Electronic Journal of Differential Equations, Vol. 2009(2009), No. 69, pp. 1–16.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR AN OLDROYD MODEL OF
VISCOELASTIC FLUIDS
GERALDO M. DE ARAÚJO, SILVANO B. DE MENEZES, ALEXANDRO O. MARINHO
Abstract. In this paper we investigate the unilateral problem for an Oldroyd
model of a viscoelastic fluid. Using the penalty method, Faedo-Galerkin’s
approximation, and basic result from the theory of monotone operators, we
establish the existence of weak solutions.
1. Introduction
It is well know that, the motion of incompressible fluids is described by the
system of Cauchy equations
∂u
∂u
+ ui
+ ∇p = div σ + f
∂t
∂xi
div u = 0,
(1.1)
where u = (u1 , . . . , un ) is the velocity, p is the pressure in the fluid, f is the density
of external forces and σ is the deviator of the stress tensor, that is, σ has the
purpose of letting us consider reactions arising in the fluid during its motion. The
∂u
vector (ui ∂xji ), j = 1, 2, . . . , n, is denoted by (u.∇)u. The Hooke’s Law establishes
a relationship between the stress tensor σ and the deformation tensor Dij (u) =
∂uj 1 ∂ui
2 ∂xj + ∂xi and their derivatives. Therefore is the Hooke’s Law that establishes
the type of fluid. Such relationship is also called of rheological equation or equation
of state (see Serrin [10] or Clifford [1]). For example, for an incompressible Stokes
fluid the relationship has the form
σ = αD + βD2
(1.2)
where α and β are scalar functions. If in (1.2) α = 2ν positive constants. and
β ≡ 0 we have the Newton’s Law σ = 2νD, which substituting in (1.1) we obtain the
equations of motion of Newtonian fluid, which is called the Navier-Stokes equations:
u0 − ν∆u + (u.∇)u + ∇p = f,
div u = 0,
2000 Mathematics Subject Classification. 74H45.
Key words and phrases. Oldroyd equation; variable viscosity; penalty method.
c
2009
Texas State University - San Marcos.
Submitted October 20, 2008. Published June 1, 2009.
Alexandro O. Marinho is partially supported by FAPEPI-Piauı́-Brazil.
1
2
G. M. DE ARAÚJO, S. B. DE MENEZES, A. O. MARINHO,
EJDE-2009/69
where ν is called the kinematic coefficient of viscosity. The Navier-Stokes model
was studied from the mathematical point of view by Leray [15] and later by Ladyzhenskaya [9]. We mention other deep contributions by Lions [16], Temam [21],
Tartar [19] and many others researchers.
The model studied in this work, introduced by Oldroyd [11, 12], was proposed
for viscous incompressible fluids whose defining equations have the form
1+λ
∂
∂
σ = 2ν 1 + kν −1
D,
∂t
∂t
(1.3)
where λ, ν, k are positive constants with ν − λk > 0. In this fluid the stress after
−1
instantaneous cessation of the motion die out like e−λ t , while the velocities of the
−1
flow after instantaneous removal of the stress die out like e−k t .
Assuming that σ(0) = 0 and D(0) = 0, we write the relationship (1.3) in the
form of integral equation
Z t
(t−ξ)
σ(x, t) = 2kλ−1 D(x, t) + 2λ−1 (ν − kλ−1 )
e− λ D(x, ξ)dξ.
(1.4)
0
Thus, the equation for the motion of Oldroyd fluid can be written by the system
of integro-differential equations
Z t
∂u
+ (u.∇)u − µ∆u −
β(t − ξ)∆u(x, ξ)dξ + ∇p = f, x ∈ Ω, t > 0 (1.5)
∂t
0
and the incompressible condition
x ∈ Ω, t > 0,
div u = 0,
with initial and boundary conditions
x ∈ Γ, t ≥ 0.
Here, µ = kλ−1 > 0 and β(t) = γe−δt , where γ = λ−1 ν − kλ−1 with δ = λ−1 .
For physical details and mathematical modelling see [2, 5, 11, 22].
The mixed problem above was investigated by Oskolkov [2], where he proves
existence of weak solution for all n ∈ N in certain Sobolev class.
In Brézis [6] we find investigation for a unilateral problem for the case of the
Navier-Stokes equations.
In the present work we consider a unilateral problem similar to Brézis [6], adding
Rt
a memory term, that is − 0 g(t−σ)∆u(σ)dσ. More precisely, in this paper we study
a unilateral problem or a variational inequality, c.f. Lions [16], for the operator
u(x, 0) = u0 ,
L=
x ∈ Ω,
and u(x, t) = 0,
∂u
+ (u.∇)u − µ∆u −
∂t
Z
t
β(t − ξ)∆u(x, ξ)dξ + ∇p − f
0
under standard hypothesis on f and u0 . Making use of the penalty method and
Galerkin’s approximations, we establish existence and uniqueness of weak solutions.
This work is organized as follows: In Section 2, we introduce the notation and
main results. In Section 3, we proof to the results. Finally, in Section 4, we prove
an simple result of uniqueness.
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OLDROYD FLUIDS
3
2. Notation and Main Results
Let Ω be a bounded domain in Rn with the boundary ∂Ω of class C 2 . For T > 0,
we denote by QT the cylinder (0, T ) × Ω, with lateral boundary ΣT = (0, T ) × ∂Ω.
By h., .i we will represent the duality pairing between X and X 0 , X 0 being the
topological dual of the space X, and by C we denote various positive constants.
