Table Of ContentOn the existence of an exponential attractor
for a planar shear flow with Tresca’s friction
2 condition
1
0
2 Grzegorz L ukaszewicz ∗†
n
a
J
7
University of Warsaw, Mathematics Department,ul.Banacha 2, 02-957 Warsaw, Poland
2
Abstract
]
h
Weconsider a two-dimensional nonstationary Navier-Stokesshear flowwith asub-
p
differential boundary condition on a part of the boundary of the flow domain,
-
h namely, with a boundary driving subject to the Tresca law. There exists a unique
t globalintimesolutionoftheconsideredproblemwhichisgovernedbyavariational
a
inequality. Our aim is to prove the existence of a global attractor of a finite frac-
m
tional dimension and of an exponentialattractor for theassociated semigroup. We
[ use the method of l-trajectories. This research is motivated by a problem from
1 lubrication theory.
v
3 Keywords: lubrication theory, Navier-Stokes equation, global solution, exponential at-
5 tractor
7
5 1991 Mathematics Subject Classification: 76D05, 76F10, 76F20
.
1
0
2 1 Introduction
1
:
v Remarking on future directions of researchin the field of contact mechanics, in their re-
i centbook[1],the authorswrote: ”Theinfinite-dimensionaldynamicalsystemsapproach
X
to contact problems is virtually nonexistent. (...) This topic certainly deserves further
r
a consideration”.
From the mathematical point of view a considerable difficulty in analysing problems
of contact mechanics, and dynamical problems in particular, comes from the presence
of involved boundary constraints which are often modelled by boundary conditions of
a dissipative subdifferential type and lead to a formulation of the considered problem
in terms of a variationalor hemivariationalinequality with, frequently, nondifferentiable
boundary functionals.
Our aim in this paper is to contribute to this topic by an examination of the large
time behaviour of solutions of a problem coming from the theory of lubrication.
∗E-mail: [email protected],Tel.: +48225544562
†ThisresearchwassupportedbyPolishGovernmentGrantNN201547638
1
Westudytheproblemofexistenceoftheglobalattractorofafinitefractaldimension
andofanexponentialattractorfora classof two-dimensionalturbulent boundary driven
flows subject to the Tresca law which naturally appears in lubrication theory. Existence
of such attractors strongly suggest that the time asymptotics of the considered flow can
be described by a finite number of parameters and then treated numerically [2, 3]. We
study the problem in its weak formulation given in terms of an evolutionary variational
inequality with a nondifferentiable boundary functional. This situation produces an
obstacle for applying directly the classical methods, presented e.g., in monographs [3, 4,
5, 6, 7], to prove that the fractal dimension of the global attractor is finite. Instead, we
apply the powerful method of l-trajectories,introduced in [8, 9] which we use further to
prove the existence of an exponential attractor. The method of l-trajectories helps to
provethe existenceofanexponentialattractorforaconsiderablylargeclassofnonlinear
problems,inparticularthatwithlackofgoodregularityproperties(c.f., e.g.,[10,11,12]
and references therein).
The problem we consider is as follows. The flow of an incompressible fluid in a
two-dimensional domain Ω is described by the equation of motion
u ν∆u+(u )u+ p=0 in Ω (1.1)
t
− ·∇ ∇
and the incompressibility condition
divu=0 in Ω. (1.2)
To define the domain Ω of the flow, let Ω be the channel,
∞
Ω = x=(x ,x ): <x < , 0<x <h(x ) ,
∞ 1 2 1 2 1
{ −∞ ∞ }
where h is a positive function, smooth, and L-periodic in x . Then we set
1
Ω= x=(x ,x ):0<x <L, 0<x <h(x )
1 2 1 2 1
{ }
and ∂Ω = Γ¯ Γ¯ Γ¯ , where Γ and Γ are the bottom and the top, and Γ is the
0 L 1 0 1 L
∪ ∪
lateral part of the boundary of Ω.
