Mathematical existence results for the Doi-Edwards polymer model

0.53 MB·English

by  Laurent Chupin

MATHEMATICAL EXISTENCE RESULTS
FOR THE DOI-EDWARDS POLYMER MODEL
LAURENT CHUPIN˚
Abstract. In this paper, we present some mathematical results on the Doi-Edwards model
describingthedynamicsofflexiblepolymersinmeltsandconcentratedsolutions. Thismodel,devel-
5 opedinthelate1970s,hasbeenusedandtestedextensivelyinmodelingandsimulationofpolymer
1 flows. From a mathematical point of view, the Doi-Edwards model consists in a strong coupling
0 betweentheNavier-Stokesequations andahighlynonlinearconstitutivelaw.
2 Theaimofthisarticleistoprovidearigorousproofofthewell-posednessoftheDoi-Edwardsmodel,
namely it has a unique regular solution. We also prove, which is generally much more difficult for
n flows of viscoelastic type, that the solution is global in time in the two dimensional case, without
a anyrestrictiononthesmallnessofthedata.
J

6 Key words. Polymer, Viscoelastic flow, Doi-Edwards model, Navier-Stokes equations, global
1 existenceresult.

] AMS subject classifications. 35A01,35B45,35Q35,76A05,76A10,76D05
P
A
1. Introduction. Numerousmodelsexistfordescribingfluidswithcomplexrhe-
h. ologicalproperties. Theygenerallyareofgreatscientificinterestandhavearichphe-
t nomenology. Their mathematical description remains challenging. We are interested
a
in this article to a model - the Doi-Edwardsmodel - which was one of the foundation
m
of the most recent physical theories but for which the mathematical theory remains
[
very poor.

1
M. Doi and S.F. Edwards wrote a series of papers [10, 11, 12, 13] expanding the
v
6 concept of reptation introduced by P.G. de Gennes in 1971. This approach was then
9 taken up in a famous book in polymer physics in 1988, see [14]. Since this model
8 was derived, numerous studies have been carried out either from a physical point of
3
view, either from a numerical point of view. Moreover, several other models were
0
born: simplifiedmodelsusingforinstancethe IndependentAlignmentapproximation
.
1 or the Currie approximation [7], more complex models like the pom-pom model [28].
0
Finally, much progress has been made on the modeling of both linear and branched
5
polymers. However,froma mathematicalpoint ofview, itseems that no justification
1
: was given even for the pioneering model.
v
i Nonetheless we can cite some recent theoretical papers on this subject, see [4, 19], in
X
which the authors are only interested in specific cases: one dimensional shear flows
r
a under the independent alignment assumption in [19], flows for which the coupling
betweenthevelocityandthestressisnottakingintoaccount,see[4]. Moregenerally,
there seems to be a real challenge to obtain global existence in time for models of
polymers. The mostcaricaturalexampleisthe Oldroydmodelforwhichthe question
of globalexistence in dimension 2 remains anopen question, see [24] for a partialan-
swer. However,there exists polymer models for which such results are proved. Thus,
for the FENE type models, N. Masmoudi [27] proved a global existence result in di-
mension 2. Similarly, for integral fluid of type K-BKZ such results hold too (see [3]).
The aimofthis paperis to proverelevantmathematicalresultsonthis relevantphys-
ical problem. The first one is the following (a more specific version of this result is
˚Universit´eBlaisePascal,Clermont-FerrandII,LaboratoiredeMath´ematiquesCNRS-UMR6620,
CampusdesC´ezeaux,F-63177Aubi`erecedex,France([email protected]).
1
2 LAURENTCHUPIN
given by Theorems 3.1 and 3.2, page 8):
Theorem 1.1. There exists a time t‹ 0 such that the Doi-Edwards model
ą
admits a unique strong solution on the interval time 0,t‹ .
r s
By the expression ”strong solution” we mean a sufficiently smooth solution so that
eachtermofthesystemiswelldefined,aswellastheinitialconditions(corresponding
to the time t 0). The lifetime of the solution, i.e. the value of time t‹, is not
“
easily quantifiable. In practiceit is wellknownthat for the Navier-Stokesequations-
modelingnewtonianbehavior,the questionoflongtimeexistenceisstillanopenone.
