2.7. Let η̄ be an isolated degenerate direction. Then the corresponding hyperbolic
eigenvalue satisfies in a small conical neighbourhood of η̄
ν−j0(ξ) =
ω1(ξ) + ω2(ξ)
(ω1(ξ)− ω2(ξ))2
iγ2(ω1(ξ)− ω2(ξ))(a21(η)− a22(η))
+O(|ξ|−1). (2.50)
Proof. We follow the treatment of the previous proof to (2, 2, 1)-block-diagonalise B(ξ) modulo
|ξ|−1. Now, we consider one of its 2× 2-blocks. (We use a similar notation as before in the hope
that it will not lead to any confusion.) Such a block is given by
B(ξ) = |ξ|B−1(η) + B0(η) +O(|ξ|−1), (2.51)
where
B−1(η) = diag
ω1(η), ω2(η)
, (2.52)
B0(η) =
a21(η) a1(η)a2(η)
a1(η)a2(η) a
. (2.53)
In the direction η̄ both diagonal entries of B−1 coincide. In a small conical neighbourhood we
denote the eigenvalues of |ξ|B−1(η) + B0(η) as δ+(ξ) and δ−(ξ). A simple calculation yields
δ±(ξ) =
ω1(ξ) + ω2(ξ)
(ω1(ξ)− ω2(ξ))2
iγ2(ω1(ξ)− ω2(ξ))(a21(η)− a22(η))
(2.54)
with δ−(ξ̄) = ω1(ξ̄) and δ+(ξ̄) = ω1(ξ̄) +
. The hyperbolic eigenvalue corresponds to δ−(ξ).
These eigenvalues are distinct in a sufficiently small neighbourhood of η̄ (and may coincide only if
a1(η) = a2(η) and (ω1(ξ)−ω2(ξ))2 = γ4/(4κ2), which gives eventually two parabolic directions).
Hence the perturbation theory of matrices implies ν+j0(ξ) = δ−(ξ) +O(|ξ|
−1) in a sufficiently
small neighbourhood of η̄ and the statement is proven. �
Remark. Note, that
ω1(ξ)− δ−(ξ)
ω1(ξ)− ω2(ξ)
= a21(η̄) (2.55)
for all fixed |ξ|, which coincides with the result (2.13) of Proposition 2.3 for the eigenvalue ν(ξ).
2.3. Collecting the results. The asymptotic expansions from Propositions 2.5 and Proposi-
tion 2.6 imply estimates for eigenvalues of B(ξ) and the proofs give representations of corre-
sponding eigenvectors. For the application of multiplier theorems and the proof of Lp–Lq decay
estimates it is essential to provide also estimates for derivatives of them.
Assume that the eigenvalues under consideration are simple. From the asymptotic expansions
we know that this is the case for small frequencies and also for large frequencies. For the middle
part it will be sufficient to know that the hyperbolic eigenvalues are separated, which follows for
sufficiently small conical neighbourhoods of these directions.
ANISOTROPIC THERMO-ELASTICITY IN 2D 13
In a first step we consider derivatives of the eigenvalues. Differentiating the characteristic
polynomial
0 = det(ν(ξ)I −B(ξ)) =
Ik(ξ)ν(ξ)
k (2.56)
with respect to ξ yields by Leibniz formula
ξ Ik(ξ)
ξ ν(ξ)
(2.57)
for all multi-indices α ∈ N20. Thus we can express the highest derivative of ν(ξ)k in terms of
lower ones and hence Faà di Bruno’s formula (see e.g. [5]) yields an expression
Dαν(ξ)
kIk(ξ)ν(ξ)
k−1 =
Ck,α,β
Dα−βIk(ξ)
Dβν(ξ)
(2.58)
with certain constants Ck,α,β . Because the eigenvalue has multiplicity one, the sum on the left-
hand side is nonzero and therefore we can calculate the derivatives of ν(ξ) by this expression.
Furthermore, it follows that for small and large frequencies the derivatives of the eigenvalue have
full asymptotic expansions and thus we are allowed to differentiate the asymptotic expansions
term by term.
It remains to consider the corresponding eigenprojections. Recall that if ν(ξ) is a eigenvalue of
multiplicity one and r(ξ) and l(ξ) are corresponding right and left eigenvectors, the correspond-
ing eigenprojection is given by the dyadic product Pν(ξ) = l(ξ) ⊗ r(ξ). Thus the constructed
diagonaliser matrices imply asymptotic expansions of these operators. Again we are only inter-
ested whether the derivatives of these eigenprojections also possess asymptotic expansions (in
order to see whether it is allowed to differentiate term by term).
For this we use the representation
Pν(ξ) =
ν̃∈specB(ξ)\{ν}
(ν̃(ξ)I −B(ξ))(ν̃(ξ)− ν(ξ))−1 (2.59)
given e.g. in [4], [7]. All terms on the right-hand side have full asymptotic expansions as |ξ| → 0
and |ξ| → ∞ together with all of their derivatives. Differentiating with respect to ξ yields the
same result for the eigenprojection. Thus we obtain
Proposition 2.8. The asymptotic expansions from Proposition 2.5 and Proposition 2.6 may be
differentiated term by term to get asymptotic expansions for the derivatives of the eigenvalues.
Furthermore, the same holds true for the corresponding eigenprojections.
From Proposition 2.1 we know that an eigenvalue ν(ξ) of the matrix B(ξ) is real if and only if
η = ξ/|ξ| is hyperbolic. We want to combine this information with the asymptotic expansions of
Proposition 2.5 and Proposition 2.6 and derive some estimates for the behaviour of the imaginary
part.
Proposition 2.9. Let c > 0 be a given constant.
(1) Let η = ξ/|ξ| ∈ S1 be parabolic. Then the eigenvalues of B(ξ) satisfy
Im ν(ξ) ≥ Cη > 0, |ξ| ≥ c, (2.60)
with a constant Cη depending on the direction η and c. Furthermore,
Im ν(ξ) ∼ b(η)|ξ|2, |ξ| ≤ c, (2.61)
where b(η) is one of the functions from Proposition 2.5.
14 MICHAEL REISSIG AND JENS WIRTH
(2) Let η̄ be non-degenerate and hyperbolic with respect to the index 1. Then ν±1 (|ξ|η̄) =
±|ξ|ω1(η̄) and ν0(ξ) and ν±2 (ξ) satisfy the statement of point 1. Furthermore,
Im ν±1 (ξ) ∼ a21(η), |ξ| ≥ c, |η − η̄| ≪ 1, (2.62)
Im ν±1 (ξ) ∼ |ξ|2a21(η), |ξ| ≤ c, |η − η̄| ≪ 1. (2.63)
Proof. The first point follows directly from the asymptotic expansions, we concentrate on the
second one. We know that the hyperbolic eigenvalues ν±1 (ξ) satisfy by Proposition 2.2
Im ν±1 (ξ) = a
1(η)N
1 (ξ) (2.64)
with a smooth non-vanishing function N±1 (ξ) defined in a neighbourhood of η̄. By Proposition 2.5
and 2.6 we see that N±1 (ξ) also has full asymptotic expansions and thus
N±1 (ξ) =
+O(|ξ|−1), |ξ| → ∞ (2.65a)
N±1 (ξ) = iκ|ξ|
2 b1(η)
a21(η)
+O(|ξ|3), |ξ| → 0. (2.65b)
Together with (2.37) we get upper and lower bounds on N±1 (ξ) and the desired statement follows.
Remark. A similar reasoning allows to replace the hyperbolic eigenvalue near degenerate di-
rections by the model expression obtained in Proposition 2.7, thus ν(ξ) ∼ δ−(ξ) uniformly in a
sufficiently small conical neighbourhood of η̄.
3. Decay estimates for solutions
Our strategy to give decay estimates for solutions to the thermo-elastic system (1.1) is to
micro-localise them. In principle we have to distinguish four different cases. On the one hand
we differentiate between small and large frequencies, on the other hand between hyperbolic
directions and parabolic ones.
We distinguish between two cases depending on the vanishing order of the coupling functions.
If the coupling functions vanish to first order at hyperbolic directions only, we rely on simple
multiplier estimates. Later on we discuss coupling functions with higher vanishing order, where
the decay rates are obtained by tools closely related to the treatment of the elasticity equation.
3.1. Coupling functions with simple zeros. In a first step we consider the first order system
DtV = B(D)V, V (0, ·) = V0 (3.1)
to Cauchy data V0 ∈ S(R2,C5). For a cut-off function χ ∈ C∞(R+) with χ(s) = 0, s ≤ ǫ, and
χ(s) = 1, s ≥ 2ǫ, we consider
Ppar(η) =
η̄ hyperbolic
χ(|η − η̄|), Phyp(η) = 1− Ppar(η). (3.2)
Then Phyp(D) localises in a conical neighbourhood of the set of hyperbolic directions, while
Ppar(D) localises to a compact set of parabolic directions.
The asymptotic formulae and representations for the characteristic roots of the full symbol
B(ξ) allow us to proof decay estimates for the solutions.
ANISOTROPIC THERMO-ELASTICITY IN 2D 15
Theorem 3.1. Assume that (A1) to (A4) are satisfied and the coupling functions vanish in
hyperbolic directions to first order.
Then the solution V (t, x) to (3.1) satisfies the following a-priori estimates:
‖χ(D)Ppar(D)V (t, ·)‖q . e−Ct‖V0‖p,r (3.3a)
‖(1− χ(D))Ppar(D)V (t, ·)‖q . (1 + t)−(
)‖V0‖p (3.3b)
‖χ(D)Phyp(D)V (t, ·)‖q . (1 + t)−
)‖V0‖p,r (3.3c)
‖(1− χ(D))Phyp(D)V (t, ·)‖q . (1 + t)−
)‖V0‖p (3.3d)
for dual indices p ∈ (1, 2], pq = p+ q, and with Sobolev regularity r > 2(1/p− 1/q).
Remark. If B(ξ) is diagonalisable for ξ 6= 0 (which is valid e.g. if B(ξ) has no double eigenvalues
for ξ 6= 0) we can make the result even more precise. To each eigenvalue ν(ξ) ∈ specB(ξ) we
have corresponding left and right eigenvectors and associated to them the eigenprojection Pν(D)
such that
Pν(D)V (t, ·) = eitν(D)Pν(D)V0. (3.4)
Thus, we can single out the influence of one eigenvalue in this way. Note, that this is only of
interest in the neighbourhood of hyperbolic directions and for the corresponding eigenvalue and
there the assumption of diagonalisability of B(ξ) may be skipped (real eigenvalues are always
simple, for small |ξ| diagonalisation works, for large |ξ| everything goes well under the assumption
of non-degeneracy and on the middle part we make the neighbourhood small enough to exclude
possible multiplicities).
Proof. We decompose the proof into four parts corresponding to the four estimates. The micro-
localised estimates are merely standard multiplier estimates. We do not use stationary phase
method.
Step 1. Parabolic directions, large frequencies. In this case we have uniformly in ξ ∈ suppχPpar ,
the estimate Im ν(ξ) ≥ C′ > 0. Taking 0 < C < C′ we obtain for these ξ
ImB(ξ) =
B(ξ)−B∗(ξ)
in the sense of self-adjoint operators and the estimate follows from the L2–L2 estimate
‖χ(D)Ppar(D)V (t, x)‖22 =
‖χ(ξ)Ppar(ξ)V̂ (t, ξ)‖22
= 2Re
χ(ξ)Ppar(ξ)V̂ (t, ξ), χ(ξ)Ppar(ξ)∂tV̂ (t, ξ)
= −2 Im
χ(ξ)Ppar(ξ)V̂ (t, ξ), χ(ξ)Ppar(ξ)B(ξ)V̂ (t, ξ)
≤ −2C‖χ(ξ)Ppar(ξ)V̂ (t, ξ)‖2 = −2C‖χ(D)Ppar(D)V (t, x)‖22,
viewed as Hs–Hs estimate and combined with Sobolev embedding.
Step 2. Parabolic directions, small frequencies. We know from Proposition 2.5 that in this case
the matrix B(ξ) has only simple eigenvalues. We will make use of the representation of solutions
V (t, x) =
ν(ξ)∈specB(ξ)
eitν(D)Pν(D)V0(x) (3.5)
with corresponding eigenprojections (amplitudes) Pν(ξ). The amplitudes are uniformly bounded
on the set of all occurring ξ and possess full asymptotic expansions in |ξ| as ξ → 0 together with
their derivatives. Especially, by Hörmander-Mikhlin multiplier theorem [10, p. 96, Theorem 3]
the operators Pν(D) are L
p-bounded for 1 < p <∞.
16 MICHAEL REISSIG AND JENS WIRTH
It remains to consider the model multiplier eitν(ξ). From Proposition 2.5 we know that
|eitν(ξ)| . e−Ct|ξ|
, |ξ| ≤ c, (3.6)
with suitable constants c and C and thus the L1–L∞ estimate
‖eitν(D)f‖∞ ≤ ‖eitν(ξ)f̂‖1 ≤ ‖eitν(ξ)‖L1({|ξ|≤c})‖f̂‖∞
. ‖f‖1
e−Ct|ξ|
|ξ|d|ξ| . (1 + t)−1 ‖f‖1
holds for all f ∈ L1(R2) with supp f̂ ⊆ {|ξ| ≤ c}. Riesz-Thorin interpolation [1, Chapter 4.2]
with the obvious L2–L2 estimate gives the desired decay result.
Step 3. Hyperbolic directions, large frequencies. We take the conical neighbourhoods small
enough to exclude all multiplicities (related to the eigenvalue which becomes real in the hyperbolic
direction). Then similar to (3.5) the solution is represented as
V (t, x) = eitν(D)P+ν (D)V0(x) + e
−itν(D)P−ν (D)V0(x) + Ṽ (t, x), (3.7)
where Ṽ (t, x) corresponds to the remaining parabolic eigenvalues of B(ξ) and satisfies the es-
timate from Step 1. Again we can use smoothness of P±ν (ξ) together with the existence of a
full asymptotic expansion as |ξ| → ∞ to get Lp-boundedness of P±ν (D) for 1 < p < ∞ from
Hörmander-Mikhlin multiplier theorem.