We areinterestedinsolutionsof(1.1)-(1.2) inΩ whichareL-periodicwith respectto
x . We assume that
1
u=0 at Γ . (1.3)
1
Moreover, we assume that there is no flux condition across Γ so that the normal com-
0
ponent of the velocity on Γ satisfies
0
u n=0 at Γ , (1.4)
0
·
and that the tangential component of the velocity u on Γ is unknown and satisfies
η 0
the Tresca friction law with a constant and positive maximal friction coefficient k. This
means that, c.f., e.g., [1, 13],
σ (u,p) k
η
| |≤
σ (u,p) <k u =U e at Γ (1.5)
| η | ⇒ η 0 1 0
σ (u,p) =k λ 0 such that u =U e λσ (u,p)
η η 0 1 η
| | ⇒∃ ≥ −
2
where σ is the tangential component of the stress tensor on Γ and U e = (U ,0),
η 0 0 1 0
U R, is the velocity of the lower surface producing the driving force of the flow.
0
∈
If n=(n ,n ) is the unit outwardnormal to Γ , and η =(η ,η ) is the unit tangent
1 2 0 1 2
vector to Γ then we have
0
σ (u,p)=σ(u,p) n ((σ(u,p) n) n)n, (1.6)
η
· − · ·
where σ(u,p)=(σ (u,p))=( pδ +ν(u +u )) is the stress tensor. In the end, the
ij ij i,j j,i
−
initial condition for the velocity field is
u(x,0)=u (x) for x Ω.
0
∈
The problem is motivated by the examination of a certain two-dimensional flow in an
infinite (rectified) journal bearing Ω ( ,+ ), where Γ ( ,+ ) represents
1
× −∞ ∞ × −∞ ∞
the outer cylinder, and Γ ( ,+ ) represents the inner, rotating cylinder. In the
0
× −∞ ∞
lubricationproblemsthe gaph betweencylindersisneverconstant. We canassumethat
therectificationdoesnotchangetheequationsasthegapbetweencylindersisverysmall
with respect to their radii.
The knowledge or the judicious choice of the boundary conditions on the fluid-solid
interfaceisofparticularinterestinlubricationareawhichisconcernedwiththinfilmflow
behaviour. Theboundaryconditionstobe employedaredeterminedbynumerousphysi-
cal parameters characterizing,for example, surface roughness and rheologicalproperties
of the fluid.
Thewidelyusedno-slipconditionwhenthefluidhasthesamevelocityassurrounding
solid boundary is not respected if the shear rate becomes too high (no-slip condition is
induced by chemical bounds between the lubricant and the surrounding surfaces and by
the action of the normal stresses, which are linked to the pressure inside the flow; on
the contrary, when tangential stressses are high they can destroy the chemical bounds
and induce slip phenomenon). We can model such situation by a transposition of the
well-known friction laws between two solids [1] to the fluid-solid interface.
The system of equations (1.1)-(1.2) with boundary conditions: (1.3) at Γ for h =
1
const and u = const on Γ , instead of (1.4)-(1.5), was intensively studied in several
0
contexts,someofthemmentionedintheintroductionof[14]. Theautonomouscasewith
h = const and with u = const on Γ was considered in [15, 16]. See also [17] where
0
6
the case h = const, u = U(t)e on Γ , was considered. The important for applications
1 0
6
dynamical problem we consider in this paper has been studied earlier in [18] in the
nonautonomouscaseforwhichtheexistenceofapullbackattractorwasestablishedwith
the use of a method that, however, did not guarantee the finite dimensionality of the
pullback attractor (or the global attractor in the reduced autonomous case).
To establish the existence of the global attractor of a finite fractal dimension we use
the method of l-trajectories as presented in [9]. This method appears very useful when
one deals with variational inequalities, c.f., [12], as it overcomes obstacles coming from
the usual methods. One needs neither compactness of the dynamics which results from
the second energy inequality nor asymptotic compactness, c.f., i.e., [7, 17], which results
from the energy equation. In the case of variational inequalities it is sometimes not
possible to get the second energy inequality and the differentiability of the associated
semigroup due to the presence of nondifferentiable boundary functionals. On the other
hand, we do not have an energy equation to prove the asymptotic compactness.
3
Whilethereareothermethodstoestablishtheexistenceoftheglobalattractorwhere
the problem of the lack of regularity appears, that, e.g., based on the notion of the
Kuratowskimeasureof noncompactnessofbounded sets,where we do notneed eventhe
continuity of the semigroup associated with a given dynamical problem, c.f., e.g., [19],
and also [18], where the nonautonomous versionof the problem consideredin this paper
was studied, the problem of a finite dimensionality of the attractor is more involved, c.f.
also [14].
