Instructions for use A uthor(s ) E ndo,Masakazu; GIGA ,Y OS HIK A Z U; Gotz,D ario; L iu,C hun
C itation Hokkaido University Preprint S eries in Mathematics, 1068: 2-35
Is s ue D ate 2015-3-20
D O I 10.14943/84212
D oc UR L http://hdl.handle.net/2115/69872
T ype bulletin (article)
F ile Information pre1068.pdf
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 con-sidered. An L2-stability of the Poiseuille-type flow is established pro-vided 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).
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 ofψ in time solves the viscous wave equation. We are interested in its stability as viscoelastic fluids. In fact, we prove that if both the Poiseuille-type 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 formu-lation 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 gener-ating 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 ofL2 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 me-chanics. 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 inR2is 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].
that correspond toProposition 3.1in 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 inR2 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⊗R2is 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 tensorF in the Lagrangian coordi-nates is defined byFij =∂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 ourF is used.
We consider the following two dimensional viscoelastic fluid system of the Oldroyd model with Dirichlet boundary condition of the form.
∂tF+u· ∇F =F∇u in (0, T)×Ω, divu= 0 in (0, T)×Ω, ∂tu−µ∆u+u· ∇u+∇π= divFTF in (0, T)×Ω,
F|t=0=F0 on Ω,
u|t=0=u0 on Ω,
u= 0 on (0, T)×∂Ω
(2.1)
with the assumption detF|t=0 = 1 and divF|t=0 = 0. In this paper, we use the following notation.
· (divG)i=∑nj=1∂jGij.
Stream function formulation. One can show that divFis subject to advection with the flow, i.e.
∂t(divF) +u· ∇(divF) = 0.
Therefore divF0= 0 implies divF = 0 for all later times. Under this assump-tion, one can find anR2-valued stream function ζ0 such thatF0=∇⊥ζ0 as in [9]. Moreover, if one letsζ be the solution of the transport equation for a divergence free functionuof the form
{
∂tζ+u· ∇ζ= 0,
ζ|t=0=ζ0 (2.2)
then one can find that for
F =∇⊥ζ=
(
−∂2ζ1 ∂1ζ1
−∂2ζ2 ∂1ζ2
)
,
the equationFt+u· ∇F =F∇uis fulfilled. It is much easier to consider the functionζ instead of F. In order to rewrite system (2.1) with respect to ζ, one can calculate
divFTF= 1 2∇|∇ζ|
2
−∆ζ1∇ζ1−∆ζ2∇ζ2.
Note that the first term is a gradient that can be absorbed in pressure term. Thus we introduce a new variable ˜π=π−12|∇ζ|2and denote that byπagain. We end up with the following new system for two dimensions with Dirichlet boundary condition.
∂tζ+u· ∇ζ= 0 in (0, T)×Ω, divu= 0 in (0, T)×Ω,
∂tu−µ∆u+u· ∇u+∇π=− 2
∑
k=1
∆ζk∇ζk in (0, T)×Ω,
ζ|t=0=ζ0 on Ω,
u|t=0=u0 on Ω,
u= 0 on (0, T)×∂Ω.
(2.3)
The corresponding assumption to the incompressibility condition detF|t=0= 1 is
∂1ζ01∂2ζ02−∂1ζ02∂2ζ01= 1. (2.4)
Incompressibility. Considering fluids with a constant density, the incom-pressibility condition takes the form divu = 0. It turns out that in terms of the deformation tensor, this means detF = 1 if detF|t=0 = 1 holds. Moreover, one can find that
∂t(detF) +u· ∇(detF) = 0.
Therefore if (2.4) holds, we have∂1ζ1∂2ζ2−∂1ζ2∂2ζ1= 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 ¯uto (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 ¯utakes the form
¯
u(t, x) =
(
ψ(t, x2) 0
)
with homogeneous Dirichlet boundary conditions. Then the divergence con-dition in (2.1) is trivially fulfilled.
In order to adequately determine the corresponding deformation tensor ¯
F or equivalently the corresponding stream functionη, 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
dtx1(t, X) = ¯u
1(t, x
1(t, X), x2(t, X)) =ψ(t, x2(t, X)), x1(0) =X1,
d
dtx2(t, X) = ¯u
2(t, x
1(t, X), x2(t, X)) = 0, x2(0) =X2, which can easily be solved by
x1(t, X) =X1+
∫ t
0
ψ(s, x2(s, X)) ds=X1+
∫ t
0
ψ(s, X2) ds,
x2(t, X) =X2,
as long asψis sufficiently regular. Let us abbreviate
ϕ(t, x2) =
∫ t
0
ψ(s, x2) ds.
Then, we can calculate the deformation tensor and the resulting elastic force
¯
F =
(
1 0
∂2ϕ 1
)
, F¯TF¯=
(
1 + (∂2ϕ)2 ∂2ϕ
∂2ϕ 1
)
and div ¯FTF¯ =
(
∂2 2ϕ 0
)
.
The stream functionη corresponding to ¯F may be chosen as
η(t, x) =
(
−x2
x1−ϕ(t, x2)
)
(3.1)
solving the system
{
∂tη+ ¯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.
∂tu¯−µ∆¯u+ ¯u· ∇u¯+∇π¯ = div ¯FTF ,¯ in (0, T)×Ω, yields the equivalent formulation
∂tψ+∂1π¯=µ∂22ψ+∂22ϕ,
∂2π¯= 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). Sinceψandϕdepend only on t and x2, the first equation implies that ∂1π¯ is a function of time only, i.e.∂1π¯(t, x1) =−h(t) for some functionh. Inserting this into the system yields
∂tψ−∂22ϕ=µ∂22ψ+h, in (0, T)×(0,1).
Finally, by the definition ofϕit isψ(t, x2) =∂tϕ(t, x2) and moreover, the ho-mogeneous Dirichlet boundary conditions for ¯ucarry over toϕ, i.e.ϕ(t,0) =
ϕ(t,1) = 0. At initial time we have ϕ(0, x2) = ∫ 0
0 ψ(s, x2) ds = 0 and
∂tϕ(0, x2) =ψ(0, x2) =ψ0(x2) for some functionψ0 that will be given satis-fying homogeneous Dirichlet conditions.
With this, we end up with a viscous wave equation in one dimension
∂t2ϕ−∂x2ϕ=µ∂t∂x2ϕ+h, in (0, T)×(0,1),
ϕ(t,0) =ϕ(t,1) = 0, for t∈(0, T), ϕ|t=0= 0, ∂tϕ|t=0=ψ0, on (0,1).
(3.2)
for someh=h(t) and initial dataψ0. Note that we use∂xinstead of∂2since we considerϕis 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 ofϕ.
Proposition 3.1. Let T > 0 and µ > 0. For ψ0 ∈ H3(0,1)∩H01(0,1) and
h ∈ H1(0, T), there exists the unique solution ϕ ∈ C3([0, T);L2(0,1))∩
C2([0, T), H3(0,1))∩C([0, T);H4(0,1)) of (3.2). Moreover, there is a
con-stantC such that the solution satisfies
∥∂tϕ(t)∥H3(0,1)+∥∂xϕ(t)∥H3(0,1)+∥∂2
tϕ(t)∥H1(0,1)
≤C(e−min(2µπ2,µ1)t∥ψ
0∥H3(0,1)+
4
∑
k=1
µ−k∥h∥ H1(0,t)
for0≤t≤T. The constantC is independent of T andµ.