It remains to understand the model multiplier e±itν(ξ) for the hyperbolic eigenvalue related
to the hyperbolic direction η̄. Using the estimate from Proposition 2.9 we conclude for r > 2
‖e±itν(D)f‖∞ ≤ ‖e±itν(ξ)f̂‖1 ≤ ‖e±itν(ξ)|ξ|−r‖L1(S1)‖ |ξ|
r f̂‖∞
. ‖〈D〉rf‖1
|ξ|1−rd|ξ|
e−C1tφ
. t−1/2 ‖〈D〉rf‖1, t ≥ 1
for all f ∈ 〈D〉−rL1(R2) with supp f̂ ⊆ S1 = {|ξ| ≥ c, |η − η̄| ≤ ǫ}. Riesz-Thorin interpolation
with the L2–L2 estimate gives the desired decay result.
Step 4. Hyperbolic directions, small frequencies. Like for large hyperbolic frequencies we make use
of the representation (3.7) to separate hyperbolic and parabolic influences. As in the previous
cases the existence of full asymptotic expansions imply that the projections P±ν (D) are L
bounded for 1 < p <∞.
It remains to understand the model multiplier e±itν(ξ). Using Proposition 2.9 we have |e±itν(ξ)| .
e−C2t|ξ|
2φ2 such that after introducing polar co-ordinates
‖e±itν(D)f‖∞ ≤ ‖e±itν(ξ)f̂‖1 ≤ ‖e±itν(ξ)‖L1(S2)‖f‖1
. ‖f‖1
e−C2tφ
2|ξ|2dφ|ξ|d|ξ|
. t−1/2‖f‖1
d|ξ| . t−1/2 ‖f‖1, t ≥ 1
for all f ∈ L1(R2) with supp f̂ ⊆ S2 = {|ξ| ≤ c, |η − η̄| ≤ ǫ}. Riesz-Thorin interpolation with
the L2–L2 estimate gives the desired decay result. �
Remark. 1. In non-degenerate hyperbolic directions where the coupling function vanishes to
order ℓ (cf. Example 2.4 case 3) we obtain by the same reasoning the weaker Lp–Lq decay rate
‖χ(D)Phyp(D)V (t, ·)‖q . (1 + t)−
)‖V0‖p,r. (3.8)
ANISOTROPIC THERMO-ELASTICITY IN 2D 17
It remains to understand whether this weaker decay rate is also sharp or whether an application
of stationary phase method may be used to improve this. We will discuss this in Section 3.2.
2. We can extend the estimate of Theorem 3.1 to the limit case p = 1, if we include the eigen-
projections Pν(ξ) into the considered model multiplier and use just their boundedness instead
of Hörmander-Mikhlin multiplier theorem. This is possible, because all the multiplier estimates
were based on Hölder inequalities.
So far we understood properties of solutions to the transformed problem (3.1) for the vector-
valued function V (t, x) given by (2.4) as
V (t, x) =
(Dt +D1/2(D))U (0)(t, x)
(Dt −D1/2(D))U (0)(t, x)
θ(t, x)
 , (3.9)
where U (0)(t, x) =MT (D)U(t, x) is the elastic displacement after transformation with the diag-
onaliser of the elastic operator. Because M(η) is unitary and homogeneous of degree zero this
diagonaliser is Lp-bounded for 1 < p <∞ with bounded inverse. Thus we have
DtU(t, x) =M(D)
V (t, x) (3.10)
A(D)U(t, x) =M(D)
0 − 1
0 − 1
V (t, x) (3.11)
such that as corollary of Theorem 3.1 we obtain
Corollary 3.2. Assume (A1) to (A4) and that the coupling functions vanish of first order in all
hyperbolic directions. Then the solution U(t, x) and θ(t, x) to (1.1) satisfy the a-priori estimates
‖DtU(t, ·)‖q + ‖
A(D)U(t, ·)‖q + ‖θ(t, ·)‖q
. (1 + t)−
) (‖U1‖p,r+1 + ‖U2‖p,r + ‖θ0‖p,r) (3.12)
for dual indices p ∈ (1, 2], pq = p+ q, and Sobolev regularity r > 2(1/p− 1/q).
Remark. Including the diagonaliser M(ξ) into the model multipliers of Theorem 3.1 allows to
overcome the restriction on p and to extend this statement up to p = 1. We preferred this way
of presenting the results because they allow to decouple both statements. Theorem 3.1 gives
deeper insight into the asymptotic behaviour of solutions than Corollary 3.2 does.
3.2. Coupling functions vanishing to higher order. We have seen that under the assump-
tion that the coupling functions aj(η) ∈ C∞(S1), aj(η) = η · rj(η) related to the symbol of the
elastic operator A(η) have only zeros of first order, we can deduce Lp–Lq decay estimates without
relying on the method of stationary phase.
Now we want to discuss how to use the method of stationary phase to deduce decay estimates
in the remaining cases. From Proposition 2.2 we know that in hyperbolic directions the imaginary
part Im ν±j0(ξ) of the hyperbolic eigenvalues vanishes like the square of the corresponding coupling
function a2j0(η), while the real part Re ν
(ξ) is essentially described by ±ωj0(ξ). First, we make
this more precise and formulate an estimate for Re νj0(ξ)
± ∓ ωj0(ξ) and its derivatives.
Proposition 3.3. Let η̄ ∈ S1 be hyperbolic with respect to the index j0 and aj0(η) vanish to
order ℓ in η̄. Then there exists a conical neighbourhood of the direction η̄ such that for all
k = 0, 1, . . . , 2ℓ− 1 the estimates
∣∂kη (Re ν
(ξ) ∓ ωj0(ξ))
∣ ≤ c(|ξ|)|η − η̄|2ℓ−k, (3.13a)
∣∂kη Im ν
∣ ≤ c(|ξ|)|η − η̄|2ℓ−k, (3.13b)
18 MICHAEL REISSIG AND JENS WIRTH
hold uniformly on it, where c(|ξ|) ∼ 1 as |ξ| → ∞ and c(|ξ|) ∼ |ξ| as |ξ| → 0.
This estimate may be used to transfer the micro-localised decay estimate for the elasticity
equation to the thermo-elastic system on a sufficiently small conical neighbourhood of η̄. We
need one further notion to prepare the main theorem of this section. Decay rates for solutions to
the elasticity equation depend heavily on the order of contact between the sheets of the Fresnel
surface and its tangents, cf. [13] or point 2 of the concluding remarks on page 21.
Proposition 3.4. For η ∈ S1 we denote by γ̄j(η) the order of contact between the j-th sheet
Sj = {ω−1j (η)η | η ∈ S
1} (3.14)
of the Fresnel curve and its tangent in the point ω−1j (η)η. Then
∂kη (∂
ηωj(η) + ωj(η)) = 0, k = 0, . . . , γ̄j(η)− 2, (3.15)
(if γ̄j(η) > 2) and
∂γ̄j(η)η ωj(η) + ∂
γ̄j(η)−2
η ωj(η) 6= 0. (3.16)
Proof. Follows by straight-forward calculation. The curvature of the j-th Fresnel curve Sj at the
point ω−1j (η)η factorises as ωj(η) + ∂
ηωj(η) and a smooth non-vanishing analytic function. �
For the following we chose the conical neighbourhood of the hyperbolic direction η̄ (with
respect to the index j0) small enough in order that the curvature of Sj0 vanishes at most in η̄.
Theorem 3.5. Let η̄ be hyperbolic with respect to the index j0 and let aj0(η) vanish in η̄ to order
ℓ > 1. Let us assume further that γ̄j0(η̄) < 2ℓ. Then
||e±itνj0 (D)f ||q . (1 + t)
)||f ||p,r (3.17)
for dual indices p ∈ [1, 2], pq = p + q, regularity r > 2(1/p − 1/q) and for all f with supp f̂
contained in a sufficiently small conical neighbourhood of η̄.
Proof. We distinguish between two cases related to the Fourier support of f , for general f we
can use linearity. In both cases we apply the method of stationary phase. Key lemma will be
the Lemma of van der Corput, cf. [11, p.334], combined for large frequencies with a dyadic
decomposition. It suffices to consider t ≥ 1, the estimate for t ≤ 1 follows from the uniform
boundedness of the Fourier multiplier together with Sobolev embedding using the regularity
imposed on the data, r > 2(1/p− 1/q).
In the following we skip the index j0 of the eigenvalue. Thus ω(η) stands for ωj0(η) and γ̄(η)
for γ̄j0(η) to shorten the notation.
Large and medium frequencies. Assume that f̂ is supported in a sufficiently small conical neigh-
bourhood of η̄ bounded away from zero, |ξ| > 1. We will use a dyadic decomposition in the radial
variable. Let for this χ ∈ C∞0 (R) be chosen in such a way that χ(R) = [0, 1], suppχ ⊆ [1/2, 2]
j∈Z χ(2
js) = 1 for all s ∈ R+. We set χj(s) = χ(2−js) such that suppχj ⊆ [2j−1, 2j+1].
We follow the treatment of Brenner [2] and use Besov spaces. Due to our assumptions on the
Fourier support of f Besov norms are given by
||f ||Brp,q =
∥2jr||χj(|D|)f ||p
ℓq(N0)
(3.18)
and corresponding Besov spaces are related to Sobolev and Lp-spaces by the embedding relations
Hp,r →֒ Brp,2, B0q,2 →֒ Lq (3.19)
ANISOTROPIC THERMO-ELASTICITY IN 2D 19
for p ∈ (1, 2] and q ∈ [2,∞). Thus it suffices to prove an Brp,2 → B0q,2 estimate for dual p and q.
The case p = q = 2, r = 0 is trivial by Plancherel and the uniform boundedness of the multiplier,
we concentrate on the B21,2 → B0∞,2 estimate. Therefore, we use
||e±itν(D)f ||B0
||χj(D)e±itν(D)f ||2∞
≤ sup
Ij(t) ||f ||Br
, (3.20)
where Ij(t) are estimates of the dyadic components of the operator,
Ij(t) = sup
|F−1[eitν(ξ)ψ(ξ)χj(|ξ|)|ξ|−r]|. (3.21)
Here ψ(ξ) localises to a small conical neighbourhood of η̄ bounded away from zero, |ξ| > 1,
containing the support of f̂ . We assume |ψ(ξ)|+ |∂ηψ(ξ)|+ |∂|ξ|ψ(ξ)| . 1 and ψ ≡ 1 on a smaller
conic set around η̄. We introduce polar co-ordinates |ξ| and φ (with φ = 0 corresponding to η̄)
and set x = tz. Using that the hyperbolic directions η̄ ∈ S1 are isolated we can assume that
the conical neighbourhood under consideration contains no further hyperbolic direction. For the
calculation of Ij(t) we integrate over the interval [−ǫ, ǫ] for φ. This yields for all j ∈ N0
Ij(t) = sup
∫ 2j+1
eit(Re ν(ξ)+z·ξ)−t Im ν(ξ)ψ(ξ)χj(|ξ|)|ξ|1−rdφd|ξ|
= 2j(2−r) sup
jt(Re ν(2jξ)2−j+z·ξ)e−t Im ν(2
jξ)ψ(2jξ)χ(|ξ|)|ξ|1−rdφd|ξ|
Due to Proposition 3.3 (let us restrict to the − case) we have
Re ν(2jξ) = 2jω(ξ) + 2j|ξ|α(2jξ), (3.22)
where
∣∂kφα(2
∣ ≤ Ck |φ|2ℓ−k, k = 0, 1, . . . , 2ℓ− 1. (3.23)
By our assumptions 2ℓ− 1 ≥ γ̄(η̄). We use this to reduce the problem to properties of the elastic
eigenvalue ωj(η) and consider
Ij(t) = 2
j(2−r) sup
j |ξ|(ω(η)+z·η+α(2jξ))e−t Im ν(2
jξ)ψ(2jξ)dφχ(|ξ|)|ξ|1−rdξ
. (3.24)
Note that, |α(2jξ)| . |φ|2ℓ is small, if we choose the neighbourhood of η̄ small enough.
If |ω(η)+z ·η| ≥ δ for some small δ the outer integral has no stationary points and integration
by parts implies arbitrary polynomial decay. We restrict to the z ∈ R2 with |ω(η) + z · η| ≤ δ
and consider only the inner integral
Ij(t, |ξ|) =
j|ξ|(ω(η)+z·η+α(2jξ))e−t Im ν(2
jξ)ψ(2jξ)dφ
. (3.25)
It can be estimated using the Lemma of van der Corput. For this we need an estimate for deriva-
tives of the phase. We start by considering the unperturbed phase ω(η) + z · η. Differentiation
yields
∂γ̄(η̄)η (ω(η) + z · η) = ∂γ̄(η̄)−2η (∂2ηω(η) + ω(η))− ∂γ̄(η̄)−4η (∂2ηω(η) + ω(η)) +−
· · ·+−
ω(η) + z · η, 2|γ̄(η̄)
∂ηω(η) + z · ηT , 26 |γ̄(η̄)
(3.26)
20 MICHAEL REISSIG AND JENS WIRTH
and by Proposition 3.4 the first term is non-zero for η̄ while the others are small in a neighbour-
hood of η̄. Choosing δ and the neighbourhood small enough and using (3.23) this implies a lower
bound on the γ̄(η̄)-th derivative of the phase,
|∂γ̄(η̄)η (ω(η) + z · η + α(2jξ))| & 1 (3.27)
uniformly on this neighbourhood of η̄ and independent of j. Thus we conclude by [11, p. 334]
for all t ≥ 1 and uniformly in z with |ω(η) + z · η| ≤ δ
Ij(t, |ξ|) ≤ Ck(t2j |ξ|)−
γ̄(η̄)
−t Im ν(2jξ)ψ(2jξ)
. (3.28)
To estimate the remaining integral we use Proposition 3.3
−t Im ν(2jξ)ψ(2jξ)
∣ dφ .
e−ctφ
|φ|2ℓ−1tdφ = 2
e−ctφ
φ2ℓ−1ctdφ
e−ctφ
Integrating over ξ ∈ [1/2, 2] and choosing r ≥ 2 we obtain for j ∈ N0
Ij(t) . (1 + t)
γ̄(η̄) , (3.29)
where the occurring constant is independent of j and (3.20) implies the desired B21,2 → B0∞,2
estimate. Interpolation with the L2–L2 estimate gives the corresponding Brp,2 → B0q,2 estimates
with r ≥ 2(1/p− 1/q) and finally the embedding relations to Sobolev and Lp-spaces the desired
estimate
||eitν(D)f ||q . (1 + t)−
γ̄(η̄) ||f ||p,r (3.30)
for all f satisfying the required Fourier support conditions.