We propose the variational inequality
Z t
g(t − σ)∆u(σ)dσ + ∇p ≥ f in QT
u0 − µ∆u + (u.∇)u −
0
div u = 0 in QT
u = 0 on ΣT
u(x, 0) = u0 (x)
(2.1)
in Ω,
where g : [0, ∞) → [0, ∞) is a function of W 1,1 (0, ∞) satisfying
Z ∞
µ
−2
g(s) ds > 0;
2
0
−C1 g ≤ g 0 ≤ −C2 g,
(2.2)
(2.3)
where C1 , C2 , C3 are positive constants;
g(0) > 0,
(2.4)
8
s
−µ
satisfies the thhree conditions above.
As an example, g(s) = e
To formulate problem (2.1) we need some notation about Sobolev spaces. We
use standard natation of L2 (Ω), Lp (Ω), W m,p (Ω) and C p (Ω) for functions that
are defined on Ω and range in R, and the notation L2 (Ω)n , Lp (Ω)n , W m,p (Ω)n
and C p (Ω)n for functions that range in Rn . Besides, we work also with the spaces
Lp (0, T ; H m (Ω)) or Lp (0, T ; H m (Ω))n . To complete this recall on functional spaces,
see for instance, Lions [16].
Also we define the following spaces
V = {ϕ ∈ D(Ω)n : div ϕ = 0},
V = V (Ω) is the closure of V in the space H01 (Ω)n with inner product and norm
denoted, respectively by
n Z
n Z 2
X
X
∂ui
∂zi
∂ui
((u, z)) =
(x)
(x) dx, kuk2 =
(x) dx,
∂xj
∂xj
∂xj
i,j=1 Ω
i,j=1 Ω
H = H(Ω) is the closure of V in the space L2 (Ω)n with inner product and norm
defined, respectively, by
n Z
n Z
X
X
2
(u, v) =
ui (x)vi (x) dx, |u|2 =
|ui (x)| dx
i=1
Ω
i=1
2
Ω
n
and V2 is the closure of V in H (Ω) with inner product and norm denoted, respectively by
n
X
((u, z))V2 =
(ui , vi )H 2 (Ω) , kuk2V2 = ((u, u))V2 ,
i=1
Remark 2.1. V , H and V2 are Hilbert’s spaces, V2 ,→ V ,→ H ,→ V 0 with
embedding dense and continuous.
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G. M. DE ARAÚJO, S. B. DE MENEZES, A. O. MARINHO,
EJDE-2009/69
Let K be a closed and convex subset of V ∩ V2 with 0 ∈ K. We introduce the
following bilinear and the trilinear forms:
n Z
X
∂vi
∂ui
a(u, v) =
(x)
(x) dx = ((u, v)),
∂xj
∂xj
i,j=1 Ω
n Z
X
∂vj
b(u, v, w) =
ui (x)
(x)wj (x) dx,
∂xi
i,j=1 Ω
We also assume that
t
Z
g(t − σ)((v, v))dσ ≥ 0 ∀ϕ ∈ K, ∀v ∈ V.
a(v, v) + b(v, ϕ, v) +
(2.5)
0
Next we shall state the main results of this paper.
Theorem 2.2. If f ∈ L2 (0, T ; H) and hypotheses (2.5) holds, then there exists a
function u such that
u ∈ L2 (0, T ; V ) ∩ L∞ (0, T ; H)
u(t) ∈ K
a.e.
(2.6)
(2.7)
T
Z
hϕ0 , ϕ − ui + µa(u, ϕ − u) + b(u, u, ϕ − u)
Z t
−
g(t − σ)∆u(σ)dσ, ϕ − u dt
0
Z
≥
(2.8)
0
T
hf, ϕ − ui dt,
∀ϕ ∈ L2 (0, T ; V ), ϕ0 ∈ L2 (0, T ; V 0 ),
0
ϕ(0) = 0,
ϕ(t) ∈ K a.e.
u(0) = u0 .
Theorem 2.3. Assumption (2.5), n = 2, and
f ∈ L2 (0, T ; V ),
f 0 ∈ L2 (0, T ; V 0 )
(2.9)
u0 ∈ K.
(2.10)
(f (0), v) − µa(u0 , v) − b(u0 , u0 , v) = (u1 , v)
for all v ∈ V some u1 ∈ V . (2.11)
Suppose also that
Then there exists a unique function u such that
u ∈ L2 (0, T ; V ∩ V2 ),
u0 ∈ L2 (0, T ; V ) ∩ L∞ (0, T ; H)
u(t) ∈ K,
∀t ∈ [0, T ]
(2.12)
(2.13)
0
(u (t), v − u(t)) + µa(u(t), v − u(t)) + b(u(t), u(t), v − u(t))
Z TZ t
+
g(t − σ)((u(σ), v − u(t)))dσdt
0
(2.14)
0
≥ (f (t), v − u(t))
∀v ∈ K, a.e. in t,
u(0) = u0 .
(2.15)
The proof of Theorems 2.2 and 2.3 will be given in Section 3 by the penalty
method. It consists in considering a perturbation of the operator L adding a singular
term called penalty, depending on a parameter > 0. We solve the mixed problem
EJDE-2009/69
OLDROYD FLUIDS
5
in Q for the penalized operator and the estimates obtained for the local solution
of the penalized equation, allow to pass to limits, when goes to zero, in order to
obtain a function u which is the solution of our problem.
First of all, let us consider the penalty operator β : V → V 0 associated to
the closed convex set K, c.f. Lions [16, p. 370]. The operator β is monotonous,
hemicontinuous, takes bounded sets of V into bounded sets of V 0 , its kernel is
K and β : L2 (0, T ; V ) → L2 (0, T ; V 0 ) is equally monotone and hemicontinous.
The penalized problem associated with the variational inequalities (2.8) and (2.14)
consists in, given 0 < < 1, find u satisfying
Z t
1
(u0 , v) + µa(u , v) + b(u , u , v) −
g(t − σ)(∆u (σ), v)dσ + (β(u ), v) = (f, v),
0
∀ v ∈ V, u ∈ L2 (0, T ; V ), u0 ∈ L2 (0, T ; V 0 )
u (x, 0) = u0 (x).
