H¨olde s abili y in Type III he moelas odynamics
M.C. Lesedua e and R. Quin anilla
Ma em`a ica Aplicada 2, ETSEIAT, Uni e si a Poli `ecnica de Ca alunya
Colom, 11. Te assa (08222). Ba celona. Spain
E-mail add esses: Ma i.Ca me.Lesedua [email protected], Ramon.Quin [email protected]
To G.A. Maugin in his 70 h bi hday
Abs ac
This no e is conce ned wi h he linea (and linea ized) Type III he moelas odynamic
heo y p oposed by G een and Naghdi. We he e assume ha he mass densi y is posi i e
and he he mal conduc i i y enso is posi i e de ini e. Howe e , we do no assume he
posi i i y o any o he enso . In his si ua ion, we ob ain H¨olde con inuous dependence
esul s on he supply e ms. We also ske ch how o p o e he con inuous dependence on
he ini ial da a.
keywo ds: Type III he moelas odynamics, H¨olde s abili y, Con inuous dependence on
ini ial da a and supply e ms, Lag ange iden i ies me hod
1 In oduc ion
The he moelas ic heo y p oposed by G een and Naghdi [5–7] has dese ed an in ense in-
es iga ion in he las yea s. Th ee sub- heo ies, whe e an en opy balance law eplaces he
cus oma y en opy inequali y, ha e been p oposed. These heo ies we e labelled as Type I, II
and III. The linea e sion o Type I ag ees wi h he usual classical heo y o he moelas ici y.
Fo Type II, he ene gy o he sys em is cons an wi h espec o he ime. Fo his eason
i is also known as “ he moelas ici y wi hou ene gy dissipa ion”. Type III is he mo e gen-
e al heo y and i con ains he o he wo as limi ing cases. We can ecall se e al pape s (see
Iesan [8,9]; Iesan and Quin anilla [10]; Lazza i and Nibbi [13]; Lesedua e e al. [14,15]; Liu and
Quin anilla [16,17]; Liu and Lin [18]; Messaoudi and Sou yane [19]; Pu i and Jo dan [20]; Qin
e al. [21]; Quin anilla [22–27]; Quin anilla and Racke [28]; Quin anilla and S aughan [29–31];
Yang and Wang [34], among o he s), whe e exis ence, uniqueness, con inuous dependence, spa-
ial and ime beha io ha e been s udied.
We ecall ha he linea sys em o he cen osymme ic Type III he moelas odyamics
can be w i en as (ρ¨ui=aijkhuk,h −aijθ,j +ρ i,
c˙
θ=−aij ˙ui,j +kijθ,i,j +bijα,i,j +ρ , (1.1)
whe e uiis he displacemen ec o , θis he empe a u e, ρis he mass densi y, cis he
he mal capaci y, (aijkh) is he elas ici y enso , aij is he coupling enso , kij is he he mal
1
conduc i i y enso and bij deno es a enso which is ypical o he Types II and III heo ies.
The cons i u i e enso s ρ,c,aijkh,aij,bij and kij a e smoo h unc ions o he posi ion. They
sa is y he symme ies
aijkh =akhij (1.2)
and
kij =kji, bij =bji.(1.3)
The he mal displacemen αis de ined by
α(x
x
x, ) = Z
0
θ(x
x
x, s)ds +α0(x
x
x),(1.4)
and iand a e he supply e ms. To gua an ee he well-posedness o he Type III he moelas-
ici y, we need o assume ha he mass densi y and he hea capaci y a e posi i e as well as he
elas ici y enso and he he mal conduc i i y. Lyapuno s abili y o he solu ions is implied
when bij is also posi i e de ini e.
In he case whe e we do no assume he posi i i y o he elas ici y enso aijkh, he p oblem
de e mined by he sys em (1.1) wi h usual ini ial and bounda y condi ions becomes ill-posed.
I is wo h ecalling ha his condi ion can be p esen in he case o p es essed solids.
