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Stability of a two-dimensional Poiseuille-type
ﬂow for a viscoelastic ﬂuid
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
Abstract. A viscoelastic ﬂow in a two-dimensional layer domain is considered. An L2 -stability of the Poiseuille-type ﬂow is established provided that both Poiseuille ﬂow and perturbation is suﬃciently small.
Our analysis is based on a stream function formulation introduced by
F.-H. Lin, C. Liu and P. Zhang (2005).
Mathematics Subject Classiﬁcation (2010). 35A01, 35A02 35Q35, 76A05,
76A10, 76D03,
1. Introduction
This paper studies the stability of a Poiseuille-type ﬂow for a viscoelastic
ﬂuid occupied in a two-dimensional layer domain Ω = R × (0, 1) with the
adherence boundary condition. We describe the motion of viscoelastic ﬂuids
in Euler’s coordinates as in [10]. We in particular consider the incompressible
Hookean model introduced by Fang-Hua Lin, Chun Liu and Ping Zhang
[9], where they construct a local-in-time smooth solution in two or three
dimensional bounded domains with smooth boundary as well as the whole
space or periodic boxes. They moreover prove global-in-time existence of
solutions with small initial data in a two-dimensional periodic box or the
whole plane which also indicates some stability of the trivial steady motion
(with zero velocity).
This work was initiated when the fourth author was a visiting professor at the University
of Tokyo. This work was partly supported by the Japan Society for the Promotion of
Science (JSPS) and the German Research Foundation through Japanese-German Graduate
Externship at IRTG 1529. The second author is partly supported by JSPS through the
Grants-in-Aid for Scientific Research No. 26220702 (Kiban S), No. 23244015 (Kiban A)
and No. 25610025 (Houga). The fourth author was partly supported by NSF grants DMS1412005, DMS-1216938 and DMS-1159937. The present address of the third author is atesio
GmbH, Bundesallee 89, 12161 Berlin, Germany. The present address of the first author is
NTT DATA Financial Solutions Corporation, Kandabashi-park-building 4th floor 1-19-1
Kandanishikicho Chiyoda Tokyo 101-0054, Japan.
2
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
In this paper we consider a Poiseuille-type ﬂow of the form u(t, x) =
(ψ(t, x2 ), 0), where x2 is the vertical variable in (0, 1). It turns out that the
integral of ψ in time solves the viscous wave equation. We are interested in
its stability as viscoelastic ﬂuids. In fact, we prove that if both the Poiseuilletype ﬂow and the initial perturbations are small, then it is exponentially
stable as the time tends to inﬁnity.
Our strategy to prove the stability is to use a stream function formulation due to [9] for a perturbed quantity from the Poiseuille ﬂow, see (4.3).
As in [9] the equation is parabolic for the velocity but not for the stream
function. Moreover, since our basic ﬂow is the Poiseuille ﬂow, there is a new
linear term of a perturbed stream function whose coeﬃcient is not small
in the momentum equation which is an extra diﬃculty compared with the
situation in [9]. As in [9] we introduce a new velocity type variable generating dissipative eﬀects and we fully take advantage of the structure of the
system to obtain energy estimates. Since there are extra linear terms with
non-small coeﬃcients, we derive several energy estimates very carefully to
cancel apparently uncontrollable terms. Except energy estimates, the way of
construction is the Galerkin method which is the same as [9]. Thus we just
concentrate on deriving energy estimates. We also established a non-trivial
behavior of the Poiseuille ﬂow, especially for higher spatial derivatives since
spatial derivatives do not fulﬁll the boundary condition.
There is also a foregoing research by Y. Giga, J. Sauer and K. Schade
[3], in which the authors established Lp exponential stability for a small
Poiseuille-type ﬂow as well as local-in-time existence for non-small initial
data if the layer is thin. Their method is completely diﬀerent since they use
Lp theory instead of L2 theory developed in this paper. We do not assume
that the thickness of the layer is small in this paper.
There is a global estimate result for incompressible viscoelastic ﬂow
subject to not necessarily Hookean elastic energy [8]. However, initial data
is assumed to be close to a trivial solution. We wonder whether our stability
results extends to such a situation but we do not pursue this problem in this
paper.
The stability of the Poiseuille ﬂow is an important topic in ﬂuid mechanics. In fact, for the incompressible Navier-Stokes ﬂow the stability of the
Couette ﬂow in a half space under small periodic perturbation is established
even if the basic ﬂow is large [4]; see also earlier work [11]. The compressible
case is also discussed in [5], where stability of a small Couette-type ﬂow is
discussed. Moreover, the stability of small steady Poiseuille-type ﬂows in a
layer domain in R2 is discussed in [6] under low Mach numbers. It is actually
unstable when the Mach number is not small as shown in a recent work by
Y. Kagei and T. Nishida [7].
For the future research, it is worth noting that this sort of stability
problem for the special solutions of the viscoelastic model can be treated in
the same way as in this paper. If an a priori estimate for the special solutions,
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
3
that correspond to Proposition 3.1 in this paper, is proved, one can use the
same estimates in this paper and obtain the stability of the perturbed ﬂow.
In Section 2 we introduce the model of a viscoelastic ﬂuid. In Section 3
we ﬁrst introduce the Poiseuille-type ﬂow in two dimensions. We then observe
that the Poiseuille-type ﬂow in two-dimension is reduced to the viscous wave
equation in the (0, 1) interval, and we investigate a priori estimates for the
viscous wave system and state our main existence and stability result. Section
4 is devoted to the introduction of a system for the perturbed Poiseuille-type
ﬂow. In Section 5, we introduce our key notion of change of variables and
discuss that the system has hidden dissipative structure. Finally in Section
6, we prove energy estimates and our main result. In Section 7, we state
some basic properties of the Stokes operator that are used in this paper.
Section 8 is dedicated to prove a priori estimates of viscous wave equation
i.e. Proposition 3.1.
2. Deformation tensor and equations of motion
Let Ω be a domain in R2 with smooth boundary and T > 0 be ﬁxed time.
We consider the viscoelastic ﬂuid in Ω described by unknown variables:
⊗
· F : (0, T ) × Ω → R2 R2 is the deformation tensor,
· π : (0, T ) × Ω → R is the pressure,
· u : (0, T ) × Ω → R2 is the velocity of the ﬂuid,
· µ > 0 is the kinematic viscosity of the ﬂuid
in Eulerian description. The deformation tensor F in the Lagrangian coordinates is deﬁned by Fij = ∂xj /∂Xi where X is the Lagrangian variables and
x = x(X, t) is the ﬂow map. In the following we always describe F in the
Eulerian coordinates. One should be careful to note our notation diﬀers from
that in [9], where the transpose of our F is used.
We consider the following two dimensional viscoelastic ﬂuid system of
the Oldroyd model with Dirichlet boundary condition of the form.

∂t F + u · ∇F = F ∇u
in (0, T ) × Ω,





div
u
=
0
in (0, T ) × Ω,




 ∂ u − µ∆u + u · ∇u + ∇π = div F T F in (0, T ) × Ω,
t
(2.1)

F |t=0 = F0
on Ω,





u|t=0 = u0
on Ω,




u=0
on (0, T ) × ∂Ω
with the assumption det F |t=0 = 1 and div F |t=0 = 0. In this paper, we use
the following notation.
· ∂t = (∂/∂t), ∂i = (∂/∂xi ),
· (∇u)ij = ∂j ui ,
4
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
· (div G)i =
∑n
j=1
∂j Gij .
Stream function formulation. One can show that div F is subject to advection
with the ﬂow, i.e.
∂t (div F ) + u · ∇(div F ) = 0.
Therefore div F0 = 0 implies div F = 0 for all later times. Under this assumption, one can ﬁnd an R2 -valued stream function ζ0 such that F0 = ∇⊥ ζ0 as
in [9]. Moreover, if one lets ζ be the solution of the transport equation for a
divergence free function u of the form
{
∂t ζ + u · ∇ζ = 0,
(2.2)
ζ|t=0 = ζ0
then one can ﬁnd that for
(
−∂2 ζ 1
F =∇ ζ=
−∂2 ζ 2
⊥
)
∂1 ζ 1
,
∂1 ζ 2
the equation Ft + u · ∇F = F ∇u is fulﬁlled. It is much easier to consider the
function ζ instead of F . In order to rewrite system (2.1) with respect to ζ,
one can calculate
div F T F =
1
∇|∇ζ|2 − ∆ζ 1 ∇ζ 1 − ∆ζ 2 ∇ζ 2 .
2
Note that the ﬁrst term is a gradient that can be absorbed in pressure term.
Thus we introduce a new variable π
˜ = π − 21 |∇ζ|2 and denote that by π again.
We end up with the following new system for two dimensions with Dirichlet
boundary condition.

∂t ζ + u · ∇ζ = 0





div u = 0





2

∑

 ∂ u − µ∆u + u · ∇u + ∇π = −
∆ζ k ∇ζ k
t
k=1




ζ|
=
ζ
t=0
0





u|t=0 = u0




u=0
in (0, T ) × Ω,
in (0, T ) × Ω,
in (0, T ) × Ω,
(2.3)
on Ω,
on Ω,
on (0, T ) × ∂Ω.
The corresponding assumption to the incompressibility condition det F |t=0 =
1 is
∂1 ζ01 ∂2 ζ02 − ∂1 ζ02 ∂2 ζ01 = 1.
Note that div F |t=0 = 0 is satisﬁed by the construction of ζ.
(2.4)
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
5
Incompressibility. Considering ﬂuids with a constant density, the incompressibility condition takes the form div u = 0. It turns out that in terms
of the deformation tensor, this means det F = 1 if det F |t=0 = 1 holds.
Moreover, one can ﬁnd that
∂t (det F ) + u · ∇(det F ) = 0.
Therefore if (2.4) holds, we have ∂1 ζ 1 ∂2 ζ 2 − ∂1 ζ 2 ∂2 ζ 1 = 1 for all time.
3. Poiseuille-type ﬂow and viscous wave equation
Let Ω = R × (0, 1) i.e. a two-dimensional layer. This section aims to construct
a suitable Poiseuille-type ﬂow solution u
¯ to (2.1) or equivalently (2.3), i.e. a
solution with horizontal ﬂow-proﬁle that is completely determined by the
vertical component. Hence, we assume that u
¯ takes the form
(
)
ψ(t, x2 )
u
¯(t, x) =
0
with homogeneous Dirichlet boundary conditions. Then the divergence condition in (2.1) is trivially fulﬁlled.
In order to adequately determine the corresponding deformation tensor
F¯ or equivalently( the corresponding
) stream function η, we introduce the ﬂow
map x(t, X) = x1 (t, X), x2 (t, X) , 0 ≤ t < T with T > 0, corresponding
to Lagrangian coordinates X. These ﬂow maps are given by the system of
ordinary diﬀerential equations

d

 x1 (t, X) = u
¯1 (t, x1 (t, X), x2 (t, X)) = ψ(t, x2 (t, X)), x1 (0) = X1 ,
dt

 d x (t, X) = u
¯2 (t, x1 (t, X), x2 (t, X)) = 0,
x2 (0) = X2 ,
2
dt
which can easily be solved by