(2.16)
We suppose n = 2. The solution of this problem is given by the followings theorems.
Theorem 2.4. If f ∈ L2 (0, T ; H) and hypotheses (2.2) holds, then, for each 0 <
< 1 and u0 ∈ H, there exists a function u with u ∈ L2 (0, T ; V ) ∩ L∞ (0, T ; H),
u0 ∈ L2 (0, T ; V 0 ) solution of (2.16).
Theorem 2.5. If f ∈ L2 (0, T ; V ) and f 0 ∈ L2 (0, T ; V 0 ) and hypotheses (2.2)
holds, then for each 0 < < 1 and u0 ∈ V , there exists a function u with
u ∈ L∞ (0, T ; V ∩ V2 ), u0 ∈ L2 (0, T ; V ) ∩ L∞ (0, T ; H) satisfying (2.16).
3. Proof of the Results
Proof of Theorem 2.2. We first prove Theorem 2.4 for the penalized problem.
comp
We employ the Faedo-Galerkin method. We note that the embedding V ,→V ,→
H ,→ V 0 are continuous and dense and that V is compactly and densely embedded
in H. Let {wν , λν }, ν ∈ N, be solutions of the spectral problem
((w, v)) = λ(w, v),
∀v ∈ V.
(3.1)
We consider (wν )ν∈N a Hilbertian basis for Faedo-Galerkin method. We represent
by Vm = [w1 , w2 , . . . , wm ] the V subspace generated by the vectors w1 , w2 , . . . , wm
and let us consider
m
X
um (t) =
gj m (t)wj
j=1
solution of approximate problem
(u0m , wj ) + µa(um , wj ) + b(um , um , wj )
Z t
1
−
g(t − σ)(∆um (σ), v)dσ + hβum , wj i
0
= hf (t), wj i, j = 1, 2, . . . m
um (x, 0) → u (x, 0)
(3.2)
strongly in V.
This system of ordinary differential equations has a solution on a interval [0, tm [,
0 < tm < T . The first estimate permits us to extend this solution to the whole
interval [0, T ].
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G. M. DE ARAÚJO, S. B. DE MENEZES, A. O. MARINHO,
EJDE-2009/69
Remark 3.1. To obtain a better notation, we omit the parameter in the approximate solutions.
First estimate. Multiplying both sides of (3.2) by gj and adding from j = 1 to
j = m, we obtain
Z t
1 d
|um (t)|2 + µkum (t)k2 +
g(t − σ)(∇um (σ), ∇um (t))dσ = (f (t), um (t)),
2 dt
0
since b(um , um , um ) = 0 (see Lions [16]) and
(βum (t), um (t)) ≥ 0 because β is monotone and 0 ∈ K. Its follows that
1 d
|um (t)|2 + µkum (t)k2
2 dt
Z
Z t
∇um (x, t)
g(s − σ)∇um (x, σ) dσdx
≤
R
Ω
0
Z
+ |f (t)||um (t)| +
|∇um (x, t)|R |g ∗ ∇um (x, t)|R dx,
(3.3)
Ω
where ∗ denotes the convolution in t. It follows from (3.3) that
d
|um (t)|2 + 2µkum (t)k2
dt Z
≤2
|∇um (x, t)|R |g ∗ ∇um (x, t)|R dx + 2|f (t)|Ckum (t)k
r
r
Z
2
3µ
C|f (t)|
kum (t)k
=2
|∇um (x, t)|R |g ∗ ∇um (x, t)|R dx + 2
3µ
2
Ω
Z
3µ
2 2
=2
|∇um (x, t)|R |g ∗ ∇um (x, t)|R dx +
kum (t)k2 +
C |f (t)|2 .
2
3µ
Ω
Ω
Remark 3.2. We note that from Cauchy-Schwarz inequality and Fubini’s theorem
we have
kg ∗ ∇um kL2 (Q) ≤ kgkL1 (0;∞) k∇um kL2 (Q) .
Thus, integrating 0 to t the inequality above, using the Remark 3.2 and using
Gronwall’s inequality we obtain
µ
2
|um |2L∞ (0,T ;H) +
− 2kgkL1 (0,∞) kum k2L2 (0,T ;V ) ≤
C + C 2 |f |2L2 (0,T ;H) .
2
3µ
Integrating these last inequality in t ∈ [0, T ] and using (2.2), we have
um
is bounded in L∞ (0, T ; H)
(3.4)
um
is bounded in L2 (0, T ; V ).
(3.5)
From (3.5), we obtain
β(um )
is bounded in L2 (0, T ; V 0 )
Second estimate. By Remark 3.2, we observe that,
Z t
2
if ξ ∈ L (0, T ; H) then
g(t − σ)ξ(σ)dσ ∈ L2 (0, T ; H).
0
(3.6)
(3.7)
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OLDROYD FLUIDS
7
Similarly we obtain
Z
t
0
Z t
g(t − σ)ξ(σ)dσ ∈ V
if ξ(t) ∈ V ,
(3.8)
g(t − σ)ξ(σ)dσ ∈ V 0
if ξ(t) ∈ V 0 .