Proof. CombiningProposition 8.1andProposition 8.3inSection 8, one can
easily obtain the result.
Notation of spaces. In this paper, we write∥f∥for∥f∥L2(U)otherwise
speci-fied, and denoteHk(U) by Sobolev spaceW2,k(U) , equipped with the norm
∥f∥Hk(U)=
√ ∑
|α|≤k ∥∂kf∥2
for some domain U ⊂Rn. We also defineH0k(U) by the closure ofC∞ c , the space of all smooth functions with compact support with respect to∥·∥Hk(U); see [2, Section 5] for more detail.
Inserting the functionψ=∂tϕinto the ansatz for ¯u, we receive a solution (¯u,π, η¯ ) of the system
∂tη+ ¯u· ∇η= 0, in (0, T)×Ω, div ¯u= 0, in (0, T)×Ω, ∂tu¯−µ∆¯u+ ¯u· ∇u¯+∇π¯ =−∆ηk∇ηk, in (0, T)×Ω,
¯
u= 0, on (0, T)×∂Ω,
η(0, x) = (−x2, x1)T, for x∈Ω, ¯
u|t=0= (ψ0,0)T, on Ω,
where−∆ηk∇ηkis a short notation for∑2
k=1−∆ηk∇ηk. Note that we choose
η by (3.1) and due to the homogeneous Dirichlet boundary conditions forϕ, it isη(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. LetΩ =R×(0,1),u0∈H2(Ω),ζ0∈H3(Ω),ψ0∈H3(0,1)∩
H1
0(0,1), andh∈H1(0,∞). There exist numbers0< δ <1, κ >0 such that
if the following three conditions hold,
1. the smallness condition for the Poiseuille-type flow
∥ψ0∥H3(0,1)+∥h∥H1(0,∞)≤κ,
2. the smallness condition for the initial perturbation
∥u0−u¯0∥H3(Ω)+∥ζ0−η0∥H3(Ω)≤κ,
3. the compatibility conditions for the initial data
divu0= 0, ζ0|∂Ω=
(
−x2
x1
)
then there exists a smooth global solution (u, ζ, π) of (2.3) with respect to initial data(u0, ζ0)satisfying
d dt
(
∥∂t(u−u¯)∥2+δ∥∇(u−u¯)∥2+δ∥∆(ζ−η)∥2+δ∥∇∆(ζ−η)∥2+∥∂t(ζ−η)∥2)
+µδ(
∥∂t∇(u−u¯)∥2+∥A(u−u¯)∥2+ 1
µ2∥∆(ζ−η)∥ 2+ 1
µ2∥∇∆(ζ−η)∥ 2)
≤0
for all timest≥0. Here,A=−P∆is the Stokes operator inΩ; seeSection 7.
Integrating the last differential inequality over (0, t) implies
δ∥∇(u−u¯)∥2(t) +µδ
∫ t
0 ∥
A(u−u¯)∥2(s) ds≤Cκ
withCκwhich tends to zero asκ→0. By (7.1) we in particular obtain
∥∇(u−u¯)∥2(t) +
∫ t
0
Cµ∥∇(u−u¯)∥2(s) ds≤Cκ
which implies
∥∇(u−u¯)∥2(t)≤Cκe−Cµt
by the Gronwall inequality. By the Poincar´e inequality this implies that ¯uis exponentially stable inH1 sense. Similar stability holds forη.
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 compatibil-ity conditions,
divu0= 0, ζ0|∂Ω=
(
−x2
x1
)
, and ∂1ζ01∂2ζ02−∂1ζ02∂2ζ01= 1. (4.1)
The second condition together with the homogeneous Dirichlet boundary condition of uguarantees ζ|∂Ω= (−x2, x1)T for all times. The third condi-tion is a reformulacondi-tion of the incompressibility condicondi-tion as we discussed in
Section 2and that holds for anyt≥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
∂2α1−∂1α2=∂1α1∂2α2−∂2α1∂1α2−∂2ϕ∂1α1. (4.2)
Then for given (¯u,π, η¯ ), (v, p, α) solves
∂tα+v· ∇α+ ¯u· ∇α=−v· ∇η in (0, T)×Ω, divv= 0 in (0, T)×Ω, ∂tv−µ∆v+v· ∇v+v· ∇u¯+ ¯u· ∇v+∇p
=−∆αk∇αk−∆ηk∇αk−∆αk∇ηk in (0, T)×Ω,
v= 0, on (0, T)×∂Ω, α(0, x) =ζ0(x)−(−x2, x1)T for x∈Ω,
v|t=0=u0−(ψ0,0)T in Ω
(4.3) Note that it isα|∂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 ifϕis small. Let us rewrite the right-hand side of the momentum equation as
−∆αk∇ηk−∆ηk∇αk=
(
−∆α2+∂
1(∂22ϕα2) ∆α1+ ∆(∂
2ϕα2)−∂23ϕα2−∂22ϕ∂2α2
)
= ∆
(
−α2
α1
)
+∂22ϕ∇α2+∇ϕ∆α2. (4.4)
Therefore the momentum equation is rewritten as follows.
∂tv−µ∆v+v· ∇v+v· ∇u¯+ ¯u· ∇v+∇p
=−∆αk∇αk+ ∆
(
−α2
α1
)
+∇ϕ∆α2+∂2
2ϕ∇α2. (4.5)
5. Change of variables and dissipation
Observing the momentum equation in (4.5), one may notice that terms like
v· ∇u¯or ¯u· ∇vcan be handled throughProposition 3.1if the Poiseuille-type flow is sufficiently small. On the other hand, ∆(−α2, α1) causes a problem. Althoughαseems 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
∂tv−µ∆
(
v+ 1
µ
(
−α2
α1
) )
+v· ∇v+v· ∇u¯+ ¯u· ∇v+∇p
=−∆αk∇αk+∇ϕ∆α2+∂2
2ϕ∇α2. (5.1) Now we will introduce a new dependent variable as in [9]:
w=v+ 1
µ
(
−α2
α1
)
or equivalently, α=µ
(
0 1
−1 0
)
Let us rewrite the transport equation ofαin (4.3), i.e.
∂tα+v· ∇α+ ¯u· ∇α=−v· ∇η. (5.2)
In the right-hand side, one can find
−v· ∇η=
(
v2
−v1
)
+v2∇ϕ
=
(
w2
−w1
)
−µ1α+v2∇ϕ.
Therefore the transport equation can be rewritten as follows
∂tα+ 1
µα+v· ∇α+ ¯u· ∇α=
(
w2
−w1
)
+v2∇ϕ. (5.3)
Hence we can see thatαhas dissipative structure. However, we must control thewterm in the right-hand side. The question is how to introduce estimates forwor v? The idea is that we regard (4.5) as a perturbed Stokes system of
wandp, i.e.