Small frequencies. Assume now that f̂ is supported in a small conical neighbourhood of η̄ with
|ξ| ≤ 2. In this case we estimate directly by the method of stationary phase. We sketch the main
ideas. It is sufficient to estimate
I(t) = sup
eitν(ξ)ψ(ξ)χ(|ξ|)
∣ , (3.31)
where the function ψ ∈ C∞(R+) localises to the small neighbourhood of η̄ with |ξ| ≤ 2. Again
we require |ψ(ξ)| + |∂ηψ(ξ)| + |∂|ξ|ψ(ξ)| . 1. In correspondence to large frequencies I(t) equals
I(t) = sup
eit|ξ|(ω(η)+z·η+α(ξ))e−t Im ν(ξ)ψ(ξ)χ(|ξ|)dφ|ξ|d|ξ|
. (3.32)
We distinguish between |ω(η) + z · η| ≥ δ, where the outer integral has no stationary points, and
|ω(η) + z · η| ≤ δ. In the first case we can apply one integration by parts and get t−1 using
|ξ| |∂|ξ|e−t Im ν(ξ)| . |ξ|2te−t|ξ|
2φ2ℓ . 1. (3.33)
In the second case we can reduce the consideration to
I(t, |ξ|) =
eit|ξ|(ω(η)+z·η+α(ξ))e−t Im ν(ξ)ψ(ξ)dφ
(3.34)
and an application of the Lemma of van der Corput. By Propositions 3.3 and 3.4 we have
γ̄(η̄)
φ (ω(η) + z · η + α(ξ)) 6= 0 (3.35)
ANISOTROPIC THERMO-ELASTICITY IN 2D 21
for all ξ in a sufficiently small conic neighbourhood of η̄. So we obtain
I(t, |ξ|) . (t|ξ|)−
γ̄(η̄)
−t Im ν(ξ)ψ(ξ)
. (t|ξ|)−
γ̄(η̄) . (3.36)
Integrating with respect to |ξ| yields the estimate for I(t)
I(t) ≤
I(t, |ξ|)|ξ|d|ξ| . t−
γ̄(η̄)
|ξ|1−
γ̄(η̄) d|ξ| . (1 + t)−
γ̄(η̄) , t ≥ 1 (3.37)
and by Hölder inequality the L1–L∞ estimate
||eitν(D)f ||∞ ≤ I(t)||f ||1 . (1 + t)−
γ̄(η̄) ||f ||1 (3.38)
for all f satisfying the required Fourier support condition.
By interpolation with the obvious L2–L2 estimate we get Lp–Lq estimates and combining
them with the estimate of the first part proves the theorem. �
If the order of contact exceeds the vanishing order of the coupling funtions, the best we can do
is to use the idea of Section 3.1 and to apply standard multiplier estimates. As already remarked
after the proof of Theorem 3.1 we obtain as decay rate in this case:
Theorem 3.6. Let η̄ be hyperbolic with respect to the index j0 and let aj0(η) vanish in η̄ of order
ℓ. Let us assume further that γ̄j0(η̄) ≥ 2ℓ, where γ̄j0(η) is defined in Proposition 3.4. Then
||e±itνj0 (D)f ||q . (1 + t)−
)||f ||p,r (3.39)
for dual indices p ∈ [1, 2], pq = p + q, regularity r > 2(1/p − 1/q) and for all f with supp f̂
contained in a sufficiently small conical neighbourhood of η̄.
4. Concluding remarks
1. In a second part of this note, [16], we will give concrete applications of the general treatment
presented so far. From the remarks and examples we made in the previous sections it follows that
we indeed cover the estimates of [3] for cubic media and [4] for rhombic media in the situations
of coupling vanishing to first order.
In [16] we will discuss the situation of higher order tangencies for rhombic media and give a
new estimate extending the results of [4]. Furthermore, we will give concrete examples of media
for all achievable decay rates in the case of a differential elastic operator A(D).
2. In general decay rates of solutions are determined by vanishing properties of the coupling
functions. If the coupling functions are nonzero, decay rates are parabolic. If they vanish to
sufficiently high order, decay rates are hyperbolic and determined by the elastic operator (micro-
localised to this direction). In the intermediate case simple multiplier estimates are sufficient.
3. In the case of isotropic media one of the coupling functions vanishes identically. In this case
one pair of eigenvalues of the matrix B(ξ) is purely real ν±(ξ) = ±(λ+µ)|ξ| and the components
related to these eigenvalues solve a wave equation. Thus they satisfy the usual Strichartz type
decay estimates, [2],
‖e±it(λ+µ)|D|P±(D)V0‖q . (1 + t)−
)‖V0‖p,r (4.1)
for p ∈ (1, 2], pq = p + q and r = 2(1/p − 1/q). More generally, if we know that one pair of
eigenvalues satisfies ν±(ξ) = ±|ξ|ω(η) for all η ∈ S1 with a smooth function ω : S1 → R+, the
decay rates for the corresponding components depend heavily on geometric properties of the
Fresnel curve S = {ω−1(η)η | η ∈ S1 } ⊂ R2. Following [8] the estimate (4.1) is valid in this case
as long as the curvature of S never vanishes. If there exist directions η where the curvature of S
vanishes, the constant 1
in the exponent has to be altered to 1
, where γ̄ denotes the maximal
order of contact of the curve S to its tangent, [13].
22 MICHAEL REISSIG AND JENS WIRTH
4. The main focus of this paper was on the treatment of non-degenerate cases, thus assumptions
(A1) to (A4) are required. Nevertheless, we showed how to obtain a control on the eigenvalues
of the matrix B(ξ) in the exceptional cases where either (A3) or (A4) is violated. In the first
case the treatment of large frequencies has to be replaced by the investigation of the expression
obtained in Proposition 2.7, while in the latter one a corresponding replacement has to be made
for small frequencies.
5. The results in this paper are essentially two-dimensional. For the study of anisotropic thermo-
elasticity in higher space dimensions there arise two essential problems. The first is that we can
not assume (A3). For example in three space dimensions and for cubic media the symbol of
the elastic operator A : S2 → C3×3 has degenerate directions with multiple eigenvalues related
to the crystal axes. Nevertheless, the multi-step diagonalisation scheme used in Section 2.2 can
be adapted to such a case. This will be done in the sequel. The second problem is that the
geometry of the set of hyperbolic directions becomes more complicated. In general we can not
expect isolated hyperbolic directions, it will be necessary to consider manifolds of hyberbolic
directions on S2.
References
[1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988
[2] P. Brenner, On Lp–Lp′ estimates for the wave equation, Math. Z. 145(3):251–254, 1975.
[3] J. Borkenstein, Lp–Lq Abschätzungen der linearen Thermoelastizitätsgleichungen für kubische Medien
im R2, Diplomarbeit, Bonn, 1993.
[4] M. S. Doll, Zur Dynamik (magneto-) thermoelastischer Systeme im R2, Dissertation, Konstanz, 2004.
[5] W.P. Johnson, The curious history of Faà di Bruno’s formula, Amer. Math. Monthly, Vol. 109(3):
217–234, 2002.
[6] T. Kato, Perturbation Theory for linear Operators, Springer, 1980.
[7] O. Liess, Decay estimates for the solutions of the system of crystal optics, Asymptot. Anal. 4:61–95,
1991.
[8] R. Racke, Lectures on Nonlinear Evolution Equations – Initial Value Problems, Aspects of Mathemat-
ics: E, Vol. 19, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1992.
[9] M. Reissig, Y.-G. Wang, Cauchy problems for linear thermoelastic systems of type III in one space
variable, Math. Meth. Appl. Sci. 28:1359–1381, 2005.
[10] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University
Press, Princeton, New Jersey, 1970.
[11] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,
Princeton University Press, Princeton, New Jersey, 1993.
[12] M. Stoth, Lp–Lq Abschätzungen für eine Klasse von Lösungen linearer Cauchy-Probleme bei
anisotropen Medien, Dissertation, Bonn, 1994.
[13] M. Sugimoto, Estimates for hyperbolic equations with non-convex characteristics, Math. Z. 222(4):521–
531, 1996.
[14] Y.-G. Wang, Microlocal analysis in nonlinear thermoelasticity, Nonlinear Anal. 54:683–705, 2003 .
[15] Y.-G. Wang, A new approach to study hyperbolic-parabolic coupled systems, in R. Picard (ed.) et al.,
Evolution equations. Propagation phenomena, global existence, influence of non-linearities. Based on
the workshop, Warsaw, Poland, July 1-July 7, 2001, Warsaw: Polish Academy of Sciences, Institute
of Mathematics, Banach Cent. Publ. 60, p. 227-236, 2003.
[16] J. Wirth, Anisotropic thermo-elasticity in 2D - Part II: Applications, Asymptotic Anal. ??:???–
???,????.
[17] K. Yagdjian, The Cauchy problem for hyperbolic operators. Multiple characteristics, micro-local ap-
proach, Akademie-Verlag, Berlin, 1997.
Michael Reissig, Institut für Angewandte Analysis, Fakultät für Mathematik und Informatik, TU
Bergakademie Freiberg, Prüferstraße 9, 09596 Freiberg, Germany
Jens Wirth, Institut für Angewandte Analysis, Fakultät für Mathematik und Informatik, TU
Bergakademie Freiberg, Prüferstraße 9, 09596 Freiberg, Germany
current address: Department of Mathematics, Imperial College, London SW7 2AZ, UK
	1. The problem under consideration
	2. General treatment of the thermo-elastic system
	2.1. Asymptotic expansion of the eigenvalues as ||0
	2.2. Asymptotic expansion of the eigenvalues as ||
	2.3. Collecting the results
	3. Decay estimates for solutions
	3.1. Coupling functions with simple zeros
	3.2. Coupling functions vanishing to higher order
	4. Concluding remarks
	References

  In this note we develop tools and techniques for the treatment of anisotropic
thermo-elasticity in two space dimensions. We use a diagonalisation technique
to obtain properties of the characteristic roots of the full symbol of the
system in order to prove $L^p$--$L^q$ decay rates for its solutions.

ANISOTROPIC THERMO-ELASTICITY IN 2D
PART I: A UNIFIED TREATMENT
MICHAEL REISSIG AND JENS WIRTH
Abstract. In this note we develop tools and techniques for the treatment of anisotropic
thermo-elasticity in two space dimensions. We use a diagonalisation technique to obtain
properties of the characteristic roots of the full symbol of the system in order to prove Lp–Lq
decay rates for its solutions.
Keywords: thermo-elasticity, a-priori estimates, anisotropic media
1. The problem under consideration
Systems of thermo-elasticity are hyperbolic-parabolic or hyperbolic-hyperbolic coupled sys-
tems (type-1, type-2 or type-3 models) describing the elastic and thermal behaviour of elastic,
heat-conducting media. The classical type-1 model of thermo-elasticity is based on Fourier’s law,
which means, that the heat flux is proportional to the gradient of the temperature. The present
paper is devoted to the study of type-1 systems for homogeneous but anisotropic media in R2.
There are different results in the literature for certain anisotropic media (cubic in [3]; rhombic
in [4]). Our goal is to present an approach, which allows to consider (an)isotropic models in R2
from a unified point of view.
We consider the type-1 system of thermo-elasticity
Utt +A(D)U + γ∇θ = 0, (1.1a)
θt − κ∆θ + γ∇TUt = 0. (1.1b)
Here A(D) denotes the elastic operator, which is assumed to be a homogeneous second order
2×2 matrix of (pseudo) differential operators and models the elastic properties of the underlying
medium. Furthermore, κ describes the conduction of heat and γ the thermo-elastic coupling of
the system. We assume κ > 0 and γ 6= 0. We solve the Cauchy problem for system (1.1) with
initial data
U(0, ·) = U1, Ut(0, ·) = U2, θ(0, ·) = θ0, (1.2)
for simplicity we assume U1, U2 ∈ S(R2,R2) and θ0 ∈ S(R2). We denote by A(ξ) the symbol of
the elastic operator and we set η = ξ/|ξ| ∈ S1. Then some basic examples for our approach are
given as follows. The material constants are always specified in such a way that the matrix A(η)
becomes positive.
Example 1.1. Cubic media in 2D are modelled by
A(η) =
(τ − µ)η21 + µ (λ+ µ)η1η2
(λ+ µ)η1η2 (τ − µ)η22 + µ
(1.3)
with constants τ, µ > 0, −2µ−τ < λ < τ . This case was treated e.g. in [3]. For the corresponding
elastic system see [12].
Example 1.2. Rhombic media in 2D are modelled by
A(η) =
(τ1 − µ)η21 + µ (λ + µ)η1η2
(λ+ µ)η1η2 (τ2 − µ)η22 + µ
(1.4)
http://arxiv.org/abs/0704.0125v3
2 MICHAEL REISSIG AND JENS WIRTH
with constants τ1, τ2, µ > 0 and −2µ−
τ1τ2 < λ <
τ1τ2. For this case we refer also to [4].