(3.9)
0
We consider ũm = um , w̃ = w in [0, T ] and ũm = 0, w̃ = 0 out of [0, T ], ge(ξ) = g(ξ)
if ξ ≥ 0 and zero if ξ < 0. Therefore, ∇e
um ∈ L2 (R; H), w
e ∈ L2 (R; V ) and
1
ge ∈ L (R). This implies
Z TZ t
g(t − σ) ((um (σ), w(t))) dσ dt
0
Z 0Z
Z
=
ge(t − σ)
∇e
um (x, σ)∇w(x,
e t) dx dσ dt
Ω
ZR ZR
=
ge ∗ ∇e
um (x, t)∇w(x,
e t) dx dt
R Ω
Z Z
=
∇e
um (x, σ)e
ğ ∗ ∇w(x,
e σ) dx dσ,
R
Ω
where e
ğ(x) = ge(−x). We observe that (3.5) implies that
Z T
Z T
((um (t), w))dt →
((u(t), w))dt, ∀ w ∈ L2 (0, T ; V ).
0
(3.10)
0
From (3.7), we have that ge ∗ w(t)
e ∈ V, ∀ w ∈ L2 (0, T ; V ), therefore (3.10) yield
Z Z e
∇e
um (σ), ğ ∗ ∇w(σ)
e
dt →
∇e
u(σ), e
ğ ∗ ∇w(σ)
e
dt.
R
R
We note that
Z Z
∇e
u(σ), e
ğ ∗ ∇w(σ)
e
dt =
(e
g ∗ ∇e
u(σ), ∇w(σ))
e
dt
R
ZR Z Z
=
ge(t − σ)∇e
u(x, σ)dσ∇w(x,
e t)dσ dt dx
Ω R R
T Z t
Z
g(t − σ) ((u(σ), w(t))) dσdt .
=
0
Then
Z TZ
0
t
Z
T
Z
g(t − σ) ((um (σ), w(t))) dσ dt →
0
0
t
g(t − σ) ((u(σ), w(t))) dσdt, (3.11)
0
0
for all w ∈ L2 (0, T ; V ).
Let Pm be the orthogonal projection H 7→ Vm ; that is,
Pm ϕ =
m
X
(ϕ, wj )wj ,
ϕ ∈ H.
j=1
By the choice of (wν )ν∈N we have
kPm kL(V,V ) ≤ 1
∗
and kPm
kL(V 0 ,V 0 ) ≤ 1.
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G. M. DE ARAÚJO, S. B. DE MENEZES, A. O. MARINHO,
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We note that Pm u0m = u0m . Multiplying both sides of the approximate equation
(3.2) by the vector wj and adding from j = 1 to j = m, we obtain using the
notations and ideas of Lions [16, pages 75-76] and (3.7), that
(u0m )
is bounded in L2 (0, T ; V 0 ).
(3.12)
The boundedness in (3.5), (3.12) and the Aubin-Lions compactness Theorem imply
that there exists a subsequence from (um ), still denoted by (um ), such that
um → u
strongly in L2 (0, T ; H) and a. e. in Q.
(3.13)
Returning to the notation um , using (3.4), (3.5) and (3.13) (see Lions [16, pages
76-77]), (3.6) and (3.11) we obtain
Z t
1
(u0 , v) + a(u , v) + b(u , u , v) −
g(t − σ)(∆u(σ), v)dσ + (ζ, v) = (f, v),
0
∀v ∈ V, u ∈ L2 (0, T ; V ), u0 ∈ L2 (0, T ; V 0 )
u (x, 0) = u0 (x).
(3.14)
It is necessary to prove that ζ = β(u ). We make this using the monotony of the
operator β (see Lions [16, Chap. 2]). Therefore, we have proved the Theorem 2.4.
Proof of Theorem 2.2. From (3.4), (3.5), (3.13) and Banach-Steinhauss theorem,
it follows that there exists a subnet (u )0<<1 , such that it converges to u as → 0,
in the weak sense . This function satisfies (2.6). On the other hand, we have from
(3.14) that
Z t
βu = [f − u0 − Au − Bu −
g(t − σ)∆u(σ)dσ].
(3.15)
0
Where hAu , vi = a(u, v) and hBu , vi = b(u , u , v).
Rt
Since 0 g(t − σ)∆u(σ)dσ ∈ V 0 and [f − u0 − Au − Bu ] is bounded, we have
βu → 0
in D0 (0, T ; V 0 ).
(3.16)
0
2
Since βu is bounded in L (0, T ; V ), we have
βu → 0 weak in L2 (0, T ; V 0 ).
On the other hand we deduce from (3.14) that
Z T
0≤
hβu , u i dt ≤ C.
(3.17)
(3.18)
0
Thus
RT
0
hβu , u idt → 0. We have that
Z T
hβu − βϕ, u − ϕi dt ≥ 0,
∀ϕin L2 (0, T ; V ),
0
because β is a monotonous operator. Thus,
Z T
Z T
Z
hβu , u i dt −
hβu , ϕi dt −
0
0
T
hβϕ, u − ϕi dt ≥ 0.
We have from (3.17) and (3.19) that
Z T
hβϕ, u(t) − ϕi dt ≤ 0.
0
(3.19)
0
(3.20)
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OLDROYD FLUIDS
9
Taking ϕ = u − λv, with v ∈ L2 (0, T ; V ) and λ > 0, we deduce using the hemicontinuity of β that
β(u(t)) = 0,
(3.21)
and this implies that u(t) ∈ K a. e.
Next, we prove that u is a solution of inequality (2.8). Let us consider X defined
by
Z T
Z T
Z T
0
X =
hϕ , ϕ − u i dt +
a(u , ϕ − u ) dt +
b(u , u , ϕ − u ) dt
0
0
0
(3.22)
Z T
Z TZ t
g(t − σ)((u (σ), ϕ − u ))dσ dt, −
+
hf, ϕ − u i dt,
0
0
0
with ϕ ∈ L2 (0, T ; V ), ϕ0 ∈ L2 (0, T ; V 0 ), ϕ(0) = 0, ϕ(t) ∈ K a.e. It follows from
(3.22) that
Z T
Z T
Z T
Z T
X =
hϕ0 , ϕi dt −
hϕ0 , u i dt +
a(u , ϕ) dt −
a(u , u ) dt
0
0
0
T
Z
b(u , u , ϕ) dt −
+
0
t
Z
−
t
0
T
Z
Z
g(t − σ)((u (σ), u ))dσ dt −
0
Z
g(t − σ)((u (σ), ϕ))dσ dt
0
T
T
Z
b(u , u , u )dt +
0
Z
0
T
Z
T
hf, ϕi dt +
0
hf, u i dt.