−µ∆w+∇p=−∂tv−v· ∇v−v· ∇u−∆αk∇αk+∂22ϕ∇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 divw first.
Divergence ofwand higher order estimate. Let us note thatwis not diver-gence free in general. However, its diverdiver-gence is quadratic inαandϕas the following calculation shows:
divw= divv+1
µdiv
(
−α2
α1
)
= 1
µ(∂2α
1−∂
1α2+∂2ϕ∂1α1−∂2ϕ∂1α1)
= 1
µ(detG−∂2ϕ∂1α
1)
= 1
µ(∂1α
1∂
2α2−∂2α1∂1α2−∂2ϕ∂1α1). (5.5)
Note that we used the incompressibility property (4.2).
Now letf andgbe the right-hand sides of (5.4), (5.5) respectively. Ifw
satisfies appropriate conditions, we can invokeLemma 7.2and obtain
µ∥w∥H3(Ω)+∥∇p∥H1(Ω)≤C(µ∥g∥H2(Ω)+∥f∥H1(Ω)). (5.6)
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 re-quired 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 imposev = 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 ¯uand uwhich vanish on the upper and lower boundary (but not on the artificial boundaries). Hence, forvas well asαthe Poincar´e inequality is still applicable. Since all the estimates do not depend on the horizontal size of the domain, one can letM tend to infinity to receive a solution of (4.3) .
The a priori estimates for the approximate solutions of the Galerkin-scheme 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 domainR×(0,1) in this section.
Notation. In order to simplify the notation, we now introduce variables cor-responding to the data, the time-derivative part and the dissipative part of the estimates respectively. We write
X(t) =∥∂tϕ∥H3(0,1)+∥∂2ϕ∥H3(0,1)+∥∂t2ϕ∥H1(0,1), Y(t) =∥∂tv∥+∥∇v∥+∥∆α∥+∥∇∆α∥,
Z(t) =∥∂t∇v∥2+∥Av∥2+ 1
µ2∥∆α∥ 2+ 1
µ2∥∇∆α∥ 2.
With the definition of Y(t) and Z(t) as it is, by the Poincar´e inequality we find
Y2(t)≤C(1 +µ2)Z(t).
Here,AinZ(t) is the Stokes operator. We useAvinstead of ∆vto annihilate the pressure term in some energy estimates with aid of the regularity of the Stokes operator∥v∥H2
2(Ω)≤C∥Av∥.
In this section, we derive five energy estimates. Then combining these results, we obtain the strong stability inequality stated inTheorem 3.2. To begin with, we need to calculate (5.6) for higher order estimates.
6.1. Spatial Estimate of the artificial variablewandv
Proposition 6.1. LetΩ =R×(0,1),u0∈H2(Ω),ζ0∈H3(Ω),ψ0∈H3(0,1)∩
H1
0(0,1),h∈H1(0,∞), and(u, π, ζ)be a solution of(2.3). If(u0, ζ0)satisfies
the compatibility conditions
divu0= 0, ζ0|∂Ω=
(
−x2
x1
)
and ∂1ζ01∂2ζ02−∂1ζ02∂2ζ01= 1,
then there exists a numerical constantC >0such that for(v, p, α) = (u, π, ζ)−
(¯u,π, η¯ ) andw=v−1 µ(−α
2, α1)T, the estimates
∥v∥H3(Ω)≤C(∥∂t∇v∥+ (1 +µ2)Z+X(∥Av∥+∥∆α∥+∥∇∆α∥))
+C
µ(∥∆α∥+∥∇∆α∥).
∥w∥H3(Ω)≤C
(
∥∂t∇v∥+ (1 +µ2)Z+X(∥Av∥+∥∆α∥+∥∇∆α∥)
)
hold for allt≥0.
Proof. Inserting (5.4) and (5.5) to (5.6), we have
µ∥w∥H3(Ω)+∥∇p∥H1(Ω)
≤C(
∂tv−v· ∇v−v· ∇u¯−u¯· ∇v −∆αk∇αk+∇ϕ∆α2+∂22ϕ∇α2
H1(Ω)
+
∂1α1∂2α2−∂2α1∂1α2−∂2ϕ∂1α1
H2(Ω)
)
≤C(∥∂tv∥H1(Ω)+∥v· ∇v∥H1(Ω)+∥v· ∇u¯∥H1(Ω)+∥u¯· ∇v∥H1(Ω)
+∥∆αk∇αk∥H1(Ω)+∥∇ϕ∆α2∥H1(Ω)+∥∂2
2ϕ∇α2∥H1(Ω)
+∥∂1α1∂2α2∥H2(Ω)+∥∂2α1∂1α2∥H2(Ω)+∥∂2ϕ∂1α1∥H2(Ω)
)
.
We investigate these terms one by one, beginning with
∥∂tv∥H1(Ω)≤ ∥∂t∇v∥ (6.1)
by the Poincar´e inequality. We next observe that
∥v· ∇v∥H1(Ω)≤ ∥v· ∇v∥+∥∇(v· ∇v)∥
≤C(
∥v∥L∞
(Ω)∥∇v∥+∥v∥L∞
(Ω)∥∇2v∥+∥∇v∥2L4(Ω)
)
≤C∥v∥2H2(Ω)
≤C∥Av∥2.
We have invoked embeddings inLemma 7.5and the regularity of the Stokes operator (7.1).
Similarly,
∥v· ∇¯u∥H1(Ω)≤C
(
∥v∥L∞(Ω)∥∇u¯∥+∥∂2u¯∥∥∇
2v
∥+∥¯u∥L∞(Ω)∥∇
2v
∥)
≤C∥Av∥∥∂tϕ∥H2(0,1)
∥u¯· ∇v∥H1(Ω)≤C(∥u¯∥L∞(Ω)∥∇v∥+∥∂2u¯∥∥∇
2v
∥+∥¯u∥L∞(Ω)∥∇
2v
∥)
Here, we have invoked the 1D-2D product estimateLemma 7.6. Note that ¯u
is a function of one space variable. Finally, we have
∥∆αk∇αk∥H1(Ω)+∥∇ϕ∆α2∥H1(Ω)+∥∂22ϕ∇α2∥H1(Ω)
≤C(
∥∆α∥∥∇α∥L∞
(Ω)+∥∇∆α∥∥∇α∥L∞
(Ω)
+∥∆α∥L4(Ω)∥∇2α∥L4(Ω)+∥∂2ϕ∥H3(0,1)∥∇α∥H3(Ω))
≤C(
∥∇∆α∥2+∥∆α∥2+∥∂2ϕ∥H3(0,1)(∥∇∆α∥+∥∆α∥)).
∥∂1α1∂2α2∥H2(Ω)+∥∂2α1∂1α2∥H2(Ω)+∥∂2ϕ∂1α1∥H2(Ω)
≤C(
∥∇α∥H2(Ω)∥∇α∥L∞
(Ω)+∥∆α∥2L4(Ω)
+∥∂2ϕ∥L∞(Ω)∥∇α∥
H2(Ω)+∥∂22ϕ∥L∞(Ω)∥∇α∥
H1(Ω)+∥∂23ϕ∥∥∇α∥L∞(Ω)
)
≤C(
∥∇∆α∥2+∥∆α∥2+∥∂2ϕ∥H3(0,1)(∥∇∆α∥+∥∆α∥)).