For the Newtonian fluids, the only existence results, for long time and for any data,
correspond to the two dimensional case. However, we have above pointed out the
difficulty to getthis kind of result even in 2D for some viscoelastic fluids. The major
point of this paper is the proof indicating that the model of Doi-Edwards admits a
strong solution for long time in 2D (a more specific version of this result is given by
Theorem 3.3, page 9):
Theorem 1.2. For all time t‹ 0, the Doi-Edwards model admits a unique
ą
strong solution on the interval time 0,t‹ .
r s
The paper is organizedasfollows. First- inSection2,we introduce the Doi-Edwards
model specifying the physical meaning of each contribution. This second section
ends by a dimensionless procedure that allows us to write the model with only three
parameters (the Reynolds number, the Weissenberg number and the ratio between
solvent viscosity and elastic viscosity). In Section 3, we present the mathematical
frameworkaswellastheassumptionswhicharephysicallydiscussed. Themainresults
aregivenattheendofthissection. Section4isdevotedtofundamentalpreliminaries
which correspond to some key points of the next proofs. The first two provide a
priori boundswhichwillimply thatthatthe stressdefinedintheDoi-Edwardsmodel
is automatically bounded. The third preliminary give a Gronwall lemma with two
time variables. The fourth preliminary result is about the maximum principle which
canbe appliedmanytimes to estimate the memoryof the fluid. The lastpreliminary
resultis abouta Cauchyproblemarisinginthe globalexistence proof. The threelast
sections(5, 6and7)aredevotedto the proofofthe three mainsresults ,namely: the
local existence result in Section 5, the uniqueness result in Section 6 and the global
existenceresultinSection7. Someopenquestionsarepresentedbywayofconclusion.
2. Governing equations.
2.1. Conservation laws. In this paper we are interestedin the flow of isother-
mal and incompressible fluids. The incompressibility implies that the mass conser-
vation is equivalent to the free-divergence of the velocity field. The isothermal as-
sumption implies that only one other conservation law suffices to describe the flow:
it corresponds to the law of conservation of the momentum (Newton’s second law of
motion). This equationis writtenas a balance betweenthe materialderivativeof the
velocity and the divergence of the Cauchy stress tensor. For a polymeric liquid, the
equations of conservationcanhence written as a system coupling the velocity field v,
the pressure p and the extra-stress tensor σ:
ρd v ∇p η ∆v divσ,
t s
` ´ “
# divv 0,
“
ExistencefortheDoi-Edwardspolymermodel 3
where ρ is the fluid density and η the solvent viscosity. The notation d corresponds
s t
to the material derivative d v ∇.
t t
“B ` ¨
2.2. Constitutive equation. A fundamental result ofthe Doi-Edwardstheory
istheexpressionforthestresstensorσ whichapplieswhenthechainswhichcompose
thefluidarerelaxedwithintheirtubes. Moreprecisely,thestresscanbededucefrom
atensorS denotedtheorientationorderparameterofthechains. Althoughthechain
tension is permanently at the equilibrium value, the orientations become anisotropi-
cally distributed as a consequence of the flow, and a stress develops accordingly. The
stress is then modelized by (see [9, page 2056]):
ℓ
σ t,x Ge 2 S t,x,s ds,
p q“ ℓ ż´2ℓ p q
where G is a characteristic modulus and ℓ is the equilibrium value of the contour
e
length of the chains. The quantity s is a arc-length coordinate along the primitive
chain.
ToevaluatethistensorS,M.DoiandS.F.EdwardswriteS u u 1δ . Here,uis
“x b ´d y
aunitvectoralongthetangenttotheprimitivechainwhichdependsontimet,spatial
position x and length s, and δ denotes the identity tensor. The entire d corresponds
to the dimension of the spatial coordinates (in practice d 2 or d 3). The average
“ “
is overthe distributionofthese vectorsinthe ensemble ofchains, i.e., moreexplicitly
1
S t,x,s f u;t,x,s u u δ du,
p q“żSd´1 p q b ´ d
` ˘
with f u;t,x,s given the orientation distribution function. To obtain an expression
p q
forthisdistribution,thehistoryofmotionmustbefound. Tothispurpose,letusfirst
recall the relevant aspects of the Doi-Edwards model. They are
X The polymer moves randomly inside the tube executing one-dimensional
Brownianmotion. Moreovertube segmentsarerandomlyorientedwhenthey
are created and deform affinely thereafter;
X If the system is macroscopically deformed, the polymer conformation is also
changed as presented on Figure 2.1;
X The macroscopic motion and Brownian motion coexist, independently from
one another.