Example 1.3. Although it is not the main point of this note, we can consider isotropic media,
where
A(η) = µI + (λ + µ)η ⊗ η
(λ+ µ)η21 + µ (λ+ µ)η1η2
(λ+ µ)η1η2 (λ+ µ)η
2 + µ
(1.5)
with Lamé constants µ > 0 and λ+ µ > 0.
We will present a unified treatment of these cases of (in general) anisotropic thermo-elasticity.
For this we assume that the homogeneous symbol A = A(ξ) = |ξ|2A(η), η = ξ/|ξ|, is given as a
function
A : S1 → C2×2 (1.6)
subject to the conditions
(A1): A is real-analytic in η ∈ S1,
(A2): A(η) is self-adjoint and positive for all η ∈ S1.
For some results we require that
(A3): A(η) has two distinct eigenvalues, # specA(η) = 2.
Under assumption (A3) the direction η ∈ S1 is called (elastically) non-degenerate. In this case we
know that the elasticity equation Utt+A(D)U = 0 is strictly hyperbolic and can be diagonalised
smoothly using a corresponding system of normalised eigenvectors rj(η) to the eigenvalues κj(η)
of A(η).
If (A3) is violated we will call the corresponding directions η ∈ S1 degenerate. For these
directions we can use the one-dimensionality of S1 in connection with the analytic perturbation
theory of self-adjoint matrices (cf. [6]). So we can always find locally smooth eigenvalues κj(η)
and corresponding locally smooth normalised eigenvectors rj(η) of A(η). For the following we
assume for simplicity that these functions extend to global smooth functions on S1.
This classification of directions is not sufficient for a precise study of Lp–Lq decay estimates
for solutions to the thermo-elastic system. It turns out that different microlocal directions
η = ξ/|ξ| ∈ S1 from the phase space have different influence on decay estimates. But how to
distinguish these directions and how to understand their influence? In general, this can be done
by a refined diagonalisation procedure applied to a corresponding first order system (first order
with respect to time). Applying a partial Fourier transform and chosing a suitable energy (of
minimal dimension) this system reads as DtV = B(ξ)V . The properties of the matrix B(ξ) are
essential for our understanding:
• the notions of hyperbolic and parabolic directions depend on the behaviour of the eigen-
values of B(ξ) (see (2.3), (2.5), Definition 1);
• the matrix B(ξ) contains spectral data of A(ξ) together with certain coupling functions
(see (2.6)) between different components of the energy. The behaviour of these coupling
functions close to hyperbolic directions has an essential influence on decay rates (see
Theorems 3.1, 3.5 and 3.6).
It turns out that we have to exclude some exceptional values of the coupling constant γ by
assuming that (see Definition 1 for the notion of hyperbolic directions)
(A4): γ2 6= 2κj0(η̄)− trA(η̄) for all hyperbolic directions η̄ with respect to κj0 .
Basically this implies the non-degeneracy of the 1-homogeneous part of B(ξ). In the following
we will call a hyperbolic direction violating (A4) a γ-degenerate direction. Assumption (A4) is
used for the treatment of small hyperbolic frequencies and plays there a similar rôle like (A3)
for large frequencies.
ANISOTROPIC THERMO-ELASTICITY IN 2D 3
In Section 2 we will give the transformation of the thermo-elastic system (1.1) to a system
of first order and the diagonalisation procedure in detail. The proposed procedure generalises
those from [14], [15], [9], [17]. The obtained results are used to represent solutions of the original
system as Fourier integrals with complex phases. Based on these representations we give micro-
localised decay estimates for solutions in Section 3. They and their method of proof depend
• the classification of directions (to be hyperbolic or parabolic);
• the order of contact between Fresnel curves (coming from the elastic part) and their
tangents for hyperbolic directions;
• the vanishing order of the coupling functions in hyperbolic directions.
Let us formulate some of the results. The first one follows from Theorem 3.1 and Corollary 3.2.
Result. Under assumptions (A1) to (A4) and if the coupling functions vanish to first order in
hyperbolic directions the solutions U(t, x) and θ(t, x) to (1.1) satisfy the Lp–Lq estimate
‖DtU(t, ·)‖q + ‖
A(D)U(t, ·)‖q + ‖θ(t, ·)‖q
. (1 + t)−
) (‖U1‖p,r+1 + ‖U2‖p,r + ‖θ0‖p,r) (1.7)
for dual indices p ∈ (1, 2], pq = p+ q, and Sobolev regularity r > 2(1/p− 1/q).
If the coupling functions vanish to higher order we have to relate their vanishing order ℓ to
the order of contact γ̄ between the Fresnel curve and its tangent in the corresponding direction
and for the corresponding sheet. In Theorems 3.5 and 3.6 we show that in this case the 1/2 in
the exponent is changed to 1/min(2ℓ, γ̄).
Our main motivation to write this paper is to provide a unified way to treat anisotropic
models of thermo-elasiticity. New analytical tools presented in these notes generalise to higher
dimensions and allow to treat especially models in 3D (outside degenerate directions). The two-
dimensional results from [3] for cubic media and [4] for rhombic media are contained / extended;
general anisotropic media can be treated. This is discussed in some detail in the second part [16]
of this note.
Acknowledgements. The first author thanks Prof. Wang Ya-Guang (Shanghai Jiao Tong
University) for the discussion about some basic ideas of the approach presented in this paper
during his stay at the TU Bergakademie Freiberg in August 2004. The stay was supported by
the German-Chinese research project 446 CHV 113/170/0-2.
2. General treatment of the thermo-elastic system
We use a partial Fourier transform with respect to the spatial variables to reduce the Cauchy
problem for (1.1) to the system of ordinary differential equations
Ûtt + |ξ|2A(η)Û + iγξθ̂ = 0, (2.1a)
θ̂t + κ|ξ|2θ̂ + iγξ · Ût = 0, (2.1b)
Û(0, ·) = Û1, Ût(0, ·) = Û2, θ̂(0, ·) = θ̂0 (2.1c)
parameterised by the frequency variable ξ.
We denote by κ1,κ2 ∈ C∞(S1) the eigenvalues of A(η) and by r1, r2 ∈ C∞(S1, S1) corre-
sponding normalised eigenvectors. Both depend in a real-analytic way on η ∈ S1. In a first step
we reduce (2.1) to a first order system. For this we use the diagonaliser of the elastic operator,
i.e. the matrix M(η) = (r1(η)|r2(η)) build up from the normalised eigenvectors, and define
U (0)(t, ξ) =MT (η)Û (t, ξ). (2.2)
4 MICHAEL REISSIG AND JENS WIRTH
Then we define by the aid of
D1/2(η) = diag(ω1(η), ω2(η)), ωj(η) =
κj(η) ∈ C∞(S1), (2.3)
the vector-valued function
V (0)(t, ξ) =
(Dt +D1/2(ξ))U (0)(t, ξ)
(Dt −D1/2(ξ))U (0)(t, ξ)
θ̂(t, ξ)
 , (2.4)
where as usual Dt = −i∂t. It satisfies a first order system with apparently simple structure. A
short calculation yields DtV
(0)(t, ξ) = B(ξ)V (0)(t, ξ) with
B(ξ) =
ω1(ξ) iγa1(ξ)
ω2(ξ) iγa2(ξ)
−ω1(ξ) iγa1(ξ)
−ω2(ξ) iγa2(ξ)
a1(ξ) − iγ2 a2(ξ) −
a1(ξ) − iγ2 a2(ξ) iκ|ξ|
, (2.5)
where we used the coupling functions
aj(ξ) = rj(η) · ξ. (2.6)
For later use we introduce the notation B1(ξ) and B2(ξ) for the homogeneous components of
B(ξ) of order 1 and 2, respectively. The coupling functions aj(η) can be understood as the
co-ordinates of η with respect to the orthonormal eigenvector basis {r1(η), r2(η)}. Therefore, it
holds a21(η) + a
2(η) = 1. Furthermore, they are well-defined and real-analytic functions on S
In the following proposition we collect some information on the characteristic polynomial of
the matrix B(ξ).
Proposition 2.1. (1) trB(ξ) = iκ|ξ|2 and detB(ξ) = iκ|ξ|6 detA(η).
(2) The characteristic polynomial of B(ξ) is given by
det(νI −B(ξ)) =(ν − iκ|ξ|2)(ν2 − κ1(ξ))(ν2 − κ2(ξ))
− νγ2|ξ|2a21(η)(ν2 − κ2(ξ))− νγ2|ξ|2a22(η)(ν2 − κ1(ξ)). (2.7)
(3) An eigenvalue ν ∈ specB(ξ), ξ 6= 0, is real if and only if ν2 = κj0 (ξ) for an index
j0 = 1, 2. If the direction is non-degenerate this is equivalent to aj0(η) = 0.
(4) If aj(η) 6= 0, j = 1, 2 the eigenvalues ν ∈ specB(ξ) satisfy
iκ|ξ|2
a21(ξ)
ν2 − κ1(ξ)
a22(ξ)
ν2 − κ2(ξ)
. (2.8)
It turns out that the property of B(ξ) to have real eigenvalues depends only on the direction
η = ξ/|ξ| ∈ S1. We will introduce a notation.
Definition 1. We call a direction η ∈ S1 hyperbolic if B(ξ) has a real eigenvalue and parabolic
if all eigenvalues of B(ξ) have non-zero imaginary part.
In hyperbolic directions we always have a pair of real eigenvalues. If η ∈ S1 is hyperbolic
with ±ωj0(ξ) ∈ specB(ξ) for ξ = |ξ|η, we call η hyperbolic with respect to the index j0 (or
with respect to the eigenvalue κj0(η) of A(η)) and ν±(ξ) = ±ωj0(ξ) the corresponding pair of
hyperbolic eigenvalues of B(ξ).
A non-degenerate direction is parabolic if and only if aj(η) 6= 0, j = 1, 2, while for non-
degenerate hyperbolic directions one of the coupling functions aj0(η) vanishes. Degenerate di-
rections are always hyperbolic (in 2D), see (2.7).
ANISOTROPIC THERMO-ELASTICITY IN 2D 5
Example 2.1. If the medium is isotropic, A(η) = µI+(λ+µ)η⊗η, the eigenvalues of A are µ and
λ+µ with corresponding eigenvectors η and η⊥. Thus all directions are hyperbolic (with respect
to the second eigenvalue). In this case the matrix B(ξ) decomposes into a diagonal hyperbolic
2 × 2-block and a parabolic 3 × 3-block. This decomposition coincides with the Helmholtz
decomposition as used in the standard treatment of isotropic thermo-elasticity.
Example 2.2. For cubic media (where we assume in addition µ 6= τ and µ + λ 6= 0) there exist
eight hyperbolic directions determined by η1η2 = 0 or η
1 = η
2 . The functions aj(η) have simple
zeros at these directions.
Example 2.3. Weakly coupled cubic media with λ+µ = 0, µ 6= τ , have the degenerate directions
η21 = η
2 , media with µ = τ , λ + µ 6= 0, for η1η2 = 0. In both cases the coupling functions aj(η)
do not vanish in these directions.
If µ = τ = −λ, the elastic system decouples directly into two wave equations with propagation
speed
µ. In this case all directions are degenerate.
Example 2.4. For rhombic media we have to distinguish between three cases.
Case 1. If the material constants satisfy (λ+2µ− τ1)(λ+2µ− τ2) > 0, we are close to the cubic
case and there exist eight hyperbolic directions given by η1η2 = 0 and
η21(λ+ 2µ− τ1) = η22(λ+ 2µ− τ2).
Case 2. If we assume on the contrary that (λ+2µ−τ1)(λ+2µ−τ2) < 0, only the four hyperbolic
directions η1η2 = 0 exist. In the Cases 1 and 2 in each hyperbolic direction one of the coupling
functions aj(η) vanishes to first order.
Case 3. In the borderline case τ1 = λ+2µ or τ2 = λ+2µ, but τ1 6= τ2, three hyperbolic directions
collapse to one. We have the four hyperbolic directions η1η2 = 0, at two of them (ηj = ±1 if
τj = λ+ 2µ) the vanishing order of the coupling function is three.
Rhombic media are degenerate if a) µ = τ1 (or µ = τ2) with degenerate direction (0, 1)
T (or
(1, 0)T ) or b) λ+µ = 0 (weakly coupled case) and (µ−τ1)(µ−τ2) > 0 with degenerate directions
determined by η21(µ− τ1) = η22(µ− τ2) or c) τi = µ = −λ (exceptional case) where all directions
are degenerate.
Proposition 2.2. Let the direction η̄ ∈ S1 be non-degenerate and hyperbolic with respect to the
index j0. Then the corresponding eigenvalues ν±(ξ) satisfy
a2j0(ξ)
ν2±(ξ) − κj0(ξ)
= qη̄(|ξ|) = Cη̄ ∓ iDη̄|ξ| (2.9)
for all non-tangential limits with real constants Cη̄, Dη̄ ∈ R, Dη̄ > 0. Furthermore, the imaginary
part of the hyperbolic eigenvalue satisfies
Im ν±(ξ)
a2j0(η)
D2η̄|ξ|2
C2η̄ + |ξ|2D2η̄
> 0, (2.10)
and thus vanishes like Im ν(|ξ|η) ∼ a2j0(η) as η → η̄ for all ξ 6= 0.
6 MICHAEL REISSIG AND JENS WIRTH
Proof. Let for simplicity j0 = 1. We use the characteristic polynomial of B(ξ) to deduce
γ2|ξ|2a21(η)
ν2± − |ξ|2κ1(η)
iκ|ξ|2
γ2|ξ|2a22(η)
ν2± − |ξ|2κ2(η)
→ 1∓ iκ|ξ|
ω1(η̄)
κ1(η̄)− κ2(η̄)
(2.11)
= 1− γ
κ1(η̄)− κ2(η̄)
︸ ︷︷ ︸
γ2Cη̄
ω1(η̄)
︸ ︷︷ ︸
γ2Dη̄
|ξ| = γ2qη̄(|ξ|). (2.12)
The existence of the limit is implied by ν± 6= 0 and ν2± 6= κ2(ξ) as consequence of ν±(ξ) →
±|ξ|ω1(η) as η → η̄ by Proposition 2.1.