The method of l-trajectories allows to prove the existence of an even more desirable
object, called exponential attractor, for many problems for which there exists a finite
dimensional global attractor [11]. An exponential attractor is a compact subset of the
phase space which is positively invariant, has finite fractal dimension, and attracts uni-
formly bounded sets at an exponential rate. It contains the global attractor and thus
its existence implies the finite dimensionality of the global attractor itself. Its crucial
property is an exponential rate of attraction of solution trajectories [10, 11]. The proof
ofthe existence ofanexponentialattractorrequiresthe solutionto be regularenoughto
ensure the Ho¨lder continuity of the semigroupin the time variable [9]. We establish this
property by providing additional a priori estimates of solutions.
Ourplanis asfollows. InSection2wehomogenizefirstthe boundarycondition(1.5)
by a smooth background flow (a simple version of the Hopf construction, c.f., e.g., [18])
andthen we presenta variationalformulationof the homogenizedproblem. In Section3
we recall briefly the proof of the existence and uniqueness of a global in time solution
of our problem and obtain some estimates of the solutions. Section 4 is devoted to a
presentation of the main definitions and elements of the theory of infinite dimensional
dynamical systems we use, in particular, of the method of l-trajectories. In Section 5
we prove the existence of the global attractor of a finite fractal dimension. At last, in
Section 6 we provethe existence of an exponentialattractorand in Section7 we provide
some final comments.
2 Variational formulation of the problem
First, we homogenize the boundary condition (1.5). To this end let
u(x ,x ,t)=U(x )e +v(x ,x ,t) (2.1)
1 2 2 1 1 2
with
U(0)=U , U(h(x ))=0, x (0,L). (2.2)
0 1 1
∈
The new vector field v is L-periodic in x and satisfies the equation of motion
1
v ν∆v+(v )v+ p=G(v) (2.3)
t
− ·∇ ∇
with
G(v)= Uv, (v) U, e +νU, e
− x1− 2 x2 1 x2x2 1
where by (v) we denoted the second component of v. As div(Ue )=0 we get
2 1
divv =0 in Ω. (2.4)
From (2.1)-(2.2) we obtain
4
v =0, on Γ , (2.5)
1
and
v n=0, on Γ . (2.6)
0
·
Moreover,we have,
∂U(x )
σ (v,p)=σ (u,p)+(ν 2 ,0).
η η ∂x |x2=0
2
Since we can define the extension U in such a way that
∂U(x )
2
=0
∂x |x2=0
2
the Tresca condition (1.5) transforms to
σ (u,p) k
η
| |≤
σ (v,p) <k v =0 at Γ (2.7)
| η | ⇒ η 0
σ (v,p) =k λ 0 such that v = λσ (v,p)
η η η
In the end, th|e initial|condi⇒tio∃n b≥ecomes −
v(x,0)=v (x)=u (x) U(x )e . (2.8)
0 0 2 1
−
TheTrescacondition(2.7)isaparticularcaseofanimportantincontactmechanicsclass
of subdifferential boundary conditions of the form, c.f., e.g. [20],
ϕ(v) ϕ(u) σn(v u) at Γ ,
0
− ≥− −
whereσnistheCauchystressvectorandvbelongstoacertainsetofadmissiblefunctions.
For ϕ(v)=k v the last condition is equivalent to (2.7).
η
| |
Nowwe canintroduce the variationalformulationof the homogenizedproblem(2.3)-
(2.8). Then, for the convenience of the readers, we describe the relations between the
classical and the weak formulations.
We begin with some basic definitions of the paper.
Let
V˜ = v ∞(Ω)2 : divv =0 in Ω, vis L-periodic in x ,
1
{ ∈C
v =0 at Γ , v n=0 at Γ
1 0
· }
and
V =closure of V˜ in H1(Ω)2, H =closure ofV˜ in L2(Ω)2.
We define scalar products in H and V, respectively, by
(u, v)= u(x)v(x)dx and ( u, v)
∇ ∇
ZΩ
and their associated norms by
1 1
v =(v,v)2 and v =( v, v)2.
| | k k ∇ ∇
5
Let, for u,v and w in V
a(u, v)=( u, v) and b(u, v, w)=((u )v, w).
∇ ∇ ·∇
In the end, let us define the functional j on V by
j(u)= k u(x ,0)dx .
1 1
| |
ZΓ0
The variational formulation of the homogenized problem (2.3)-(2.8) is as follows.