retraction
affine
transformation
Fig.2.1. (inspiredfromthebook[9,page2058])–Whenamacroscopic deformationisapplied,
a polymer chain is transformed into a new conformation. The new chain is on the curve which
is the affine transformation of the initial curve. The new position of each segment is obtained by
retraction, preserving the initial lengths.
4 LAURENTCHUPIN
Alltheseconsiderationsbeingtakenintoaccount,itispossibletowrite(see [14,page
277])
`8
S t,x,s K t,T,x,s S G t,T,x dT. (2.1)
T
p q“´ B p q p p qq
ż0
It makes appear the deformation gradient tensor G which depends not only on the
currenttime t andspatial variable x but also on another time T. The time T allows
totakeintoaccountallthe historyofthemotion. ThedeformationgradientGfulfills
the differential equation (see [2]):
d G G G ∇v.
t T
`B “ ¨
Finally, the integral kernel K satisfies (the coefficient D is a curvilinear diffusion
e
coefficient):
s
d K K ∇v : Sds K D 2K 0. (2.2)
t `BT ` Bs ´ eBs “
´ ż0 ¯
Remark 2.1.
1. Note that the equation (2.1) does not explicitly defined the orientation tensor
since S again appears in the equation (2.2). It is one of the difficulties to
obtain existence results.
2. The usualformulation usethetime t and a other timet1 in thepast. Morally,
the deformation gradient tensor G measures the deformation between these
two times. In the present paper we select as independent variable the age
T t t1, which is measured relative to the current time t. This viewpoint
“ ´
is relatively classical in the numerical framework for integral models, see for
instance [20, 21, 30].
The model is closed with the expression of the function S, see [14, eq. (7.115)]:
1 G u G u 1
S G p ¨ qbp ¨ q δ, (2.3)
p q“ G u 0 G u 0´ d
x| ¨ |y A | ¨ | E
where the brackets correspond to the average over the isotropic distribution of
0
unit vectors u Sd´x1¨.y
P
2.3. Boundary and initial conditions. The previous equations are supple-
mentedbyboundaryandinitialconditions. Throughoutthis articlewerestricttothe
case where the macroscopic field is assumed to be periodic. Thus the only condition
that we impose on the unknowns v, p, K and G with respect to the variable x is to
be periodic. Clearly this “simplification” is purely mathematical and it will be inter-
esting to treat a more physical case imposing, for instance, the value of the velocity
at the macroscopic boundary.
By definition of the integral kernel K, we impose the following conditions:
ℓ ℓ
K t,0,x,s 1 and K t,T,x, K t,T,x, 0. (2.4)
p q“ p ´2q“ p 2q“
In the same way, the quantity G t,T,x which corresponds to the deformation gra-
p q
dient from a past times t T to the current time t must naturally satisfies
´
G t,0,x δ. (2.5)
p q“
ExistencefortheDoi-Edwardspolymermodel 5
The initial conditions correspond to the value that we impose at time t 0. We
“
assume that we know the velocity at this initial time and at any point x of the
domain. In the same way we assume that we know all the past of the flow before the
initial time: we then know the value of G and K at t 0, for any age T and at any
“
point x. To summarize, we assume that there exists an initial velocity v , an initial
0
deformation G and an initial function K such that
0 0
v 0,x v x ,
0
p q“ p q
G 0,T,x G T,x , (2.6)
0
p q“ p q
K 0,T,x,s K T,x,s .