∫ t
∫ t

 x (t, X) = X +
ψ(s, x2 (s, X)) ds = X1 +
ψ(s, X2 ) ds,
1
1
0
0

 x (t, X) = X ,
2
2
as long as ψ is suﬃciently regular. Let us abbreviate
∫ t
ϕ(t, x2 ) =
ψ(s, x2 ) ds.
0
Then, we can calculate the deformation tensor and the resulting elastic
force
F¯ =
)
1
0
,
∂2 ϕ 1
(
(
F¯ T F¯ =
1 + (∂2 ϕ)2
∂2 ϕ
)
∂2 ϕ
1
(
and div F¯ T F¯ =
)
∂22 ϕ
.
0
Note here, that with x2 (t, X) = X2 it is also (∂/∂X2 ) = (∂/∂x2 ) = ∂2 . Let
us also remark at this point, that div F¯ = 0.
6
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
The stream function η corresponding to F¯ may be chosen as
(
)
−x2
η(t, x) =
x1 − ϕ(t, x2 )
(3.1)
solving the system
{
∂t η + u
¯ · ∇η = 0,
in (0, T ) × Ω,
η(0, x) = (−x2 , x1 )T , for x ∈ Ω.
Inserting the elastic force into the balance of momentum for u
¯, i.e.
T ¯
¯
∂t u
¯ − µ∆¯
u+u
¯ · ∇¯
u + ∇¯
π = div F F , in (0, T ) × Ω,
yields the equivalent formulation
}
∂t ψ + ∂1 π
¯ = µ∂22 ψ + ∂22 ϕ,
∂2 π
¯ = 0.
in (0, T ) × Ω.
We conclude from the second equation that the pressure is a function
depending only on the horizontal variable π
¯=π
¯ (t, x1 ). Since ψ and ϕ depend
only on t and x2 , the ﬁrst equation implies that ∂1 π
¯ is a function of time
only, i.e. ∂1 π
¯ (t, x1 ) = −h(t) for some function h. Inserting this into the system
yields
∂t ψ − ∂22 ϕ = µ∂22 ψ + h,
in (0, T ) × (0, 1).
Finally, by the deﬁnition of ϕ it is ψ(t, x2 ) = ∂t ϕ(t, x2 ) and moreover, the homogeneous Dirichlet boundary conditions for u
¯ carry over to ϕ, i.e. ϕ(t, 0) =
∫0
ϕ(t, 1) = 0. At initial time we have ϕ(0, x2 ) = 0 ψ(s, x2 ) ds = 0 and
∂t ϕ(0, x2 ) = ψ(0, x2 ) = ψ0 (x2 ) for some function ψ0 that will be given satisfying homogeneous Dirichlet conditions.
With this, we end up with a viscous wave equation in one dimension
 2
2
2

 ∂t ϕ − ∂x ϕ = µ∂t ∂x ϕ + h, in (0, T ) × (0, 1),
ϕ(t, 0) = ϕ(t, 1) = 0,
for t ∈ (0, T ),
(3.2)


ϕ|t=0 = 0, ∂t ϕ|t=0 = ψ0 , on (0, 1).
for some h = h(t) and initial data ψ0 . Note that we use ∂x instead of ∂2 since
we consider ϕ is the function with two variables (t, x) here.
We state an a priori estimate for the Poiseuille-type ﬂow in the following
proposition. It enables us to control the norms of higher spatial derivatives
of ϕ.
Proposition 3.1. Let T > 0 and µ > 0. For ψ0 ∈ H 3 (0, 1) ∩ H01 (0, 1) and
h ∈ H 1 (0, T ), there exists the unique solution ϕ ∈ C 3 ([0, T ); L2 (0, 1)) ∩
C 2 ([0, T ), H 3 (0, 1)) ∩ C([0, T ); H 4 (0, 1)) of (3.2). Moreover, there is a constant C such that the solution satisfies
∥∂t ϕ(t)∥H 3 (0,1) + ∥∂x ϕ(t)∥H 3 (0,1) + ∥∂t2 ϕ(t)∥H 1 (0,1)
4
(
)
∑
2 1
≤ C e− min(2µπ , µ )t ∥ψ0 ∥H 3 (0,1) +
µ−k ∥h∥H 1 (0,t)
k=1
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
7
for 0 ≤ t ≤ T . The constant C is independent of T and µ.
Proof. Combining Proposition 8.1 and Proposition 8.3 in Section 8, one can
easily obtain the result.
Notation of spaces. In this paper, we write ∥f ∥ for ∥f ∥L2 (U ) otherwise speciﬁed, and denote H k (U ) by Sobolev space W 2,k (U ) , equipped with the norm
√∑
∥f ∥H k (U ) =
∥∂ k f ∥2
|α|≤k
for some domain U ⊂ Rn . We also deﬁne H0k (U ) by the closure of Cc∞ , the
space of all smooth functions with compact support with respect to ∥·∥H k (U ) ;
see [2, Section 5] for more detail.
Inserting the function ψ = ∂t ϕ into the ansatz for u
¯, we receive a solution
(¯
u, π
¯ , η) of the system

∂t η + u
¯ · ∇η = 0,





div u
¯ = 0,




 ∂t u
¯ − µ∆¯
u+u
¯ · ∇¯
u + ∇¯
π = −∆η k ∇η k ,










in (0, T ) × Ω,
in (0, T ) × Ω,
in
(0, T ) × Ω,
on (0, T ) × ∂Ω,
u
¯ = 0,
η(0, x) = (−x2 , x1 ) , for x ∈ Ω,
T
u
¯|t=0 = (ψ0 , 0)T ,
on Ω,
∑2
where −∆η k ∇η k is a short notation for k=1 −∆η k ∇η k . Note that we choose
η by (3.1) and due to the homogeneous Dirichlet boundary conditions for ϕ,
it is η(t, x)|∂Ω = (−x2 , x1 )T for any 0 ≤ t ≤ T .
We are now in a position to state our main result.
Theorem 3.2. Let Ω = R × (0, 1), u0 ∈ H 2 (Ω), ζ0 ∈ H 3 (Ω), ψ0 ∈ H 3 (0, 1) ∩
H01 (0, 1), and h ∈ H 1 (0, ∞). There exist numbers 0 < δ < 1, κ > 0 such that
if the following three conditions hold,
1. the smallness condition for the Poiseuille-type flow
∥ψ0 ∥H 3 (0,1) + ∥h∥H 1 (0,∞) ≤ κ,
2. the smallness condition for the initial perturbation
∥u0 − u
¯0 ∥H 3 (Ω) + ∥ζ0 − η0 ∥H 3 (Ω) ≤ κ,
3. the compatibility conditions for the initial data
(
)
−x2
div u0 = 0, ζ0 |∂Ω =
and ∂1 ζ01 ∂2 ζ02 − ∂1 ζ02 ∂2 ζ01 = 1
x1
8
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
then there exists a smooth global solution (u, ζ, π) of (2.3) with respect to
initial data (u0 , ζ0 ) satisfying
)
d(
∥∂t (u−¯
u)∥2 +δ∥∇(u−¯
u)∥2 +δ∥∆(ζ−η)∥2 +δ∥∇∆(ζ−η)∥2 +∥∂t (ζ−η)∥2
dt
(
)
1
1
+µδ ∥∂t ∇(u− u
¯)∥2 +∥A(u− u
¯)∥2 + 2 ∥∆(ζ −η)∥2 + 2 ∥∇∆(ζ −η)∥2 ≤ 0
µ
µ
for all times t ≥ 0. Here, A = −P ∆ is the Stokes operator in Ω; see Section 7.
Integrating the last diﬀerential inequality over (0, t) implies
∫ t
δ∥∇(u − u
¯)∥2 (t) + µδ
∥A(u − u
¯)∥2 (s) ds ≤ Cκ
0
with Cκ which tends to zero as κ → 0. By (7.1) we in particular obtain
∫ t
2
Cµ∥∇(u − u
¯)∥2 (s) ds ≤ Cκ
∥∇(u − u
¯)∥ (t) +
0
which implies
∥∇(u − u
¯)∥2 (t) ≤ Cκ e−Cµt
by the Gronwall inequality. By the Poincar´e inequality this implies that u
¯ is
exponentially stable in H 1 sense. Similar stability holds for η.
4. Perturbation of the ﬂow through the layer
We are interested in the solution (u, π, ζ) of the system (2.1) and its stability
around the Poiseuille-type ﬂow (¯
u, π
¯ , η). Let (u0 , ζ0 ) satisﬁes the compatibility conditions,
(
)
−x2
div u0 = 0, ζ0 |∂Ω =
, and ∂1 ζ01 ∂2 ζ02 − ∂1 ζ02 ∂2 ζ01 = 1.
(4.1)
x1
The second condition together with the homogeneous Dirichlet boundary
condition of u guarantees ζ|∂Ω = (−x2 , x1 )T for all times. The third condition is a reformulation of the incompressibility condition as we discussed in
Section 2 and that holds for any t ≥ 0.
Now let us introduce the perturbation
(v, p, α) = (u, π, ζ) − (¯
u, π
¯ , η).
By a simple calculation one can show that the decomposition ζ = α + η
implies
∂2 α1 − ∂1 α2 = ∂1 α1 ∂2 α2 − ∂2 α1 ∂1 α2 − ∂2 ϕ∂1 α1 .
This will be a crucial identity later on.
(4.2)
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
Then for given (¯
u, π
¯ , η), (v, p, α) solves