I is wo h no ing se e al e e ences o his si ua ion. Fo ins ance, esul s conce ning
uniqueness and g ow h o solu ions ha e been ob ained a [25, 29] unde he condi ion ha
he enso s kij and bij a e posi i e. I is also wo h men ioning ha he only con ibu ion
conce ning he case when aijkh and bij a e no de ini e was ob ained ecen ly (see [15]). The e,
i was showed he uniqueness o solu ions. We he e wan o show how o ob ain he con inuous
dependence wi h espec o he supply e ms and ini ial da a unde simila es ic ions.
In 1960, John [11] showed how o ob ain con inuous dependence esul s in he sense o
H¨olde . This concep is weake han he usual de ini ion o he con inuous dependence. The
basic idea consis s o impose ha he solu ions belong o a sui able cons ain class. Since his
con ibu ion, many in es iga ions ha e been di ec ed o his kind o esul s. We may ci e he
wo ks o Ames and Payne [1] and Knops and Payne [12] conce ning iso he mal elas odynamics.
We can also ecall ha H¨olde s abili y esul s can be ound in di e en amewo ks om he
he moelas ici y (see Cimmelli and dell’Isola [3] and dell’Isola [4]). Fo he classical he moe-
las odynamics, we may ecall he con ibu ions o Wilkes [33], Ames and S aughan [2] and
Rione o and Chi i a [32]. I is wo h ecalling and compa ing se e al ela ed esul s. In [15],
we p oposed a uniqueness esul unde simila assump ions o he ones conside ed he e and
in [24] he au ho p oposed a H¨olde s abili y esul by using he loga i hmic con exi y a gu-
men unde he assump ion ha bij is posi i e de ini e. We he e ob ain a new esul p o ing he
H¨olde s abili y o he solu ions when we do no assume ha he elas ici y enso nei he he bij
enso a e posi i e de ini e. I is known ha wo usual echniques o s udy ill-posed p oblems
in he moelas ici y a e he loga i hmic con exi y and he Lag ange iden i ies me hod. We he e
conside he second one. We wan o emphasize ha since he enso bij is no posi i e de ini e,
i does no seem possible o apply he loga i hmic con exi y a gumen s. Howe e , we he e a e
able o use he Lag ange iden i ies me hod. In ac , Lag ange iden i y me hod allows us o
ob ain equali y (3.5) ha implies he inequali y (3.12). This inequali y is he s a ing poin o
de elop ou app oach. The e o e, Lag ange iden i y is he undamen al ing edien o ob ain ou
esul s.
2
The plain o his no e is he ollowing. In he nex sec ion we ecall he basic assump ions
and he condi ions de ining he p oblem. A basic inequali y is ob ained in Sec ion 3. H¨olde ’s
con inuous dependence wi h espec o he supply e ms is ob ained in Sec ion 4. In he las
sec ion we ske ch how o ex end he a gumen o p o e con inuous dependence wi h espec o
he ini ial da a.
2 P elimina ies
In his sec ion we p opose he basic assump ions whe e we a e going o wo k wi h. We s udy
smoo h solu ions o (1.1) on B×I, whe e Iis a bounded ime in e al and Bis a egula
domain wi h bounda y Γ smoo h enough o apply he Di e gence Theo em.
In wha ollows we suppose ha he cons i u i e enso s a e bounded abou and ha e he
ollowing p ope ies:
(A1) The mass densi y ρand he hea capaci y ca e posi i e unc ions. Tha is,
ρ(x
x
x)≥ρ0>0, c(x
x
x)≥c0>0, x
x
x∈B.
(A2) The he mal conduc i i y enso kij is posi i e de ini e. Tha is, he e exis s a posi i e
cons an Csuch ha
kijξiξj≥Cξiξi,(2.1)
o e e y ec o (ξi).
The physical meaning o condi ion (A1) is ob ious. Condi ion (A2) gua an ees ha he
dissipa ion o he sys em is no nega i e and hen, he ene gy does no inc ease. I is an
usual assump ion in he he momechanical s udies. We also no e ha om (2.1), he ollowing
inequali y
|bijξiξj| ≤ C1kijξiξj(2.2)
is sa is ied o e e y ec o (ξi), whe e C1is a calculable cons an 1which depends on he enso s
kij and bij.