0
p q“ p q
2.4. Remark: The I.A. approximation. A common approximation for the
Doi-Edwards model, called the independent alignment approximation (I.A. approxi-
mation, see [14, section 7.7.2]), is to neglect the transport term ∇v : sSds K in
the equation (2.2), and also to simplify the expression (2.3) of the func0tion SBussing
` ş ˘
G u G u 1
SpIAq G p ¨ qbp ¨ q δ.
p q“ G u2 0´ d
A | ¨ | E
The I.A. approximation is actually quite popular in the rheology literature (see e.g.,
[22, 23, 26]) since the corresponding configurational equation for K can be explicitly
solved using the Fourier series. The stress tensor σ is then more simply given by
(see [14, Equation 7.195]):
`8
σpIAq t,x G m T SpIAq G t,T,x dT,
e
p q“ p q p p qq
ż0 (2.7)
8D TD
where m T eexp ´ ep2 .
p q“ π2ℓ2 ℓ2
pÿodd ´ ¯
For such a model the global existence result is a consequence of a general result on
viscoelastic flows with memory, see [3]. Nevertheless, it is also well known that this
approximationcausesseriouserrorincertainsituations,thisisclearlyspecifiedinthe
seminal book [14]. More precisely, it is proved that I.A. predicts a negative Weis-
senbergeffect(see[18])whiletheversionwithoutI.A.predictsapositiveWeissenberg
effect (see [25]). To paraphrase M. Doi [9, page 2064]: ”Mathematically [...] there
seems no a priori reason why the term ∇v : sSds can be neglected compared
0 Bs
with the term D 2”.
eBs ` ş ˘
2.5. Dimensionless procedure. In order to recover characteristic properties
of the system, we use a nondimensionalization procedure. We denote by L a char-
acteristic macroscopic length, by V a characteristic velocity of the flow. It is then
natural to define a dimensionless coordinates x‹, a dimensionless velocity v‹ and a
dimensionless time t‹ by the following relations
L
x Lx‹, v Vv‹, t t‹.
“ “ “ V
For polymer flow, there exists also two microscopic characteristic sizes which corre-
spond to the length ℓ and to the diffusion coefficient D . They allow to define a
e
dimensionless microscopic length s‹ and another dimensionless time T‹:
ℓ2
s ℓs‹, T T‹.
“ “ De
6 LAURENTCHUPIN
Finally, in a dilute polymer solution, two viscosities naturally appear: the solvent
viscosity η and the elastic one defined using the characteristic modulus: η LGe.
s e “ V
If we denote by η η η the total viscosity, then we defined the dimensionless
s e
“ `
pressure and stress as follows
ηV ηV
p p‹, σ σ‹.
“ L “ L
Taking into account all these new unknowns and new variables, the complete system
reads (without in the notations):
‹
Red v ∇p 1 ω ∆v divσ, (2.8a)
t
` ´p ´ q “
divv 0, (2.8b)
“
1
2
σ t,x ω S t,x,s ds, (2.8c)
p q“ ż´12 p q
`8
S t,x,s K t,T,x,s S G t,T,x dT, (2.8d)
T
p q“´ B p q p p qq
ż0
1
d G G G ∇v, (2.8e)
t ` WeBT “ ¨
1 s 1
d K K ∇v : S K 2K 0. (2.8f)
t ` WeBT ` Bs ´ WeBs “
´ ż0 ¯
Inthis setofequations, Reis the usualReynolds number, ω standsfor the viscosities
ratioandWeistheWeissenbergnumberdefinedbytheratiobetweenthemacroscopic
time and the microscopic time. More precisely we have
ρVL η ℓ2 D
Re , ω e, We { e.
“ η “ η “ L V
{
The functions S is always defined by the relation (2.3).
The goal of the rest of the paper is to analyze, from a mathematical point of view,
the existence of a solution to the system (2.8). More exactly, by given initial data
v ,G ,K ), isthereatripletoffunctions v,G,K whichcoincideswiththedataat
0 0 0
p p q
initial time and such that the previous system holds for any future time?
3. Mathematical framework, assumptions and main results.
3.1. Notations. The integer d stands for the spatial dimension of the flow. It
will be equal to 2 or 3 in the first parts and exclusively equal to 2 in the Section 7
where we prove a global existence result.
Notations for functional analysis –
The d dimensional torus is denoted T.
´
Forallrealn 0andallintegerq 1,the setWn,q correspondstotheusual
x
´ Sobolevspaceěswithrespecttothesěpacevariablex T. Weclassicallydenote
Lq W0,q and Hn Wn,2 and we do not take inPto account the dimension
x x x x
“ “
in the notations, for instance the space W1,q 3 will be denoted W1,q.
x x
p q
´ All the norms will be denoted by index, like }v}Wx1,q.