∂t α + v · ∇α + u
¯ · ∇α = −v · ∇η





div v = 0





∂
v
−
µ∆v
+
v
·
∇v
+
v
· ∇¯
u+u
¯ · ∇v + ∇p

 t

= −∆αk ∇αk − ∆η k ∇αk − ∆αk ∇η k



v = 0,






α(0, x) = ζ0 (x) − (−x2 , x1 )T




v|t=0 = u0 − (ψ0 , 0)T
9
in (0, T ) × Ω,
in (0, T ) × Ω,
in (0, T ) × Ω,
on (0, T ) × ∂Ω,
for x ∈ Ω,
in Ω
(4.3)
Note that it is α|∂Ω = 0 for all times since ζ|∂Ω = η|∂Ω = (−x2 , x1 ).
The stream function of the Poiseuille-type ﬂow is given by η(t, x) =
(−x2 , x1 − ϕ(t, x2 )). We note that derivatives of η contain constant parts
which may not be small even if ϕ is small. Let us rewrite the right-hand side
of the momentum equation as
(
)
−∆α2 + ∂1 (∂22 ϕα2 )
k
k
k
k
−∆α ∇η − ∆η ∇α =
∆α1 + ∆(∂2 ϕα2 ) − ∂23 ϕα2 − ∂22 ϕ∂2 α2
( 2)
−α
=∆
+ ∂22 ϕ∇α2 + ∇ϕ∆α2 .
(4.4)
α1
Therefore the momentum equation is rewritten as follows.
∂t v − µ∆v + v · ∇v + v · ∇¯
u+u
¯ · ∇v + ∇p
( 2)
−α
k
k
= −∆α ∇α + ∆
+ ∇ϕ∆α2 + ∂22 ϕ∇α2 . (4.5)
α1
5. Change of variables and dissipation
Observing the momentum equation in (4.5), one may notice that terms like
v · ∇¯
u or u
¯ · ∇v can be handled through Proposition 3.1 if the Poiseuille-type
ﬂow is suﬃciently small. On the other hand, ∆(−α2 , α1 ) causes a problem.
Although α seems to have no dissipative structure so far, this term produces
linear terms. That calls a particular method.
Taking a closer look at right-hand side in (4.5), one can ﬁnd another
dissipative structure. Let us focus on the term ∆(−α2 , α1 )T in (4.4), and
rewrite whole equation as follows
(
)
(
1 −α2 )
∂t v − µ∆ v +
+ v · ∇v + v · ∇¯
u+u
¯ · ∇v + ∇p
µ α1
= −∆αk ∇αk + ∇ϕ∆α2 + ∂22 ϕ∇α2 . (5.1)
Now we will introduce a new dependent variable as in [9]:
(
)
(
)
1 −α2
0 1
w=v+
or equivalently, α = µ
(w − v).
−1 0
µ α1
10
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
Let us rewrite the transport equation of α in (4.3), i.e.
∂t α + v · ∇α + u
¯ · ∇α = −v · ∇η.
(5.2)
In the right-hand side, one can ﬁnd
( 2 )
v
−v · ∇η =
+ v 2 ∇ϕ
−v 1
( 2 )
1
w
− α + v 2 ∇ϕ.
=
−w1
µ
Therefore the transport equation can be rewritten as follows
(
)
1
w2
¯ · ∇α =
∂t α + α + v · ∇α + u
+ v2 ∇ϕ.
−w1
µ
(5.3)
Hence we can see that α has dissipative structure. However, we must control
the w term in the right-hand side. The question is how to introduce estimates
for w or v? The idea is that we regard (4.5) as a perturbed Stokes system of
w and p, i.e.
−µ∆w + ∇p = −∂t v − v · ∇v − v · ∇u − ∆αk ∇αk + ∂22 ϕ∇α2 + ∇ϕ∆α2
(5.4)
and invoke a higher regularity estimates of the Stokes system (Lemma 7.2).
For this purpose, we need to calculate div w ﬁrst.
Divergence of w and higher order estimate. Let us note that w is not divergence free in general. However, its divergence is quadratic in α and ϕ as the
following calculation shows:
( 2)
1
−α
div w = div v + div
α1
µ
1
= (∂2 α1 − ∂1 α2 + ∂2 ϕ∂1 α1 − ∂2 ϕ∂1 α1 )
µ
1
= (det G − ∂2 ϕ∂1 α1 )
µ
1
(5.5)
= (∂1 α1 ∂2 α2 − ∂2 α1 ∂1 α2 − ∂2 ϕ∂1 α1 ).
µ
Note that we used the incompressibility property (4.2).
Now let f and g be the right-hand sides of (5.4), (5.5) respectively. If w
satisﬁes appropriate conditions, we can invoke Lemma 7.2 and obtain
µ∥w∥H 3 (Ω) + ∥∇p∥H 1 (Ω) ≤ C(µ∥g∥H 2 (Ω) + ∥f ∥H 1 (Ω) ).
(5.6)
We can easily obtain the estimate for v by the deﬁnition of w. We will state
the result of these estimates in the next section as a proposition.
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
11
6. A priori estimate with Energy method
The existence of approximate solutions to (4.3) may be proved using a Galerkin
approximation scheme similarly to [9]. Since compact embeddings are required for this approach in order to pass to the limit, the problem then needs
to be considered on a sequence of domains ΩM = (−M, M ) × (0, 1).
We impose v = 0 on the artiﬁcial left and right boundaries. For the stream
function α no boundary conditions may be imposed and it will in general
not vanish on the artiﬁcial boundaries. It vanishes on the lower and upper
boundary however, since by deﬁnition α = ζ − η and the stream functions
η and ζ are transported by u
¯ and u which vanish on the upper and lower
boundary (but not on the artiﬁcial boundaries). Hence, for v as well as α the
Poincar´e inequality is still applicable. Since all the estimates do not depend
on the horizontal size of the domain, one can let M tend to inﬁnity to receive
a solution of (4.3) .
The a priori estimates for the approximate solutions of the Galerkinscheme are of the same structure as for the original equations. Let us therefore
concentrate on the formal a priori estimates for system (4.3).
Let Ω be a layer domain R × (0, 1) in this section.
Notation. In order to simplify the notation, we now introduce variables corresponding to the data, the time-derivative part and the dissipative part of
the estimates respectively. We write
X(t) = ∥∂t ϕ∥H 3 (0,1) + ∥∂2 ϕ∥H 3 (0,1) + ∥∂t2 ϕ∥H 1 (0,1) ,
Y (t) = ∥∂t v∥ + ∥∇v∥ + ∥∆α∥ + ∥∇∆α∥,
1
1
Z(t) = ∥∂t ∇v∥2 + ∥Av∥2 + 2 ∥∆α∥2 + 2 ∥∇∆α∥2 .
µ
µ
With the deﬁnition of Y (t) and Z(t) as it is, by the Poincar´e inequality we
ﬁnd
Y 2 (t) ≤ C(1 + µ2 )Z(t).
Here, A in Z(t) is the Stokes operator. We use Av instead of ∆v to annihilate
the pressure term in some energy estimates with aid of the regularity of the
Stokes operator ∥v∥H22 (Ω) ≤ C∥Av∥.
In this section, we derive ﬁve energy estimates. Then combining these
results, we obtain the strong stability inequality stated in Theorem 3.2. To
begin with, we need to calculate (5.6) for higher order estimates.
6.1. Spatial Estimate of the artiﬁcial variable w and v
It is diﬃcult to estimate higher spatial derivatives directly. We cannot use
integration by parts since higher spatial derivatives would not vanish on the
boundary in general. Therefore we consider a priori estimates for higher order terms in time, and then transfer them into spatial estimates using the
regularity of the Stokes system Lemma 7.2.
12
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
Proposition 6.1. Let Ω = R×(0, 1), u0 ∈ H 2 (Ω), ζ0 ∈ H 3 (Ω), ψ0 ∈ H 3 (0, 1)∩
H01 (0, 1), h ∈ H 1 (0, ∞), and (u, π, ζ) be a solution of (2.3). If (u0 , ζ0 ) satisfies
the compatibility conditions
(
)
−x2
div u0 = 0, ζ0 |∂Ω =
and ∂1 ζ01 ∂2 ζ02 − ∂1 ζ02 ∂2 ζ01 = 1,
x1
then there exists a numerical constant C > 0 such that for (v, p, α) = (u, π, ζ)−
(¯
u, π
¯ , η) and w = v − µ1 (−α2 , α1 )T , the estimates
(
)
∥v∥H 3 (Ω) ≤ C ∥∂t ∇v∥ + (1 + µ2 )Z + X(∥Av∥ + ∥∆α∥ + ∥∇∆α∥)
C
+ (∥∆α∥ + ∥∇∆α∥).
µ
(
)
∥w∥H 3 (Ω) ≤ C ∥∂t ∇v∥ + (1 + µ2 )Z + X(∥Av∥ + ∥∆α∥ + ∥∇∆α∥)
hold for all t ≥ 0.
Proof. Inserting (5.4) and (5.5) to (5.6), we have
µ∥w∥H 3 (Ω) + ∥∇p∥H 1 (Ω)
(
≤ C ∂t v − v · ∇v − v · ∇¯
u−u
¯ · ∇v
− ∆αk ∇αk + ∇ϕ∆α2 + ∂22 ϕ∇α2 H 1 (Ω)
)
+ ∂1 α1 ∂2 α2 − ∂2 α1 ∂1 α2 − ∂2 ϕ∂1 α1 H 2 (Ω)
(
≤ C ∥∂t v∥H 1 (Ω) + ∥v · ∇v∥H 1 (Ω) + ∥v · ∇¯
u∥H 1 (Ω) + ∥¯
u · ∇v∥H 1 (Ω)
+ ∥∆αk ∇αk ∥H 1 (Ω) + ∥∇ϕ∆α2 ∥H 1 (Ω) + ∥∂22 ϕ∇α2 ∥H 1 (Ω)
)
+ ∥∂1 α1 ∂2 α2 ∥H 2 (Ω) + ∥∂2 α1 ∂1 α2 ∥H 2 (Ω) + ∥∂2 ϕ∂1 α1 ∥H 2 (Ω) .
We investigate these terms one by one, beginning with
∥∂t v∥H 1 (Ω) ≤ ∥∂t ∇v∥
(6.1)
by the Poincar´e inequality. We next observe that
∥v · ∇v∥H 1 (Ω) ≤ ∥v · ∇v∥ + ∥∇(v · ∇v)∥
(
)
≤ C ∥v∥L∞ (Ω) ∥∇v∥ + ∥v∥L∞ (Ω) ∥∇2 v∥ + ∥∇v∥2L4 (Ω)
≤ C∥v∥2H 2 (Ω)
≤ C∥Av∥2 .
We have invoked embeddings in Lemma 7.5 and the regularity of the Stokes
operator (7.1).
Similarly,
(
)
∥v · ∇¯
u∥H 1 (Ω) ≤ C ∥v∥L∞ (Ω) ∥∇¯
u∥ + ∥∂2 u
¯∥∥∇2 v∥ + ∥¯
u∥L∞ (Ω) ∥∇2 v∥
∥¯
u · ∇v∥H 1 (Ω)
≤ C∥Av∥∥∂t ϕ∥H 2 (0,1)
(
)
≤ C ∥¯
u∥L∞ (Ω) ∥∇v∥ + ∥∂2 u
¯∥∥∇2 v∥ + ∥¯
u∥L∞ (Ω) ∥∇2 v∥
≤ C∥Av∥∥∂t ϕ∥H 2 (0,1) .
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
13
Here, we have invoked the 1D-2D product estimate Lemma 7.6. Note that u
¯
is a function of one space variable.
Finally, we have
∥∆αk ∇αk ∥H 1 (Ω) + ∥∇ϕ∆α2 ∥H 1 (Ω) + ∥∂22 ϕ∇α2 ∥H 1 (Ω)
(
≤ C ∥∆α∥∥∇α∥L∞ (Ω) + ∥∇∆α∥∥∇α∥L∞ (Ω)
)
+ ∥∆α∥L4 (Ω) ∥∇2 α∥L4 (Ω) + ∥∂2 ϕ∥H 3 (0,1) ∥∇α∥H 3 (Ω)
(
)
≤ C ∥∇∆α∥2 + ∥∆α∥2 + ∥∂2 ϕ∥H 3 (0,1) (∥∇∆α∥ + ∥∆α∥) .
∥∂1 α1 ∂2 α2 ∥H 2 (Ω) + ∥∂2 α1 ∂1 α2 ∥H 2 (Ω) + ∥∂2 ϕ∂1 α1 ∥H 2 (Ω)
(
≤ C ∥∇α∥H 2 (Ω) ∥∇α∥L∞ (Ω) + ∥∆α∥2L4 (Ω)
+ ∥∂2 ϕ∥L∞ (Ω) ∥∇α∥H 2 (Ω) + ∥∂22 ϕ∥L∞ (Ω) ∥∇α∥H 1 (Ω) + ∥∂23 ϕ∥∥∇α∥L∞ (Ω)
(
)
≤ C ∥∇∆α∥2 + ∥∆α∥2 + ∥∂2 ϕ∥H 3 (0,1) (∥∇∆α∥ + ∥∆α∥) .
Combining these results, we obtain,
µ∥w∥H 3 (Ω) + ∥∇p∥H 1 (Ω)
(
≤ C ∥∂t ∇v∥2 + ∥Av∥2 + ∥∆α∥2 + ∥∇∆α∥2
)
+ (∥∂t ϕ∥H 2 (0,1) + ∥∂2 ϕ∥H 3 (0,1) )(∥Av∥ + ∥∆α∥ + ∥∇∆α∥)
)
≤ C(∥∂t ∇v∥ + (1 + µ2 )Z + X(∥Av∥ + ∥∆α∥ + ∥∇∆α∥)
(6.2)
We immediately ﬁnd a similar estimate for higher regularity of v with v =
w − µ1 (−α2 , α1 )T i.e.
(
)
∥v∥H 3 (Ω) ≤ C ∥∂t ∇v∥ + (1 + µ2 )Z + X(∥Av∥ + ∥∆α∥ + ∥∇∆α∥)
C
+ (∥∆α∥ + ∥∇∆α∥).
(6.3)
µ
Later on, we will use these results to estimate higher derivatives in time
as mentioned above.
6.2. Summary of the energy estimates
Let us state the result of the energy estimates ﬁrst:
Proposition 6.2. Under the same assumption as in Proposition 6.1, we have
the following estimates:
1 d
∥∇v∥2 + µ∥Av∥2 ≤ ∥∆α∥∥Av∥ + C(1 + µ)(X + Y )Z,
2 dt
( 2)
(
)
1 d
−α
2
2
∥∂t v∥ + µ∥∂t ∇v∥ ≤ − ∂t ∇
, ∂t ∇v
1
α
2 dt
+ C(1 + µ)(1 + X + Y )(X + Y )Z,
(6.4)
(6.5)
)
14
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
1 d
1
1
∥∆α∥2 + ∥∆α∥2 ≤ C ∥∂t ∇v∥∥∆α∥
2 dt
µ
µ
1
+ C( + 1 + µ + µ2 )(X + Y )Z,
µ
1 d
1
1
∥∇∆α∥2 + ∥∇∆α∥2 ≤ C ∥∂t ∇v∥∥∇∆α∥
2 dt
µ
µ
1
+ C( + 1 + µ + µ2 )(X + Y )Z,
µ
( 2)
)
(
1 d
−α
2
, ∂t ∇v
∥∂t ∇α∥ ≤ ∂t ∇
1
α
2 dt
1
+ C( + 1 + µ + µ2 )(1 + X + Y )3 (1 + Y )Z.
µ
Here, C > 0 is a numerical constant (independent of µ).
(6.6)
(6.7)
(6.8)
The estimate (6.4) is obtained by taking the inner product of momentum equation for v with Av, for short we write (v-momentum, Av). Let
us summarize the estimates and corresponding inner products in the table
below.
Estimate
Inner product
(6.4)