Obviously, Im qη̄(|ξ|) = ∓κ|ξ|/γ2ω1(η̄) = ∓Dη̄|ξ| is non-zero for ξ 6= 0 and considering the
imaginary part of the first limit expression
Im qη̄(|ξ|) = lim
a21(ξ)
ν2±(ξ)− κ1(ξ)
= lim
−2Re ν±(ξ) Im ν±(ξ) a21(ξ)
|ν2±(ξ)− κ1(ξ)|2
= ∓2ω1(|ξ|η̄)|qη̄(|ξ|)|2 lim
Im ν±(|ξ|η)
|ξ|2a21(η)
proves the second statement, limη→η̄
Im ν±(ξ)
a21(η)
Dη̄ |ξ|
2ω1(η̄)(C
η̄+|ξ|
2D2η̄)
In the case of isolated degenerate directions (others are not of interest, because then the
system is decoupled) we can find a replacement for Proposition 2.1.
Proposition 2.3. Let η̄ ∈ S1 be an isolated degenerate direction, κ1(η̄) = κ2(η̄). Then the
corresponding hyperbolic eigenvalues ν±(ξ) satisfy
ω1(ξ)− ν2±(ξ)
ω1(ξ)− ω2(ξ)
= a21(η̄) > 0, (2.13)
and, therefore,
Im ν±(ξ)
ω1(ξ)− ω2(ξ)
= 0, lim
ω1(ξ)− Re ν±(ξ)
ω1(ξ)− ω2(ξ)
= a21(η̄). (2.14)
Thus, if a1(η̄) 6= 0 then the eigenvalues ν±(ξ) approach ±ω1(ξ) at the contact order between
ω1(ξ) and ω2(ξ) (while they approach ±ω1(ξ) with a higher order if a1(ξ) = 0).
2.1. Asymptotic expansion of the eigenvalues as |ξ| → 0. If |ξ| is small the first order part
B1(ξ) dominates B2(ξ), so the properties of the eigenvalues are governed by spectral properties
of B1(ξ).
Proposition 2.4. (1) trB1(ξ) = 0 and detB1(ξ) = 0.
(2) If the direction η ∈ S1 is parabolic the nonzero eigenvalues ν̃ of B1(η) satisfy
γ−2 =
a21(η)
ν̃2 − κ1(η)
a22(η)
ν̃2 − κ2(η)
(2.15)
and are thus real and related to κj(η) by
0 < κ1(η) < ν̃
1(η) < κ2(η) < ν̃
2(η) (2.16)
(if κ1(η) < κ2(η)).
ANISOTROPIC THERMO-ELASTICITY IN 2D 7
(3) If η is non-degenerate and hyperbolic with respect to the index 1 we have κ1(η) = ν̃
1(η),
while for hyperbolic directions with respect to the index 2 three cases occur depending on
the size of the coupling constant γ:
γ2 < κ2(η̄)− κ1(η̄) : κ2(η) = ν̃22 (η),
γ2 = κ2(η̄)− κ1(η̄) : ν̃21(η) = κ2(η) = ν̃22 (η),
γ2 > κ2(η̄)− κ1(η̄) : ν̃21(η) = κ2(η).
(2.17)
(4) If the direction is degenerate, κ1(η) = κ2(η), we have the eigenvalues ±
κ1(η) and
κ1(η) + γ2.
The existence of five distinct eigenvalues of the homogeneous principal part B1(η) for all
parabolic and most hyperbolic directions allows us to calculate the full asymptotic expansion of
the eigenvalues ν(ξ) of B(ξ) as |ξ| → 0. We will give only the first terms in detail, but provide
the whole diagonalisation procedure. Assumption (A4) guarantees the non-degeneracy of B1(ξ).
Note that, even if (A4) is violated the matrix B1(ξ) is diagonalisable (as consequence of its block
structure).
Proposition 2.5. As |ξ| → 0 the eigenvalues of the matrix B(ξ) behave as
ν0(ξ) = iκ|ξ|2b0(η) +O(|ξ|3), (2.18a)
ν±j (ξ) = ±|ξ|ν̃j(η) + iκ|ξ|
2bj(η) +O(|ξ|3) (2.18b)
for all non-γ-degenerate directions, where the functions bj ∈ C∞(S1) are given by
b0(η) =
γ2a21(η)
κ1(η)
γ2a22(η)
κ2(η)
> 0 (2.19)
bj(η) =
1 + γ2a21(η)
ν̃2j + κ1(η)
(ν̃2j − κ1(η))2
+ γ2a22(η)
ν̃2j + κ2(η)
(ν̃2j − κ2(η))2
≥ 0. (2.20)
Furthermore, bj(η) > 0 if η is parabolic and bj0(η) = 0 if η is hyperbolic with respect to the index
Proof. We apply a diagonalisation scheme in order to extract the spectral information for B(ξ).
We assume that the eigenvalues are denoted such that κ1(η) ≤ κ2(η).
Step 1. By Proposition 2.4 we know that the homogeneous first order part B1(η) has the
distinct eigenvalues ν̃0 = 0 and ν̃
j (η) = ±ν̃j(η), which are ordered as κ1(η) ≤ ν̃1(η) ≤ κ2(η) ≤
ν̃2(η) (where equality holds only under the exceptions stated in Proposition 2.4). We denote
corresponding normalised and bi-orthogonal left and right eigenvectors of the matrix B1(η) by
±(η) and e±j (η). If we collect them in the matrices
L(η) = (0e(η)|1e+(η)|1e−(η)|2e+(η)|2e−(η)), (2.21a)
R(η) = (e0(η)|e+1 (η)|e
1 (η)|e
2 (η)|e
2 (η)), (2.21b)
we have L∗(η)R(η) = I and
L∗(η)B1(η)R(η)) = D1(η) = diag(0, ν̃1(η),−ν̃1(η), ν̃2(η),−ν̃2(η)). (2.22)
Further we get
L∗(η)B2(η)R(η) = iκb∗(η)⊗ ∗b(η), (2.23)
where b∗(η) and ∗b(η) are vectors collecting the last entries b0(η), b
j (η) and 0b(η), jb
±(η) of the
eigenvectors e0(η), e
j (η) and 0e(η), je
±(η), respectively.
8 MICHAEL REISSIG AND JENS WIRTH
The matrix
B(0)(ξ) = L∗(η)B(ξ)R(η) = |ξ|D1(η) + |ξ|2iκb∗(η)⊗ ∗b(η) (2.24)
is diagonalised modulo O(|ξ|2) as |ξ| → 0 and has a main part with distinct entries. We denote
R(2)(ξ) = |ξ|2iκb∗ ⊗ ∗b.
Step 2. We construct a diagonaliser of B(0)(ξ) as |ξ| → 0 of the form
Nk(ξ) = I +
|ξ|jN (j)(η). (2.25)
For this we denote the k-homogeneous part of R(k)(ξ) by R̃(k)(ξ) and its entries by R̃
ij (η).
Then we set for k = 1, 2, . . .
Dk+1(η) = diag R̃(k+1)(η), (2.26)
N (k)(η) =
(k+1)
12 (η)
d1(η)−d2(η)
· · · R̃
(k+1)
15 (η)
d1(η)−d5(η)
(k+1)
21 (η)
d2(η)−d1(η)
0 · · · R̃
(k+1)
25 (η)
d2(η)−d5(η)
. . .
(k+1)
51 (η)
d5(η)−d1(η)
(k+1)
52 (η)
d5(η)−d2(η)
· · · 0
, (2.27)
where dj(η) are the entries of D1(η). By construction we have the commutator relation
[D1(η), N (k)(η)] = Dk+1(η)− R̃(k+1)(η), (2.28)
such that
R(k+2)(ξ) = B(0)(ξ)Nk(ξ)−Nk(ξ)
|ξ|jDj(η)
= R(k+1)(ξ)
+ |ξ|kB(0)(ξ)N (k)(η)− |ξ|kN (k)(η)
|ξ|jDj(η) −Nk(ξ)|ξ|k+1Dk+1(η)
= R(2)(ξ)|ξ|kN (k)(η)− |ξ|kN (k)(η)
|ξ|jDj(η)− (Nk(ξ)− I)|ξ|k+1Dk+1(η)
= O(|ξ|k+2).
Using Nk(ξ) − I = O(|ξ|) we see that for |ξ| ≤ ck, ck sufficiently small, the matrix Nk(ξ) is
invertible and
N−1k (ξ)B
(0)(ξ)Nk(ξ) =
|ξ|jDj(η) +O(|ξ|k+2). (2.29)
Thus, the entries of Dj(η) contain the asymptotic expansion of the eigenvalues, while the rows
of Nk(ξ)R(η) (and L(η)N
k (ξ)) give asymptotic expansions of the right (and left) eigenvectors
of B(ξ).
Furthermore, the construction implies that all occurring matrices are smooth functions of
η ∈ S1.
Step 3. We calculate the first terms explicitly. For this we need the diagonal entries of the
matrix b∗ ⊗ ∗b. Therefore, we determine the left and right eigenvectors of B1(η). If we assume
ANISOTROPIC THERMO-ELASTICITY IN 2D 9
that the direction η is non-degenerate we get for e0(η) = (r
1 , r
1 , r
2 , r
2 , r0)
T and 0e(η) =
(ℓ+1 , ℓ
1 , ℓ
2 , ℓ
2 , ℓ0)
T the equations
±r±j ωj + iγajr0 = 0, ±ℓ
j ωj −
ajℓ0 = 0, (2.30a)
1 + a2r
2 + a1r
1 + a2r
2 = 0, a1ℓ
1 + a2ℓ
2 + a1ℓ
1 + a2ℓ
2 = 0, (2.30b)
together with the normalisation condition
r+1 ℓ
1 + · · · r
2 + r0ℓ0 = 1. (2.31)
The first equations imply the representation
±r±j (η) = −
iγaj(η)
ωj(η)
r0(η), ±ℓ±j (η) =
iγaj(η)
2ωj(η)
ℓ0(η), (2.32)
the second line of equations follows from the first, while the normalisation condition yields
b0(η) = r0(η) ℓ0(η) =
γ2a21(η)
κ1(η)
γ2a22(η)
κ2(η)
6= 0. (2.33)
To calculate the eigenvectors we can further require r0(η) = ℓ0(η) =
b0(η) > 0.
Similarly, we obtain for the eigenvectors e+k (η) = (r
1 , . . . , r
2 , r0)
T and ke
+(η) = (ℓ+1 , . . . , ℓ
2 , ℓ0)
the equations (we use the same notation as above in the hope that this will not lead to confusion
here)
±r±j ωj + iγajr0 = ±ν̃kr
j , ±ℓ
j ωj −
ajℓ0 = ±ν̃k(η)ℓ±j (2.34)
together with the normalisation condition. Thus for parabolic directions we get
±r±j (η) =
iγaj(η)
ν̃k(η)− ωj(η)
r0(η), ±ℓ±j (η) = −
iγaj(η)
2(ν̃k(η)− ωj(η))
ℓ0(η), (2.35)
and hence
b+k (η) = r0(η) ℓ0(η) =
j=1,2
γ2a2j(η)
2(ν̃k(η)− ωj(η))2
j=1,2
γ2a2j(η)
2(ν̃k(η) + ωj(η))2
1 + γ2a21(η)
ν̃2k(η) + κ1(η)
(ν̃2k(η) − κ1(η))2
+ γ2a22(η)
ν̃2k(η) + κ2(η)
(ν̃2k(η)− κ2(η))2
. (2.36)
For e−k (η) and ke
−(η) we have to replace ν̃k(η) by −ν̃k(η) and obtain b−k (η) = b
k (η) = bk(η).
If the direction η is non-degenerate and hyperbolic with respect to the index j0, the entries r
and ℓ±j0 are undetermined by (2.34), while the other entries of the vectors are zero. Together with
the normalisation condition this determines the eigenvectors and gives bj0(η) = 0. It remains to
consider degenerate directions. Then we have ν̃1 = ω1 and ν̃2 > ω1 such that for k = 1 we have
ℓ0 = r0 = 0, r
1 and ℓ
1 are non-zero while r
2 = ℓ
2 = 0, especially b1(η) = 0. For k = 2 we get
from the above expression for b2(η) = (2 + 2κ
2/γ2)−1 > 0. �
Remark. 1. Note that for all non-degenerate hyperbolic directions η̄ ∈ S1 with respect to the
index 1 the limit
a21(η)b
1 (η) =
κ1(η̄)
κ1(η̄)− κ2(η̄)
(2.37)
is taken and non-zero, while for hyperbolic directions with respect to the index 2 the corre-
sponding limit is non-zero only if γ2 6= κ2(η̄) − κ1(η̄), i.e. if the direction is not γ-degenerate.
Near γ-degenerate directions Step 1 of the previous proof is still valid. Similar to Step 2 we
10 MICHAEL REISSIG AND JENS WIRTH
can diagonalise to a (2, 2, 1) block structure. The eigenvalues of these blocks can be calculated
explicitely.
2. For degenerate directions we obtain similarly
b1(η)
(κ1(η)− κ2(η))2
a21(η̄)a
2(η̄)
2γ2κ1(η̄)
(2.38)
and b1(η) vanishes to the double contact order.
2.2. Asymptotic expansion of the eigenvalues as |ξ| → ∞. If we consider large frequencies
the second order part B2(ξ) dominates B1(ξ). This makes it necessary to apply a different two-
step diagonalisation scheme. We follow partly ideas from [9], [14], [15] adapted to our special
situation.
