Stochastic homogenization of rate-dependent models of monotone type in plasticity

0.33 MB·

by  Martin Heida

Stochastic homogenization of rate-dependent
models of monotone type in plasticity
Martin Heida∗, Sergiy Nesenenko†
7
1 January 16, 2017
0
2

n
a Abstract
J
2 Inthis work we deal with thestochastic homogenization of theinitial
1 boundary value problems of monotone type. The models of monotone
type under consideration describe the deformation behaviour of inelas-

] tic materials with a microstructure which can be characterised by ran-
P
dom measures. Based on the Fitzpatrick function concept we reduce the
A
studyoftheasymptoticbehaviourofmonotoneoperatorsassociatedwith
. our models to the problem of the stochastic homogenization of convex
h
functionalswithinanergodicandstationarysetting. TheconceptofFitz-
t
a patrick’sfunctionhelpsustointroduceandshowtheexistenceoftheweak
m
solutionsforrate-dependentsystems. Thederivationsofthehomogeniza-
[ tion results presented in this work are based on the stochastic two-scale

convergence in Sobolev spaces. For completeness, we also present some
1
two-scalehomogenizationresultsforconvexfunctionals,whicharerelated
v
tothe classical Γ-convergencetheory.
5
0 Key words: stochastic homogenization, random measures, plasticity, stochas-
5
tic two-scale convergence, Γ-convergence, monotone operator method, Fitz-
3
0 patrick’s function, Palm measures, random microstructure.
.
1
0
AMS 2000 subject classification: 74Q15, 74C05, 74C10, 74D10, 35J25,
7
34G20, 34G25, 47H04, 47H05
1
:
v
i 1 Introduction
X
r
a In this work we are concerned with the homogenization of the initial boundary
value problemdescribing the deformationbehaviorofinelastic materials with a
microstructure which can be characterisedby random measures.
While the periodic homogenization theory for elasto/visco-plastic models
is sufficiently well established (see [2, 11, 17, 18, 19, 26, 27, 30, 31] and ref-
erences therein), some improvement in the development of the techniques for
∗MartinHeida,WeierstrassInstituteforAppliedAnalysisandStochastics, Mohrenstrasse
39,10117Berlin,Germany,email: [email protected],Tel.: [phone]
†Corresponding author: Sergiy Nesenenko, Fakult¨at II, Institut fu¨r Mathematik, Tech-
nische Universit¨at Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany, email:
[email protected], Tel.: [phone]
1
the stochastic homogenization of the quasi-static initial boundary value prob-
lems of monotone type has to be achieved yet. To the best knowledge of the
authors, there are only two works ([13, 14]) available on the market which
are concerned with the homogenization problem of rate-independent systems
in plasticity within an ergodic and stationary setting. In this work we extend
the resultsobtainedin[14]forperfectly elasto-plasticmodels torate-dependent
plasticity. Our main ingredient in the construction of the stochastic homoge-
nizationtheoryforrate-dependentmodelsofmonotonetype isthe combination
of the Fitzpatrick function concept and the two-scale convergence technique in
spaces equipped with random measures due to V.V. Zhikov and A.L. Pyatnit-
skii (see [34]). The Fitzpatrick function is used here to reduce the study of
the asymptotic behaviour of monotone operators associated with the models
under consideration to the problem of the stochastic homogenizationof convex
functionals defined on Sobolev spaces with random measures.
Setting of the problem. Let Q R3 be an open bounded set, the set of
⊂
material points of the solid body, with a Lipschitz boundary ∂Q, the number
η > 0 denote the scaling parameter of the microstructure and T be some
e
positive number (time of existence). For 0<t T
e
≤
Q =Q (0,t).