(v-momentum, Av)
(6.5)
(∂t (v-momentum), ∂t v)
(6.6)
(∆(α-transport), ∆α)
(6.7)
(∇∆(α-transport), ∇∆α)
(6.8)
(∂t ∇(α-transport), ∂t α)
Corresponding subsection
6.3
6.4
6.5
6.6
6.7
In the following subsections, we shall show these estimates one by one.
Before going into the detail, let us note what are the aims of each estimate.
The ﬁrst estimate (6.4) is the core estimate, although the estimate produces
a linear term ∥∆α∥∥∆Av∥. This problem will lead us to the energy estimates
of α i.e. (6.6) and (6.7). One can immediately notice that these estimates
produce the term ∥∂t ∇v∥ in the right-hand side. In order to manage these
terms, we derive the estimate (6.5) to absorb ∥∂t ∇v∥ in the right-hand side
of (6.6) and (6.7). However, we receive another linear term again as one can
see in (6.5). Therefore we derive another estimate (6.8) to cancel out this
linear term.
6.3. A priori estimate for the velocity gradient
For receiving an estimate for spatial derivatives, we want to test the equation with second derivatives of v. A simple way would be using −∆v which,
unfortunately, does not vanish on the boundary. Hence, the pressure term
would not vanish in the estimate and must be estimated explicitly. We will
therefore employ Av instead of −∆v in the estimate.
Taking the inner product of the momentum equation in (4.5), i.e.
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
15
∂t v − µ∆v + v · ∇v + v · ∇¯
u+u
¯ · ∇v + ∇p
( 2)
−α
k
k
= −∆α ∇α + ∆
+ ∂22 ϕ∇α2 + ∇ϕ∆α2 ,
α1
with Av. We can use the boundary condition for ∂t v to integrate by parts.
Since the Helmholtz-projection is self-adjoint, we have
(∂t v, Av) = −(∂t v, P ∆v) = −(P ∂t v, ∆v)
= −(∂t v, ∆v) = (∂t ∇v, ∇v) =
1 d
∥∇v∥2 ,
2 dt
and
(−µ∆v, Av) = µ∥Av∥2 .
For the convection terms we use the embedding H 2 (Ω) ,→ L∞ (Ω), the 1D-2D
product estimates and the usual Stokes regularity to estimate
|(v · ∇v, Av)| ≤ ∥v∥L∞ (Ω) ∥∇v∥∥Av∥ ≤ C∥v∥H 2 (Ω) ∥∇v∥∥Av∥
≤ C∥∇v∥∥Av∥2 ≤ CY Z,
|(v · ∇¯
u, Av)| ≤ C∥∇v∥∥∇¯
u∥∥Av∥ ≤ C∥∂t ϕ∥H 1 (0,1) ∥Av∥2 ≤ CXZ,
and
|(¯
u · ∇v, Av)| ≤ ∥¯
u∥L∞ (Ω) ∥∇v∥∥Av∥ ≤ C∥∂t ϕ∥H 1 (0,1) ∥Av∥2 ≤ CXZ.
Since the Stokes operator maps into L2σ (Ω), the L2 closure of all smooth
solenoidal vector ﬁelds with compact support in Ω, the pressure term ∇p
vanishes in the a priori estimate.
The quadratic form in α gives
|(−∆αk ∇αk , Av)| ≤ ∥∆α∥∥∇α∥L∞ (Ω) ∥Av∥
≤ C∥∆α∥(∥∆α∥ + ∥∇∆α∥)∥Av∥
≤ CµY Z,
and for the last terms it is
( 2
)
∂2 ϕ∇α2 + ∇ϕ∆α2 , −Av ≤ C(∥∂22 ϕ∥∥∇2 α∥ + ∥∂2 ϕ∥L∞ (Ω) ∥∆α∥)∥Av∥
≤ C∥∂2 ϕ∥H 1 (0,1) ∥∆α∥∥Av∥
≤ CµXZ.
( 2)
−α
Linear term. The term ∆
contains linear parts with non-small coefα1
ﬁcients. Due to the presence of the Helmholtz-projection, it is not possible
to cancel this term with a corresponding term (v 2 , −v 1 )T in the α-estimate.
We have
( 2)
(
)
∆ −α1 , Av ≤ ∥∆α∥∥Av∥.
α
16
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
Altogether we have for the estimate of ∇v
1 d
∥∇v∥2 + µ∥Av∥2 ≤ ∥∆α∥∥Av∥ + C(1 + µ)(X + Y )Z.
2 dt
6.4. A priori estimate for the time derivative of the velocity
Here, we investigate higher derivatives in time, since we are able to transfer
higher time regularity to higher space regularity by using the Stokes regularity.
We apply ∂t to equation (4.5) and take the inner product with ∂t v. Since
v vanishes on the boundary, so does ∂t v and integration by parts gives the
terms
1 d
(∂t2 v, ∂t v) =
∥∂t v∥2 and (−µ∂t ∆v, ∂t v) = µ∥∂t ∇v∥2 .
2 dt
With div ∂t v = 0 and ∂t v = 0 on the boundary, we ﬁnd (v·∇∂t v, ∂t v) = 0
and therefore the Poincar´e inequality as well as the usual Stokes regularity
give
(∂t (v · ∇v), ∂t v) = (∂t v · ∇v, ∂t v)
≤ ∥∂t v∥L4 (Ω) ∥∇v∥L4 (Ω) ∥∂t v∥
≤ C∥∂t v∥H 1 (Ω) ∥∇v∥H 1 (Ω) ∥∂t v∥
≤ C∥∂t ∇v∥∥Av∥∥∂t v∥
≤ CY Z.
Similarly, we have after employing the product estimates for one- and twodimensional functions
(∂t (¯
u · ∇v), ∂t v) = (∂t u
¯ · ∇v, ∂t v)
≤ ∥∂t u
¯∥L∞ (Ω) ∥∇v∥∥∂t v∥
≤ C∥∂t2 ϕ∥H 1 (0,1) ∥Av∥∥∂t ∇v∥
≤ CXZ
and
(∂t (v · ∇¯
u), ∂t v) ≤ (∥∂t v · ∇¯
u∥ + ∥v · ∇∂t u
¯∥)∥∂t v∥
≤ C(∥∂t ∇v∥∥∇¯
u∥ + ∥∇v∥∥∂t ∇¯
u∥)∥∂t ∇v∥
≤ C(∥∂t ϕ∥H 1 + ∥∂t2 ϕ∥H 1 )(∥∂t ∇v∥ + ∥Av∥)∥∂t ∇v∥
≤ CXZ.
The pressure term vanishes due
( to
)div ∂t v = 0 and ∂t v = 0 on the
(
)
−α2
boundary. In the linear term ∂t ∆
, ∂t v we integrate by parts once,
1
α
using ∂t v = 0 on the boundary:
( 2)
( 2)
(
(
)
)
−α
−α
∂t ∆
, ∂t v = − ∂t ∇
, ∂t ∇v .
α1
α1
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
17
This term is going to be absorbed in the estimate for the time derivative of
the gradient of the stream function α. Considering the quadratic α-term, we
note with Einstein’s sum convention
[∆αk ∇αk ]i = ∂l (∂l αk ∂i αk ) − ∂l αk ∂i ∂l αk = [div (∇αk ⊗ ∇αk )]i − [∇(|∇αk |2 )]i
and therefore in the a priori estimate, the second part vanishes being a gradient and we have after integrating by parts
(−∂t (∆αk ∇αk ), ∂t v) = −(∂t div (∇αk ⊗ ∇αk ), ∂t v) + (∂t ∇(|∇αk |2 ), ∂t v)
= (∂t (∇αk ⊗ ∇αk ), ∂t ∇v)
≤ C∥∂t ∇α∥∥∇α∥L∞ (Ω) ∥∂t ∇v∥
≤ C∥∂t ∇α∥(∥∆α∥ + ∥∇∆α∥)∥∂t ∇v∥
≤ Cµ∥∂t ∇α∥Z.
In the remaining term, we estimate after integrating by parts
(
)
∂t (∂22 ϕ∇α2 + ∇ϕ∆α2 ), ∂t v
)
( (
)
∂1 (∂22 ϕα2 )
= ∂t
, ∂t v
2
div (∂2 ϕ∇α )
≤ C(∥∂t (∂22 ϕα2 )∥ + ∥∂t (∂2 ϕ∇α2 )∥)∥∂t ∇v∥
(
≤ C ∥∂22 ∂t ϕ∥∥∇α∥ + ∥∂22 ϕ∥∥∂t ∇α∥
)
+ ∥∂2 ∂t ϕ∥∥∇2 α∥ + ∥∂2 ϕ∥L∞ (Ω) ∥∂t ∇α∥ ∥∂t ∇v∥
≤ C(∥∂t ϕ∥H 2 (0,1) ∥∆α∥ + ∥∂2 ϕ∥H 1 (0,1) ∥∂t ∇α∥)∥∂t ∇v∥
)
( 1
µ
∥∆α∥2 + ∥∂t ∇v∥2 +
≤ C∥∂t ϕ∥H 2 (0,1)
2µ
2
C∥∂2 ϕ∥H 1 (0,1) ∥∂t ∇α∥∥∂t ∇v∥
≤ CµXZ + C∥∂2 ϕ∥H 1 (0,1) ∥∂t ∇α∥∥∂t ∇v∥.
For the estimation of the remaining terms including ∥∂t ∇α∥ in the two
foregoing estimates, we employ the transport equation for the stream function
α in (4.3). Note, that one can write −v · ∇η = (v 2 , −v 1 )T + (0, ∂2 ϕv 2 )T and
hence
∥∂t ∇α∥ ≤ ∥∇(v · ∇α)∥ + ∥∇(¯
u · ∇α)∥ + ∥∇(v · ∇η)∥
≤ ∥∇v∥∥∇α∥L∞ (Ω) + ∥v∥L4 (Ω) ∥∇2 α∥L4 (Ω) + ∥∇¯
u∥∥∇2 α∥ + ∥¯
u∥L∞ (Ω) ∥∇2 α∥
+ ∥∇v∥ + ∥∂22 ϕ∥∥∇v∥ + ∥∂2 ϕ∥L∞ (Ω) ∥∇v∥
≤ ∥∇v∥
+ C(∥∆α∥ + ∥∇∆α∥ + ∥∂2 ϕ∥H 1 + ∥∂t ϕ∥H 1 )(∥∇v∥ + ∥∆α∥)
≤ C(1 + X + Y )Y.
(6.9)
Applying this inequality yields
(−∂t (∆αk ∇αk ), ∂t v) ≤ Cµ∥∂t ∇α∥Z
≤ Cµ(1 + X + Y )Y Z
18
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
and with ∥∂t ∇v∥Y ≤ C(1 + µ)Z, we have
)
(
∂t (∂22 ϕ∇α2 + ∇ϕ∆α2 ), ∂t v
≤ CµXZ + C∥∂2 ϕ∥H 1 (0,1) ∥∂t ∇v∥(1 + X + Y )Y
≤ C(1 + µ)(1 + X + Y )XZ.
Summarizing the foregoing estimates, we receive
( 2)
(
)
1 d
−α
2
2
∥∂t v∥ + µ∥∂t ∇v∥ + ∂t ∇
, ∂t ∇v
1
α
2 dt
≤ C(1 + µ)(1 + X + Y )(X + Y )Z.
6.5. A priori estimate for the Laplacian of the stream function
We aim to control the H 3 (Ω)-norm of α with an a priori estimate of the
Laplacian(Lemma 7.4) and it is therefore necessary to estimate ∆α as well
as ∇∆α. We will be able to produce a regularizing term ∥∆α∥2 on the lefthand side. This, however, comes at the cost of linear error terms involving
the artiﬁcial variable w = v + µ1 (−α2 , α1 )T . These error terms will later on
be handled with higher estimates of w (6.2).
We apply ∆ to the transport equation of the stream function (5.3), i.e.
( 2 )
1
w
∂t α + α + v · ∇α + u
¯ · ∇α =
+ v 2 ∇ϕ.
−w1
µ
and take the inner product with ∆α. Then the time derivative gives (∂t ∆α, ∆α) =
1 d
2
2
2 dt ∥∆α∥ . The second term gives (∆α, ∆α) = ∥∆α∥ . For the third term
we note (v · ∇∆α, ∆α) = 0 and therefore with Einstein’s sum convention
(∆(v · ∇α), ∆α) = (∆v · ∇α + 2∂i v · ∇∂i α, ∆α)
≤ C(∥∆α∥∥∇α∥L∞ (Ω) + ∥∇v∥L4 (Ω) ∥∇2 α∥L4 (Ω) )∥∆α∥
≤ C∥Av∥(∥∆α∥ + ∥∇∆α∥)∥∆α∥
(µ
)
1
≤ C(∥∆α∥ + ∥∇∆α∥) ∥Av∥2 +
∥∆α∥2
2
2µ
≤ CµY Z.
Similarly the second advection term yields
(∆(¯
u · ∇α), ∆α) = (∆¯
u · ∇α + 2∂i u
¯ · ∇∂i α, ∆α)
≤ C(∥∂22 ∂t ϕ∥∥∇2 α∥ + ∥∂2 ∂t ϕ∥L∞ (Ω) ∥∇2 α∥)∥∆α∥
≤ C∥∂t ϕ∥H 2 (0,1) ∥∆α∥2
≤ Cµ2 XZ.
(
Let us take care of the right-hand side.
( ( w2 )
)
)
( ( w2 )
)
2
−∆
+
v
∇ϕ
,
∆α
=
−
∆
,
∆α
+ (∆(∂2 ϕv 2 ), ∆α2 )
−w1
−w1
≤ ∥∆w∥∥∆α∥ + C∥∂2 ϕ∥H 2 (0,1) ∥Av∥∥∆α∥
≤ ∥∆w∥∥∆α∥ + CµXZ.
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
19
Now invoking the estimate in Proposition 6.1, we have
1(
∥∆w∥∥∆α∥ ≤ C∥∆α∥ ∥∂t ∇v∥ + (1 + µ2 )Z + X(∥Av∥ + ∥∆α∥
µ
)
+ ∥∇∆α∥)
1
1
1
≤ C ∥∆α∥∥∂t ∇v∥ + C (1 + µ2 )Y Z + C (µ + µ2 )XZ
µ
µ
µ
1
1
≤ C ∥∆α∥∥∂t ∇v∥ + C( + 1 + µ)(X + Y )Z.
µ
µ
The complete estimate is of the form
1 d
1
1
1
∥∆α∥2 + ∥∆α∥2 ≤ C ∥∂t ∇v∥∥∆α∥ + C( + 1 + µ + µ2 )(X + Y )Z.
2 dt
µ
µ
µ
6.6. A priori estimate for gradient of the Laplacian of the stream function
We now want to estimate terms ∥∇∆α∥ such that together with the foregoing
estimate, we can control the H 3 (Ω)-norm of α. Once again, we receive error
terms including higher space derivatives of the artiﬁcial variable w.
For the corresponding estimate we apply ∇∆ to (5.3) and then take the
L2 -inner product with ∇∆α. Then the ﬁrst two terms give us
1 d
(∂t ∇∆α + ∇∆α, ∇∆α) =
∥∇∆α∥2 + ∥∇∆α∥2 .
2 dt
We employ the identity (v · ∇∇∆α, ∇∆α) = 0 and estimate (using
Einstein’s sum convention)
(∇∆(v · ∇α), ∇∆α)
= (∂l (∆v · ∇α + 2∂i v · ∇∂i α + v · ∇∆α), ∂l ∆α)
= (∂l ∆v · ∇α + ∆v · ∇∂l α
+ 2∂l ∂i v · ∇∂i α + 2∂i v · ∇∂l ∂i α + ∂l v · ∇∆α, ∂l ∆α)
≤ (∥∇∆v∥∥∇α∥L∞ (Ω) + 3∥∇2 v∥L4 (Ω) ∥∇2 α∥L4 (Ω) + 2∥∇v∥L∞ (Ω) ∥∇3 α∥)∥∇∆α∥
≤ C∥∇v∥H 2 (Ω) ∥∇α∥H 2 (Ω) ∥∇∆α∥
≤ C∥∇v∥H 2 (Ω) (∥∆α∥2 + ∥∇∆α∥2 )
≤ C∥∇v∥H 2 (Ω) Y 2 .
This is the ﬁrst place, where higher spacial derivatives of v are appearing as
error terms.
Once again using (¯
u · ∇∇∆α, ∇∆α) = 0 we obtain more easily
(∇∆(¯
u · ∇α), ∇∆α)
= (∂l ∆¯
u · ∇α + ∆¯
u · ∇∂l α
+ 2∂l ∂i u
¯ · ∇∂i α + 2∂i u
¯ · ∇∂l ∂i α + ∂l u
¯ · ∇∆α, ∂l ∆α)
≤ C(∥∂23 ∂t ϕ∥∥∇2 α∥ + ∥∂22 ∂t ϕ∥L∞ (Ω) ∥∇2 α∥ + ∥∂2 ∂t ϕ∥L∞ (Ω) ∥∇3 α∥)∥∇∆α∥
≤ C∥∂t ϕ∥H 3 (0,1) ∥∇∆α∥(∥∆α∥ + ∥∇∆α∥)
≤ Cµ2 XZ.
20
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
Linear term. Now let us focus on the right-hand side. Once again, we receive
higher spatial derivatives of w and v.
(
( ( w2 )
)
)
2
− ∇∆