Combining these results, we obtain,
µ∥w∥H3(Ω)+∥∇p∥H1(Ω)
≤C(
∥∂t∇v∥2+∥Av∥2+∥∆α∥2+∥∇∆α∥2
+ (∥∂tϕ∥H2(0,1)+∥∂2ϕ∥H3(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− 1 µ(−α
2, α1)T i.e.
∥v∥H3(Ω)≤C
(
∥∂t∇v∥+ (1 +µ2)Z+X(∥Av∥+∥∆α∥+∥∇∆α∥)
)
+C
µ(∥∆α∥+∥∇∆α∥). (6.3)
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 inProposition 6.1, we have the following estimates:
1 2
d dt∥∇v∥
2+µ
∥Av∥2≤ ∥∆α∥∥Av∥+C(1 +µ)(X+Y)Z, (6.4)
1 2
d dt∥∂tv∥
2+µ
∥∂t∇v∥2≤ −
(
∂t∇
(
−α2
α1
)
, ∂t∇v
)
(6.5)
1 2
d dt∥∆α∥
2+ 1
µ∥∆α∥
2
≤C1
µ∥∂t∇v∥∥∆α∥ (6.6)
+C(1
µ+ 1 +µ+µ
2)(X+Y)Z,
1 2
d
dt∥∇∆α∥
2+ 1
µ∥∇∆α∥
2≤C1
µ∥∂t∇v∥∥∇∆α∥ (6.7)
+C(1
µ+ 1 +µ+µ
2)(X+Y)Z,
1 2
d
dt∥∂t∇α∥
2≤(∂ t∇
(
−α2
α1
)
, ∂t∇v
)
(6.8)
+C(1
µ+ 1 +µ+µ
2)(1 +X+Y)3(1 +Y)Z.
Here,C >0 is a numerical constant (independent ofµ).
The estimate (6.4) is obtained by taking the inner product of momen-tum 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 Corresponding subsection (6.4) (v-momentum, Av) 6.3
(6.5) (∂t(v-momentum), ∂tv) 6.4 (6.6) (∆(α-transport), ∆α) 6.5
(6.7) (∇∆(α-transport), ∇∆α) 6.6
(6.8) (∂t∇(α-transport), ∂tα) 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 equa-tion with second derivatives ofv. 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 employAv instead of−∆vin the estimate.
∂tv−µ∆v+v· ∇v+v· ∇u¯+ ¯u· ∇v+∇p
=−∆αk∇αk+ ∆
(
−α2
α1
)
+∂22ϕ∇α2+∇ϕ∆α2,
with Av. We can use the boundary condition for∂tv to integrate by parts. Since the Helmholtz-projection is self-adjoint, we have
(∂tv, Av) =−(∂tv, P∆v) =−(P ∂tv,∆v)
=−(∂tv,∆v) = (∂t∇v,∇v) = 1 2
d dt∥∇v∥
2,
and
(−µ∆v, Av) =µ∥Av∥2.
For the convection terms we use the embeddingH2(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∥H2(Ω)∥∇v∥∥Av∥
≤C∥∇v∥∥Av∥2≤CY Z,
|(v· ∇u, Av¯ )| ≤C∥∇v∥∥∇u¯∥∥Av∥ ≤C∥∂tϕ∥H1(0,1)∥Av∥2≤CXZ,
and
|(¯u· ∇v, Av)| ≤ ∥u¯∥L∞(Ω)∥∇v∥∥Av∥ ≤C∥∂tϕ∥H1(0,1)∥Av∥2≤CXZ.
Since the Stokes operator maps intoL2
σ(Ω), theL2closure of all smooth solenoidal vector fields with compact support in Ω, the pressure term ∇p
vanishes in the a priori estimate. The quadratic form inαgives
|(−∆αk∇αk, Av)| ≤ ∥∆α∥∥∇α∥
L∞(Ω)∥Av∥
≤C∥∆α∥(∥∆α∥+∥∇∆α∥)∥Av∥ ≤CµY Z,
and for the last terms it is
(
∂22ϕ∇α2+∇ϕ∆α2,−Av
)
≤C(∥∂22ϕ∥∥∇2α∥+∥∂2ϕ∥L∞(Ω)∥∆α∥)∥Av∥
≤C∥∂2ϕ∥H1(0,1)∥∆α∥∥Av∥
≤CµXZ.
Linear term. The term ∆
(
−α2
α1
)
contains linear parts with non-small coef-ficients. Due to the presence of the Helmholtz-projection, it is not possible to cancel this term with a corresponding term (v2,−v1)T in the α-estimate. We have
(
∆
(
−α2
α1
)
, Av)
Altogether we have for the estimate of∇v
1 2
d dt∥∇v∥
2+µ∥Av∥2≤ ∥∆α∥∥Av∥+C(1 +µ)(X+Y)Z.
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 regular-ity.
We apply∂tto equation (4.5) and take the inner product with∂tv. Since
v vanishes on the boundary, so does ∂tv and integration by parts gives the terms
(∂t2v, ∂tv) = 1 2
d dt∥∂tv∥
2 and (
−µ∂t∆v, ∂tv) =µ∥∂t∇v∥2.
With div∂tv= 0 and∂tv= 0 on the boundary, we find (v·∇∂tv, ∂tv) = 0 and therefore the Poincar´e inequality as well as the usual Stokes regularity give
(∂t(v· ∇v), ∂tv) = (∂tv· ∇v, ∂tv)
≤ ∥∂tv∥L4(Ω)∥∇v∥L4(Ω)∥∂tv∥
≤C∥∂tv∥H1(Ω)∥∇v∥H1(Ω)∥∂tv∥
≤C∥∂t∇v∥∥Av∥∥∂tv∥ ≤CY Z.
Similarly, we have after employing the product estimates for one- and two-dimensional functions
(∂t(¯u· ∇v), ∂tv) = (∂tu¯· ∇v, ∂tv)
≤ ∥∂tu¯∥L∞(Ω)∥∇v∥∥∂tv∥
≤C∥∂t2ϕ∥H1(0,1)∥Av∥∥∂t∇v∥
≤CXZ
and
(∂t(v· ∇u¯), ∂tv)≤(∥∂tv· ∇u¯∥+∥v· ∇∂tu¯∥)∥∂tv∥
≤C(∥∂t∇v∥∥∇u¯∥+∥∇v∥∥∂t∇u¯∥)∥∂t∇v∥
≤C(∥∂tϕ∥H1+∥∂t2ϕ∥H1)(∥∂t∇v∥+∥Av∥)∥∂t∇v∥
≤CXZ.