To he ield equa ions we adjoin he bounda y condi ions
ui(x
x
x, ) = ¯ui(x
x
x, ), α(x
x
x, ) = ¯α(x
x
x, ), x
x
x∈Γ, ∈I, (2.3)
oge he wi h he ini ial condi ions
ui(x
x
x, 0) = u0
i(x
x
x),˙ui(x
x
x, 0) = 0
i(x
x
x), α(x
x
x, 0) = α0(x
x
x),˙α(x
x
x, 0) = θ0(x
x
x), x
x
x∈B. (2.4)
To s udy he con inuous dependence o he solu ions wi h espec o he supply e ms, we deno e
by u(1)
i, α(1) he solu ion co esponding o he ex e nal da a  (1)
i, (1)and by u(2)
i, α(2),
he solu ion co esponding o he supply e ms  (2)
i, (2).
We in oduce he no a ion
ui=u(2)
i−u(1)
i, α =α(2) −α(1), Fi= (2)
i− (1)
i, q = (2) − (1).(2.5)
1We no e ha C1=b∗/k∗, whe e b∗is he maximum o he absolu e alues o he eigen alues o he ma ix
bij and k∗is he minimum o he eigen alues o kij .
3
I ollows ha (ui, α) sa is ies he p oblem de e mined by he sys em
(ρ¨ui=aijkhuk,h −aijθ,j +ρFi,
c˙
θ=−aij ˙ui,j +kijθ,i,j +bijα,i,j +ρq, (2.6)
wi h he homogeneous bounda y condi ions
ui(x
x
x, ) = α(x
x
x, )=0, x
x
x∈Γ, ∈I, (2.7)
and he null ini ial condi ions
ui(x
x
x, 0) = ˙ui(x
x
x, 0) = α(x
x
x, 0) = ˙α(x
x
x, 0) = 0, x
x
x∈B. (2.8)
In ou s udies i will be use ul he ollowing inequali y
Z
0
kijα,iα,j ds ≤4C2
2 2
π2Z
0
kij ˙α,i ˙α,j ds, (2.9)
which is sa is ied o e e y unc ion αsuch ha α(0) = 0 and whe e C2is a calculable cons an 2.
3 A basic inequali y
In his sec ion we ob ain an inequali y sa is ied by he solu ions o he p oblem de ined by he
sys em (2.6) wi h bounda y and ini ial condi ions (2.7) and (2.8), espec i ely. This inequali y
will be ele an o ob ain ou esul s.
As we assume null Di ichle bounda y condi ions and null ini ial condi ions, we ob ain ha
he ene gy equali y
ZBρ˙ui˙ui+cθ2+aijkhui,juk,h +bijα,iα,jd
+ 2 Z
0ZB
kijθ,iθ,j d ds −2Z
0ZB
(ρFi˙ui+ρqθ)d ds = 0
(3.1)
is sa is ied o e e y solu ion o ou p oblem.
On he o he hand, om he equali ies
d
ds [ρ˙ui(s) ˙ui(2 −s)] = ρ¨ui(s) ˙ui(2 −s)−ρ˙ui(s)¨ui(2 −s),(3.2)
d
ds [cθ(s)θ(2 −s)] = c˙
θ(s)θ(2 −s)−cθ(s)˙
θ(2 −s) (3.3)
and he null ini ial and bounda y condi ions, we ob ain
ZBρ˙ui˙ui+bijα,iα,j −aijkhui,juk,h −cθ2d
=Z
0ZB
ρ(Fi(s) ˙ui(2 −s)−Fi(2 −s) ˙ui(s)) d ds
−Z
0ZB
ρ(q(s)θ(2 −s)−q(2 −s)θ(s)) d ds.
(3.4)
2We no e ha C2
2=k∗/k∗, whe e k∗is he maximum o he eigen alues o he symme ic ma ix kij and k∗
is he minimum o he eigen alues o kij .