Since we are interested in the incompressible flows, we introduce
´
H v Lq ; divv 0 .
q x
“t P “ u
ExistencefortheDoi-Edwardspolymermodel 7
The Stokes operator A is introduced, with domain D A W2,q H ,
q q x q
´ p q “ X
whereas we denote (see [8, Section 2.3] for some details on this space)
`8
Dqr “tv PHq; }v}Lqx ` }Aqe´tAqv}rLqxdt 1{r ă`8u.
ż0
` ˘
The notations of kind Lr 0,t‹;X denote the space of r-integrable functions
´ p q
on 0,t‹ withvaluesinthespaceX. ForinstanceG Lr 0,t‹;L8Lq means
p q P p T xq
that
t‹ r
q
G r : sup G t,x,T qdx dt .
} }Lrp0,t‹;L8TLqxq “ż0 TPR`ˆżT| p q| ˙ ă`8
Notations for tensorial analysis – In System (2.8), the first equation (2.8a) is
a vectorial equation (the velocity v is a function with values in Rd). The equa-
tions (2.8c), (2.8d) and (2.8e) are tensorialequations (the stress σ, the orientationS
andthedeformationtensorGarefunctionswithvaluesinthesetofthe2-tensors). In
the followingproofs,weneedtoworkwiththegradientofsuch2-tensors,thatiswith
3-tensors, and even with 4-tensors. We introduce here some notations for tensors.
The set of linear applications on the d-dimensional space is denoted L Rd .
´ p q
The products A B, A B and A:B between two tensors of order p and q
´ b ¨
are respectively defined component by component by
A B a b ,
b i1,...ip,j1,...,jq “ i1,...,ip j1,...,jq
`A¨B i˘1,...ip´1,j2,...,jq “ai1,...,ip´1,kbk,j2,...,jq ,
`A:B˘i1,...ip´2,j3,...,jq “ai1,...,ip´2,k,ℓbk,ℓ,js`1,...,jq ,
` ˘
where we use the Einstein convention for the summations with indexes k
and ℓ.
Notealsothatalltheseproductsareinnerproductsonthesetofthep-tensors.
´
It allows us to define a generalized Froebenius norm:
A2 a2 .
| | “ i1,...,ip
i1,ÿ...,ip
We conclude this section introducing the constant C. This constant stands for any
constant depending on the data of the problem: initial conditions, physical parame-
ters... In some cases, informations will be given on the dependence of this constant
(seeforexampleSection7whereweexplainthatthisconstantmaydependontimet‹
but must remain bounded when t‹ is bounded).
3.2. Assumptions. Theresultsprovedinthisarticlerequiressomeassumptions
about the data. In addition to the assumptions on the regularity of the initial condi-
tions that will be specified in each theorem statement, we will need some ”natural”
assumptions.
X The first assumption relates to the initial deformation G :
0
γ 0 ; detG γ. (3.1)
0
D ą ě
We note that in many applications, the fluid is assumed to be initially quiescent. In
that case, we have G δ and detG 1. Moreover, we will see (equation (4.5)
0 0
“ “
8 LAURENTCHUPIN
in the preliminary section 4.2) that the quantity detG is only convected by the flow.
If the fluid is assumed to be at rest in the past (that is for T large enough), then
we always have detG 1. The assumption on the positiveness of detG allows us
0 0
“
consider, for instance, such cases.
X The second assumption relates to the initial memory m K :
0 T 0
“´B
m 0, (3.2a)
0
ě
m 0. (3.2b)
T 0
B ď
The assumption (3.2a) corresponds to the fact that the quantity m describes the
0
memory of the fluid, that is the weight that must have the quantity S in the flow
via therelation(2.8d). Itisphysicallypositive. The assumption(3.2b)indicatesthat
the memory decreases with the age T: It is linked to the principle of fading memory,
see [5].
