R E S E A R C H
Open Access
Pullback attractors for a two-phase flow
model in an infinite delay case
Min Yang
**Correspondence: [email protected]
College of Aerospace Engineering, Nanjing University Of Aeronautics and Astronautics, Nanjing, 210016, China
Department of Mathematics, College of Sciences, Hohai University, Nanjing, 210098, China
Abstract
In this paper we study the existence of solutions for a coupled
Allen-Cahn-Navier-Stokes model in two dimensions with an external force containing infinite delay effects in the weighted spaceCδ(Y). We prove the existence of pullback
attractors for the dynamical system associated to the problem under more general assumptions.
Keywords: Allen-Cahn-Navier-Stokes; infinite delays; pullback attractors
1 Introduction
Diffuse interface models are well-known tools to describe dynamics of complex fluids ([, ]). For instance, this approach is used in [] to describe cavitation phenomena in a flow-ing liquid. The resultflow-ing model essentially consists of the Navier-Stokes equations suitably coupled with the well-known phase-field system. In the isothermal compressible case, ex-istence of a global weak solution for such a system has been recently proved in []. In the incompressible isothermal case, neglecting chemical reactions and other forces, the model reduces to an evolution system which governs the fluid velocityuand the order parame-terφ. This system can be written as a Navier-Stokes equation coupled with a convective Allen-Cahn equation.
In [, ], Gal and Grasselli proved that the initial and boundary value problem gener-ates a strongly continuous semigroup on a suitable phase space which possesses the global attractorAand establish the existence of an exponential attractorEwhich entails thatA has finite fractal dimension. Medjo in [] studied the pullback asymptotic behavior of so-lutions for a non-autonomous homogeneous two-phase flow model in a two-dimensional domain.
Recently the appearance of delay effects in partial difference equations has been inten-sively treated. In [–], the authors studied the D Navier-Stokes equations in which additional external forces were included in the model. The existence of an attractor for a D Navier-Stokes system with delays is proved in []. The authors proved the existence and uniqueness of a stationary solution and the exponential decay of the solutions of the evolutionary problem to this stationary solution in [] and strengthened some results on the existence and properties of pullback attractors in []. It is a natural generalization to the Allen-Cahn-Navier-Stokes equations with delays. Medjo in [, ] studied a coupled Cahn-Hilliard-Navier-Stokes model with delays in a two-dimensional domain and proved
the existence and uniqueness of the weak and strong solution when the external force con-tains some delays. He also discusses the asymptotic behavior of the weak solutions and the stability of the stationary solutions.
Our purpose is to study a coupled Allen-Cahn-Navier-Stokes model with infinite delays. The paper is organized as follows. Section we describe the model and some functional spaces useful for the problem. Section we prove the existence and uniqueness of the solution. Also we analyze the continuity properties of the solutions with respect to ini-tial data. Section is devoted to generalizing the results on asymptotic behavior, proving under more general assumptions the existence of pullback attractors.
2 A two-phase flow model
We consider a model of homogeneous incompressible two-phase flow with singularly os-cillating forces. We assume that the domain⊂R be an open and bounded set with smooth enough boundary∂and consider (arbitrary) valuesτ <T inR. Then we con-sider the following system:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂u
∂t –νu+ (u· ∇)u+∇p–Kμ∇φ=g(t) +G(t, (u,φ)t),
divu= ,
∂φ
∂t +u· ∇φ+μ= ,
μ= – φ+αf(φ),
()
in×(τ,T).
In (), the unknown functions are the velocityu= (u,u) of the fluid, its pressurepand
the order parameterφ. The quantityμis the variational derivative of the following free energy functional:
F(φ) =
|∇φ|
+αF(φ)
ds,
whereF(r) = rf(ζ)dζ, the constantsν> andK> the kinematic viscosity of the fluid and capillarity coefficient, respectively, ,α> are two physical parameters describing the interaction between the two phases. In particular, is related with the thickness of the interface separating the two fluid. The numberτ∈Ris the initial time. We endow () with the boundary condition
u= , ∂φ
∂η = on∂×(τ,T),
where∂is the boundary ofandηis its outward normal. The initial condition is given by
(u,φ)(t+τ) =ϑ(t) = (ϑ,ϑ)(t), t∈(–∞, ]. ()
The termsg(t) is a non-delayed external force field,G(t, (u,φ)t) is another external force
containing some hereditary characteristics and we denote by (u,φ)tthe function defined
We assume thatf∈C(R) satisfies
lim
|r|→+∞f
(r) > , f(r)≤c
f
+|r|k, ∀r∈R, ()
wherecf is some positive constant andk∈[, +∞) is fixed. So we get
f(r)≤cf
+|r|k+, ∀r∈R. ()
IfXis a real Hilbert space with inner product (·,·)X, we will denote the induced norm
by| · |XwhileX∗will indicate its dual with·,·for the duality betweenX∗andXand the
norm by · ∗. We set
V=
u∈C∞() :divu= in.
We denote byHandVthe closure ofVin (L())and (H()), respectively. The scalar
product inHis denoted by (·,·)Land the associated norm by| · |L. Moreover, the space Vis endowed with the scalar product
(u,v)=
i=
(∂xiu,∂xiv)L, u=(u,u)/.
We define the operatorAby
Au=Pu, ∀u∈D(A) =H()∩V,
wherePis the Leray-Helmholtz projector fromL() ontoH
. ThenAis a self-adjoint
positive unbounded operator inH which is associated with the scalar product defined
above. Furthermore,A– is a compact linear operator onH and|A· |L is a norm on D(A) that is equivalent to theH-norm. From (), we can findγ > such that
lim
|r|→+∞f
(r) > γ> . ()
Then we can define the linear positive unbounded operatorAγ onL() by
Aγφ= –φ+γ φ, ∀φ∈D(Aγ),
where
D(Aγ) =
ρ∈H();∂ρ
∂η= on∂
.