Proposition 2.6. As |ξ| → ∞ the eigenvalues of the matrix B(ξ) behave as
ν0(ξ) = iκ|ξ|2 −
+O(|ξ|−1), (2.39a)
ν±j (ξ) = ±|ξ|ωj(η) +
a2j (η) +O(|ξ|−1). (2.39b)
for all non-degenerate directions ξ/|ξ| ∈ S1.
Remark. 1. Note that, while in hyperbolic directions we always have ν±j0(ξ) = ±|ξ|ωj0(η) for
one index j0, in degenerate hyperbolic directions all aj(η) may be non-zero. Hence the statement
of the above theorem cannot be valid in such directions in general.
2. Degenerate directions play for large frequencies a similar rôle as γ-degenerate directions play
for small frequencies.
Proof. The proof will be decomposed into several steps. In a first step we use the main part
B2(ξ) = iκ|ξ|2 diag(0, 0, 0, 0, 1) to block-diagonalise B(ξ). In a second step we diagonalise the
upper 4× 4 block for all non-degenerate directions.
Step 1. For a matrix B ∈ C5×5 we denote by b-diag4,1B the block diagonal of B consisting
of the upper 4 × 4 block and the lower corner entry. We construct a diagonalisation scheme to
block-diagonalise B(ξ) as |ξ| → ∞.
We set R(−1)(ξ) = B(ξ) − B2(ξ) = B1(ξ) and B−2(ξ) = B2(ξ) and construct recursively a
diagonaliser modulo the upper 4× 4 block,
Mk(ξ) = I +
|ξ|−jM (j)(η). (2.40)
Again we denote by R̃(k)(η) the (−k)-homogeneous part of R(k)(ξ) (which exists because it exists
for R(−1)(ξ) and the existence is transfered by the construction). Then we introduce the recursive
scheme
Bk−2(η) = b-diag4,1 R̃(k−2)(η), (2.41)
M (k)(η) =
(k−2)
15 (η)
(k−2)
45 (η)
−R̃(k−2)51 (η) · · · −R̃
(k−2)
54 (η) 0
(2.42)
for k = 1, 2, . . ., such that the commutator relation
[B−2(η),M (k)(η)] = Bk−2(η)− R̃(k−2)(η), (2.43)
ANISOTROPIC THERMO-ELASTICITY IN 2D 11
holds. Thus it follows
R(k−1)(ξ) = B(ξ)Mk(ξ)−Mk(ξ)
|ξ|−jBj(η) = O(|ξ|1−k) (2.44)
and using that Mk(ξ) is invertible for |ξ| ≥ Ck, Ck sufficiently large, we obtain the block diago-
nalisation
M−1k (ξ)B(ξ)Mk(ξ) =
|ξ|−jBj(η) +O(|ξ|1−k), (2.45)
where Bj(η) = b-diag4,1 Bj(η) is (4, 1)-block diagonal.
Step 2. By Step 1 we constructed Mk(ξ) such that M
k (ξ)B(ξ)Mk(ξ) is (4, 1)-block diagonal
modulo O(|ξ|1−k). The upper 4×4 block has already diagonal main part |ξ|−1D−1(η) = B1(ξ). If
the direction η = ξ/|ξ| is non-degenerate, the diagonal entries ±ω1(η) and ±ω2(η) are mutually
distinct and thus we can apply the standard diagonalisation procedure (cf. proof of Proposi-
tion 2.5) in the corresponding subspace. This does not alter the lower corner entry and gives
only combinations of the entries of the last column and of the last row (without changing their
asymptotics).
Thus we can construct a matrix Nk−1(ξ) = I +
j=1 |ξ|−jN (j)(ξ), which is invertible for
|ξ| > C̃k−1, C̃k−1 sufficiently large, such that
N−1k−1(ξ)M
k (ξ)B(ξ)Mk(ξ)Nk−1(ξ) = |ξ|
2B−2(η) +
|ξ|−jDj(η) +O(|ξ|1−k) (2.46)
is diagonal modulo O(|ξ|1−k).
Step 3. We give the first matrices explicitly. Following Step 1 we get
M (1)(η) =
γa1(η)
γa2(η)
γa1(η)
γa2(η)
γa1(η)
γa2(η)
γa1(η)
γa2(η)
(2.47)
together with B−1(η) = diag(ω1(η), ω2(η),−ω1(η),−ω2(η), 0) and
R̃(0)(η) = B1(η)M
(1)(η)−M (1)(η)B−1(η)
γ2a21(η)
γ2a1(η)a2(η)
· · · γa1(η)ω1(η)
γ2a1(η)a2(η)
γ2a22(η)
· · · γa2(η)ω2(η)
. . .
γa1(η)ω1(η)
γa2(η)ω2(η)
· · · −iγ
, (2.48)
B0(η) = b-diag4,1 R̃(0)(η) (2.49)
in the first diagonalisation step. Applying a second step alters only the last row and column to
O(|ξ|−1). Following Step 2 we diagonalise the upper 4× 4 block to |ξ|B−1(η) +B0(η) +O(|ξ|−1)
modulo O(|ξ|−1) and the statement is proven. �
Remark. If the direction η is degenerate, i.e. ω1(η) = ω2(η), we can block-diagonalise in Step
2 to (2, 2, 1)-block form. To diagonalise further we have to know that the 0-homogeneous part
of these 2× 2-blocks has distinct eigenvalues.
12 MICHAEL REISSIG AND JENS WIRTH
One possible treatment of degenerate directions is given in the following proposition. Note
that a corresponding statement can be obtained for γ-degenerate directions as |ξ| → 0.
Proposition 2.7. Let η̄ be an isolated degenerate direction. Then the corresponding hyperbolic
eigenvalue satisfies in a small conical neighbourhood of η̄
ν−j0(ξ) =
ω1(ξ) + ω2(ξ)
(ω1(ξ)− ω2(ξ))2
iγ2(ω1(ξ)− ω2(ξ))(a21(η)− a22(η))
+O(|ξ|−1). (2.50)
Proof. We follow the treatment of the previous proof to (2, 2, 1)-block-diagonalise B(ξ) modulo
|ξ|−1. Now, we consider one of its 2× 2-blocks. (We use a similar notation as before in the hope
that it will not lead to any confusion.) Such a block is given by
B(ξ) = |ξ|B−1(η) + B0(η) +O(|ξ|−1), (2.51)
where
B−1(η) = diag
ω1(η), ω2(η)
, (2.52)
B0(η) =
a21(η) a1(η)a2(η)
a1(η)a2(η) a
. (2.53)
In the direction η̄ both diagonal entries of B−1 coincide. In a small conical neighbourhood we
denote the eigenvalues of |ξ|B−1(η) + B0(η) as δ+(ξ) and δ−(ξ). A simple calculation yields
δ±(ξ) =
ω1(ξ) + ω2(ξ)
(ω1(ξ)− ω2(ξ))2
iγ2(ω1(ξ)− ω2(ξ))(a21(η)− a22(η))
(2.54)
with δ−(ξ̄) = ω1(ξ̄) and δ+(ξ̄) = ω1(ξ̄) +
. The hyperbolic eigenvalue corresponds to δ−(ξ).
These eigenvalues are distinct in a sufficiently small neighbourhood of η̄ (and may coincide only if
a1(η) = a2(η) and (ω1(ξ)−ω2(ξ))2 = γ4/(4κ2), which gives eventually two parabolic directions).
Hence the perturbation theory of matrices implies ν+j0(ξ) = δ−(ξ) +O(|ξ|
−1) in a sufficiently
small neighbourhood of η̄ and the statement is proven. �
Remark. Note, that
ω1(ξ)− δ−(ξ)
ω1(ξ)− ω2(ξ)
= a21(η̄) (2.55)
for all fixed |ξ|, which coincides with the result (2.13) of Proposition 2.3 for the eigenvalue ν(ξ).
2.3. Collecting the results. The asymptotic expansions from Propositions 2.5 and Proposi-
tion 2.6 imply estimates for eigenvalues of B(ξ) and the proofs give representations of corre-
sponding eigenvectors. For the application of multiplier theorems and the proof of Lp–Lq decay
estimates it is essential to provide also estimates for derivatives of them.
Assume that the eigenvalues under consideration are simple. From the asymptotic expansions
we know that this is the case for small frequencies and also for large frequencies. For the middle
part it will be sufficient to know that the hyperbolic eigenvalues are separated, which follows for
sufficiently small conical neighbourhoods of these directions.
ANISOTROPIC THERMO-ELASTICITY IN 2D 13
In a first step we consider derivatives of the eigenvalues. Differentiating the characteristic
polynomial
0 = det(ν(ξ)I −B(ξ)) =
Ik(ξ)ν(ξ)
k (2.56)
with respect to ξ yields by Leibniz formula
ξ Ik(ξ)
ξ ν(ξ)
(2.57)
for all multi-indices α ∈ N20. Thus we can express the highest derivative of ν(ξ)k in terms of
lower ones and hence Faà di Bruno’s formula (see e.g. [5]) yields an expression
Dαν(ξ)
kIk(ξ)ν(ξ)
k−1 =
Ck,α,β
Dα−βIk(ξ)
Dβν(ξ)
(2.58)
with certain constants Ck,α,β . Because the eigenvalue has multiplicity one, the sum on the left-
hand side is nonzero and therefore we can calculate the derivatives of ν(ξ) by this expression.
Furthermore, it follows that for small and large frequencies the derivatives of the eigenvalue have
full asymptotic expansions and thus we are allowed to differentiate the asymptotic expansions
term by term.
It remains to consider the corresponding eigenprojections. Recall that if ν(ξ) is a eigenvalue of
multiplicity one and r(ξ) and l(ξ) are corresponding right and left eigenvectors, the correspond-
ing eigenprojection is given by the dyadic product Pν(ξ) = l(ξ) ⊗ r(ξ). Thus the constructed
diagonaliser matrices imply asymptotic expansions of these operators. Again we are only inter-
ested whether the derivatives of these eigenprojections also possess asymptotic expansions (in
order to see whether it is allowed to differentiate term by term).
For this we use the representation
Pν(ξ) =
ν̃∈specB(ξ)\{ν}
(ν̃(ξ)I −B(ξ))(ν̃(ξ)− ν(ξ))−1 (2.59)
given e.g. in [4], [7]. All terms on the right-hand side have full asymptotic expansions as |ξ| → 0
and |ξ| → ∞ together with all of their derivatives. Differentiating with respect to ξ yields the
same result for the eigenprojection. Thus we obtain
Proposition 2.8. The asymptotic expansions from Proposition 2.5 and Proposition 2.6 may be
differentiated term by term to get asymptotic expansions for the derivatives of the eigenvalues.
Furthermore, the same holds true for the corresponding eigenprojections.
From Proposition 2.1 we know that an eigenvalue ν(ξ) of the matrix B(ξ) is real if and only if
η = ξ/|ξ| is hyperbolic. We want to combine this information with the asymptotic expansions of
Proposition 2.5 and Proposition 2.6 and derive some estimates for the behaviour of the imaginary
part.
Proposition 2.9. Let c > 0 be a given constant.
(1) Let η = ξ/|ξ| ∈ S1 be parabolic. Then the eigenvalues of B(ξ) satisfy
Im ν(ξ) ≥ Cη > 0, |ξ| ≥ c, (2.60)
with a constant Cη depending on the direction η and c. Furthermore,
Im ν(ξ) ∼ b(η)|ξ|2, |ξ| ≤ c, (2.61)
where b(η) is one of the functions from Proposition 2.5.
14 MICHAEL REISSIG AND JENS WIRTH
(2) Let η̄ be non-degenerate and hyperbolic with respect to the index 1. Then ν±1 (|ξ|η̄) =
±|ξ|ω1(η̄) and ν0(ξ) and ν±2 (ξ) satisfy the statement of point 1. Furthermore,
Im ν±1 (ξ) ∼ a21(η), |ξ| ≥ c, |η − η̄| ≪ 1, (2.62)
Im ν±1 (ξ) ∼ |ξ|2a21(η), |ξ| ≤ c, |η − η̄| ≪ 1. (2.63)
Proof. The first point follows directly from the asymptotic expansions, we concentrate on the
second one. We know that the hyperbolic eigenvalues ν±1 (ξ) satisfy by Proposition 2.2
Im ν±1 (ξ) = a
1(η)N
1 (ξ) (2.64)
with a smooth non-vanishing function N±1 (ξ) defined in a neighbourhood of η̄. By Proposition 2.5
and 2.6 we see that N±1 (ξ) also has full asymptotic expansions and thus
N±1 (ξ) =
+O(|ξ|−1), |ξ| → ∞ (2.65a)
N±1 (ξ) = iκ|ξ|
2 b1(η)
a21(η)
+O(|ξ|3), |ξ| → 0. (2.65b)
Together with (2.37) we get upper and lower bounds on N±1 (ξ) and the desired statement follows.
Remark. A similar reasoning allows to replace the hyperbolic eigenvalue near degenerate di-
rections by the model expression obtained in Proposition 2.7, thus ν(ξ) ∼ δ−(ξ) uniformly in a
sufficiently small conical neighbourhood of η̄.
3. Decay estimates for solutions
Our strategy to give decay estimates for solutions to the thermo-elastic system (1.1) is to
micro-localise them. In principle we have to distinguish four different cases. On the one hand
we differentiate between small and large frequencies, on the other hand between hyperbolic
directions and parabolic ones.
We distinguish between two cases depending on the vanishing order of the coupling functions.
If the coupling functions vanish to first order at hyperbolic directions only, we rely on simple
multiplier estimates. Later on we discuss coupling functions with higher vanishing order, where
the decay rates are obtained by tools closely related to the treatment of the elasticity equation.