0
0
(3.23)
On the other hand, taking v = ϕ − u in (2.16) and integrating in QT , we obtain
that
Z T
Z T
Z T
Z T
−
hu0 , ϕi dt +
hu0 , u i dt −
a(u , ϕ) dt +
a(u , u ) dt
0
Z
0
0
T
−
Z
Z
0
t
Z
Z
1
+
g(t − σ)((u (σ), u ))dσ dt −
0
0
Z T
Z T
+
hf, ϕi dt −
hf, u i dt = 0,
0
Z
t
g(t − σ)((u (σ), ϕ))dσ dt
0
T
T
b(u , u , u )dt −
b(u , u , ϕ) dt +
0
0
T
Z
0
T
hβu − βϕ, ϕ − u i dt
0
0
(3.24)
because βϕ = 0. Adding member to member (3.23) and (3.24), we obtain
Z T
Z T
Z T
X =
hϕ0 , ϕi dt −
hϕ0 , u i dt −
hu0 , ϕi dt
0
0
Z
+
T
hu0 , u i dt
0
1
+
0
Z
(3.25)
T
hβϕ − βu , ϕ − u i dt ≥ 0,
0
because
Z
T
hϕ0 , ϕi dt −
0
Z
T
hϕ0 , u i dt −
0
Z
=
0
T
hϕ0 − u0 , ϕ − u i ≥ 0.
Z
0
T
hu0 , ϕi dt +
Z
0
T
hu0 , u i dt
10
G. M. DE ARAÚJO, S. B. DE MENEZES, A. O. MARINHO,
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On the other hand, b(u , u , u ) = 0. From (3.22)-(3.23) it follows that
Z T
Z T
X =
hϕ0 , ϕ − u i dt +
a(u , ϕ) dt
0
0
T
Z
t
Z
≥
hf, ϕ − u i dt
0
0
T
Z
(3.26)
0
T
Z
g(t − σ)((u , u ))dσ dt.
0
0
0
T
Z
T
Z
t
Z
g(t − σ)((u , u ))dσ dt.. (3.27)
b(u , ϕ, u ) dt +
a(u , u )dt +
0
0
0
0
t
Z
b(u , ϕ, u ) dt +
0
T
T
Z
a(u , u )dt +
Consider
Z
Y =
T
Z
g(t − σ)((u , ϕ))dσ dt −
+
It follows from (2.5) with v = u − u that
t
Z
a(u − u , u − u ) + b(u − u , ϕ, u − u ) +
g(t − σ)((u − u , u − u ))dσ ≥ 0.
0
On the other hand, we can write
Z T
Z
Y =
a(u − u, u − u) dt +
0
T
b(u − u, ϕ, u − u) dt
0
T
Z
Z
T
a(u, u − u) dt +
+
T
Z
0
T Z t
Z
0
0
T
0
g(t − σ)((u − u, u − u))dσ dt
b(u , ϕ, u) dt +
Z
0
t
Z
b(u, ϕ, u − u) dt
a(u , u) dt +
0
+
T
Z
T
Z
t
Z
g(t − σ)((u, u − u))dσ dt +
+
0
g(t − σ)((u , u))dσ dt.
0
0
0
This implies
T
Z
Y ≥
T
Z
a(u, u − u) dt +
a(u , u) dt +
0
T
Z
0
Z
T
Z
0
0
T
Z
Z
t
g(t − σ)((u, u − u))dσ dt
b(u , ϕ, u) dt +
0
t
0
Taking lim sup in (3.28) we obtain
Z T
Z T
Z
lim sup Y ≥
a(u, u) dt +
b(u, ϕ, u) dt +
0
0
0
T
Z
Z
≥
g(t − σ)((u, u))dσ dt. (3.29)
0
a(u , ϕ) dt
Z
0
T
hf, ϕ − u i dt
o
0
Z
a(u, u) dt +
0
t
0
t
T
Z
T
g(t − σ)((u , u))dσ dt −
0
T
0
It follows from (3.26) and (3.29) that
Z
nZ T
lim sup
hϕ0 , ϕ − u i dt +
Z
(3.28)
0
g(t − σ)((u , u))dσ dt.
+
+
b(u, ϕ, u − u) dt
0
+
T
Z
T
Z
T
Z
0
t
g(t − σ)((u, u))dσ dt.
b(u, ϕ, u) dt +
0
0
(3.30)
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OLDROYD FLUIDS
It follows from (3.30) that
Z
Z T
hϕ0 , ϕ − ui dt +
T
Z
T
Z
Z
T
b(u, u, ϕ − u) dt
a(u, ϕ − u) dt +
0
0
0
11
t
g(t − σ)((u, ϕ − u))dσ dt
+
0
0
Z
≥
T
hf, ϕ − ui dt
0
for all ϕ ∈ L2 (0, T ; V ), ϕ0 ∈ L2 (0, T ; V 0 ), ϕ(0) = 0, ϕ(t) ∈ K a.e.