1 + v ∇ϕ , ∇∆α
−w
( 2 )
)
(
w
2
= ∇∆
1 , ∇∆α + (∇∆(∂2 ϕv ), ∇∆α)
−w
≤ ∥∇∆w∥∥∇∆α∥ + C∥∂2 ϕ∥H 3 (0,1) ∥∇v∥H 2 (Ω) ∥∇∆α∥
≤ ∥∇∆w∥∥∇∆α∥ + C∥∇v∥H 2 (Ω) XY.
Here again, we have invoked Proposition 6.1 for the ﬁrst term
∥∇∆w∥∥∇∆α∥
)
1(
≤ C∥∇∆α∥ ∥∂t ∇v∥ + (1 + µ2 )Z + X(∥Av∥ + ∥∆α∥ + ∥∇∆α∥)
µ
1
1
≤ C ∥∇∆α∥∥∂t ∇v∥ + C( + 1 + µ)(X + Y )Z.
µ
µ
We deal with the second term in the same way.
∥∇v∥H 2 (Ω) Y
(
)
≤ C ∥∂t ∇v∥Y + (1 + µ2 )Y Z + XY (∥Av∥ + ∥∆α∥ + ∥∇∆α∥)
C
+ (∥∆α∥ + ∥∇∆α∥)Y
µ
(
)
≤ C (1 + µ)Z + (1 + µ2 )Y Z + (1 + µ)XZ
≤ C(1 + µ + µ2 )(1 + X + Y )Z.
(6.10)
We summarize the above estimates to obtain
1 d
1
1
∥∇∆α∥2 + ∥∇∆α∥2 ≤ C ∥∂t ∇v∥∥∇∆α∥+
2 dt
µ
µ
1
C( + 1 + µ + µ2 )(1 + X + Y )(X + Y )Z.
µ
6.7. A priori estimate for the time derivative of the gradient of the stream
function
The following estimate has the role of absorbing the linear error term that
appeared when estimating the time-derivative of v. In contrast to the two
foregoing estimates on ∆α and ∇∆α we will not produce a stabilizing term
on the left-hand side of the estimate since this would come at the cost of a
linear error term ∂t ∇w. For that term, the higher Stokes-regularity result in
Proposition 6.1 is not applicable. The result would not be easier to estimate
than ∂t ∇α itself. Therefore, the main goal of the following estimate is simply
to absorb the linear remainder from the ∂t v-estimate.
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
21
Application of ∂t ∇ to (5.2) and taking the inner product with ∂t ∇α
yields for the ﬁrst term
1 d
∥∂t ∇α∥2 .
2 dt
In the following, we need to estimate the term ∥∂t ∇α∥ several times.
We remind, that as calculated in (6.9), we have
(∂t2 ∇α, ∂t α) =
∥∂t ∇α∥ ≤ C(1 + X + Y )Y.
Similarly to the foregoing a priori estimates for α, the term
(v · ∇∂t ∇α, ∂t ∇α) vanishes. This way, it is
(∂t ∇(v · ∇α), ∂t ∇α)
= (∂t ∂i v · ∇α + ∂i v · ∇∂t α + ∂t v · ∇∂i α, ∂t ∂i α)
≤ C(∥∂t ∇v∥∥∇α∥L∞ (Ω) + ∥∇v∥L∞ (Ω) ∥∂t ∇α∥ + ∥∂t v∥L4 (Ω) ∥∇2 α∥L4 (Ω) )∥∂t ∇α∥
≤ C(∥∂t ∇v∥∥∇α∥H 2 (Ω) ∥∂t ∇α∥ + ∥∇v∥H 2 (Ω) ∥∂t ∇α∥2 )
≤ C∥∂t ∇v∥(∥∆α∥ + ∥∇∆α∥)(1 + X + Y )Y
+ C∥∇v∥H 2 (Ω) (1 + X + Y )2 Y 2
≤ Cµ(1 + X + Y )Y Z + C(1 + µ + µ2 )(1 + X + Y )3 Y Z.
Note that in the last line, the estimate of ∇v (6.10) was invoked.
The second advection term can be estimated similarly with Y 2 ≤ C(1 +
2
µ )Z
(∂t ∇(¯
u · ∇α), ∂t ∇α)
= (∂t ∂i u
¯ · ∇α + ∂i u
¯ · ∇∂t α + ∂t u
¯ · ∇∂i α, ∂t ∂i α)
≤ C(∥∂2 ∂t2 ϕ∥∥∇2 α∥ + ∥∂2 ∂t ϕ∥L∞ (Ω) ∥∂t ∇α∥ + ∥∂t2 ϕ∥L∞ (Ω) ∥∇2 α∥)∥∂t ∇α∥
≤ C(∥∂t2 ϕ∥H 1 (0,1) ∥∆α∥∥∂t ∇α∥ + ∥∂t ϕ∥H 2 (0,1) ∥∂t ∇α∥2 )
(
≤ C ∥∂t2 ϕ∥H 1 (0,1) ∥∆α∥(1 + X + Y )Y
)
+ ∥∂t ϕ∥H 2 (0,1) (1 + X + Y )2 Y 2
≤ C(1 + µ2 )(1 + X + Y )2 XZ.
Linear term. We split the linear term into
( 2 ) (
)
v
0
−v · ∇η =
+
−v 1
∂2 ϕv 2
and estimate the second part by
(∂t ∇(∂2 ϕv 2 ), ∂t ∇α2 )
= (∂t ∂22 ϕv 2 + ∂22 ϕ∂t v 2 + ∂t ∂2 ϕ∇v 2 + ∂2 ϕ∂t ∇v 2 , ∂t ∇α2 )
≤ C(∥∂t ϕ∥H 2 (0,1) ∥∇v∥ + ∥∂2 ϕ∥H 1 (0,1) ∥∂t ∇v∥)(1 + X + Y )Y
≤ C(1 + X + Y )XY 2 + C(1 + µ)(1 + X + Y )XZ
≤ C(1 + µ + µ2 )(1 + X + Y )XZ.
22
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
The remaining linear term is used to cancel a corresponding term appearing in the estimate of the time-derivative of v. It is
( 2 )
(
)
v
2
1
1
2
∂t ∇
1 , ∂t ∇α = (∂t ∇v , ∂t ∇α ) − (∂t ∇v , ∂t ∇α )
−v
( 2)
(
)
−α
= ∂t ∇
, ∂t ∇v .
1
α
Hence, the combined estimate is
( 2)
(
)
1 d
−α
∥∂t ∇α∥2 − ∂t ∇
, ∂t ∇v
1
α
2 dt
≤ C(1 + µ + µ2 )(1 + X + Y )3 (1 + Y )Z.
6.8. Combining the estimates
In this subsection, we are going to combine all the estimates in Proposition 6.2
and derive the key estimate for the stability argument.
With equation (6.4) we proceed using Young’s inequality to absorb the
term ∥Av∥ in the right-hand side. This yields
1 d
µ
1
∥∇v∥2 + ∥Av∥2 ≤
∥∆α∥2 + C(1 + µ)(X + Y )Z.
2 dt
2
2µ
Applying Young’s inequality for (6.6) and (6.7) in a similar way, we receive
1 d
3
C
1
∥∆α∥2 +
∥∆α∥2 ≤ ∥∂t ∇v∥2 + C( + 1 + µ + µ2 )(X + Y )Z,
2 dt
4µ
µ
µ
(6.11)
1 d
1
C
∥∇∆α∥2 +
∥∇∆α∥2 ≤ ∥∂t ∇v∥2
(6.12)
2 dt
2µ
µ
1
+ C( + 1 + µ + µ2 )(1 + X + Y )(X + Y )Z.
µ
Now we can add these three inequalities above to ﬁnd a combined estimate
) µ
1 d(
1
1
∥∇v∥2 + ∥∆α∥2 + ∥∇∆α∥2 + ∥Av∥2 +
∥∆α∥2 +
∥∇∆α∥2
2 dt
2
4µ
2µ
C
1
≤ ∥∂t ∇v∥2 + C( + 1 + µ + µ2 )(1 + X + Y )(X + Y )Z.
(6.13)
µ
µ
We now turn to (6.5) and (6.8). Adding these inequalities leads to a
cancellation of the remainders of linear terms of the time-derivative estimates
for v and α. The resulting estimate is
1 d
(∥∂t v∥2 + ∥∂t ∇α∥2 ) + µ∥∂t ∇v∥2
2 dt
1
≤ C( + 1 + µ + µ2 )(1 + X + Y )3 (X + Y )Z. (6.14)
µ
We aim to absorb the remaining quadratic term C∥∂t ∇v∥2 /µ in the
right-hand side of (6.13) with the one on the left-hand side of (6.14). For
that it is necessary to multiply (6.13) by a suﬃciently small constant δ > 0.
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
23
The sum of (6.14) with δ×(6.13) gives
)
1 d(
∥∂t v∥2 + δ∥∇v∥2 + δ∥∆α∥2 + δ∥∇∆α∥2 + ∥∂t ∇α∥2
2 dt
C
µ
1
1
+ (µ − δ )∥∂t ∇v∥2 + δ ∥Av∥2 + δ ∥∆α∥2 + δ ∥∇∆α∥2
µ
2
4µ
2µ
1
≤ C(1 + δ)( + 1 + µ + µ2 )(1 + X + Y )3 (X + Y )Z.
µ
We choose 0 < δ < 1 such that µ − δ C
µ ≥
µ
2
and hence receive
)
d(
∥∂t v∥2 + δ∥∇v∥2 + δ∥∆α∥2 + δ∥∇∆α∥2 + ∥∂t ∇α∥2
dt
δ
δ
∥∆α∥2 + ∥∇∆α∥2
+ µ∥∂t ∇v∥2 + δµ∥Av∥2 +
2µ
µ
1
≤ C(1 + δ)( + 1 + µ + µ2 )(1 + X + Y )3 (X + Y )Z.
µ
Finally we modify the left-hand side so that Z appears, and put Cµ,δ =
C(1 + δ)(1/µ + 1 + µ + µ2 ) to simplify the estimate. We obtain
)
d(
∥∂t v∥2 + δ∥∇v∥2 + δ∥∆α∥2 + δ∥∇∆α∥2 + ∥∂t ∇α∥2 + 2µδZ
dt
≤ Cµ,δ (1 + X + Y )3 (X + Y )Z.
(6.15)
6.9. Stability argument
In this subsection, we give a proof of the estimate in our main result Theorem 3.2
with the aid of estimate (6.15). As we mentioned at the beginning of Section 6,
the actual existence proof is by the Galerkin method which we skipped in this
paper. We also note that the solution is unique, which is similarly proved by
the estimate in this paper.
Proof. It is obvious that Proposition 6.2 applies under the assumption of
Theorem 3.2. Thus we can employ (6.15). Fix some a > 0 such that Cµ,δ (1 +
a)3 a < µδ. Note that as long as Y (t), X(t) ≤ a/2 hold for such a, we have
)
d(
∥∂t v∥2 + δ∥∇v∥2 + δ∥∆α∥2 + δ∥∇∆α∥2 + ∥∂t ∇α∥2 (t) + µδZ(t) ≤ 0.
dt
Thus we can conclude corresponding norm of the solution is decreasing except
the case where all X, Y, Z equal zero which is a trivial case.
Therefore, the proof is reduced to show that Y (t), X(t) ≤ a/2 hold for
all t ≥ 0 if we choose initial data and ﬂow data suﬃciently small.
We now invoke the a priori estimate for the Poiseuille ﬂow Proposition 3.1
and obtain
(
)
X(t) ≤ C ∥ψ0 ∥H 3 (0,1) + ∥h∥H 1 (0,∞)
for some C = Cµ . Therefore if we choose initial data ψ0 and pressure data h
suﬃciently small, we have X(t) < a/2 for all t ≥ 0. In what follows, we may
assume X(t) < a/2 for all t ≥ 0.
24
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
Now let us focus on Y (t). For this, we investigate the initial data Y (0)
ﬁrst.
Y (0) = ∥∂t v(0)∥ + ∥∇v0 ∥ + ∥∆α0 ∥ + ∥∇∆α0 ∥.
Let us recall the momentum equation (4.5) and ﬁnd
∂t v(0) = µ∆v0 − v0 · ∇v0 − v0 · ∇¯
u(0) − u
¯(0) · ∇v0 − ∇p
( 2)
−α0
− ∆α0k ∇α0k + ∂22 ϕ(0)∇α02 + ∇ϕ(0)∆α02 .
+∆
α01
Applying the Helmholtz projection to remove the pressure term, we obtain
∥∂t v(0)∥ ≤ C(µ∥∆v0 ∥ + ∥v0 · ∇v0 ∥ + ∥v0 · ∇¯
u∥+
∥¯
u0 · ∇v0 ∥ + ∥∆α0k ∇α0k ∥ + ∥∂22 ϕ(0)∇α02 ∥)
≤ C(µ + ∥∇v0 ∥ + ∥ψ0 ∥H 1 (0,1) )∥Av0 ∥
+ C(1 + ∥∆α0 ∥ + ∥∇∆α0 ∥)∥∆α0 ∥.
Similarly, we recall the transport equation of α (5.2) for ∂t ∇α(0).
∥∂t ∇α(0)∥ ≤ ∥∇(v0 · ∇α0 )∥ + ∥∇(¯
u(0) · ∇α0 )∥ + ∥∇(v0 · ∇η0 )∥
≤ ∥∇2 α0 ∥∥v0 ∥L∞ (Ω) + ∥∇v0 ∥L4 (Ω) ∥∇α0 ∥L4 (Ω) + ∥∇2 α0 ∥∥¯
u(0)∥L∞ (Ω)
+ ∥∇¯
u(0)∥L4 (Ω) ∥∇α0 ∥L4 (Ω) + ∥∇v0 ∥∥∇η0 ∥L∞ (Ω)
≤ C(∥∇v0 ∥ + (∥Av0 ∥ + ∥ψ0 ∥H 1 (0,1) )∥∆α0 ∥).
Therefore we choose v0 , α0 and retake ψ0 if necessary so small that
√
δa
Y (0), ∥∂t ∇α(0)∥ <
8
holds. Then we deﬁne a time T∗ = inf{t > 0; Y (t) > a/2}. To prove T∗ = ∞,
suppose T∗ < ∞ and seek a contradiction. Y (t) ≤ a/2 for all t ∈ [0, T∗ ] holds,
since Y is continuous. Thus,
)
d(
∥∂t v∥2 + δ∥∇v∥2 + δ∥∆α∥2 + δ∥∇∆α∥2 + ∥∂t ∇α∥2 (t) + µδZ(t) ≤ 0
dt
holds in the interval [0, T∗ ]. Integrating over [0, T∗ ], we receive
∥∂t v(T∗ )∥2 + δ∥∇v(T∗ )∥2 + δ∥∆α(T∗ )∥2 + δ∥∇∆α(T∗ )∥2 + ∥∂t ∇α(T∗ )∥2
≤ ∥∂t v(0)∥2 + δ∥∇v0 ∥2 + δ∥∆α0 ∥2 + δ∥∇∆α0 ∥2 + ∥∂t ∇α0 ∥2 .
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
25
Now evaluating Y (T∗ ) with the above estimate yields,
(
)2
(Y (T∗ ))2 = ∥∂t v(T∗ )∥ + ∥∇v(T∗ )∥ + ∥∆α(T∗ )∥ + ∥∇∆α(T∗ )∥
4
≤ (∥∂t v(T∗ )∥2 + δ∥∇v(T∗ )∥2 + δ∥∆α(T∗ )∥2 + δ∥∇∆α(T∗ )∥2 )
δ
)
4(
≤ ∥∂t v(0)∥2 + δ∥∇v0 ∥2 + δ∥∆α0 ∥2 + δ∥∇∆α0 ∥2 + ∥∂t ∇α0 ∥2
δ
)
4(
< (Y (0))2 + ∥∂t ∇α(0)∥2
δ √
4 ( δY0 )2
≤
×2
δ
8
2
Y
≤ 0.
8
This leads to a contradiction to the deﬁnition of T∗ and therefore T∗ = ∞. 7. Basic properties of the Stokes operator
In this section, we recall some basic estimates for the Stokes and the Laplacian
operator, which are frequently used in this paper for the reader’s convenience.
Assume that Ω is either the layer R×(0, 1) or one of the approximations
(−M, M ) × (0, 1) in this section.
7.1. Stokes operator
Deﬁnition 7.1. Let P be the Helmholtz projection on Ω. The Stokes operator A is deﬁned as a closed linear operator in L2σ (Ω) with domain D(A) =
H 2 (Ω) ∩ H01 (Ω) ∩ L2σ (Ω) such that Au = −P ∆.
Note that 0 ∈ ρ(A) and D(A) = H 2 (Ω) ∩ H01 (Ω) ∩ L2σ (Ω) in either case
that Ω is layer domain or bounded domain [12, III.2]. Therefore we have the
estimate
∥v∥H 2 (Ω) ≤ C∥Av∥
(7.1)
for v ∈ H (Ω) with v = 0 on the boundary and div v = 0.
2
7.2. Stokes system
We use the regularity of Stokes system to obtain higher spatial estimates.
Lemma 7.2. Let Ω = R × (0, 1) or Ω = (−M, M ) × (0, 1) for M > 1. Let
f ∈ H 1 (Ω), g ∈ H 2 (Ω) and (u, p) solve