4
F om he equali y (3.1) and he ela ion (3.4), we conclude ha
ZB
(ρ˙ui˙ui+bijα,iα,j)d +Z
0ZB
kijθ,iθ,j d ds
=Z
0ZB
(ρFi˙ui+ρqθ)d ds +1
2Z
0ZB
(ρFi(s) ˙ui(2 −s)−ρFi(2 −s) ˙ui(s)) d ds
+1
2Z
0ZB
(ρq(2 −s)θ(s)−ρq(s)θ(2 −s)) d ds.
(3.5)
We now conside se e al es ima es:
Z
0ZB
[ρFi˙ui+ρqθ]d ds ≤Z
0ZB
ρFiFid ds1/2Z
0ZB
ρ˙ui˙uid ds1/2
+Z
0ZB
ρq2d ds1/2Z
0ZB
ρθ2d ds1/2
≤Z
0ZBρFiFi+ρq2d ds1/2Z
0ZBρ˙ui˙ui+ρθ2d ds1/2
,
(3.6)
whe e we ha e used he inequali y
√a√b+√c√d≤√a+c√b+d. (3.7)
In a simila way, we ha e ha
Z
0ZB
[ρFi(s) ˙ui(2 −s)−ρFi(2 −s) ˙ui(s)] d ds
≤Z
0ZB
ρFiFid ds1/2Z2
ZB
ρ˙ui˙uid ds1/2
+Z2
ZB
ρFiFid ds1/2Z
0ZB
ρ˙ui˙uid ds1/2
≤Z2
0ZB
ρFiFid ds1/2Z2
0ZB
ρ˙ui˙uid ds1/2
.
(3.8)
We can also ob ain ha
Z
0ZB
[ρq(2 −s)θ(s)−ρq(s)θ(2 −s)] d ds
≤Z2
0ZB
ρq2d ds1/2Z2
0ZB
ρθ2d ds1/2
.
(3.9)
The e o e, we see ha
ZB
(ρ˙ui˙ui+bijα,iα,j)d +Z
0ZB
kijθ,iθ,j d ds
≤3
2Z2
0ZBρFiFi+ρq2d ds1/2Z2
0ZBρ˙ui˙ui+ρθ2d ds1/2
.
(3.10)
5

Le us assume ha
sup
∈[0,T]ZBρ˙ui˙ui+ρθ2d ≤N2
1(3.11)
and ha ≤T/2. We ob ain ha he inequali y
ZB
(ρ˙ui˙ui+bijα,iα,j)d +Z
0ZB
kijθ,iθ,j d ds
≤3
2T1/2N1ZT
0ZBρFiFi+ρq2d ds1/2(3.12)
is sa is ied o e e y ≤T/2. Es ima e (3.12) is undamen al in ou app oach. Howe e , as we
do no assume ha bij is a posi i e de ini e ma ix, we do no ha e a posi i e quad a ic o m
on he le side o he es ima e. Thus, we will need o manipula e his e m o wo k wi h a
posi i e quad a ic o m. This will be he main aim o he nex sec ions.
4 The main esul
The aim o his sec ion is o ob ain an es ima e o he solu ions o he p oblem de e mined by
(2.6), (2.7) and (2.8). F om he inequali y (3.12) and he use o he Poinca ´e ype inequali y
(2.9) we will be able o ge such kind o es ima e.
We a e going o decompose he in e al [0, T/2] in a ini e sequence o subin e als I0,I1,
I2, . . . , Imsuch ha [0, T/2] ⊂
m
[
j=0
Ijand such ha each in e sec ion Ii∩Ii+1 has a unique
elemen , deno ed by { i}, o i= 1, . . . , m −1. Then, we will ob ain ha
ZB
ρ˙ui˙uid +1
2Z
0ZB
kijθ,iθ,j d ds ≤En,(4.1)
whene e ∈In∩[0, T/2] whe e he es ima e Enis de ined by he ecu ence
E0=3
2T1/2N1Z
0ZBρFiFi+ρq2d ds1/2
(4.2)
and
En+1 =16C1C2T
πEn+3
2T1/2N1Z
0ZBρFiFi+ρq2d ds1/2
.(4.3)
Ou i s s ep is o ob ain ha
ZB
ρ˙ui˙uid +1
2Z
0ZB
kijθ,iθ,j d ds ≤E0,(4.4)
whene e ∈I0=h0,π
16C1C2i.