In the integralmodels, that is to say when the memory is explicitly given in terms of
age T, it is a combination of exponentially decreasing functions (see for instance the
Doi-Edwards model under the I.A. approximation, subsection 2.4 and more precisely
the expression (2.7) of the memory). Such decreasing behaviors will be prescribed
in the functional spaces with exponential weight. For example, we will impose that
there exists µ 0 such that m Ce´WeµT.
0
ą ď
3.3. Main results. The firstresultconcernsanexistence resultforstrongsolu-
tion. It is a local in time result:
Theorem 3.1 (local existence). Let r 2, , q d, and µ 0.
Ps `8r Ps `8r ą
Ifthedata v , G and K satisfy theassumptions (3.1), (3.2a)andhave the following
0 0 0
regularity
v Dr, G L8W1,q, G L8Lq,
0 P q 0 P T x BT 0 P T x
eWeµT K L8L8 , eWeµT{2 ∇K L2Lq , K L8L2 ,
BT 0 P T x,s BT 0 P T x,s Bs 0 P T x,s
then there exists t‹ 0 and a strong solution u,G,K to System (2.8) in 0,t‹ ,
ą p q r s
which satisfies the initial and boundary conditions (2.4), (2.5) and (2.6). Moreover
we have
v Lr 0,t‹;W2,q , v Lr 0,t‹;Lq ,
x t x
GP L8p 0,t‹,L8Wq 1,q , B GP, Gp Lr 0,qt‹;L8Lq ,
K,P K,peµpWeTT´tqx Kq L8 0,t‹;L8L8 , BsK BLt2 0P,t‹;pL8L2 T, xq
BT BT P p T x,sq Bt P p T x,sq
∇K L8 0,t‹;L1Lq L2Lq , K L8 0,t‹;L8L2 ,
BT P p T x,sX T x,sq Bs P p T x,sq
and the memory m K remains non negative.
T
“´B
Remark 3.1. In this article, we will not give any result on the pressure p. In
practice, the latter is regarded as a Lagrange multiplier associated to the divergence
free constraint. It can be solved usingthe Riesz transforms. More precisely, taking the
divergence of the first equation (2.8a) of System (2.8) we use the periodic boundary
conditions to have
p ∆ ´1divdiv σ v v . (3.3)
“´p´ q p ´ b q
From Theorem 3.1, we can prove that the solutions of System (2.8) discussed in this
paper have σ v v in L8 0,t‹;L2 . The pressure in the solution of (2.8) is meant
x
´ b p q
ExistencefortheDoi-Edwardspolymermodel 9
to be given by (3.3).
We will see during the proof of this theorem 3.1 that one of the key point is the
behavior of the memory m K for large value of the age T: if the memory
T
“ ´B
is exponentially decreasing at t 0 (with respect to the age variable T) then the
“
solution will be exponentially decreasing for any time t 0.
ą
In the same way, it is possible to prove that if the memory m is initially decreas-
ing1 (with respect to the age variable T) then the solution will be decreasing for any
time t 0. The proof - which is not given in the proof of Theorem 3.1 - consists in
ą
derivating twice the equation(2.8f) with respectto T,andnext inapplying the max-
imum principle (see the subsection 4.4, page 13) to the function m. Therefore, the
T
B
assumption (3.2b), which is not necessary to obtain local existence, is also preserved
in time.
We willshow thatthe solutionobtainedin Theorem3.1 withthis additionalassump-
tion (3.2b) is the only one in the class of regularsolutions. Precisely,the result reads
as follows.
Theorem 3.2 (uniqueness). Let t‹ 0.
ą
Let u ,G ,K and u ,G ,K be two solutions to System (2.8) satisfying the
1 1 1 2 2 2
p q p q
initial and boundary conditions (2.4), (2.5) and (2.6). If we have, for i 1,2 ,
Pt u
∇v L2 0,t‹;L8 ,
i x
P p q
G L2 0,t‹;L8 L8 W1,d ,
i P p Tp x X x qq (3.4)
K L8 0,t‹;L8L8 L1L8 ,
BT i P p T x,sX T x,sq
∇K L8 0,t‹;L2Ld ,
BT i P p T x,sq
andifeachm K isdecreasingwithrespecttoT thenthetwosolutionscoincide.
i T i
“´B
Obviously, the solution obtained in Theorem 3.1 satisfies the regularity requested
in (3.4). Combining Theorems 3.1 and 3.2 we get a local and uniqueness result.