 −µ∆u + ∇p = f, in Ω,
div u = g, in Ω,
(7.2)

u = 0, on ∂Ω.
Then there holds
µ∥u∥H 3 (Ω) + ∥∇p∥H 1 (Ω) ≤ C(µ∥g∥H 2 (Ω) + ∥f ∥H 1 (Ω) ).
The constant C is independent of the assumption to Ω.
26
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
Proof. See [12, Theorem1.5.3].
7.3. Elliptic regularity and implications
Lemma 7.3. Under the same assumptions for Ω of Lemma 7.2, there exists
a constant C such that
∥f ∥H 2 (Ω) ≤ C∥∆f ∥
(7.3)
holds for all f ∈ H 2 (Ω) which vanish on both upper and lower boundaries.
Proof. It follows by the elliptic regularity. For more details, see [1, Cor6.31].
Moreover, by standard elliptic regularity arguments, we can estimate
the H 3 (Ω)-norm by only terms of the Laplacian as stated below.
Lemma 7.4. Assume the same hypotheses of Lemma 7.2. Then there exists
constant C such that
C −1 ∥f ∥H 3 (Ω) ≤ ∥∆f ∥ + ∥∇∆f ∥ ≤ C∥f ∥H 3 (Ω)
(7.4)
holds for all f ∈ H 3 (Ω) which vanishes on the upper and lower boundary.
Proof. See [2, Section 6.3].
We will use some embeddings to deal with products of functions.
Lemma 7.5. Assume the same hypotheses of Lemma 7.2. The following embeddings hold:
H 2 (Ω) ,→ L∞ (Ω), H 1 (Ω) ,→ L4 (Ω), and
H 1 (0, 1) ,→ L∞ (0, 1).
(7.5)
Proof. see [2, Section 5.6]
The following lemma are used to control the order of estimates.
Lemma 7.6. Assume the same hypotheses of Lemma 7.2. Let g ∈ H 1 (Ω)
vanish on the upper and lower boundaries and f ∈ L2 (0, 1), one can estimate
using the embedding above,
∥f g∥L2 (Ω) ≤ C∥f ∥L2 (0,1) ∥∇g∥L2 (Ω) .
(7.6)
Proof. We invoke the embeddings Lemma 7.5 to conclude
(∫ ∫
) 12
∥f g∥L2 (Ω) =
|f (y)|2 |g(x, y)|2 dy dx
≤
(∫
R
R
(0,1)
∫
) 21
|f (y)|2 dy dx
∥g(x, ·)∥2L∞ (0,1)
≤ ∥f ∥L2 (0,1)
(0,1)
(∫
R
) 12
C 2 ∥g(x, ·)∥2H 1 (0,1) dx
(7.7)
≤ C∥f ∥L2 (0,1) ∥∇g∥L2 (Ω) .
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
27
8. Viscous wave equation
This section is dedicated to a proof of Proposition 3.1. We split the initialboundary value problem (3.2) into two parts for sharper estimation. The ﬁrst
part is the homogeneous case, i.e.,
 2
2
2
in (0, 1),