6
F om (2.2), he a i hme ic-geome ic mean inequali y and (2.9), we know ha
ZB
bijα,iα,j d +Z
0ZB
kijθ,iθjd ds
≥ −C1ZB
kijα,iα,j d +Z
0ZB
kijθ,iθ,j d ds
=−2C1Z
0ZB
kijα,iθ,j d ds +Z
0ZB
kijθ,iθ,j d ds
≥ −2C1Z
0ZB
kijα,iα,j d ds1/2Z
0ZB
kijθ,iθ,jd ds1/2
+Z
0ZB
kijθ,iθ,j d ds
≥1−4C1C2
πZ
0ZB
kijθ,iθ,jd ds.
(4.5)
I we assume ha < π
16C1C2
, we see ha
ZB
bijα,iα,j d +Z
0ZB
kijθ,iθ,jd ds ≥1
2Z
0ZB
kijθ,iθ,jd ds. (4.6)
Thus, in iew o he es ima e (3.12), we ob ain he inequali y (4.4).
Le us assume ha we ha e ob ained an es ima e o he ype
ZB
ρ˙ui˙uid +1
2Z
0ZB
kijθ,iθ,j d ds ≤En,(4.7)
o nπ
16C1C2
< < (n+ 1)π
16C1C2
<T
2, and we wan o ge a simila bound o
(n+ 1)π
16C1C2
< < min (n+ 2)π
16C1C2
,T
2.
The analysis s a s by conside ing he ela ions
ZB
ρ˙ui˙uid +Z
0ZB
kijθ,iθ,j d ds
=ZB
ρ˙ui˙uid +ZB
bijα,iα,j d +Z
0ZB
kijθ,iθ,j d ds −ZB
bijα,iα,j d
≤3
2T1/2N1ZT
0ZBρFiFi+ρq2d ds1/2
+C1ZB
kijα,iα,j d .
(4.8)
He e we ha e applied he es ima es (2.2) and (3.12). We no e ha
C1ZB
kijα,iα,j d ≤2C1ZB
kij (α,i( )−α,i( n+1)) (α,j( )−α,j( n+1)) d
+ 2C1ZB
kijα,i( n+1)α,j( n+1)d ,
(4.9)
7
whe e n+1 =(n+ 1)π
16C1C2
. We ha e ha
C1ZB
kijα,iα,j d
≤4C1Z
n+1 ZB
kij (α,i(s)−α,i( n+1)) θ,j d ds + 4C1Z n+1
0ZB
kijα,iθ,j d ds
≤4C1Z
n+1 ZB
kij (α,i(s)−α,i( n+1)) (α,j(s)−α,j( n+1)) d ds1/2Z
n+1 ZB
kijθ,iθ,j d ds1/2
+ 4C1Z n+1
0ZB
kijα,iα,j d ds1/2Z n+1
0ZB
kijθ,iθ,j d ds1/2
≤8C1C2( − n+1)
πZ
n+1 ZB
kijθ,iθ,j d ds +8C1C2 n+1
πZ n+1
0ZB
kijθ,iθ,j d ds.
(4.10)
F om (4.8) and (4.9), i ollows ha
ZB
ρ˙ui˙uid +Z
0ZB
kijθ,iθ,j d ds
≤3
2T1/2N1ZT
0ZBρFiFi+ρq2d ds1/2
+8C1C2( − n+1)
πZ
0ZB
kijθ,iθ,j d ds +16C1C2 n+1
πEn.