3.1. Coupling functions with simple zeros. In a first step we consider the first order system
DtV = B(D)V, V (0, ·) = V0 (3.1)
to Cauchy data V0 ∈ S(R2,C5). For a cut-off function χ ∈ C∞(R+) with χ(s) = 0, s ≤ ǫ, and
χ(s) = 1, s ≥ 2ǫ, we consider
Ppar(η) =
η̄ hyperbolic
χ(|η − η̄|), Phyp(η) = 1− Ppar(η). (3.2)
Then Phyp(D) localises in a conical neighbourhood of the set of hyperbolic directions, while
Ppar(D) localises to a compact set of parabolic directions.
The asymptotic formulae and representations for the characteristic roots of the full symbol
B(ξ) allow us to proof decay estimates for the solutions.
ANISOTROPIC THERMO-ELASTICITY IN 2D 15
Theorem 3.1. Assume that (A1) to (A4) are satisfied and the coupling functions vanish in
hyperbolic directions to first order.
Then the solution V (t, x) to (3.1) satisfies the following a-priori estimates:
‖χ(D)Ppar(D)V (t, ·)‖q . e−Ct‖V0‖p,r (3.3a)
‖(1− χ(D))Ppar(D)V (t, ·)‖q . (1 + t)−(
)‖V0‖p (3.3b)
‖χ(D)Phyp(D)V (t, ·)‖q . (1 + t)−
)‖V0‖p,r (3.3c)
‖(1− χ(D))Phyp(D)V (t, ·)‖q . (1 + t)−
)‖V0‖p (3.3d)
for dual indices p ∈ (1, 2], pq = p+ q, and with Sobolev regularity r > 2(1/p− 1/q).
Remark. If B(ξ) is diagonalisable for ξ 6= 0 (which is valid e.g. if B(ξ) has no double eigenvalues
for ξ 6= 0) we can make the result even more precise. To each eigenvalue ν(ξ) ∈ specB(ξ) we
have corresponding left and right eigenvectors and associated to them the eigenprojection Pν(D)
such that
Pν(D)V (t, ·) = eitν(D)Pν(D)V0. (3.4)
Thus, we can single out the influence of one eigenvalue in this way. Note, that this is only of
interest in the neighbourhood of hyperbolic directions and for the corresponding eigenvalue and
there the assumption of diagonalisability of B(ξ) may be skipped (real eigenvalues are always
simple, for small |ξ| diagonalisation works, for large |ξ| everything goes well under the assumption
of non-degeneracy and on the middle part we make the neighbourhood small enough to exclude
possible multiplicities).
Proof. We decompose the proof into four parts corresponding to the four estimates. The micro-
localised estimates are merely standard multiplier estimates. We do not use stationary phase
method.
Step 1. Parabolic directions, large frequencies. In this case we have uniformly in ξ ∈ suppχPpar ,
the estimate Im ν(ξ) ≥ C′ > 0. Taking 0 < C < C′ we obtain for these ξ
ImB(ξ) =
B(ξ)−B∗(ξ)
in the sense of self-adjoint operators and the estimate follows from the L2–L2 estimate
‖χ(D)Ppar(D)V (t, x)‖22 =
‖χ(ξ)Ppar(ξ)V̂ (t, ξ)‖22
= 2Re
χ(ξ)Ppar(ξ)V̂ (t, ξ), χ(ξ)Ppar(ξ)∂tV̂ (t, ξ)
= −2 Im
χ(ξ)Ppar(ξ)V̂ (t, ξ), χ(ξ)Ppar(ξ)B(ξ)V̂ (t, ξ)
≤ −2C‖χ(ξ)Ppar(ξ)V̂ (t, ξ)‖2 = −2C‖χ(D)Ppar(D)V (t, x)‖22,
viewed as Hs–Hs estimate and combined with Sobolev embedding.
Step 2. Parabolic directions, small frequencies. We know from Proposition 2.5 that in this case
the matrix B(ξ) has only simple eigenvalues. We will make use of the representation of solutions
V (t, x) =
ν(ξ)∈specB(ξ)
eitν(D)Pν(D)V0(x) (3.5)
with corresponding eigenprojections (amplitudes) Pν(ξ). The amplitudes are uniformly bounded
on the set of all occurring ξ and possess full asymptotic expansions in |ξ| as ξ → 0 together with
their derivatives. Especially, by Hörmander-Mikhlin multiplier theorem [10, p. 96, Theorem 3]
the operators Pν(D) are L
p-bounded for 1 < p <∞.
16 MICHAEL REISSIG AND JENS WIRTH
It remains to consider the model multiplier eitν(ξ). From Proposition 2.5 we know that
|eitν(ξ)| . e−Ct|ξ|
, |ξ| ≤ c, (3.6)
with suitable constants c and C and thus the L1–L∞ estimate
‖eitν(D)f‖∞ ≤ ‖eitν(ξ)f̂‖1 ≤ ‖eitν(ξ)‖L1({|ξ|≤c})‖f̂‖∞
. ‖f‖1
e−Ct|ξ|
|ξ|d|ξ| . (1 + t)−1 ‖f‖1
holds for all f ∈ L1(R2) with supp f̂ ⊆ {|ξ| ≤ c}. Riesz-Thorin interpolation [1, Chapter 4.2]
with the obvious L2–L2 estimate gives the desired decay result.
Step 3. Hyperbolic directions, large frequencies. We take the conical neighbourhoods small
enough to exclude all multiplicities (related to the eigenvalue which becomes real in the hyperbolic
direction). Then similar to (3.5) the solution is represented as
V (t, x) = eitν(D)P+ν (D)V0(x) + e
−itν(D)P−ν (D)V0(x) + Ṽ (t, x), (3.7)
where Ṽ (t, x) corresponds to the remaining parabolic eigenvalues of B(ξ) and satisfies the es-
timate from Step 1. Again we can use smoothness of P±ν (ξ) together with the existence of a
full asymptotic expansion as |ξ| → ∞ to get Lp-boundedness of P±ν (D) for 1 < p < ∞ from
Hörmander-Mikhlin multiplier theorem.
It remains to understand the model multiplier e±itν(ξ) for the hyperbolic eigenvalue related
to the hyperbolic direction η̄. Using the estimate from Proposition 2.9 we conclude for r > 2
‖e±itν(D)f‖∞ ≤ ‖e±itν(ξ)f̂‖1 ≤ ‖e±itν(ξ)|ξ|−r‖L1(S1)‖ |ξ|
r f̂‖∞
. ‖〈D〉rf‖1
|ξ|1−rd|ξ|
e−C1tφ
. t−1/2 ‖〈D〉rf‖1, t ≥ 1
for all f ∈ 〈D〉−rL1(R2) with supp f̂ ⊆ S1 = {|ξ| ≥ c, |η − η̄| ≤ ǫ}. Riesz-Thorin interpolation
with the L2–L2 estimate gives the desired decay result.
Step 4. Hyperbolic directions, small frequencies. Like for large hyperbolic frequencies we make use
of the representation (3.7) to separate hyperbolic and parabolic influences. As in the previous
cases the existence of full asymptotic expansions imply that the projections P±ν (D) are L
bounded for 1 < p <∞.
It remains to understand the model multiplier e±itν(ξ). Using Proposition 2.9 we have |e±itν(ξ)| .
e−C2t|ξ|
2φ2 such that after introducing polar co-ordinates
‖e±itν(D)f‖∞ ≤ ‖e±itν(ξ)f̂‖1 ≤ ‖e±itν(ξ)‖L1(S2)‖f‖1
. ‖f‖1
e−C2tφ
2|ξ|2dφ|ξ|d|ξ|
. t−1/2‖f‖1
d|ξ| . t−1/2 ‖f‖1, t ≥ 1
for all f ∈ L1(R2) with supp f̂ ⊆ S2 = {|ξ| ≤ c, |η − η̄| ≤ ǫ}. Riesz-Thorin interpolation with
the L2–L2 estimate gives the desired decay result. �
Remark. 1. In non-degenerate hyperbolic directions where the coupling function vanishes to
order ℓ (cf. Example 2.4 case 3) we obtain by the same reasoning the weaker Lp–Lq decay rate
‖χ(D)Phyp(D)V (t, ·)‖q . (1 + t)−
)‖V0‖p,r. (3.8)
ANISOTROPIC THERMO-ELASTICITY IN 2D 17
It remains to understand whether this weaker decay rate is also sharp or whether an application
of stationary phase method may be used to improve this. We will discuss this in Section 3.2.
2. We can extend the estimate of Theorem 3.1 to the limit case p = 1, if we include the eigen-
projections Pν(ξ) into the considered model multiplier and use just their boundedness instead
of Hörmander-Mikhlin multiplier theorem. This is possible, because all the multiplier estimates
were based on Hölder inequalities.
So far we understood properties of solutions to the transformed problem (3.1) for the vector-
valued function V (t, x) given by (2.4) as
V (t, x) =
(Dt +D1/2(D))U (0)(t, x)
(Dt −D1/2(D))U (0)(t, x)
θ(t, x)
 , (3.9)
where U (0)(t, x) =MT (D)U(t, x) is the elastic displacement after transformation with the diag-
onaliser of the elastic operator. Because M(η) is unitary and homogeneous of degree zero this
diagonaliser is Lp-bounded for 1 < p <∞ with bounded inverse. Thus we have
DtU(t, x) =M(D)
V (t, x) (3.10)
A(D)U(t, x) =M(D)
0 − 1
0 − 1
V (t, x) (3.11)
such that as corollary of Theorem 3.1 we obtain
Corollary 3.2. Assume (A1) to (A4) and that the coupling functions vanish of first order in all
hyperbolic directions. Then the solution U(t, x) and θ(t, x) to (1.1) satisfy the a-priori estimates
‖DtU(t, ·)‖q + ‖
A(D)U(t, ·)‖q + ‖θ(t, ·)‖q
. (1 + t)−
) (‖U1‖p,r+1 + ‖U2‖p,r + ‖θ0‖p,r) (3.12)
for dual indices p ∈ (1, 2], pq = p+ q, and Sobolev regularity r > 2(1/p− 1/q).
Remark. Including the diagonaliser M(ξ) into the model multipliers of Theorem 3.1 allows to
overcome the restriction on p and to extend this statement up to p = 1. We preferred this way
of presenting the results because they allow to decouple both statements. Theorem 3.1 gives
deeper insight into the asymptotic behaviour of solutions than Corollary 3.2 does.
3.2. Coupling functions vanishing to higher order. We have seen that under the assump-
tion that the coupling functions aj(η) ∈ C∞(S1), aj(η) = η · rj(η) related to the symbol of the
elastic operator A(η) have only zeros of first order, we can deduce Lp–Lq decay estimates without
relying on the method of stationary phase.
Now we want to discuss how to use the method of stationary phase to deduce decay estimates
in the remaining cases. From Proposition 2.2 we know that in hyperbolic directions the imaginary
part Im ν±j0(ξ) of the hyperbolic eigenvalues vanishes like the square of the corresponding coupling
function a2j0(η), while the real part Re ν
(ξ) is essentially described by ±ωj0(ξ). First, we make
this more precise and formulate an estimate for Re νj0(ξ)
± ∓ ωj0(ξ) and its derivatives.
Proposition 3.3. Let η̄ ∈ S1 be hyperbolic with respect to the index j0 and aj0(η) vanish to
order ℓ in η̄. Then there exists a conical neighbourhood of the direction η̄ such that for all
k = 0, 1, . . . , 2ℓ− 1 the estimates
∣∂kη (Re ν
(ξ) ∓ ωj0(ξ))
∣ ≤ c(|ξ|)|η − η̄|2ℓ−k, (3.13a)
∣∂kη Im ν
∣ ≤ c(|ξ|)|η − η̄|2ℓ−k, (3.13b)
18 MICHAEL REISSIG AND JENS WIRTH
hold uniformly on it, where c(|ξ|) ∼ 1 as |ξ| → ∞ and c(|ξ|) ∼ |ξ| as |ξ| → 0.
This estimate may be used to transfer the micro-localised decay estimate for the elasticity
equation to the thermo-elastic system on a sufficiently small conical neighbourhood of η̄. We
need one further notion to prepare the main theorem of this section. Decay rates for solutions to
the elasticity equation depend heavily on the order of contact between the sheets of the Fresnel
surface and its tangents, cf. [13] or point 2 of the concluding remarks on page 21.
Proposition 3.4. For η ∈ S1 we denote by γ̄j(η) the order of contact between the j-th sheet
Sj = {ω−1j (η)η | η ∈ S
1} (3.14)
of the Fresnel curve and its tangent in the point ω−1j (η)η. Then
∂kη (∂
ηωj(η) + ωj(η)) = 0, k = 0, . . . , γ̄j(η)− 2, (3.15)
(if γ̄j(η) > 2) and
∂γ̄j(η)η ωj(η) + ∂
γ̄j(η)−2
η ωj(η) 6= 0. (3.16)
Proof. Follows by straight-forward calculation. The curvature of the j-th Fresnel curve Sj at the
point ω−1j (η)η factorises as ωj(η) + ∂
ηωj(η) and a smooth non-vanishing analytic function. �
For the following we chose the conical neighbourhood of the hyperbolic direction η̄ (with
respect to the index j0) small enough in order that the curvature of Sj0 vanishes at most in η̄.
Theorem 3.5. Let η̄ be hyperbolic with respect to the index j0 and let aj0(η) vanish in η̄ to order
ℓ > 1. Let us assume further that γ̄j0(η̄) < 2ℓ. Then
||e±itνj0 (D)f ||q . (1 + t)
)||f ||p,r (3.17)
for dual indices p ∈ [1, 2], pq = p + q, regularity r > 2(1/p − 1/q) and for all f with supp f̂
contained in a sufficiently small conical neighbourhood of η̄.