 ∂t ϕ1 − ∂x ϕ1 = µ∂t ∂x ϕ1 ,
ϕ1 (t, 0) = ϕ1 (t, 1) = 0,
for t ∈ (0, T ),
(8.1)


ϕ1 (0) = 0, ∂t ϕ1 (0) = ψ0 , in (0, 1).
The second one is the inhomogeneous case, i.e.,
 2
2
2

 ∂t ϕ2 − ∂x ϕ2 = µ∂t ∂x ϕ2 + h, in (0, 1),
ϕ2 (t, 0) = ϕ2 (t, 1) = 0,
for t ∈ (0, T ),


ϕ2 (0) = 0, ∂t ϕ2 (0) = 0,
in (0, 1).
(8.2)
Again, here h is some given function which depends only on t. We shall show
a priori estimates for each of them. Note that we only need estimates for
∂x4 ϕi , ∂t ∂x3 ϕi , ∂t2 ∂x ϕi
i = 1, 2
in both cases by the Poincar´e inequality.
8.1. Homogeneous case
In this case, we can use the separation of variables method and derive the
solution explicitly. For the readability, let us denote the solution ϕ1 of (8.1)
by ϕ in this subsection.
Separation of variables. To begin with, we consider the simple ansatz with
the form ϕ(t, x) = T (t)X(x) with the boundary condition X(0) = X(1) = 0.
Then inserting this ansatz in the system (8.1) yields,
X(x)T ′′ (t) − X ′′ (x)T (t) = µX ′′ (x)T ′ (t),
x ∈ (0, 1),
t > 0.
This leads to the following equation
T ′′ (t)
X ′′ (x)
=
=λ
+ T (t)
X(x)
µT ′ (t)
for some λ ∈ R unless X(x) and µT ′ (t) + T (t) vanish.
Let us focus on X(x) ﬁrst. The equation X ′′ (x) = λX(t) with initial
condition X(0) = X(1) = 0 gives us a solution Xn (x) = an sin(nπx) and here
λ = −(nπ)2 . We simply regard an = 1 and take care of those coeﬃcients in
T (t) side.
Now let us turn to Tn′′ (t) + µ(nπ)2 Tn′ (t) + (nπ)2 Tn (t) = 0. Solving the
characteristic equation y 2 + µ(nπ)2 y + (nπ)2 = 0 yields
√
µ
µ2
±
2
yn = − (nπ) ± nπ
(nπ)2 − 1.
2
4
28
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
We set Bn = nπ
√
µ2 (nπ)2 − 1 for simplicity. Then yn± is written as
4
 µ
2
2

− 2 (nπ) ± iBn , n < µπ ,
2
yn± = − µ2 (nπ)2 ,
n = µπ
,

 µ
2
2
− 2 (nπ) ± Bn , n > µπ .
2
Now we assume N := µπ
∈ N in the following. Otherwise, we just ignore the
±
terms with respect to yN = −µ(N π)2 /2. Hence the solution Tn (t) is written
as
 µ
2
− (nπ)2 t
(an sin(Bn t) + bn cos(Bn t)), n < µπ
,