t ×
Let 3 denote the set of symmetric 3 3-matrices,and let u (x,t) R3 be the
η
S × ∈
unknown displacement of the material point x at time t, σ (x,t) 3 be the
η
unknown Cauchy stress tensor and z (x,t) RN denote the unkn∈owSn vector
η
∈
of internal variables. The model equations of the problem (the microscopic
problem) are
div σ (x,t) = b(x,t), (1)
x η
−
σ (x,t) = C [x](ε( u (x,t)) Bz (x,t)), (2)
η η x η η
∇ −
∂ z (x,t) g x,BTσ (x,t) L [x]z (x,t) , (3)
t η η η η η
∈ −
together with the homogeneous Diri(cid:0)chlet boundary condition (cid:1)
u (x,t)=0, (x,t) ∂Q (0, ), (4)
η
∈ × ∞
and the initial condition
z (x,0)=z(0)(x), x Q. (5)
η η ∈
In model equations (1) - (5)
1
ε( u (x,t))= ( u (x,t)+( u (x,t))T) 3
x η x η x η
∇ 2 ∇ ∇ ∈S
denotesthestraintensor(themeasureofdeformation),B :RN 3 isalinear
→S
mapping, which assigns to each vector of internal variables z (x,t) the plastic
η
strain tensor ε (x,t) 3, i.e. the following relation ε (x,t) = Bz (x,t)
p,η p,η η
∈ S
holds. We recall that the space 3 can be isomorphically identified with the
spaceR6 (see[1,p. 31]). ThereforSe,thelinearmappingB :RN 3 isdefined
asa compositionofa projectorfromRN ontoR6 andthe isomor→phSismbetween
R6 and 3. The transpose BT : 3 RN is given by
S S →
BTv =(zˆ,0)T
2
for v 3 and z =(zˆ,z˜)T RN, zˆ R6, z˜ RN−6.
Fo∈reSveryx Qweden∈ote byC∈[x]: 3∈ 3 alinearsymmetric mapping,
η
the elasticity te∈nsor. It is assumed that tShe →maSpping x C [x] is measurable.
η
→
Further,wesupposethatthereexisttwopositiveconstants0<α<β suchthat
the two-sided inequality
αξ 2 C [x]ξ ξ β ξ 2 for any ξ 3.
η
| | ≤ · ≤ | | ∈S
is satisfied uniformly with respect to x Q and η > 0. The given function
b:Q [0, ) R3 is the volume force. ∈The (N N)-matrix L [x] represents
η
× ∞ → ×
hardeningeffects. Itisassumedtobe positivesemi-definite,only. Forallx Q
the function z g (x,z) : RN 2RN is maximal monotone and satisfies∈the
η
→ →
following condition
0 g (x,0), x Q.
η
∈ ∈
The mapping x (L [x],g (x, )) is measurable.
η η
→ ·
Remark 1.1. Visco-plasticity is typically included in the former conditions by
choosing the function g to be in Norton-Hoff form, i.e.
η
Σ
g (x,Σ)=[Σ σ (x)]rη(x) , Σ 3, x Q,
η | |− y + Σ ∈S ∈
| |
where σ : Q (0, ) is the flow stress function and r : Q (0, ) is some
y η
→ ∞ → ∞
material function together with [x] :=max(x,0).
+
In order to specify the dependence of the model coefficients in (1) - (5)
on the microstructure scaling parameter η > 0, we introduce the concept of
a spatial dynamical system. Throughout this paper, we follow the setting of
Papanicolaouand Varadhan [22] and make the following assumptions.
Assumption1.1. Let(Ω, , )beaprobabilityspacewithcountablygenerated
Ω
F P
σ-algebra Ω. Further, we assumewe are given afamily (τx)x∈Rn of measurable
F
bijective mappings τ :Ω Ω, having the properties of a dynamical system on
x
7→
(Ω, , ), i.e. they satisfy (i)-(iii):
Ω
F P
(i) τ τ =τ , τ =id (Group property)
x y x+y 0
◦
(ii) (τ B)= (B) x Rn, B (Measure preserving)
−x Ω
P P ∀ ∈ ∈F
(iii) A:Rn Ω Ω, (x,ω) τ ω ismeasurable(Measurablilityofevaluation)
x
× → 7→
We finally assume that the system (τx)x∈Rn is ergodic. This means that for
every measurable function f :Ω R there holds
→
[f(ω)=f(τ ω) x Rn, a.e. ω Ω] [f(ω)=const -a.e.ω Ω] . (6)
x
∀ ∈ ∈ ⇒ P ∈
For reader’s convenience, we recall the following well-known result (see [9,
Section VI.15]).
Lemma 1.1. Let (A, ,µ) be a finite measure space with countably generated
F
σ-algebra . Then, for all 1 p< , Lp(A;µ) contains a countable dense set
F ≤ ∞
of simple functions.