Proof of Theorem 2.3. We first prove Theorem 2.5 for the penalized problem. As
in the proof of Theorem 2.2, we employ the Faedo-Galerkin Method. Let (wν )ν∈N
be a Hilbertian basis of V . By Vm = [w1 , w2 , . . . wm ] we represent the subspace
generated by the m first vectors of (wν ). Consider
u m =
m
X
gj m wj
j=1
solution of approximate penalized problem
(u0m , wj ) + µa(um , wj ) + b(um , um , wj )
Z t
1
−
g(t − σ)(∆um (σ), v)dσ + hβum , wj i
0
= hf (t), wj i, j = 1, 2, . . . m
um (x, 0) → u (x, 0)
(3.31)
strongly in V.
First estimate. As in the proof of Theorem 2.4, omitting the parameter and
taking v = um in the approximate equation (3.31) we obtain
(um )
(um )
is bounded in L∞ (0, T ; H),
2
is bounded in L (0, T ; V ),
(3.32)
(3.33)
Second estimate. In both sides of (3.31) we take the derivatives with respect t
and consider v = u0m (t). We obtain
(u00m (t), u0m (t)) + µa(u0m (t), u0m (t))
+ b(u0m (t), um (t), u0m (t)) + b(um (t), u0m (t), u0m (t))
Z t
1
g 0 (t − σ)((um (t), u0m ))dσ
+ ((βum (t))0 , u0m (t)) +
0
1
0
0
+ g(0)((um (t), um (t))) +
(βum ) (t), u0m (t)
= (f 0 (t), u0m (t)),
(3.34)
because
Z
Z t
d t
g(t − σ)∆um (σ) dσ = g(0)∆um (t) +
g 0 (t − σ)∆um (σ) dσ.
dt 0
0
We note that
u0m (0) → u1
strongly in H,
um (0) → u0
strongly in V.
(3.35)
12
G. M. DE ARAÚJO, S. B. DE MENEZES, A. O. MARINHO,
EJDE-2009/69
Indeed, (3.35)1 is obtained using (3.31) with t = 0 and (2.11). Note that β(u0 ) = 0.
Then
1 d 0
|u (t)|2 + µku0m (t)k2 + b(u0m (t), um (t), u0m (t))
2 dt m
Z t
1 d
(3.36)
g 0 (t − σ)((um (t), u0m ))dσ + g(0)
+
kum (t)k2
2 dt
0
= (f 0 (t), u0m (t)),
because b(um (t), u0m (t), u0m (t)) = 0 and ((βum )0 (t), u0m (t)) ≥ 0 (see Lions [16, page
399]).
Remark 3.3. The derivative with respect to t of (β(v(t)), w) is only formal. The
correct method is to consider the difference equation in t+h and t, divided by h and
take the limits when h → 0. Here is fundamental the operator β to be monotonous.
This justify the formal procedure of taking the derivative with respect to t, on both
sides of (3.31) and take v = u0m (t). See Brezis [6], Browder [8] or Lions [17] for
details.
As n = 2, we have (see Lions [16, page 70])
kuk2L4 (Ω) ≤ Ckuk|u|,
∀ u ∈ H01 (Ω).
(3.37)
Moreover, H01 (Ω) ,→ L2 (Ω); therefore,
|b(u, v, u)| ≤
2 Z
X
i,j=1
∂vj
|ui (x)|
(x)|uj (x)| ≤ kuk(L4 (Ω))2 kvk.
∂xi
Ω
This and (3.37) imply
|b(u0m (t), um (t), u0m (t))|
r
≤
≤
µ 0
ku (t)kC
2 m
r
2 0
|u (t)|kum (t)k
µ m
µ 0
C2
kum (t)k2 +
kum (t)k2 |u0m (t)|2 .
4
µ
(3.38)
Therefore,
1 d 0
1 d
|u (t)|2 + µku0m (t)k2 + g(0)
kum (t)k2
2 dt m
2 dt
Z t
≤
g 0 (t − σ)((um (t), u0m ))dσ + |b(u0m (t), um (t), u0m (t))|R
R
0
r
r
2 0
µ 0
kf (t)kV 0
ku (t)k,
+
µ
2 m
(3.39)
Therefore, from (3.39), (3.37) and Remark 3.2 we obtain
1 d 0
1 d
|u (t)|2 + g(0)
kum (t)k2 + µku0m (t)k2
2 dt m
2 dt
µ
C2
≤ ku0m (t)k2 +
kum (t)k|u0m (t)|2
4
2µ
C1 0
µ
+ kg 0 kL1 (0,∞) ku0m (t)kkum (t)k +
kf (t)k2 + ku0m (t)k2 ,
µ
4
(3.40)
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OLDROYD FLUIDS
13
Integrating (3.40) from 0 to t and using the hypothesis (2.3), (2.4) we obtain
µ
Z t
0
2
|um (t)| +
− 2kgkL1 (0,∞)
ku0m (s)k2 ds
2
0
Z T
Z t
Z T
kf 0 (t)k2 dt.
kum (s)k2 |u0m (s)|2 ds + C
kum (t)k2 dt + C
≤ C22 kgkL1 (0,∞)
0
0
0
(3.41)
From (3.4) and hypothesis on f we obtain
Z t
µ
Z t
|u0m (t)|2 +
− 2kgkL1 (0,∞)
ku0m (s)k2 ds ≤ C + C
kum (s)k2 |u0m (s)|2 ds.
2
0
0
(3.42)
Being (um ) is bounded in L2 (0, T ; V ) we have, using Gronwall’s inequality in (3.41)
and hypothesis H1, that
(u0m )
(u0m )
is bounded in L2 (0, T ; V )
(3.43)
∞
is bounded in L (0, T ; H).
(3.44)
Third estimate. Let (wν ) be the orthonormal system of V ∩ V2 formed by the
eigenfunctions of the Laplace operator.