In the two-dimensional case, it is possible to show that the solution v,G,K of
p q
problem (2.8) exists for any time t‹ 0. More precisely we have the following result:
ą
Theorem 3.3 (global existence in 2D). Let r 2, , q 2, and µ 0.
We assume that 1 1 1 and that the data v ,PsG `a8ndr KPssat`is8fyrthe samąe as-
r ` q ă 2 0 0 0
sumptions that in Theorem 3.1.
Let t‹ 0 be arbitrary.
ą
There exists a constant C depending only on the data with C bounded for bounded t‹,
and a solution v,G,K of (2.8) satisfying the initial and boundary conditions (2.4),
p q
(2.5) and (2.6) such that
}∇2v}Lrp0,t‹;Lqxq ďC, }∇v}L8p0,t‹;L8xq ďC, (3.5)
}∇S}Lrp0,t‹;Lqx,sq ďC, }S}L8p0,t‹;L8xq ďC.
Theestimates(3.5)announcedinthetheorem3.3abovearesufficienttoproveaglobal
in time existence of a solution v,G,K . In fact if (3.5) holds then it is possible to
p q
prove - principally using the lemmas introduced in the proof of the local existence
result - that the solution v,G,K of (2.8) at time t‹ have the same regularity that
p q
1Be careful not to confuse the terms ”exponentially decreasing” and ”decreasing”. The first
means that m is bounded by a function of the form e´T, while the second means that BTm ď 0.
Moreoverthefirstisaglobalproperty,whilethesecondisalocalproperty.
10 LAURENTCHUPIN
at time 0. Applying the local result (theorem 3.1), we deduce that the solution can
not blow up in finite time.
4. Preliminaries. Inthissectionwegivesomeresultswhichwillbeusingduring
the different proofs of the previous theorems.
4.1. Some bounds for the function S. Oneofthe key points ofthe proofof
globalexistence lies in the factthat the stressσ, defined by (2.8c)-(2.8d)is bounded.
The first result in this direction is the following result concerning the function S:
Proposition 4.1. The function S defined by the relation (2.3) is of class C1
on L Rd 0 and satisfies the following properties:
p qzt¯u
S 0 ; G L Rd 0 , S G S ; (4.1a)
8 8
D ě @ P p qzt¯u | p q|ď
S1 0 ; G L Rd 0 , G S1 G S1 . (4.1b)
D 8 ě @ P p qzt¯u | || p q|ď 8
Proof. Recall the definition (2.3) of the function S:
1 G u G u 1
S G p ¨ qbp ¨ q δ.
p q“ G u 0 G u 0´ d
x| ¨ |y A | ¨ | E
X We first notice that the following inequality is obvious
f f u du f u du f , (4.2)
0 0
|x y |“ˇżSd´1 p q ˇďżSd´1| p q| “x| |y
ˇ ˇ
so that the first point of tˇhe propositionˇ4.1 is a direct consequence of the inequality
A B A B foralltensorsAandB,andoftherelation δ ?d. Moreprecisely
|webobt|aďin,|fo|r|al|l G L Rd 0 : | |“
P p qzt¯u
1
S G 1 ,
| p q|ď ` ?d
which corresponds to (4.1a).
X For the second point (4.1b), we write the tensor S G component by component:
p q
for any k,ℓ 1,...,d 2,
p qPt u
1 G u G u 1
S G p ¨ qkp ¨ qℓ δ .
p qkℓ “ G u 0 G u 0´ d kℓ
x| ¨ |y A | ¨ | E
It is then not difficult to evaluate each components of the derivative2 tensor S1 G .
p q
For instance for i,j,k,ℓ 1,...,d 4 we have
p qPt u
G u u
G G u p ¨ qi j.
B ij | ¨ | “ G u
| ¨ |
` ˘
2TheapplicationS beingdefinedonanopensetofLpRdqwithvalues inLpRdq, itsdifferential
isanapplicationwithvaluesinLpLpRdq,LpRdqq. Consequently, S1pGqcanbeidentifytoa4-order
tensorwhosecomponents areS1pGqi,j,k,ℓ“BGijSpGqkℓ.