(4.11)
I we assume ha − n+1 <π
16C1C2
, we ob ain he desi ed es ima e (4.1)–(4.3). We no e ha
om he ecu ence (4.3), we ha e
Em≤16C1C2T
π16C1C2T
πEm−2+E0+E0
≤ ··· ≤ 16C1C2T
πm
E0+16C1C2T
πm−1
E0+···+E0.
(4.12)
Hence,
Em≤"m
X
k=0 16C1C2T
πk#E0.(4.13)
Thus, we can conclude ha he es ima e
ZB
ρ˙ui˙uid +1
2Z
0ZB
kijθ,iθ,j d
≤3
2T1/2N1"m
X
k=0 16C1C2T
πk#ZT
0ZBρFiFi+ρq2d ds1/2(4.14)
is sa is ied whene e mis he i s na u al numbe such ha m > 8C1C2T
π. The e o e, we ha e
p o ed he ollowing esul .
8
Theo em 4.1 Le u(1)
i, α(1)and u(2)
i, α(2)be he solu ions o he sys em (1.1) co espond-
ing o he supply e ms  (1)
i, (1)and  (2)
i, (2), espec i ely. Then, he di e ence deno ed
by (2.5) sa is ies he es ima e
ZB
ρ˙ui˙uid +1
2Z
0ZB
kijθ,iθ,j d
≤3
2T1/2N1




1−16C1C2T
πm+1
1−16C1C2T
π




ZT
0ZBρFiFi+ρq2d ds1/2
,
(4.15)
whe e mis he i s na u al numbe such ha m > 8C1C2T/π.
5 Con inuous dependence on ini ial da a
The analysis p oposed in Sec ion 4 can be adap ed o s udy he s abili y wi h espec o
he ini ial da a. Le us assume ha we ha e wo solu ions u(1)
i, α(1)and u(2)
i, α(2) o he
homogeneous e sion ( i= 0, = 0) o he sys em (1.1) wi h he same bounda y condi ions, bu
wi h di e en ini ial condi ions. Using he no a ion p oposed p e iously, we deno e by (ui, α) he
solu ion o he homogeneous e sion o he sys em (2.6) wi h homogeneous bounda y condi ions
(2.7) and he ini ial condi ions
ui(x
x
x, 0) = u(2)
i(x
x
x, 0) −u(1)
i(x
x
x, 0) = u∗
i(x
x
x),
˙ui(x
x
x, 0) = ˙u(2)
i(x
x
x, 0) −˙u(1)
i(x
x
x, 0) = ∗
i(x
x
x),
α(x
x
x, 0) = α(2)(x
x
x, 0) −α(1)(x
x
x, 0) = α∗(x
x
x),
˙α(x
x
x, 0) = ˙α(2)(x
x
x, 0) −˙α(1)(x
x
x, 0) = θ∗(x
x
x).
(5.1)
In his case, he ene gy equa ion gi es
E( ) = ZBρ˙ui˙ui+cθ2+aijklui,juk,l +bijα,iα,jd + 2 Z
0ZB
kijθ,iθ,j d ds =E(0),(5.2)
whe e
E(0) = ZBρ ∗
i ∗
i+c(θ∗)2+aijklu∗
i,ju∗
k,l +bijα∗
,iα∗
,jd . (5.3)
As we conside he homogeneous sys em, he Lag ange iden i ies a gumen implies ha
ZBρ˙ui˙ui+bijα,iα,j −aijklui,juk,l −cθ2d
=ZBρ ∗
i˙ui(2 ) + bijα∗
,iα,j(2 )−aijklu∗
i,juk,l(2 )−cθ∗θ(2 )d .
(5.4)
F om (5.2) and (5.4), we ob ain ha
ZB
(ρ˙ui˙ui+bijα,iα,j)d +Z
0ZB
kijθ,iθ,j d ds
=E(0)
2+1
2ZBρ ∗
i˙ui(2 ) + bijα∗
,iα,j(2 )−aijklu∗
i,juk,l(2 )−cθ∗θ(2 )d .
(5.5)
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