The pressure term vanishes due to div∂tv = 0 and ∂tv = 0 on the boundary. In the linear term(∂t∆
(
−α2
α1
)
, ∂tv
)
we integrate by parts once, using∂tv= 0 on the boundary:
(
∂t∆
(
−α2
α1
)
, ∂tv)=−
(
∂t∇
(
−α2
α1
)
This term is going to be absorbed in the estimate for the time derivative of the gradient of the stream functionα. Considering the quadraticα-term, we note with Einstein’s sum convention
[∆αk∇αk]
i=∂l(∂lαk∂iαk)−∂lαk∂i∂lαk = [div (∇αk⊗ ∇αk)]i−[∇(|∇αk|2)]i and therefore in the a priori estimate, the second part vanishes being a gra-dient and we have after integrating by parts
(−∂t(∆αk∇αk), ∂tv) =−(∂tdiv (∇αk⊗ ∇αk), ∂tv) + (∂t∇(|∇αk|2), ∂tv) = (∂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(∂22ϕ∇α2+∇ϕ∆α2), ∂tv)
=(∂t
(
∂1(∂22ϕα2) div (∂2ϕ∇α2)
)
, ∂tv
)
≤C(∥∂t(∂22ϕα2)∥+∥∂t(∂2ϕ∇α2)∥)∥∂t∇v∥ ≤C(
∥∂22∂tϕ∥∥∇α∥+∥∂22ϕ∥∥∂t∇α∥
+∥∂2∂tϕ∥∥∇2α∥+∥∂2ϕ∥L∞(Ω)∥∂t∇α∥
)
∥∂t∇v∥ ≤C(∥∂tϕ∥H2(0,1)∥∆α∥+∥∂2ϕ∥H1(0,1)∥∂t∇α∥)∥∂t∇v∥
≤C∥∂tϕ∥H2(0,1)(
1 2µ∥∆α∥
2+µ 2∥∂t∇v∥
2)
+
C∥∂2ϕ∥H1(0,1)∥∂t∇α∥∥∂t∇v∥
≤CµXZ+C∥∂2ϕ∥H1(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
αin (4.3). Note, that one can write−v· ∇η = (v2,−v1)T + (0, ∂
2ϕv2)T and hence
∥∂t∇α∥ ≤ ∥∇(v· ∇α)∥+∥∇(¯u· ∇α)∥+∥∇(v· ∇η)∥
≤ ∥∇v∥∥∇α∥L∞(Ω)+∥v∥L4(Ω)∥∇2α∥L4(Ω)+∥∇u¯∥∥∇2α∥+∥¯u∥L∞(Ω)∥∇
2α
∥
+∥∇v∥+∥∂22ϕ∥∥∇v∥+∥∂2ϕ∥L∞(Ω)∥∇v∥
≤ ∥∇v∥
+C(∥∆α∥+∥∇∆α∥+∥∂2ϕ∥H1+∥∂tϕ∥H1)(∥∇v∥+∥∆α∥)
≤C(1 +X+Y)Y. (6.9) Applying this inequality yields
and with∥∂t∇v∥Y ≤C(1 +µ)Z, we have
(
∂t(∂22ϕ∇α2+∇ϕ∆α2), ∂tv)
≤CµXZ+C∥∂2ϕ∥H1(0,1)∥∂t∇v∥(1 +X+Y)Y
≤C(1 +µ)(1 +X+Y)XZ.
Summarizing the foregoing estimates, we receive 1
2
d dt∥∂tv∥
2+µ∥∂
t∇v∥2+
(
∂t∇
(
−α2
α1
)
, ∂t∇v
)
≤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 H3(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 left-hand side. This, however, comes at the cost of linear error terms involving the artificial variablew=v+ 1
µ(−α
2, α1)T. These error terms will later on be handled with higher estimates ofw(6.2).
We apply ∆ to the transport equation of the stream function (5.3), i.e.
∂tα+ 1
µα+v· ∇α+ ¯u· ∇α=
(
w2
−w1
)
+v2∇ϕ.
and take the inner product with ∆α. Then the time derivative gives (∂t∆α,∆α) = 1
2 d dt∥∆α∥
2. The second term gives (∆α,∆α) =
∥∆α∥2. For the third term we note (v· ∇∆α,∆α) = 0 and therefore with Einstein’s sum convention
(∆(v· ∇α),∆α) = (∆v· ∇α+ 2∂iv· ∇∂iα,∆α) ≤C(∥∆α∥∥∇α∥L∞
(Ω)+∥∇v∥L4(Ω)∥∇2α∥L4(Ω))∥∆α∥
≤C∥Av∥(∥∆α∥+∥∇∆α∥)∥∆α∥
≤C(∥∆α∥+∥∇∆α∥)(µ 2∥Av∥
2+ 1 2µ∥∆α∥
2)
≤CµY Z.
Similarly the second advection term yields
(∆(¯u· ∇α),∆α) = (∆¯u· ∇α+ 2∂iu¯· ∇∂iα,∆α) ≤C(∥∂2
2∂tϕ∥∥∇2α∥+∥∂2∂tϕ∥L∞(Ω)∥∇2α∥)∥∆α∥
≤C∥∂tϕ∥H2(0,1)∥∆α∥2
≤Cµ2XZ.
Let us take care of the right-hand side.
(
−∆(
(
w2
−w1
)
+v2∇ϕ),∆α)=−(∆
(
w2
−w1
)
,∆α)+ (∆(∂2ϕv2),∆α2)
≤ ∥∆w∥∥∆α∥+C∥∂2ϕ∥H2(0,1)∥Av∥∥∆α∥
Now invoking the estimate inProposition 6.1, we have
∥∆w∥∥∆α∥ ≤C∥∆α∥µ1(
∥∂t∇v∥+ (1 +µ2)Z+X(∥Av∥+∥∆α∥
+∥∇∆α∥))
≤C1
µ∥∆α∥∥∂t∇v∥+C
1
µ(1 +µ
2)Y Z+C1
µ(µ+µ
2)XZ
≤C1
µ∥∆α∥∥∂t∇v∥+C(
1
µ+ 1 +µ)(X+Y)Z.
The complete estimate is of the form 1
2
d dt∥∆α∥
2+1
µ∥∆α∥
2
≤C1
µ∥∂t∇v∥∥∆α∥+C(
1
µ + 1 +µ+µ
2)(X+Y)Z.
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 theH3(Ω)-norm ofα. Once again, we receive error terms including higher space derivatives of the artificial variablew.
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
(∂t∇∆α+∇∆α,∇∆α) = 1 2
d
dt∥∇∆α∥
2+
∥∇∆α∥2.
We employ the identity (v· ∇∇∆α,∇∆α) = 0 and estimate (using Einstein’s sum convention)
(∇∆(v· ∇α),∇∆α)
= (∂l(∆v· ∇α+ 2∂iv· ∇∂iα+v· ∇∆α), ∂l∆α) = (∂l∆v· ∇α+ ∆v· ∇∂lα
+ 2∂l∂iv· ∇∂iα+ 2∂iv· ∇∂l∂iα+∂lv· ∇∆α, ∂l∆α) ≤(∥∇∆v∥∥∇α∥L∞
(Ω)+ 3∥∇2v∥L4(Ω)∥∇2α∥L4(Ω)+ 2∥∇v∥L∞
(Ω)∥∇3α∥)∥∇∆α∥
≤C∥∇v∥H2(Ω)∥∇α∥H2(Ω)∥∇∆α∥
≤C∥∇v∥H2(Ω)(∥∆α∥2+∥∇∆α∥2)
≤C∥∇v∥H2(Ω)Y2.