Problem 2.1. Given v H, find v :(0, ) H such that:
0
∈ ∞ →
(i) for all T >0,
v ([0,T];H) L2(0,T;V), with v L2(0,T;V′)
t
∈C ∩ ∈
where V′ is the dual space to V.
(ii) for all Θ in V, all T > 0, and for almost all t in the interval [0,T], the following
variational inequality holds
<v (t),Θ v(t)> +νa(v(t),Θ v(t)) + b(v(t),v(t),Θ v(t)) (2.9)
t
− − −
+ j(Θ) j(v(t)) ( (v(t)),Θ v(t))
− ≥ L −
(iii) the initial condition
v(x,0)=v (x) (2.10)
0
holds.
In (2.9) the functional (v(t)) is defined for almost all t 0 by,
L ≥
( (v(t)),Θ)= νa(ξ,Θ) b(ξ,v(t),Θ) b(v(t),ξ,Θ),
L − − −
where ξ =Ue is a suitable smooth background flow.
1
We have the following relations between classical and weak formulations.
Proposition 2.1. Every classical solution of Problem (2.3)-(2.8) is also a solution of
Problem 2.1. On the other hand, every solution of Problem 2.1 which is smooth enough
is also a classical solution of Problem (2.3)-(2.8).
Proof. Let v be a classical solution of Problem (2.3)-(2.8). As it is (by assumption)
sufficiently regular, we have to check only (2.9). Remark first that (2.3) can be written
as
v Divσ(v,p)+(v )v =G(v(t)). (2.11)
t
− ·∇
Let Θ V. Multiplying (2.11) by Θ v(t) and using Green’s formula we obtain
∈ −
v (Θ v(t))dx + σ (v,p)(Θ v(t)) dx+b(v(t), v(t), Θ v(t))
t ij i,j
− − −
ZΩ ZΩ
= σ (v,p)n (Θ v(t)) + G(v(t))(Θ v(t))dx(2.12)
ij j i
− −
Z∂Ω ZΩ
6
for t (0,T). As v(t) and Θ are in V, after some calculations we obtain
∈
σ (v,p)(Θ v(t)) dx=νa(v(t), Θ v(t)), (2.13)
ij i,j
− −
ZΩ
and using (1.6) we obtain
σ (v,p)n (Θ v(t)) = σ (v,p) (Θ v (t)) (σ n n )n (Θ v (t)) .
ij j i η η ij j i i η i
− · − − −
Z∂Ω ZΓ0 ZΓ0
As n (Θ v (t)) =0 on Γ , we get
i η i 0
−
σ (v,p)n (Θ v(t)) = σ (v,p) (Θ v (t))= (σ Θ+k Θ)
ij j i η η η
− · − · | |
Z∂Ω ZΓ0 ZΓ0
k(Θ v (t)) (k v (t) +σ v (t)).
η η η η
− | |−| | − | | ·
ZΓ0 ZΓ0
Remark that σ Θ+k Θ 0 for σ k, and the Tresca condition (2.7) is equivalent
η η
· | |≥ | |≤
to
k v (t) +σ v (t)=0 a.e. on Γ .
η η η 0
| | ·
Thus
σ (v,p)n (Θ v(t)) k(Θ v (t))= j(Θ)+j(v(t)). (2.14)
ij j i η
− ≥− | |−| | −
Z∂Ω ZΓ0
As
G(v(t))(Θ v(t))dx= ( (v(t)), Θ v(t)),
− L −
ZΩ
from (2.13) and (2.14) we see that (2.12) becomes (2.9), and (2.10) is the same as (2.8).
Conversely,suppose that v is a sufficiently smooth solution to Problem 2.1 and let ϕ be
in the space (H1 (Ω))2 = ϕ V : ϕ = 0 on Γ . We take Θ = v(t) ϕ in (2.9), and
div { ∈ } ±
using the Green formula, we obtain
<v (t) ν∆v(t)+(v(t) )v(t) G(v(t)), ϕ>=0 ϕ (H1 (Ω))2.
t − ·∇ − ∀ ∈ div
Thus, there exists a distribution p(t) on Ω such that
v (t) ν∆v(t)+(v(t) )v(t) G(v(t))= p(t) in Ω
t
− ·∇ − ∇
so that (2.3) holds. We obtain (2.7) as in [21], and we have immediately (2.4)-(2.6) and
(2.8).