3
The coefficients in (1) - (5) are defined as follows. First, we define the
stationary random fields through the relations
C[x,ω]=C˜[τ ω], L[x,ω]=L˜[τ ω],
x x
and for every fixed v RN
∈
g(x,ω,v)=g˜(τ ω,v),
x
whereC˜,L˜ aremeasurablefunctionsoverΩandω g˜(ω, )ismeasurableinthe
7→ ·
sense of Definition 2.2. Then, given the specified assumptions on the random
fields, the coefficients C [x], L [x] and the mapping x g (x, ) are defined as
η η η
7→ ·
x x
C [x]=C ,ω , L [x]=L ,ω ,
η η
η η
(cid:20) (cid:21) (cid:20) (cid:21)
and for each fixed v RN
∈
x
g (x,v)=g ,ω,v .
η
η
(cid:18) (cid:19)
Furthermore, we assume that
zη(0)(x)=z˜(0) x,τxηω , x∈Q.
(cid:16) (cid:17)
for some ergodic function z˜(0) L2(Q Ω; µ).
∈ × L⊗
Fromamodellingperspective,thisconstructionisequivalenttotheassump-
tion that the coefficients and the given functions in (1) - (5) are statistically
homogeneous (see [7], for example).
Notation. The symbols and(, ) will denote a normand a scalarproduct
in Rk, respectively. Let S|b·e| a me·as·urable set in Rs. For m N, q [1, ],
we denote by Wm,q(S,Rk) the Banach space of Lebesgue inte∈grable f∈uncti∞ons
havingq-integrableweakderivativesuptoorderm. Thisspaceisequippedwith
the norm . If m = 0, we write ; and if (additionally) q = 2, we
m,q,S q,S
also writek·k . We set Hm(S,Rk) = kW·mk,2(S,Rk). We choose the numbers
S
k·k
p,q satisfying 1<p,q < and 1/p+1/q=1. For such p and q one can define
the bilinear form on the ∞product space Lp(S,Rk) Lq(S,Rk) by
×
(ξ,ζ) = (ξ(s),ζ(s))ds.
S
ZS
ForfunctionsvdefinedonΩ [0, )wedenotebyv(t)themappingx v(x,t),
× ∞ 7→
which is defined on Ω. The space Lq(0,T ;X) denotes the Banach space of
e
all Bochner-measurable functions u : [0,T ) X such that t u(t) q is
e → 7→ k kX
integrableon[0,T ). Finally,wefrequentlyusethespacesWm,q(0,T ;X),which
e e
consistofBochnermeasurablefunctionshavingq-integrableweakderivativesup
to order m.
4
2 Preliminaries.
In this section we briefly recall some basic facts from convex analysis and non-
linear functional analysis which are needed for further discussions. For more
details see [5, 15, 23, 33], for example.
Let V be a reflexive Banach space with the norm , V∗ be its dual space
k·k
with the norm . The brackets , denote the duality pairing between V
∗
k·k h· ·i
and V∗. By V we shall always mean a reflexive Banach space throughout this
section.
For a function φ:V R the sets
→
dom(φ)= v V φ(v)< , epi(φ)= (v,t) V φ(v) t
{ ∈ | ∞} { ∈ ×R| ≤ }
arecalledtheeffectivedomainandtheepigraphofφ,respectively. Onesaysthat
the function φ is proper if dom(φ) = and φ(v) > for every v V. The
6 ∅ −∞ ∈
epigraphisanon-emptyclosedconvexsetiffφisaproperlowersemi-continuous
convex function or, equivalently, iff φ is a proper weakly lower semi-continuous
convex function (see [33, Theorem 2.2.1]).
The Legendre-Fenchel conjugate of a proper convex lower semi-continuous
function φ:V R is the function φ∗ defined for each v∗ V∗ by
→ ∈
φ∗(v∗)= sup v∗,v φ(v) .
{h i− }
v∈V
TheLegendre-Fenchelconjugateφ∗ isconvex,lowersemi-continuousandproper
on the dual space V∗. Moreover,the Young-Fenchel inequality holds
v V, v∗ V∗ : φ∗(v∗)+φ(v) v∗,v , (7)
∀ ∈ ∀ ∈ ≥h i
and the inequality φ ψ implies ψ∗ φ∗ for any two proper convex lower
semi-continuous functi≤ons ψ,φ:V R≤(see [33, Theorem 2.3.1]).