This is the first place, where higher spacial derivatives ofv are appearing as error terms.
Once again using (¯u· ∇∇∆α,∇∆α) = 0 we obtain more easily (∇∆(¯u· ∇α),∇∆α)
= (∂l∆¯u· ∇α+ ∆¯u· ∇∂lα
+ 2∂l∂iu¯· ∇∂iα+ 2∂iu¯· ∇∂l∂iα+∂lu¯· ∇∆α, ∂l∆α) ≤C(∥∂32∂tϕ∥∥∇2α∥+∥∂22∂tϕ∥L∞(Ω)∥∇
2α
∥+∥∂2∂tϕ∥L∞(Ω)∥∇
3α
∥)∥∇∆α∥ ≤C∥∂tϕ∥H3(0,1)∥∇∆α∥(∥∆α∥+∥∇∆α∥)
Linear term. Now let us focus on the right-hand side. Once again, we receive higher spatial derivatives ofwandv.
(
− ∇∆(
(
w2
−w1
)
+v2∇ϕ),∇∆α)
=(∇∆
(
w2
−w1
)
,∇∆α)+ (∇∆(∂2ϕv2),∇∆α)
≤ ∥∇∆w∥∥∇∆α∥+C∥∂2ϕ∥H3(0,1)∥∇v∥H2(Ω)∥∇∆α∥
≤ ∥∇∆w∥∥∇∆α∥+C∥∇v∥H2(Ω)XY.
Here again, we have invokedProposition 6.1for the first term
∥∇∆w∥∥∇∆α∥
≤C∥∇∆α∥1µ(
∥∂t∇v∥+ (1 +µ2)Z+X(∥Av∥+∥∆α∥+∥∇∆α∥)
)
≤C1
µ∥∇∆α∥∥∂t∇v∥+C(
1
µ+ 1 +µ)(X+Y)Z.
We deal with the second term in the same way.
∥∇v∥H2(Ω)Y
≤C(
∥∂t∇v∥Y + (1 +µ2)Y Z+XY(∥Av∥+∥∆α∥+∥∇∆α∥)
)
+C
µ(∥∆α∥+∥∇∆α∥)Y
≤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
2
d
dt∥∇∆α∥
2+ 1
µ∥∇∆α∥
2
≤C1
µ∥∂t∇v∥∥∇∆α∥+
C(1
µ+ 1 +µ+µ
2)(1 +X+Y)(X+Y)Z.
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 ∆αand∇∆αwe 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
Application of ∂t∇ to (5.2) and taking the inner product with ∂t∇α yields for the first term
(∂2t∇α, ∂tα) = 1 2
d
dt∥∂t∇α∥
2.
In the following, we need to estimate the term ∥∂t∇α∥ several times. We remind, that as calculated in (6.9), we have
∥∂t∇α∥ ≤C(1 +X+Y)Y.
Similarly to the foregoing a priori estimates forα, the term (v· ∇∂t∇α, ∂t∇α) vanishes. This way, it is
(∂t∇(v· ∇α), ∂t∇α)
= (∂t∂iv· ∇α+∂iv· ∇∂tα+∂tv· ∇∂iα, ∂t∂iα) ≤C(∥∂t∇v∥∥∇α∥L∞
(Ω)+∥∇v∥L∞
(Ω)∥∂t∇α∥+∥∂tv∥L4(Ω)∥∇2α∥L4(Ω))∥∂t∇α∥
≤C(∥∂t∇v∥∥∇α∥H2(Ω)∥∂t∇α∥+∥∇v∥H2(Ω)∥∂t∇α∥2)
≤C∥∂t∇v∥(∥∆α∥+∥∇∆α∥)(1 +X+Y)Y
+C∥∇v∥H2(Ω)(1 +X+Y)2Y2
≤Cµ(1 +X+Y)Y Z+C(1 +µ+µ2)(1 +X+Y)3Y Z.
Note that in the last line, the estimate of∇v (6.10) was invoked.
The second advection term can be estimated similarly withY2≤C(1 +
µ2)Z
(∂t∇(¯u· ∇α), ∂t∇α)
= (∂t∂iu¯· ∇α+∂iu¯· ∇∂tα+∂tu¯· ∇∂iα, ∂t∂iα) ≤C(∥∂2∂t2ϕ∥∥∇2α∥+∥∂2∂tϕ∥L∞
(Ω)∥∂t∇α∥+∥∂t2ϕ∥L∞
(Ω)∥∇2α∥)∥∂t∇α∥ ≤C(∥∂2tϕ∥H1(0,1)∥∆α∥∥∂t∇α∥+∥∂tϕ∥H2(0,1)∥∂t∇α∥2)
≤C(
∥∂2tϕ∥H1(0,1)∥∆α∥(1 +X+Y)Y
+∥∂tϕ∥H2(0,1)(1 +X+Y)2Y2)
≤C(1 +µ2)(1 +X+Y)2XZ.
Linear term. We split the linear term into
−v· ∇η=
(
v2
−v1
)
+
(
0
∂2ϕv2
)
and estimate the second part by (∂t∇(∂2ϕv2), ∂t∇α2)
= (∂t∂22ϕv2+∂22ϕ∂tv2+∂t∂2ϕ∇v2+∂2ϕ∂t∇v2, ∂t∇α2) ≤C(∥∂tϕ∥H2(0,1)∥∇v∥+∥∂2ϕ∥H1(0,1)∥∂t∇v∥)(1 +X+Y)Y
≤C(1 +X+Y)XY2+C(1 +µ)(1 +X+Y)XZ
The remaining linear term is used to cancel a corresponding term ap-pearing in the estimate of the time-derivative ofv. It is
(
∂t∇
(
v2
−v1
)
, ∂t∇α
)
= (∂t∇v2, ∂t∇α1)−(∂t∇v1, ∂t∇α2)
=(∂t∇
(
−α2
α1
)
, ∂t∇v
)
.
Hence, the combined estimate is 1
2
d
dt∥∂t∇α∥
2
−(∂t∇
(
−α2
α1
)
, ∂t∇v
)
≤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 inProposition 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 2
d dt∥∇v∥
2+µ 2∥Av∥
2
≤ 21µ∥∆α∥2+C(1 +µ)(X+Y)Z.
Applying Young’s inequality for (6.6) and (6.7) in a similar way, we receive 1
2
d dt∥∆α∥
2+ 3 4µ∥∆α∥
2
≤ Cµ∥∂t∇v∥2+C( 1
µ+ 1 +µ+µ
2)(X+Y)Z,
(6.11) 1
2
d
dt∥∇∆α∥
2+ 1
2µ∥∇∆α∥
2
≤ Cµ∥∂t∇v∥2 (6.12)
+C(1
µ+ 1 +µ+µ
2)(1 +X+Y)(X+Y)Z.