3 Existence and uniqueness of a global in time solu-
tion
In this section we establish, following [18], the existence and uniqueness of a global in
time solution for Problem 2.1. First, we present two lemmas.
7
Lemma 3.1. ([17]) There exists a smooth extension
ξ(x )=U(x )e
2 2 1
of U e from Γ to Ω satisfying: (2.2),
0 1 0
∂U(x )
2 =0,
∂x |x2=0
2
and such that
ν
b(v, ξ, v) v 2 for all v V.
| |≤ 4k k ∈
Moreover,
ξ 2+ ξ 2 = U(x )2dx dx + U, (x )2dx dx F,
| | |∇ | | 2 | 1 2 | x2 2 | 1 2 ≤
ZΩ ZΩ
where F depends on ν,Ω, and U .
0
Lemma 3.2. ([18]) For all v in V we have the Ladyzhenskaya inequality
1 1
v L4(Ω) C(Ω)v 2 v 2. (3.1)
k k ≤ | | k k
Proof. Let v V and r C1(( L,L)) such that r =1 on [0, L] and r =0 at x = L.
1
∈ ∈ − −
Defineϕ=rv,andextendϕby0toΩ =( L, L) (0, h),whereh=max h(x ).
1 − × 0≤x1≤L 1
We obtain
x1 ∂ϕ L ∂ϕ
ϕ2(x ,x ) = 2 ϕ(t ,x ) (t ,x )dt 2 ϕ(x ,x ) (x ,x )dx
1 2 1 2 ∂t 1 2 1 ≤ | 1 2 ||∂x 1 2 | 1
Z−L 1 Z−L 1
and
h ∂ϕ h ∂ϕ
ϕ2(x ,x )= 2 ϕ(x ,t ) (x ,t )dt 2 ϕ(x ,x ) (x ,x )dx ,
1 2 − 1 2 ∂t 1 2 2 ≤ | 1 2 ||∂x 1 2 | 2
Zx2 2 Z0 2
whence
ϕ 4 = ϕ2(x ,x )ϕ2(x ,x )dx dx
k kL4(Ω1) 1 2 1 2 1 2
ZΩ1
h L
sup ϕ2(x ,x )dx sup ϕ2(x ,x )dx
1 2 2 1 2 1
≤ Z0 −L≤x1≤L ! Z−L 0≤x2≤h !
h L ∂ϕ L h ∂ϕ
4 ϕ dx dx ϕ dx dx .
≤ Z0 Z−L| ||∂x1| 1 2!× Z−LZ0 | ||∂x2| 2 1!
By the Cauchy-Schwartz inequality,
∂ϕ ∂ϕ
ϕ 4 4ϕ2
k kL4(Ω1) ≤ | |L2(Ω1)|∂x |L2(Ω1)|∂x |L2(Ω1)
1 2
∂ϕ ∂ϕ
2ϕ2 2 + 2
≤ | |L2(Ω1) |∂x |L2(Ω1) |∂x |L2(Ω1)
(cid:18) 1 2 (cid:19)
2ϕ2 ϕ2 .
≤ | |L2(Ω1)|∇ |L2(Ω1)
8
We use r 1 and the Poincar´einequality to get
| |≤
v ϕ , ϕ 2v and ϕ C v
k kL4(Ω) ≤k kL4(Ω1) | |L2(Ω1) ≤ | |L2(Ω) |∇ |L2(Ω1) ≤ k kV
for some constant C, whence (3.1) holds.
Theorem 3.1. For any v H and U R there exists a solution of Problem 2.1.
0 0
∈ ∈
Proof. We provide only the main steps of the proof as it is quite standard and, on the
other hand, long. The estimates we obtain will be used further in the paper.
Observe that the functional j is convex, lower semicontinuous but nondifferentiable.
To overcomethis difficulty we use the following approach(see, i.e., [20], [22]). For δ >0
let j :V R be a functional defined by
δ
→
1
ϕ j (ϕ)= k ϕ1+δdx
7→ δ 1+δ | |
ZΓ0
which is convex, lower semicontinuous and finite on V, and has the following properties
(i) there exists χ V′ and µ R such that j (ϕ) <χ,ϕ> +µ for all ϕ V,
δ
∈ ∈ ≥ ∈
(ii) lim j (ϕ)=j(ϕ) for all ϕ V,
δ→0+ δ
∈
(iii) v ⇀v (weakly) in V lim j (v ) j(v).