→
Due to PropositionII.2.5 in[5]a proper convexlowersemi-continuousfunc-
tion φ satisfies the following identity
intdom(φ)=intdom(∂φ), (8)
where∂φ:V 2V∗ denotesthesubdifferentialofthefunctionφ. Wenotethat
→
the equality in (7) holds iff v∗ ∂φ(v).
∈
Remark 2.1. We recall that the subdifferential of a lower semi-continuous
proper and convex function is maximal monotone (see [5, Theorem II.2.1]) in
the sense of Definition 2.1 below.
Convex integrands. Let the numbers p,q satisfy 1<q 2 p< ,1/p+
1/q = 1. For a proper convex lower semi-continuous funct≤ion φ≤: Rk∞ R we
define a functional I on Lp(G,Rk) by →
φ
φ(v(x))dx, φ(v) L1(G,Rk)
I (v)= G ∈ ,
φ
(+R ∞, otherwise
where G is a bounded domain in RN with some N N. Due to Proposition
∈
II.8.1 in [28], the functional I is proper, convex, lower semi-continuous, and
φ
v∗ ∂I (v) iff
φ
∈
v∗ Lq(G,Rk), v Lp(G,Rk) and v∗(x) ∂φ(v(x)), a.e.
∈ ∈ ∈
5
Due to the result of Rockafellar in [24, Theorem 2], the Legendre-Fenchel con-
jugate of Iφ is equal to Iφ∗, i.e.
∗
Iφ =Iφ∗,
where φ∗ is the Legendre-Fenchel(cid:0)con(cid:1)jugate of φ.
Maximal monotone operators. For a multivalued mapping A : V 2V∗
→
the sets
D(A)= v V Av = , GrA= [v,v∗] V V∗ v D(A), v∗ Av
{ ∈ | 6 ∅} { ∈ × | ∈ ∈ }
are called the effective domain and the graph of A, respectively.
Definition 2.1. A mapping A:V 2V∗ is called monotone if and only if the
→
following inequality holds
v∗ u∗,v u 0 [v,v∗],[u,u∗] GrA.
h − − i≥ ∀ ∈
A monotone mapping A : V 2V∗ is called maximal monotone iff the
→
inequality
v∗ u∗,v u 0 [u,u∗] GrA
h − − i≥ ∀ ∈
implies [v,v∗] GrA.
∈
It is well known ([23, p. 105]) that if A is a maximal monotone operator,
then for any v D(A) the image Av is a closed convex subset of V∗ and the
∈
graph GrA is demi-closed1.
Canonical extensions of maximal monotone operators. In this subsec-
tionwebrieflypresentsomefactsaboutmeasurablemulti-valuedmappings(see
[4, 6, 15, 21], for example). We assume that V, and hence V∗, is separable and
denote the set of maximal monotone operators from V to V∗ by M(V V∗).
×
Further, let (S,Σ(S),µ) be a σ finite µ complete measurable space.
− −
Definition 2.2. A mapping A : S M(V V∗) is measurable iff for every
→ ×
open set U V V∗ (respectively closed set, Borel set, open ball, closed ball),
∈ ×
x S A(x) U =
{ ∈ | ∩ 6 ∅}
is measurable in S.
The fact that the closed or Borel sets can be equivalently used in Defini-
tion 2.2 follows from the closedness of the values of the mapping A : S
M(V V∗) (see [4, Theorem 8.1.4]). →
×
Remark 2.2. Theorem 8.1.4 in [4] also implies that under theabove conditions
themeasurabilityofamappingA:S M(V V∗)isequivalenttotheexistence
→ ×
of a countable dense subset consisting of measurable selectors, i.e. there exists
a sequence of measurable functions vn n∈N : S V V∗ such that for any
{ } → ×
x S the image A(x) can be represented as follows
∈
A(x)= n∈Nvn(x).
∪
1A set A ∈ V ×V∗ is demi-closed if vn converges strongly to v0 in V and vn∗ converges
weakly to v0∗ in V∗ (or vn converges weakly to v0 in V and vn∗ converges strongly to v0∗ in
V∗)and[vn,vn∗]∈GrA,then[v,v∗]∈GrA
6
The following lemma will be used in the sequel (see [32, Lemma 3.1]).