Now we can add these three inequalities above to find a combined estimate 1
2
d dt
(
∥∇v∥2+∥∆α∥2+∥∇∆α∥2)
+µ 2∥Av∥
2+ 1 4µ∥∆α∥
2+ 1
2µ∥∇∆α∥
2
≤Cµ∥∂t∇v∥2+C( 1
µ + 1 +µ+µ
2)(1 +X+Y)(X+Y)Z. (6.13)
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 forvandα. The resulting estimate is
1 2
d dt(∥∂tv∥
2+
∥∂t∇α∥2) +µ∥∂t∇v∥2
≤C(1
µ+ 1 +µ+µ
2)(1 +X+Y)3(X+Y)Z. (6.14)
The sum of (6.14) withδ×(6.13) gives 1
2
d dt
(
∥∂tv∥2+δ∥∇v∥2+δ∥∆α∥2+δ∥∇∆α∥2+∥∂t∇α∥2
)
+ (µ−δC
µ)∥∂t∇v∥
2+δµ 2∥Av∥
2+δ 1 4µ∥∆α∥
2+δ 1
2µ∥∇∆α∥
2
≤C(1 +δ)(1
µ+ 1 +µ+µ
2)(1 +X+Y)3(X+Y)Z.
We choose 0< δ <1 such thatµ−δC µ ≥
µ
2 and hence receive
d dt
(
∥∂tv∥2+δ∥∇v∥2+δ∥∆α∥2+δ∥∇∆α∥2+∥∂t∇α∥2)
+µ∥∂t∇v∥2+δµ∥Av∥2+
δ
2µ∥∆α∥
2+δ
µ∥∇∆α∥
2
≤C(1 +δ)(1
µ+ 1 +µ+µ
2)(1 +X+Y)3(X+Y)Z.
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 dt
(
∥∂tv∥2+δ∥∇v∥2+δ∥∆α∥2+δ∥∇∆α∥2+∥∂t∇α∥2
)
+ 2µδZ
≤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 resultTheorem 3.2
with the aid of estimate (6.15). As we mentioned at the beginning ofSection 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 somea >0 such thatCµ,δ(1 +
a)3a < µδ. Note that as long asY(t), X(t)≤a/2 hold for such a, we have
d dt
(
∥∂tv∥2+δ∥∇v∥2+δ∥∆α∥2+δ∥∇∆α∥2+∥∂t∇α∥2
)
(t) +µδZ(t)≤0.
Thus we can conclude corresponding norm of the solution is decreasing except the case where allX, Y, Z equal zero which is a trivial case.
Therefore, the proof is reduced to show thatY(t), X(t)≤a/2 hold for allt≥0 if we choose initial data and flow data sufficiently small.
We now invoke the a priori estimate for the Poiseuille flowProposition 3.1
and obtain
X(t)≤C(
∥ψ0∥H3(0,1)+∥h∥H1(0,∞)
)
Now let us focus onY(t). For this, we investigate the initial dataY(0) first.
Y(0) =∥∂tv(0)∥+∥∇v0∥+∥∆α0∥+∥∇∆α0∥.
Let us recall the momentum equation (4.5) and find
∂tv(0) =µ∆v0−v0· ∇v0−v0· ∇u¯(0)−¯u(0)· ∇v0− ∇p
+ ∆
(
−α2 0
α10
)
−∆αk0∇αk0+∂22ϕ(0)∇α20+∇ϕ(0)∆α20.
Applying the Helmholtz projection to remove the pressure term, we obtain
∥∂tv(0)∥ ≤C(µ∥∆v0∥+∥v0· ∇v0∥+∥v0· ∇¯u∥+
∥u¯0· ∇v0∥+∥∆αk0∇αk0∥+∥∂22ϕ(0)∇α20∥)
≤C(µ+∥∇v0∥+∥ψ0∥H1(0,1))∥Av0∥
+C(1 +∥∆α0∥+∥∇∆α0∥)∥∆α0∥.
Similarly, we recall the transport equation ofα(5.2) for∂t∇α(0).
∥∂t∇α(0)∥ ≤ ∥∇(v0· ∇α0)∥+∥∇(¯u(0)· ∇α0)∥+∥∇(v0· ∇η0)∥
≤ ∥∇2α0∥∥v0∥L∞
(Ω)+∥∇v0∥L4(Ω)∥∇α0∥L4(Ω)+∥∇2α0∥∥u¯(0)∥L∞
(Ω)
+∥∇u¯(0)∥L4(Ω)∥∇α0∥L4(Ω)+∥∇v0∥∥∇η0∥L∞(Ω)
≤C(∥∇v0∥+ (∥Av0∥+∥ψ0∥H1(0,1))∥∆α0∥).
Therefore we choosev0, α0 and retakeψ0 if necessary so small that
Y(0), ∥∂t∇α(0)∥< √
δa
8
holds. Then we define a timeT∗= inf{t >0;Y(t)> a/2}. To proveT∗=∞, supposeT∗<∞and seek a contradiction.Y(t)≤a/2 for allt∈[0, T∗] holds, sinceY is continuous. Thus,
d dt
(
∥∂tv∥2+δ∥∇v∥2+δ∥∆α∥2+δ∥∇∆α∥2+∥∂t∇α∥2
)
(t) +µδZ(t)≤0
holds in the interval [0, T∗]. Integrating over [0, T∗], we receive
∥∂tv(T∗)∥2+δ∥∇v(T∗)∥2+δ∥∆α(T∗)∥2+δ∥∇∆α(T∗)∥2+∥∂t∇α(T∗)∥2
Now evaluatingY(T∗) with the above estimate yields,
(Y(T∗))2=(
∥∂tv(T∗)∥+∥∇v(T∗)∥+∥∆α(T∗)∥+∥∇∆α(T∗)∥) 2
≤4δ(∥∂tv(T∗)∥2+δ∥∇v(T∗)∥2+δ∥∆α(T∗)∥2+δ∥∇∆α(T∗)∥2)
≤4δ(
∥∂tv(0)∥2+δ∥∇v0∥2+δ∥∆α0∥2+δ∥∇∆α0∥2+∥∂t∇α0∥2)
<4 δ
(
(Y(0))2+∥∂t∇α(0)∥2
)
≤4δ( √
δY0 8
)2
×2
≤Y
2 0 8 .
This leads to a contradiction to the definition ofT∗and thereforeT∗=∞.
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 layerR×(0,1) or one of the approximations (−M, M)×(0,1) in this section.
7.1. Stokes operator
Definition 7.1. LetP be the Helmholtz projection on Ω. The Stokes opera-tor A is defined as a closed linear operator inL2
σ(Ω) with domain D(A) =
H2(Ω)∩H1
0(Ω)∩L2σ(Ω) such thatAu=−P∆.
Note that 0∈ρ(A) andD(A) =H2(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∥H2(Ω)≤C∥Av∥ (7.1)
forv∈H2(Ω) withv= 0 on the boundary and divv= 0.
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 ∈H1(Ω),g∈H2(Ω) and(u, p)solve
−µ∆u+∇p =f, in Ω,
divu =g, inΩ, u = 0, on∂Ω.
(7.2)
Then there holds
µ∥u∥H3(Ω)+∥∇p∥H1(Ω)≤C(µ∥g∥H2(Ω)+∥f∥H1(Ω)).