e µ2
2
2
−
(nπ)
t
Tn (t) = e 2
(tan + bn ),
n = µπ ,

 − µ2 (nπ)2 t an Bn t bn −Bn t
2
e
(2e
+ 2e
),
n > µπ
for some coeﬃcients an , bn ∈ R.
Determining the coeﬃcients. We now consider the solution ansatz for (8.1)
of the form
ϕ(t, x) =
N
−1
∑
µ
sin(nπx)e− 2 (nπ) t (an sin(Bn t) + bn cos(Bn t))
2
n=1
µ
+ sin(N πx)e− 2 (N π) t (taN + bN )
∞
(a
)
∑
µ
2
bn
n Bn t
+
sin(nπx)e− 2 (nπ) t
e
+ e−Bn t .
2
2
2
n=N +1
We determine an , bn so that ϕ(0, x) = 0 and ϕt (0, x) = ψ0 are satisﬁed.
Decay in time. At this point, we would like to remark, that each of the
functions Tn decay exponentially in time. This observation is directly clear
2
2
in the cases n ≤ µπ
and in the latter case, we note Bn < µ2 (nπ)2 for n > µπ
.
In fact, it is
√
µ
µ2
µ
2
2
− (nπ) + Bn = − (nπ) + nπ
(nπ)2 − 1
2
2 (
4
)
√
√
2
µ2
µ
= −nπ
(nπ)2 −
(nπ)2 − 1
4
4
π
√
= −√
(8.3)
2
µ
µ2 2
1
2
4 π +
4 π − n2
π
≤− √
2
2 µ4 π 2
1
=− .
µ
Hence, the solution decays exponentially to zero at inﬁnity at least as fast as
1
2
e− µ t for n > µπ
.
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
29
We superpose these solutions for n ∈ N receiving the solution ansatz for
ϕ of the form
ϕ(t, x) =
N
−1
∑
µ
sin(nπx)e− 2 (nπ) t (an sin(Bn t) + bn cos(Bn t))
2
n=1
µ
+ sin(N πx)e− 2 (N π) t (taN + bN )
∞
(a
)
∑
µ
2
bn
n Bn t
+
sin(nπx)e− 2 (nπ) t
e
+ e−Bn t .
2
2
2
n=N +1
Determining the coeﬃcients. Our next step is to exploit the initial conditions
on ϕ to determine the proper values for the coeﬃcients an and bn . It is
0 = ϕ(0, x)
=
N
−1
∑
∞
∑
sin(nπx)bn + sin(N πx)bN +
n=1
sin(nπx)
n=N +1
(a + b )
n
n
2
( a − b ))
n
n
.
=
sin(nπx) bn + χ{N +1,... } (n)
2
n=1
(
∞
∑
It is then easy to conclude that for n = 1, . . . , N we have bn = 0 and for
n = N + 1, . . . it is bn = −an and therefore
ϕ(t, x) =
N
−1
∑
µ
sin(nπx)e− 2 (nπ) t (an sin(Bn t))
2
n=1
µ
+ sin(N πx)e− 2 (N π) t (taN )
+
∞
∑
2
µ
sin(nπx)e− 2 (nπ)
n=N +1
2
t an
2
(
)
eBn t − e−Bn t .
The other initial condition ϕt (0, x) = ψ0 (x) enables us to uniquely determine
the remaining coeﬃcients via Fourier-series. We calculate the derivative in
time
ϕt (t, x) =
N
−1
∑
n=1
sin(nπx)
((
−
) µ
2
µ
(nπ)2 e− 2 (nπ) t (an sin(Bn t))
2
)
µ
2
+ e− 2 (nπ) t (an Bn cos(Bn t))
(( µ
)
) µ
µ
2
2
+ sin(N πx) − (N π)2 e− 2 (N π) t (taN ) + e− 2 (N π) t aN
2
∞
(( µ
∑
) µ
)
2 an (
+
eBn t − e−Bn t
sin(nπx) − (nπ)2 e− 2 (nπ) t
2
2
n=N +1
))
µ
2 an Bn (
eBn t + e−Bn t .
+ e− 2 (nπ) t
2
(8.4)
30
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
At t = 0 then holds with BN = 0
ψ0 (x) = ϕt (0, x) =
N
−1
∑
sin(nπx)an Bn + sin(N πx)aN +
n=1
=
∞
∑
∞
∑
sin(nπx)an Bn
n=N +1
sin(nπx)(an Bn + χ{N } (n)aN ).
n=1
Now we will assume that ψ0 ∈ H 3 (0, 1) with ψ0 (0) = ψ0 (1) = 0 and
therefore we can write it as a Fourier-series
∞
ψ0 (x) =
β0 ∑
+
αk sin(2πkx) + βk cos(2πkx)
2
k=1
=
∞
∑
αk sin(2πkx),
k=1
where the coeﬃcients βk vanish due to the boundary conditions. Moreover,
ψ ∈ H 3 (0, 1) implies with Plancherel’s identity, that the series ((2kπ)3 αk )k∈N
is in l2 . Comparing this representation with condition (8.4), the uniqueness
of the Fourier-series implies that a2k+1 = 0 and
{
∫
−1
−1 1
B2k
αk = B2k
ψ0 (x) sin(2πkx) dx, 2k ̸= N,
0
∫1
a2k =
αk = 0 ψ0 (x) sin(2πkx) dx,
2k = N.
Altogether, for N =
2
µπ
∈ N even, the solution ϕ of system (8.1) is given
by
−1
∑
N
2
ϕ(t, x) =
−1
αk B2k
sin(2kπx)e− 2 (2kπ) t sin(B2k t)
µ
2
k=1
µ
+ α N sin(N πx)e− 2 (N π) t (taN )
2
+
∞
∑
k= N
2 +1
2
(
)
µ
2
αk −1
B2k sin(2kπx)e− 2 (2kπ) t eB2k t − e−B2k t ,
2
∫1
where αk = 0 ψ0 (x) sin(2πkx) dx. In the case, where N is odd, the second
term vanishes, more precisely
N −1
2
ϕ(t, x) =
∑
−1
αk B2k
sin(2kπx)e− 2 (2kπ) t sin(B2k t)
µ
k=1
+
2
∞
(
)
∑
µ
2
αk −1
B2k sin(2kπx)e− 2 (2kπ) t eB2k t − e−B2k t .
2
N +1
k=
2
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
Continuity of the solution. Looking at the coeﬃcients, we see that
2
31
µ
√
(kπ)2
8
≤
B2k ≤
for n such that 1 ≤
(πn) . Therefore, the term B2k may
absorb (2πk) coming from a time-derivative or two derivatives in space.
With ψ0 ∈ H 3 (0, 1) we see that ϕtxxx (t) is continuous in time (up to zero)
as a function taking values in L2 (0, 1) and we can estimate with Plancherel’s
identity
µ
2
2 (kπ)
2
µ
8
2
∥ϕtxxx (t)∥2
N −1
2
≤
∑ )2
µ
2 (
µ
−1
(2kπ)3 e− 2 (2kπ) t B2k cos(B2k t) − (2kπ)2 sin(B2k t) αk B2k
2
k=1
∞
α
∑
µ
2
k −1
+
B2k (2kπ)3 e− 2 (2kπ) t
2
N +1
k=
2
)2
(
µ
× B2k (eB2k t + e−B2k t ) − (2kπ)2 (eB2k t − e−B2k t ) 2
N −1
2
∑
)2
µ
2 (
µ (2kπ)2
≤
sin(B2k t) αk (2kπ)3 e− 2 (2kπ) t cos(B2k t) −
2 B2k
k=1
∞
α
∑
µ
2
k
+
(2kπ)3 e− 2 (2kπ) t
2
N +1
k=
2
(
)2
µ (2kπ)2 B2k t
× (eB2k t + e−B2k t ) −
(e
− e−B2k t ) 2 B2k
∞ ∞ 2
2
µ
2 ∑
1 ∑
≤ Ce− 2 (2π) t
αk (2kπ)3 αk (2kπ)3 + Ce− µ t
k=1
≤ Ce
1
)t
− min(2µπ 2 , µ
k=1
∥ψ0 ∥2H 3 (0,1) .
2
Here we used the exponential decay in time for high frequency part n > µπ
calculated in (8.3).
With the same arguments we ﬁnd similar estimates for the case where
N is even as well as ϕxxxx and ϕttx . We have now proved the following
Proposition:
Proposition 8.1. For ψ0 ∈ H 3 (0, 1) ∩ H01 (0, 1) there exists a unique solution
ϕ ∈ C 2 ([0, ∞); H 1 (0, 1)) ∩ C 1 ([0, ∞), H 3 (0, 1)) ∩ C([0, ∞); H 4 (0, 1)) of (8.1).
The solution satisfies
∥ϕt (t)∥H 3 (0,1) + ∥ϕx (t)∥H 3 (0,1) + ∥ϕtt (t)∥H 1 (0,1) ≤ Ce− min(2µπ
2 1
, µ )t
∥ψ0 ∥H 3 (0,1) .
for t ≥ 0. The constant C > 0 is independent of t and µ.
8.2. Inhomogeneous case
Now let us consider inhomogeneous case (8.2) and its solution ϕ2 . Note that
for the readability, we denote ϕ2 by ϕ again.
32
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
Instead of solving the equation explicitly, we take advantage of the structure of the equations. First, we shall show the following point wise estimates
in time.
Proposition 8.2 (A priori estimate in time). Let k ∈ N, T > 0 and ϕ be
the solution in C k+1 ([0, T ); H 3 (0, 1))of system (8.2) with some given h ∈
H2k (0, T ). Then for each t ∈ [0, T ] the following estimates holds.
∫ t
∥∂tk ϕt (t)∥2L2 (0,1) + ∥∂tk ϕx (t)∥2L2 (0,1) + µ
∥∂tk ϕtx (s)∥2L2 (0,1) ds
0
1
∥∂ k h∥L2 (0,T )
3µ t
≤ ∥∂tk ϕt (0)∥L2 (0,1) + ∥∂tk ϕx (0)∥L2 (0,1) +
and
∫
t
∥∂tk ϕtx (t)∥2L2 (0,1) + ∥∂tk ϕxx (t)∥2L2 (0,1) + µ
∥∂tk ϕtxx (s)∥2L2 (0,1) ds
0
≤ ∥∂tk ϕtx (0)∥L2 (0,1) + ∥∂tk ϕxx (0)∥L2 (0,1) +
1 k
∥∂ h∥L2 (0,T ) .
µ t
Proof. Let us focus on the ﬁrst estimate. We take derivatives ∂tk of (8.2) and
take inner products with ∂tk+1 ϕ(s) in L2 (0, 1). Then we have,
∫ 1
∫ 1
∂tk+1 ϕt (s, x)∂tk ϕt (s, x) dx −
∂tk ϕxx (s, x)∂tk+1 ϕ(s, x) dx
0
0
∫
∫
1
∂tk+1 ϕxx (s, x)∂tk+1 ϕ(s, x) dx + ∂tk h(s)
=µ
0
1
∂tk+1 ϕ(s, x) dx.
0
We can employ integration by parts in the second, third and fourth term
since ∂tk ϕ(s, 0) = ∂tk ϕ(s, 1) = 0, and that gives us
)
1 d( k
∥∂t ϕt (s)∥2L2 (0,1) + ∥∂tk ϕx (s)∥2L2 (0,1)
2 dt
∫ 1
k
2
k
= −µ∥∂t ϕtx (s)∥L2 (0,1) − ∂t h(s)
x ∂tk+1 ϕx (s, x) dx.
0
We estimate the fourth term by means of the Cauchy-Schwarz inequality and
Young’s inequality:
∫ 1
(∫ 1
) 12
∂tk h(s)
x ∂tk+1 ϕx (s, x) dx ≤ |∂tk h(s)|
∥∂tk ϕtx (s)∥L2 (0,1)
x2 dx
0
0
1 k
µ
≤
|∂ h(s)|2 + ∥∂tk ϕtx (s)∥2L2 (0,1) .
6µ t
2
Absorbing the last term and integrating from 0 to some t > 0 leads to the
intended result:
∫ t
∥∂tk ϕt (t)∥2L2 (0,1) + ∥∂tk ϕx (t)∥2L2 (0,1) + µ
∥∂tk ϕtx (s)∥2L2 (0,1) ds
0
≤
∥∂tk ϕt (0)∥2L2 (0,1)
+
∥∂tk ϕx (0)∥2L2 (0,t)
+
1
∥∂ k h∥2 2
.
3µ t L (0,t)
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
33
One can obtain the second estimate by taking inner products with
∂tk+1 ∂x2 ϕ instead of ∂tk+1 ϕ in the above calculation.
We somehow need to introduce a spatial a priori estimate. However,
the same trick we used in the proposition above doesn’t work since there
is no boundary condition given for higher space-derivatives of ϕ. Therefore
we take advantage of the structure of the equation and eliminate temporal
derivatives. Let us introduce new variable z = ϕ+ µϕt for that purpose. Then
z satisﬁes the following.


 ∂t z = µ∂x2 z + z − ϕ + µh, in (0, 1),


µ µ
(8.5)
z(t,
0)
=
z(t,
1)
= 0,
for t ∈ (0, T ),



 z(0) = 0,
in (0, 1).
Additionally, by solving z = ϕ + µϕt for ϕ, we have
∫
1 t − t−s
ϕ(t) =
e µ z(s) ds.
µ 0
(8.6)
Estimate of ∂x4 ϕ. Taking derivatives ∂x2 in (8.5), we have
∂x4 z =
1
∂2z
∂2ϕ
(∂t ∂x2 z − x − x ).
µ
µ
µ
Inserting this into (8.6) will produce
∫
(
∂ 2 z(s) ∂x2 ϕ(s) )
1 t − t−s
ds
∂x4 ϕ(t) =
−
e µ ∂t ∂x2 z(s) − x
µ 0
µ
µ
∫ t
t−s (
t
1
2
∂ 2 ϕ(s) )
= 2
ds + ∂x2 z(t) − e− µ ∂x2 z(0).
e− µ − ∂x2 z(s) − x
µ 0
µ
µ
Note that we used integration by parts for ∂t ∂x2 z to eliminate the derivative
in time. Finally we take norm in L2 (0, 1) in both sides.
(
1)
C
∥∂x4 ϕ(t)∥L2 (0,1) ≤ C 1 + 2 sup ∥∂x2 z(s)∥L2 (0,1) + 2 sup ∥∂x2 ϕ(s)∥L2 (0,1)
µ 0≤s≤t
µ 0≤s≤t
(
)
1 )(
≤C 1+ 2
sup ∥∂t ∂x2 ϕ(s)∥L2 (0,1) + sup ∥∂x2 ϕ(s)∥L2 (0,1) .
µ
0≤s≤t
0≤s≤t
Finally, we invoke the a priori estimate in time above and obtain the intended
result.
With the same arguments we ﬁnd similar estimates for ∂t ∂x3 ϕ, ∂t2 ∂x ϕ.
Proposition 8.3. Let T > 0. For h ∈ H 1 (0, T ), there exists the unique solution
ϕ ∈ C 2 ([0, ∞); H 1 (0, 1)) ∩ C 1 ([0, ∞), H 3 (0, 1)) ∩ C([0, ∞); H 4 (0, 1)) of the
34
Masakazu Endo, Yoshikazu Giga, Dario G¨otz and Chun Liu
system (8.2). The solution satisfies
∥∂t ϕ(t)∥H 3 (0,1) + ∥∂x ϕ(t)∥H 3 (0,1) + ∥∂t2 ϕ(t)∥H 1 (0,1)
≤C
4
∑
µ−k ∥h∥H 1 (0,t)
k=1
for 0 ≤ t ≤ T . The constant C is independent of µ and t.
References
[1] R. Adams and J. Fournier, Sobolev Spaces, Pure and Applied Mathematics,
Elsevier Science, 2003.
[2] L. Evans, Partial Diﬀerential Equations, Graduate studies in mathematics,
American Mathematical Society, 2010.
[3] Y. Giga, J. Sauer and K. Schade, Strong stability of 2D viscoelastic Poiseuilletype ﬂows, preprint (submitted), 2014.
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Masakazu Endo
University of Tokyo, Department of Mathematics, Graduate School of Mathematical
Sciences, 3-8-1 Komaba Meguro-ku Tokyo 153-8941, Japan
e-mail: [email protected]
Stability of Poiseuille-type ﬂow for a viscoelastic ﬂuid
35
Yoshikazu Giga
University of Tokyo, Department of Mathematics, Graduate School of Mathematical
Sciences, 3-8-1 Komaba Meguro-ku Tokyo 153-8941, Japan
e-mail: [email protected]
Dario G¨
otz
Technische Universit¨
at Darmstadt, Fachbereich Mathematik, Schlossgartenstr. 7,
D-64298 Darmstadt, Germany
e-mail: [email protected]
Chun Liu
Department of Mathematics, Pennsylvania State University, State College, PA
16802, USA
e-mail: [email protected]