As in the proof of Theorem 2.4, omitting the parameter and taking wj = −∆um
in the approximate equation (3.31) we obtain
1 d
kum (t)k2 + µ|∆um (t)|2
2 dt
≤ |b(um (t), um (t), −∆um (t))|R
Z
Z t
(3.45)
∆um (x, t)
g(t − σ)∆um (x, σ)dσ dx
+
Ω
R
0
1
µ
+ |f (t)|2 + |∆um (t)|2
µ
4
because hβum , −∆um i ≥ 0 (see Haraux [4, page 58]). We note that
2 Z
X
∂umj |umj (t)
|b(um (t), um (t), −∆um (t)| ≤
(t)|∆umj (t)|
∂xi
i,j=1
(3.46)
∂um ≤ kum (t)k2(L3 (Ω))2 (t) |∆um (t)|,
∂xi
H01 (Ω) ,→ L6 (Ω), with 21 + 13 + 61 = 1.
because H01 (Ω) ,→ L3 (Ω),
Substituting (3.46) in (3.45) and using the Remark 3.2, we obtain
µ
d
kum (t)k2 +
− 2kgkL1 (0,∞) |∆um (t)|2
dt
2
≤ Ckf (t)k2 + Ckum (t)k2(L3 (Ω))2 − C kum (t)k2 .
Integrating the above inequality from 0 to t, observing that um ∈ L2 (0, T ; V ) ⊂
L2 (0, T ; (L3 (Ω))2 and using the Gronwall’s Lemma, it follows that
um
um
is bounded in L∞ (0, T ; V )
2
is bounded in L (0, T ; V2 ).
(3.47)
(3.48)
To complete the proof of Theorem 2.5, we use the same argument used in the proof
of Theorem 2.4.
14
G. M. DE ARAÚJO, S. B. DE MENEZES, A. O. MARINHO,
EJDE-2009/69
We shall now prove Theorem 2.3. From the previous convergence, and BanachSteinhauss theorem, it follows that there exists a subnet (u )0<<1 , such that it
converges to u as → 0, in the sense of previous convergence.
This function satisfies (2.12) and (2.13). Using the same arguments used in
Theorem 2.2 we obtain that βu = 0. Therefore, u satisfy (2.15) of Theorem 2.3.
We need to show only that u is a solution of inequality (2.14) a.e. in t. In fact,
we have that u satisfies
Z t
1
0
(u , ve) + µa(u , ve) + b(u , u , ve) +
g(t − σ)((u , ve))dσ + (βu , ve) = (f, ve),
0
u (0) = u0 .
(3.49)
for all ve ∈ V . Then from (3.49), with ve = v − u , v ∈ K, we have
Z t
0
(u , v − u ) + µa(u , v) + b(u , u , v) +
g(t − σ)((u , v))dσ − (f, v − u )
0
(3.50)
Z t
≥ µa(u , u ) +
g(t − σ)((u , u ))dσ, ∀v ∈ K,
0
because (βu − βv, u − v) ≥ 0. Let us denote
Z t
Xv = (u0 , v − u ) + µa(u , v) + b(u , u , v) +
g(t − σ)((u , v))dσ − (f, v − u ).
0
We obtain
t
Z
Xv ≥ µa(u , u ) +
g(t − σ)((u , u ))dσ,
∀v ∈ V.
(3.51)
0
Let ψ ∈ C 0 ([0, T ]) with ψ(t) ≥ 0. Then vψ ∈ C 0 ([0, T ]; V ) for all v ∈ V .
ui uj → ui uj weakly in L2 (0, T, L2 (Ω))
It follows from (3.51) that
Z T
Z
ψ(u0 , v − u ) dt + µ
0
T
Z
ψa(u , v) dt +
0
T
Z
T
Z
g(t − σ)((u , v))dσ dt −
+ψ
0
0
T
Z
≥µ
(3.52)
ψ(f, v − u ) dt
0
Z
ψb(u , u , v) dt
0
t
Z
T
t
g(t − σ)((u , u ))dσ.
ψa(u , u ) dt +
0
0
Taking lim sup in both side of inequality (3.52) we obtain
Z T
Z T
Z T
ψ(u0 , v − u) dt + µ
ψa(u, v) dt −
ψb(u, u, v) dt
0
0
T
Z
Z
0
T
Z
0
t
Z
g(t − σ)((u, v))dσ dt −
+
0
≥µ
(3.53)
ψ(f, v − u) dt
0
Z
T
Z
t
g(t − σ)((u, u))dσ dt,
ψa(u, u) dt +
0
T
0
0
because
Z
T
Z
ψa(u , u ) dt ≥ lim inf µ
lim sup µ
0
T
Z
ψa(u , u ) dt ≥ µ
0
T
ψa(u, u) dt
0
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OLDROYD FLUIDS
15
and
Z
T
t
Z
T
Z
t
Z
g(t − σ)(−∆u , u )dσ dt
g(t − σ)((u, u ))dσ dt = lim sup
lim sup
0
0
0
0
Z
T
t
Z
g(t − σ)(−∆u, u)dσ dt
=
0
0
Z
T
t
Z
g(t − σ)((u, u))dσ dt
=
0
0
From (3.53) we obtain finally
Z
0
t
(u , v − u) + µa(u, v − u) + b(u, u, v − u) +
g(t − σ)((u, v − u))dσ
0
(3.54)
≥ (f, v − u) ∀v ∈ K, a.e. in t.
4. Uniqueness
We now prove that when n = 2 we have uniqueness in Theorem 2.3. Indeed,
suppose that u1 , u2 are two solutions of (2.14) and set w = u2 − u1 and t ∈ (0, T ).
Taking v = u1 (resp. u2 ) in the inequality (2.14) relative to v2 (resp. v1 ) and
adding up the results we obtain
Z t
Z t
Z t
−
(w0 , w) dt − µ
a(w, w) dt +
b(u1 , u1 , w) dt
0
0
0
Z t
Z tZ t
−
b(u2 , u2 , w) dt −
g(t − σ)((w, w)) ≥ 0.