Proof. We distinguish between two cases related to the Fourier support of f , for general f we
can use linearity. In both cases we apply the method of stationary phase. Key lemma will be
the Lemma of van der Corput, cf. [11, p.334], combined for large frequencies with a dyadic
decomposition. It suffices to consider t ≥ 1, the estimate for t ≤ 1 follows from the uniform
boundedness of the Fourier multiplier together with Sobolev embedding using the regularity
imposed on the data, r > 2(1/p− 1/q).
In the following we skip the index j0 of the eigenvalue. Thus ω(η) stands for ωj0(η) and γ̄(η)
for γ̄j0(η) to shorten the notation.
Large and medium frequencies. Assume that f̂ is supported in a sufficiently small conical neigh-
bourhood of η̄ bounded away from zero, |ξ| > 1. We will use a dyadic decomposition in the radial
variable. Let for this χ ∈ C∞0 (R) be chosen in such a way that χ(R) = [0, 1], suppχ ⊆ [1/2, 2]
j∈Z χ(2
js) = 1 for all s ∈ R+. We set χj(s) = χ(2−js) such that suppχj ⊆ [2j−1, 2j+1].
We follow the treatment of Brenner [2] and use Besov spaces. Due to our assumptions on the
Fourier support of f Besov norms are given by
||f ||Brp,q =
∥2jr||χj(|D|)f ||p
ℓq(N0)
(3.18)
and corresponding Besov spaces are related to Sobolev and Lp-spaces by the embedding relations
Hp,r →֒ Brp,2, B0q,2 →֒ Lq (3.19)
ANISOTROPIC THERMO-ELASTICITY IN 2D 19
for p ∈ (1, 2] and q ∈ [2,∞). Thus it suffices to prove an Brp,2 → B0q,2 estimate for dual p and q.
The case p = q = 2, r = 0 is trivial by Plancherel and the uniform boundedness of the multiplier,
we concentrate on the B21,2 → B0∞,2 estimate. Therefore, we use
||e±itν(D)f ||B0
||χj(D)e±itν(D)f ||2∞
≤ sup
Ij(t) ||f ||Br
, (3.20)
where Ij(t) are estimates of the dyadic components of the operator,
Ij(t) = sup
|F−1[eitν(ξ)ψ(ξ)χj(|ξ|)|ξ|−r]|. (3.21)
Here ψ(ξ) localises to a small conical neighbourhood of η̄ bounded away from zero, |ξ| > 1,
containing the support of f̂ . We assume |ψ(ξ)|+ |∂ηψ(ξ)|+ |∂|ξ|ψ(ξ)| . 1 and ψ ≡ 1 on a smaller
conic set around η̄. We introduce polar co-ordinates |ξ| and φ (with φ = 0 corresponding to η̄)
and set x = tz. Using that the hyperbolic directions η̄ ∈ S1 are isolated we can assume that
the conical neighbourhood under consideration contains no further hyperbolic direction. For the
calculation of Ij(t) we integrate over the interval [−ǫ, ǫ] for φ. This yields for all j ∈ N0
Ij(t) = sup
∫ 2j+1
eit(Re ν(ξ)+z·ξ)−t Im ν(ξ)ψ(ξ)χj(|ξ|)|ξ|1−rdφd|ξ|
= 2j(2−r) sup
jt(Re ν(2jξ)2−j+z·ξ)e−t Im ν(2
jξ)ψ(2jξ)χ(|ξ|)|ξ|1−rdφd|ξ|
Due to Proposition 3.3 (let us restrict to the − case) we have
Re ν(2jξ) = 2jω(ξ) + 2j|ξ|α(2jξ), (3.22)
where
∣∂kφα(2
∣ ≤ Ck |φ|2ℓ−k, k = 0, 1, . . . , 2ℓ− 1. (3.23)
By our assumptions 2ℓ− 1 ≥ γ̄(η̄). We use this to reduce the problem to properties of the elastic
eigenvalue ωj(η) and consider
Ij(t) = 2
j(2−r) sup
j |ξ|(ω(η)+z·η+α(2jξ))e−t Im ν(2
jξ)ψ(2jξ)dφχ(|ξ|)|ξ|1−rdξ
. (3.24)
Note that, |α(2jξ)| . |φ|2ℓ is small, if we choose the neighbourhood of η̄ small enough.
If |ω(η)+z ·η| ≥ δ for some small δ the outer integral has no stationary points and integration
by parts implies arbitrary polynomial decay. We restrict to the z ∈ R2 with |ω(η) + z · η| ≤ δ
and consider only the inner integral
Ij(t, |ξ|) =
j|ξ|(ω(η)+z·η+α(2jξ))e−t Im ν(2
jξ)ψ(2jξ)dφ
. (3.25)
It can be estimated using the Lemma of van der Corput. For this we need an estimate for deriva-
tives of the phase. We start by considering the unperturbed phase ω(η) + z · η. Differentiation
yields
∂γ̄(η̄)η (ω(η) + z · η) = ∂γ̄(η̄)−2η (∂2ηω(η) + ω(η))− ∂γ̄(η̄)−4η (∂2ηω(η) + ω(η)) +−
· · ·+−
ω(η) + z · η, 2|γ̄(η̄)
∂ηω(η) + z · ηT , 26 |γ̄(η̄)
(3.26)
20 MICHAEL REISSIG AND JENS WIRTH
and by Proposition 3.4 the first term is non-zero for η̄ while the others are small in a neighbour-
hood of η̄. Choosing δ and the neighbourhood small enough and using (3.23) this implies a lower
bound on the γ̄(η̄)-th derivative of the phase,
|∂γ̄(η̄)η (ω(η) + z · η + α(2jξ))| & 1 (3.27)
uniformly on this neighbourhood of η̄ and independent of j. Thus we conclude by [11, p. 334]
for all t ≥ 1 and uniformly in z with |ω(η) + z · η| ≤ δ
Ij(t, |ξ|) ≤ Ck(t2j |ξ|)−
γ̄(η̄)
−t Im ν(2jξ)ψ(2jξ)
. (3.28)
To estimate the remaining integral we use Proposition 3.3
−t Im ν(2jξ)ψ(2jξ)
∣ dφ .
e−ctφ
|φ|2ℓ−1tdφ = 2
e−ctφ
φ2ℓ−1ctdφ
e−ctφ
Integrating over ξ ∈ [1/2, 2] and choosing r ≥ 2 we obtain for j ∈ N0
Ij(t) . (1 + t)
γ̄(η̄) , (3.29)
where the occurring constant is independent of j and (3.20) implies the desired B21,2 → B0∞,2
estimate. Interpolation with the L2–L2 estimate gives the corresponding Brp,2 → B0q,2 estimates
with r ≥ 2(1/p− 1/q) and finally the embedding relations to Sobolev and Lp-spaces the desired
estimate
||eitν(D)f ||q . (1 + t)−
γ̄(η̄) ||f ||p,r (3.30)
for all f satisfying the required Fourier support conditions.
Small frequencies. Assume now that f̂ is supported in a small conical neighbourhood of η̄ with
|ξ| ≤ 2. In this case we estimate directly by the method of stationary phase. We sketch the main
ideas. It is sufficient to estimate
I(t) = sup
eitν(ξ)ψ(ξ)χ(|ξ|)
∣ , (3.31)
where the function ψ ∈ C∞(R+) localises to the small neighbourhood of η̄ with |ξ| ≤ 2. Again
we require |ψ(ξ)| + |∂ηψ(ξ)| + |∂|ξ|ψ(ξ)| . 1. In correspondence to large frequencies I(t) equals
I(t) = sup
eit|ξ|(ω(η)+z·η+α(ξ))e−t Im ν(ξ)ψ(ξ)χ(|ξ|)dφ|ξ|d|ξ|
. (3.32)
We distinguish between |ω(η) + z · η| ≥ δ, where the outer integral has no stationary points, and
|ω(η) + z · η| ≤ δ. In the first case we can apply one integration by parts and get t−1 using
|ξ| |∂|ξ|e−t Im ν(ξ)| . |ξ|2te−t|ξ|
2φ2ℓ . 1. (3.33)
In the second case we can reduce the consideration to
I(t, |ξ|) =
eit|ξ|(ω(η)+z·η+α(ξ))e−t Im ν(ξ)ψ(ξ)dφ
(3.34)
and an application of the Lemma of van der Corput. By Propositions 3.3 and 3.4 we have
γ̄(η̄)
φ (ω(η) + z · η + α(ξ)) 6= 0 (3.35)
ANISOTROPIC THERMO-ELASTICITY IN 2D 21
for all ξ in a sufficiently small conic neighbourhood of η̄. So we obtain
I(t, |ξ|) . (t|ξ|)−
γ̄(η̄)
−t Im ν(ξ)ψ(ξ)
. (t|ξ|)−
γ̄(η̄) . (3.36)
Integrating with respect to |ξ| yields the estimate for I(t)
I(t) ≤
I(t, |ξ|)|ξ|d|ξ| . t−
γ̄(η̄)
|ξ|1−
γ̄(η̄) d|ξ| . (1 + t)−
γ̄(η̄) , t ≥ 1 (3.37)
and by Hölder inequality the L1–L∞ estimate
||eitν(D)f ||∞ ≤ I(t)||f ||1 . (1 + t)−
γ̄(η̄) ||f ||1 (3.38)
for all f satisfying the required Fourier support condition.
By interpolation with the obvious L2–L2 estimate we get Lp–Lq estimates and combining
them with the estimate of the first part proves the theorem. �
If the order of contact exceeds the vanishing order of the coupling funtions, the best we can do
is to use the idea of Section 3.1 and to apply standard multiplier estimates. As already remarked
after the proof of Theorem 3.1 we obtain as decay rate in this case:
Theorem 3.6. Let η̄ be hyperbolic with respect to the index j0 and let aj0(η) vanish in η̄ of order
ℓ. Let us assume further that γ̄j0(η̄) ≥ 2ℓ, where γ̄j0(η) is defined in Proposition 3.4. Then
||e±itνj0 (D)f ||q . (1 + t)−
)||f ||p,r (3.39)
for dual indices p ∈ [1, 2], pq = p + q, regularity r > 2(1/p − 1/q) and for all f with supp f̂
contained in a sufficiently small conical neighbourhood of η̄.
4. Concluding remarks
1. In a second part of this note, [16], we will give concrete applications of the general treatment
presented so far. From the remarks and examples we made in the previous sections it follows that
we indeed cover the estimates of [3] for cubic media and [4] for rhombic media in the situations
of coupling vanishing to first order.
In [16] we will discuss the situation of higher order tangencies for rhombic media and give a
new estimate extending the results of [4]. Furthermore, we will give concrete examples of media
for all achievable decay rates in the case of a differential elastic operator A(D).
2. In general decay rates of solutions are determined by vanishing properties of the coupling
functions. If the coupling functions are nonzero, decay rates are parabolic. If they vanish to
sufficiently high order, decay rates are hyperbolic and determined by the elastic operator (micro-
localised to this direction). In the intermediate case simple multiplier estimates are sufficient.
3. In the case of isotropic media one of the coupling functions vanishes identically. In this case
one pair of eigenvalues of the matrix B(ξ) is purely real ν±(ξ) = ±(λ+µ)|ξ| and the components
related to these eigenvalues solve a wave equation. Thus they satisfy the usual Strichartz type
decay estimates, [2],
‖e±it(λ+µ)|D|P±(D)V0‖q . (1 + t)−
)‖V0‖p,r (4.1)
for p ∈ (1, 2], pq = p + q and r = 2(1/p − 1/q). More generally, if we know that one pair of
eigenvalues satisfies ν±(ξ) = ±|ξ|ω(η) for all η ∈ S1 with a smooth function ω : S1 → R+, the
decay rates for the corresponding components depend heavily on geometric properties of the
Fresnel curve S = {ω−1(η)η | η ∈ S1 } ⊂ R2. Following [8] the estimate (4.1) is valid in this case
as long as the curvature of S never vanishes. If there exist directions η where the curvature of S
vanishes, the constant 1
in the exponent has to be altered to 1
, where γ̄ denotes the maximal
order of contact of the curve S to its tangent, [13].
22 MICHAEL REISSIG AND JENS WIRTH
4. The main focus of this paper was on the treatment of non-degenerate cases, thus assumptions
(A1) to (A4) are required. Nevertheless, we showed how to obtain a control on the eigenvalues
of the matrix B(ξ) in the exceptional cases where either (A3) or (A4) is violated. In the first
case the treatment of large frequencies has to be replaced by the investigation of the expression
obtained in Proposition 2.7, while in the latter one a corresponding replacement has to be made
for small frequencies.
5. The results in this paper are essentially two-dimensional. For the study of anisotropic thermo-
elasticity in higher space dimensions there arise two essential problems. The first is that we can
not assume (A3). For example in three space dimensions and for cubic media the symbol of
the elastic operator A : S2 → C3×3 has degenerate directions with multiple eigenvalues related
to the crystal axes. Nevertheless, the multi-step diagonalisation scheme used in Section 2.2 can
be adapted to such a case. This will be done in the sequel. The second problem is that the
geometry of the set of hyperbolic directions becomes more complicated. In general we can not
expect isolated hyperbolic directions, it will be necessary to consider manifolds of hyberbolic
directions on S2.
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Michael Reissig, Institut für Angewandte Analysis, Fakultät für Mathematik und Informatik, TU
Bergakademie Freiberg, Prüferstraße 9, 09596 Freiberg, Germany
Jens Wirth, Institut für Angewandte Analysis, Fakultät für Mathematik und Informatik, TU
Bergakademie Freiberg, Prüferstraße 9, 09596 Freiberg, Germany
current address: Department of Mathematics, Imperial College, London SW7 2AZ, UK
	1. The problem under consideration
	2. General treatment of the thermo-elastic system
	2.1. Asymptotic expansion of the eigenvalues as ||0
	2.2. Asymptotic expansion of the eigenvalues as ||
	2.3. Collecting the results
	3. Decay estimates for solutions
	3.1. Coupling functions with simple zeros
	3.2. Coupling functions vanishing to higher order
	4. Concluding remarks
	References