Lemma 2.1. Leta mapping A:S M(V V∗)bemeasurable. For any (S)-
measurable function v :S V, the→multiva×lued mapping Aˆ: x A(x,v(Lx)) is
→ 7→
then closed-valued and measurable.
Given a mapping A : S M(V V∗), one can define a monotone graph
→ ×
from Lp(S,V) to Lq(S,V∗), where 1/p+1/q=1, as follows:
Definition 2.3. Let A : S M(V V∗). The canonical extension of A from
→ ×
Lp(S,V) to Lq(S,V∗), where 1/p+1/q =1, is defined by:
Gr = [v,v∗] Lp(S,V) Lq(S,V∗) [v(x),v∗(x)] GrA(x)fora.e.x S .
p
A { ∈ × | ∈ ∈ }
In the following, we will drop the index p for readability. Since we always
work fix p at the beginning of a statement, there cannot occur confusion with
this notation. Monotonicity of defined in Definition 2.3 is obvious, while its
A
maximality follows from the next proposition (see [8, Proposition 2.13]).
Proposition 2.1. Let A : S M(V V∗) be measurable. If Gr = , then
→ × A 6 ∅
is maximal monotone.
A
Remark 2.3. We point out that the maximality of A(x) for almost every x S
∈
does not imply the maximality of as the latter can be empty (see [8]).
A
Fitzpatrick’s function. For a proper operator β : V 2V∗ the Fitzpatrick
→
function is defined as the convex and lower semicontinuous function given by
f (v,v∗)=sup v∗,v v∗,v v v∗ β(v ) , (v,v∗) V V∗. (9)
β {h 0i−h 0 0− i| 0 ∈ 0 } ∀ ∈ ×
It is known ([10]) that, whenever β is maximal monotone,
f (v,v∗) v∗,v , (v,v∗) V V∗, (10)
β
≥h i ∀ ∈ ×
f (v,v∗)= v∗,v v∗ β(v). (11)
β
h i ⇔ ∈
Any measurable maximal monotone operator A : S M(V V∗) can be
represented by its Fitzpatrick function f : S V V∗→ R, wh×ich is Σ(S)
A
(V V∗)-measurable. Namely, the graph of×a ma×pping→A : S M(V V∗⊗)
B × → ×
can be written in the form (see [32, Proposition 3.2])
for a.e. x S GrA(x) = [v,v∗] V V∗ f (x,v,v∗)= v,v∗ .
A
∈ { ∈ × | h i}
WenotethatthemeasurabilityoftheFitzpatrickfunctionf :S V V∗ R
A
× × →
follows directly from its definition and Remark 2.2.
The graph of the canonical extension of a measurable operator A : S
M(V V∗) canbe equivalently representedin terms of its Fitzpatrick functio→n
F :×Lp(S,V) Lq(S,V∗) R, i.e.
Ap × →
Gr = [v,v∗] Lp(S,V) Lq(S,V∗) F (v,v∗)= v,v∗ .
Ap ∈ × | Ap h i
Again, we omit p(cid:8)if no confusion occurs. Moreover, the following re(cid:9)sult holds
(see [32, Proposition 3.3])
the functional F is convex and lower semi-continuous;
A
•
7
for any [v,v∗] Lp(S,V) Lq(S,V∗), the integral
• ∈ ×
F (v,v∗)= f (x,v(x),v∗(x))dx
A A
ZS
exists either finite or equal to + ;
∞
if there exists a pair [v,v∗] Lp(S,V) Lq(S,V∗) such that F (v,v∗)<
A
• ∈ ×
+ , then
∞
F∗(v∗,v)= f∗(x,v∗(x),v(x))dx
A A
ZS
holds for all [v∗,v] Lq(S,V∗) Lp(S,V).
∈ ×
3 Existence of solutions
In this section we introduce and show the existence of weak solutions for the
initialboundaryvalue(1)-(5). Tosimplifythenotations,throughoutthewhole
section we ignore the fact the coefficients and the given functions in (1) - (5)
depend on ω Ω. The results proved below hold for a.e. ω Ω.