0
0
0
Therefore,
1
2
Z
0
t
d
|w(t)|2 dt + µ
dt
Z
t
2
Z
kw(t)k dt ≤
0
t
|b(w, u2 , w)| dt,
(4.1)
0
RtRt
because 0 0 g(t − σ)((w, w)) ≥ 0 and b(u2 , u2 , w) − b(u1 , u1 , w) = b(w, u2 , w). On
the other hand, if n = 2, we have (see Lions [16, page 70])
|b(w(t), u2 (t), w(t))| ≤ Ckw(t)k|w(t)|ku2 (t)k.
(4.2)
It follows from (4.1) and (4.2) that
Z
Z t
µ t
kw(t)k2 dt ≤ C
|w(t)|2 ku2 (t)k2 dt.
|w(t)|2 +
2 0
0
This implies, using Gronwall’s inequality that w = 0, because u2 ∈ L2 (0, T ; V ),
therefore u1 (t) = u2 (t), for all t ∈ [0, T ].
References
[1] Clifford Ambrose Truesdell; A first course in rational continuum mechanics, Vol. I(Part 1),
Academic Press, 1977; Vols. II(Parts 2,3), III(Parts 4-6)(to appear); French transl. of Parts
1-4, Masson, Paris, 1973; Russian transl. of Parts 1-5, ”Mir”, Moscow, 1975.
[2] A. P. Oskolkov; initial boundary value problems for the equations of motion of Kelvin-Voigt
fluids and Oldroyd fluids, Proc. of the Steklov, Institute of Mathematics, 1989, Issue 2.
[3] A. P. Oskolkov and M. M. Akhmatov; Convergent difference schemes for equations of motion
of Oldroyd fluids, J. Sov. Math. 2, No. 4 (1974).
[4] A. Haraux; Nonlinear Evolution Equations-Global Behavior of Solutions. Lectures Notes in
Mathematics 841, Springer Verlag, 1981.
16
G. M. DE ARAÚJO, S. B. DE MENEZES, A. O. MARINHO,
EJDE-2009/69
[5] G. Astarita and G. Marruci; Principles of non-Newtonian fluid mechanics, McGraw-Hill,
1974
[6] H. Brézis; Inequations Variationelles Relatives a L’Operateur de Navier-Stokes, Journal of
Mathematical Analysis ans Applications, 19 (1972), 159-165.
[7] H. Brézis; Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.
[8] F. E. Browder; Nonlinear wave equations, Math. Zeitschr., 80 (1962), 249-264.
[9] O. A. Ladyzhenskaya; The Mathematical Theory of Viscous Incompressible Flow, Gordon
and Breach-London (Second Edition), 1989.
[10] James Serrin; Mathematical principles of classical fluid mechanics, Handbuch der Physik,
Vol. 8/1, Springer-Verlag, 1959, pp. 125-263.
[11] J. G. Oldroyd; Non-Newtonisn flow of liquids and solids, Rheology: Theory and Applications.
Vol. 1(F.R. Eirich Editor), Academic Press, 1959, pp. 653-682.
[12] J. G. Oldroyd; On the formulation of rheological equations of state, Proc. Roy. Soc. London
Ser. A200(1950), 253-541.
[13] J. G. Oldroyd; The elastic and viscous proprieties of emulsions and suspensions, Proc. Roy.
Soc. London Ser. A218(1953), 122-132.
[14] J. G. Oldroyd; Non-linear stress, rate of strain relations at finite rates of shear in so-called
linear elastico-viscous liquids, Second-order Effects in Elasticity, Plasticity and Fluid Dynamics (Internat. Sympos., Haifa, 1962; M. Reiner and D. Abir Editors), Jerusalem Academic
Press, Jerusalem, and Pergamon Press, Oxford, 1964, pp. 520-529.
[15] J. Leray; Essai sur les Mouvements Plans d’un Liquide Visqueux que limitent des Parois, J.
Math Pures Appl. t. XIII (1934), 331-418.
[16] J. L. Lions; Quelques Méthodes de Resolution Des Problémes Aux Limites Non Linéaires,
Dunod, Paris, 1969.
[17] J. L. Lions; Équations differentielles operationelles, Spring-Verlag, Berlin, 1961.
[18] J. L. Lions and G. Prodi; Um Théoréme d’Existence et Unicité dans les Equations de NavierStokes en Dimension 2, C. R. Acad. Sci. Paris, 248(1959), 319-321, Ouvres Choisies de
Jacques-Louis Lions, Vol.1-EDP-sciences, Paris, 2003.
[19] L. Tartar; Partial Differential Equations Models in Oceanography, Carnegie Mellon University, 1999.
[20] O. A. Ladyzhenskaya; Mathematical problems inthe dynamics of a viscous incompressible
mathematical theory of viscous incompressible flow, Gordon and Breach, New York, 1963;
rev. 1969.
[21] R. Temam; Navier- Stokes Equations, Theory and Numerical Analysis, North-Holland Publishing Company, 1979.
[22] W. L. Wilkinson; Non-Newtonian fluids, Pergamon Press, Oxford, 1960.
Geraldo M. de Araújo
Instituto de Matemática, Universidade Federal do Pará, Belém-PA, Brazil
E-mail address: gera@ufpa.br
Silvano B. de Menezes
Departamento Matemática, Universidade Federal do Ceará, Fortaleza-CE, Brazil
E-mail address: silvano@ufpa.br
Alexandro O. Marinho
Departamento de Matemática, Universidade Federal do Piauı́ - Campus Ministro Reis
Veloso, Parnaiba-PI, Brazil
E-mail address: marinho@ufpi.edu.br