Proof. See [12, Theorem1.5.3].
7.3. Elliptic regularity and implications
Lemma 7.3. Under the same assumptions forΩ of Lemma 7.2, there exists a constantC such that
∥f∥H2(Ω)≤C∥∆f∥ (7.3)
holds for allf ∈H2(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 theH3(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 constantC such that
C−1∥f∥H3(Ω)≤ ∥∆f∥+∥∇∆f∥ ≤C∥f∥H3(Ω) (7.4)
holds for allf ∈H3(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 ofLemma 7.2. The following em-beddings hold:
H2(Ω)֒→L∞(Ω), H1(Ω)֒→L4(Ω), and H1(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 ∈ H1(Ω)
vanish on the upper and lower boundaries andf ∈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 embeddingsLemma 7.5to conclude
∥f g∥L2(Ω)=
(∫
R
∫
(0,1)|
f(y)|2|g(x, y)|2dydx) 1 2
≤(
∫
R∥
g(x,·)∥2L∞(0,1)
∫
(0,1)|
f(y)|2dydx) 1 2
≤ ∥f∥L2(0,1)
(∫
R
C2∥g(x,·)∥2H1(0,1)dx
)12
≤C∥f∥L2(0,1)∥∇g∥L2(Ω).
(7.7)
8. Viscous wave equation
This section is dedicated to a proof ofProposition 3.1. We split the initial-boundary value problem (3.2) into two parts for sharper estimation. The first part is the homogeneous case, i.e.,
∂2tϕ1−∂x2ϕ1=µ∂t∂x2ϕ1, in (0,1),
ϕ1(t,0) =ϕ1(t,1) = 0, for t∈(0, T),
ϕ1(0) = 0, ∂tϕ1(0) =ψ0, in (0,1).
(8.1)
The second one is the inhomogeneous case, i.e.,
∂t2ϕ2−∂2xϕ2=µ∂t∂x2ϕ2+h, in (0,1),
ϕ2(t,0) =ϕ2(t,1) = 0, for t∈(0, T),
ϕ2(0) = 0, ∂tϕ2(0) = 0, in (0,1).
(8.2)
Again, herehis some given function which depends only ont. We shall show a priori estimates for each of them. Note that we only need estimates for
∂x4ϕi, ∂t∂x3ϕi, ∂t2∂xϕi 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 solutionϕ1 of (8.1) byϕin this subsection.
Separation of variables. To begin with, we consider the simple ansatz with the formϕ(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)
µT′(t) +T(t)=
X′′(x)
X(x) =λ
for someλ∈RunlessX(x) andµT′(t) +T(t) vanish.
Let us focus on X(x) first. The equation X′′(x) = λX(t) with initial conditionX(0) =X(1) = 0 gives us a solutionXn(x) =ansin(nπx) and here
λ=−(nπ)2. We simply regarda
n = 1 and take care of those coefficients in
T(t) side.
Now let us turn toT′′
n(t) +µ(nπ)2Tn′(t) + (nπ)2Tn(t) = 0. Solving the characteristic equationy2+µ(nπ)2y+ (nπ)2= 0 yields
yn±=−
µ
2(nπ) 2
±nπ
√
µ2 4 (nπ)
We setBn=nπ
√ µ
2
4(nπ)2−1
for simplicity. Thenyn± is written as
yn±=
−µ2(nπ) 2±iB
n, n < µπ2 , −µ2(nπ)
2, n= 2
µπ, −µ2(nπ)2±Bn, n > µπ2 .
Now we assumeN := 2
µπ ∈Nin the following. Otherwise, we just ignore the terms with respect toyN± =−µ(N π)2/2. Hence the solutionT
n(t) is written as
Tn(t) =
e−µ
2(nπ) 2
t(a
nsin(Bnt) +bncos(Bnt)), n < µπ2 ,
e−µ2(nπ) 2
t(ta
n+bn), n= µπ2 ,
e−µ
2(nπ) 2
t(an 2 eBnt+
bn
2 e−Bnt), n > 2 µπ
for some coefficientsan, bn∈R.
Determining the coefficients. We now consider the solution ansatz for (8.1) of the form
ϕ(t, x) = N−1
∑
n=1
sin(nπx)e−µ2(nπ) 2
t(a
nsin(Bnt) +bncos(Bnt))
+ sin(N πx)e−µ2(N π) 2
t(ta
N+bN)
+ ∞
∑
n=N+1
sin(nπx)e−µ2(nπ) 2
t(an 2 e
Bnt+bn 2 e
−Bnt).
We determinean, bn so thatϕ(0, x) = 0 andϕt(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 in the casesn≤ 2
µπ and in the latter case, we noteBn< µ 2(nπ)
2 forn > 2 µπ. In fact, it is
−µ2(nπ)2+Bn =−
µ
2(nπ) 2+nπ
√
µ2 4 (nπ)
2−1
=−nπ
(√
µ2 4 (nπ)
2−
√
µ2 4 (nπ)
2−1
)
=−√ π
µ2
4π2+
√
µ2
4π2− 1 n2
≤ − π
2
√
µ2
4π2
=−1
µ.
(8.3)
Hence, the solution decays exponentially to zero at infinity at least as fast as
We superpose these solutions forn∈Nreceiving the solution ansatz for
ϕof the form
ϕ(t, x) = N−1
∑
n=1
sin(nπx)e−µ2(nπ) 2
t(a
nsin(Bnt) +bncos(Bnt))
+ sin(N πx)e−µ2(N π) 2
t(ta
N+bN)
+ ∞
∑
n=N+1
sin(nπx)e−µ2(nπ) 2
t(an 2 e
Bnt+bn 2 e
−Bnt).
Determining the coefficients. Our next step is to exploit the initial conditions onϕto determine the proper values for the coefficients an andbn. It is
0 =ϕ(0, x)
= N−1
∑
n=1
sin(nπx)bn+ sin(N πx)bN + ∞
∑
n=N+1
sin(nπx)(an+bn 2 ) = ∞ ∑ n=1
sin(nπx)(bn+χ{N+1,...}(n)
(an−bn
2
))
.
It is then easy to conclude that for n = 1, . . . , N we have bn = 0 and for
n=N+ 1, . . . it isbn=−an and therefore
ϕ(t, x) = N−1
∑
n=1
sin(nπx)e−µ2(nπ) 2
t(a
nsin(Bnt))
+ sin(N πx)e−µ2(N π) 2
t(ta N)
+ ∞
∑
n=N+1
sin(nπx)e−µ2(nπ) 2
tan 2
(
eBnt−e−Bnt).
The other initial conditionϕt(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)((
−µ2(nπ)2)
e−µ2(nπ) 2
t(a
nsin(Bnt))
+e−µ2(nπ) 2
t(a
nBncos(Bnt))
)
+ sin(N πx)((
−µ2(N π)2)
e−µ2(N π) 2
t(ta
N) +e− µ
2(N π) 2 ta N ) + ∞ ∑
n=N+1
sin(nπx)((
−µ2(nπ)2)
e−µ2(nπ) 2
tan 2
(
eBnt−e−Bnt)
+e−µ2(nπ) 2
tanBn 2
(