∈ ∈
Solvability concept. We startthis sectionwiththe presentationofthe intu-
itiveideaswhichleadtothedefinitionofweaksolutionsfortheinitialboundary
value problem (1) - (5). To give a meaning for the solvability of problem (1) -
(5) we are going to use the concept of Fitzpatrick functions defined in (9).
Weassumefirstthatatripleoffunctions(u ,σ ,z )isgivenwiththefollow-
η η η
ingproperties: foreveryt (0,T )the function(u (t),σ (t))isaweaksolution
e η η
∈
of the boundary value problem
div σ (x,t) = b(x,t), (12)
x η
−
σ (x,t) = C [x](ε( u (x,t)) Bz (x,t)), (13)
η η x η η
∇ −
u (x,t) = 0, x ∂Q. (14)
η
∈
(0)
This particularly holds for z (0) = z and the corresponding initial values
η η
(u (0),σ (0)) = (u(0),σ(0)). The equations (3) - (5) are satisfied pointwise for
η η η η
almostevery(x,t),andbaswellas(u ,σ ,z )aresmoothenough. Then,based
η η η
on equivalence (11), we can rewrite equation (3) as follows
f x,BTσ (x,t) L [x]z (x,t),∂ z (x,t)
gη η − η η t η
=(cid:0)BTσ (x,t) L [x]z (x,t),∂ z (x,t) ,(cid:1)
η η η t η
−
which holds for almos(cid:0)t every (x,t) Q (0,Te). Integrat(cid:1)ing the last equality
∈ ×
over Q gives
f x,BTσ L z ,∂ z dx= BTσ L z ,∂ z dx. (15)
gη η− η η t η η− η η t η
ZQ ZQ
(cid:0) (cid:1) (cid:0) (cid:1)
8
Using (1), (2) and (4) the right hand side in (15) becomes (A :=C−1)
η η
1 d 2
BTσ L z ,∂ z dx= BTσ ,∂ z L1/2z
ZQ η − η η t η η t η Q− 2dt(cid:13) η η(cid:13)Q
(cid:0) (cid:1) (cid:0) (cid:1) 1 d (cid:13) (cid:13)2
=(σ ,ε(∂ u )) (A σ ,∂ σ ) (cid:13)L1/2z (cid:13)
η t∇x η Q− η η t η Q− 2dt η η Q
(cid:13) (cid:13)
=(b,∂tuη)Q− 21ddt A1η/2ση 2Q+ L(cid:13)(cid:13)1η/2zη 2Q(cid:13)(cid:13) . (16)
(cid:26)(cid:13) (cid:13) (cid:13) (cid:13) (cid:27)
(cid:13) (cid:13) (cid:13) (cid:13)
Integrating relations (15) and (16) with re(cid:13)spect to(cid:13)t lea(cid:13)ds to (cid:13)
(A [x]σ (x,t),σ (x,t))dx+ (L [x]z (x,t),z (x,t))dx
η η η η η η
ZQ ZQ
t
+ f x,BTσ (x,τ) L [x]z (x,τ),∂ z (x,τ) dxdτ (17)
gη η − η η τ η
Z0 ZQ
(cid:0) (cid:1)
= (A [x]σ (x,0),σ (x,0))dx+ L [x]z(0)(x),z(0)(x) dx+(b,∂ u ) .
η η η η η η τ η Q
ZQ ZQ(cid:16) (cid:17) t
Takingintoaccounttheinequality(10),weconcludethatthetripleoffunctions
(u ,σ ,z ) satisfies equality (17) if and only if the inequality
η η η
(A [x]σ (x,t),σ (x,t))dx+ (L [x]z (x,t),z (x,t))dx
η η η η η η
ZQ ZQ
t
+ f x,BTσ (x,τ) L [x]z (x,τ),∂ z (x,τ) dxdτ (18)
gη η − η η τ η
Z0 ZQ
(cid:0) (cid:1)
A [x]σ(0)(x),σ(0)(x) dx+ L [x]z(0)(x),z(0)(x) dx+(b,∂ u )
≤ZQ(cid:16) η η η (cid:17) ZQ(cid:16) η η η (cid:17) τ η Qt
holds for all t (0,T ) and some function σ(0) L2(Q, 3) solving the elliptic
e η
∈ ∈ S
boundary value problem (12) - (14).