δ ⇒ δ→0+ δ δ ≥
The functional j is Gˆateaux differentiable in V, with
δ
(j′(v), Θ)= k v δ−1vΘdx , Θ V.
δ | | 1 ∈
ZΓ0
Let us consider the following equation
dv
( δ,Θ)+νa(v (t),Θ) + b(v (t),v (t),Θ)+(j′(v ),Θ)
dt δ δ δ δ δ
= νa(ξ,Θ) b(ξ,v (t),Θ) b(v (t),ξ,Θ) (3.2)
δ δ
− − −
with initial condition
v (0)=v . (3.3)
δ 0
For δ >0, we establish an a prioriestimates of v . Since (j′(v ),v ) 0, v (t) V, and
δ δ δ δ ≥ δ ∈
b(v (t),v (t),v (t))=b(ξ,v (t),v (t))=0 then taking Θ=v (t) in (3.2) we get
δ δ δ δ δ δ
1 d
v (t)2+ν v (t) 2 νa(ξ,v (t)) b(v (t),ξ,v (t))
2dt| δ | k δ k ≤− δ − δ δ
In view of Lemma 3.1 we obtain
1 d ν
v (t)2+ v (t) 2 ν ξ 2.
2dt| δ | 2k δ k ≤ k k
We estimate the right hand side in terms of the data using Lemma 3.1 to get
1 d ν
v (t)2+ v (t) 2 F. (3.4)
2dt| δ | 2k δ k ≤
9
with F =F(ν,Ω,U ). From (3.4) we conclude that
0
t
v (t)2+ν v (s) 2ds v(0)2+2tF, (3.5)
δ δ
| | k k ≤| |
Z0
whence
v is bounded in L2(0,T;V) L∞(0,T;H), independently of δ. (3.6)
δ
∩
The existence of v satisfying (3.2)-(3.3) is based on inequality (3.4), the Galerkin ap-
δ
proximations, and the compactness method. Moreover,from (3.5) we can deduce that
dv
δ is bounded in L2(0,T;V′). (3.7)
dt
From(3.6)and(3.7)weconcludethatthereexistsvsuchthat(possiblyforasubsequence)
dv dv
v ⇀v in L2(0,T;V), and δ ⇀ in L2(0,T;V′). (3.8)
δ dt dt
In view of (3.8), v ([0T];H), and
∈C
v v in L2(0,T;H) strongly.
δ
→
We can now pass to the limit δ 0 in (3.2)-((3.3) as in [13] to obtain the variational
→
inequality(2.9)foralmosteveryt (0,T). ThustheexistenceofasolutionofProblem2.1
∈
is established.
Theorem 3.2. Under the hypotheses of Theorem 3.1, the solution v of Problem 2.1 is
unique and the map v(τ) v(t), for t>τ 0, is Lipschitz continuous in H.
→ ≥
Proof. Letv andw be twosolutionsofProblem2.1. Thenforu(t)=w(t) v(t)wehave
−
1 d
u(t)2+ν u(t) 2 b(u(t),w(t),u(t))+b(u(t),ξ,u(t)).
2dt| | k k ≤
By Lemma 3.1 and the Ladyzhenskaya inequality (3.1) we obtain
d ν 2
u(t)2+ u(t) 2 C(Ω)4 w(t) 2 u(t)2, (3.9)
dt| | 2k k ≤ ν k k | |
and in view of the Poincar´e inequality we conclude
d σ 2
u(t)2+ u(t)2 C(Ω)4 w(t) 2 u(t)2.
dt| | 2| | ≤ ν k k | |
Using again the Gronwall lemma, we obtain
t σ 2
u(t)2 u(τ)2exp C(Ω)4 w(s) 2 ds . (3.10)
| | ≤| | {− 2 − ν k k }
Zτ (cid:18) (cid:19)
From(3.8)itfollowsthatthe solutionw ofProblem2.1belongsto L2(τ,t;V). By(3.10)
the map v(τ) v(t), t>τ 0, in H is Lipschitz continuous, with
→ ≥
w(t) v(t) C w(τ) v(τ) (3.11)
| − |≤ | − |
uniformly for t,τ in a given interval [0,T] and initial conditions w(0),v(0) in a given
bounded set B in H.
In particular, as u(0) = w(0) v(0) = 0, the solution v of Problem 2.1 is unique.
−
This ends the proof of Theorem 3.2.
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