The above computations suggest the following notion of weak solutions for
the initial boundary value problem (1) - (5).
Definition 3.1. Let the numbers p,q satisfy 1<q 2 p< ,1/p+1/q=1.
≤ ≤ ∞
A function (u ,σ ,z ) such that
η η η
(u ,σ ) W1,q(0,T ;W1,q(Q,R3) Lq(Q, 3)),
η η ∈ e 0 × S
z W1,q(0,T ;Lq(Q,RN)), Σ :=BTσ L z Lp(Q ,RN)
η ∈ e η η− η η ∈ Te
with
(σ ,L1/2z ) L∞(0,T ;L2(Q, 3 RN))
η η η ∈ e S ×
is called a weak solution of the initial boundary value problem (1) - (5), if for
every t (0,T ) the function (u (t),σ (t)) is a weak solution of the boundary
e η η
∈
value problem (1) - (2), (4) for every given Bz (t) Lq(Q, 3), the initial
η
∈ S
condition (5) is satisfied pointwise for almost every (x,t) and the inequality
(18) holds for all t (0,T ) and the function σ(0) L2(Q, 3) determined by
e η
∈ ∈ S
equations (12) - (14).
9
Now,weshowthattheabovedefinitionofweaksolutionsfor(1)-(5)iscon-
sistent. Namely,we aregoingto provethat ifa triple offunctions (u ,σ ,z )is
η η η
aweaksolutionof(1)-(5)inthesenseofDefinition3.1andpossessesadditional
regularity,thenthistripleoffunctionsisasolutionoftheinitialboundaryvalue
problem(1)-(5),i.e. theconstitutiveinclusion(3)issatisfiedpointwisefora.e.
(x,t) Q . To this end, we assumethat the weaksolution(u ,σ ,z )has the
∈ Te η η η
following regularity
(u ,σ ) W1,1(0,T ;H1(Q,R3) L2(Q, 3)),
η η ∈ e 0 × S
z W1,1(0,T ;L2(Q,RN)).
η e
∈
Then, it is immediately seen that the function σ(0) L2(Q, 3) as a unique
η
∈ S
solution of the problem (12) - (14) satisfies the relation σ(0)(x) = σ (x,0) for
η η
a.e. x Q and the following identity
∈
∂
A σ (t),σ (t) A σ(0),σ(0) = (A σ (x,τ),σ (x,s))dsdx
η η η Q− η η η Q ∂τ η η η
ZQt
(cid:0) (cid:1) (cid:0) (cid:1)
Moreover,we have that
L1/2z (t) 2 L1/2z(0) 2 = t ∂ L1/2z (τ) 2 dτ.
η η Q− η η Q ∂τk η η kQ
Z0
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
Then, the inequality (18) can be rewritten as follows
(A ∂ σ ,σ )+(L z ,∂ z )+f x,BTσ L z ,∂ z dτdx (b,∂ u ) .
η τ η η η η τ η gη η− η η τ η ≤ τ η Qt
ZQt(cid:16) (cid:0) (cid:1)(cid:17)
Handling the equations (1) - (2) as above we obtain that the last inequality
takes the following form
(L z ,∂ z )+f x,BTσ L z ,∂ z dτdx (BTσ ,∂ z ) .
η η τ η gη η− η η τ η ≤ η τ η Qt
ZQt(cid:16) (cid:0) (cid:1)(cid:17)
or, equivalently,
f x,BTσ L z ,∂ z dτdx (BTσ L z ,∂ z )dxdτ.
gη η− η η τ η ≤ η− η η τ η
ZQt ZQt
(cid:0) (cid:1)
Therefore, by (10) and the standard localization argument we get that
f x,BTσ (x,t) L [x]z (x,t),∂ z (x,t)
gη η − η η t η
=(cid:0)BTσ (x,t) L [x]z (x,t),∂ z (x,t) ,(cid:1)
η η η t η
−
which holds for a.e. ((cid:0)x,t) Q . Now, based on the equiv(cid:1)alence result (11) we
∈ Te
concludethattheinclusion(3)issatisfiedpointwisefromtheassumedtemporal
regularity of (u ,σ ,z ). The pointwise meaning of (5) follows.
η η η
10