Note thatA–γ is a compact linear operator on L() and|Aγ · |L is a norm onD(Aγ) that is equivalent to theH-norm. We setV
=H(). Furthermore we denote byλ> a
positive constant satisfying
λ|ω|L≤ ω ∀ω∈V; λA/γ ψ
We also introduce the bilinear operatorsB,B and their associated trilinear formsb,
b as well as the coupling mappingR, which are defined fromD(A)×D(A) intoH,
D(A)×D(Aγ) intoL() andL()×D(A/γ ) intoH, respectively. More precisely, we
set
B(u,v),w
=
(u· ∇)v·wdx=b(u,v,w), ∀u,v,w∈D(A),
B(u,φ),ρ
=
(u· ∇)φρdx=b(u,φ,ρ), ∀u∈D(A),φ,ρ∈D(Aγ),
R(μ,φ),w
=
μ[∇φ·w]dx=b(w,φ,μ), ∀w∈D(A), (μ,φ)∈L()×D
A/γ .
Note thatR(μ,φ) =Pμ∇φandB,BandRsatisfy the following estimates:
B(u,v)V∗
≤c|u|
/
Lu/|v|/Lv/, ∀u,v∈V, ()
B(u,φ)V∗
≤c| u|/
Lu/|φ|/Lφ/, ∀u∈V,φ∈V, ()
R(Aγφ,ρ)V∗
≤cρ
/|Aγρ|/
L|Aγφ|L, ∀φ,ρ∈D(Aγ). ()
Now we define the Hilbert spacesYandVbyY=H×H(),V=V×D(Aγ) endowed
with the scalar products whose associated norms are
(u,φ)Y=K–|u|L+
|∇φ|L+γ|φ|L
=K–|u|L+ A/γ φ
L; (u,φ)V=u+|Aγφ|L.
Letfγ(r) =f(r) –α– γrand observe thatfγ still satisfies () withγ in place of γ since
≤α, andFγ(r) = rfγ(ζ)dζis bounded from below.
There are several phase spaces which allow us to deal with infinite delays. For instance, for a givenδ> , and a given Banach spaceX, we may consider the space
Cδ(X) =
ϕ∈C(–∞, ];X:∃ lim
s→–∞e
δsϕ(s)∈X
with the norm
ϕδ:= sup
s∈(–∞,]
eδsϕ(s)X.
In order to state the problem in the correct framework, we assume that G: [τ,T]× Cδ(Y)→(L())satisfies
(g) for any(u,φ)∈Cδ(Y),t∈[τ,T]→G(t, (u,φ))is measurable; (g) for anyt∈R,G(t, ) = ;
(g) there exists a constantLg> such that, for anyt∈[τ,T],(u,φ), (u,φ)∈Cδ(Y),
|G(t, (u,φ)) –G(t, (u,φ))|L≤Lg(u,φ) – (u,φ)δ.
We will assume that the distributed delay term on the Banach spaceCδ(Y) is given by
Gt, (u,φ)t
:=
–∞G
t,s, (u,φ)(t+s)ds,
where the functionG: [τ,T]×(–∞, ]×Y→(L())satisfies
G(t,s, ) = , G
t,s, (u,φ)
–G
t,s, (u,φ)R≤L(s)(u,φ) – (u,φ)Y,
for the functionL> . We assume thatL(·)e–(δ+θ)·∈L(–∞, ) for certainθ> . We can
prove as in [] that (g)-(g) are all satisfied.
Hereafter, we will use the above space and the distributed delay for our problem. Using the notations above, we rewrite () in the form
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
du
dt +νAu+B(u,u) –KR( Aγφ,φ) =g(t) +G(t, (u,φ)t), dφ
dt +B(u,φ) +μ= ,
μ= Aγφ+αfγ(φ),
(u,φ)(t+τ) =ϑ(t) = (ϑ,ϑ)(t), t∈(–∞, ].
()
Remark In the weak formulation (), the termμ∇φis replaced by Aγ∇φ. This is jus-tified sincefγ(φ) is the gradientFγ(φ) and can be incorporated into the pressure gradient; see []. To simplify the notation, we setα=K= .
Remark Set∀(ω,ψ)∈Y
E(t) =ε(ω,ψ) =(ω,ψ)Y+ Fγ(ψ), L+α,
whereαis a constant large enough and independent on (ω,ψ) such thatE(t) is
nonneg-ative.
Definition A pair (u,φ) is called a weak solution to () if
(u,φ)∈C((–∞,T];Y)∩L[τ,T];V, du dt ∈L
[τ,T];V∗
,
dφ
dt,μ∈L
[τ,T];V∗
,
and (u,φ) satisfies ()and ()inV∗andV∗, respectively.
Remark If (u,φ) is a weak solution of () in the sense given above, then (u,φ) satisfies an energy equality. Namely,
E(t) –E(s) + t
s
νu+ |μ|L
dr
= t
s
g(r),u(r)+Gr, (u,φ)r
,u(r)dr ∀s,t∈[τ,T],
3 Existence of solutions
In this section we establish existence of weak and strong solution for problem () as addi-tional assumptions are satisfied and some related properties.
Theorem Assume thatg∈L(τ,T;V∗
),G: [τ,T]×Cδ(Y)→(L())satisfies(g)-(g).
Then,forϑ∈Cδ(Y),there exists a unique weak solution(u,φ)of().
Proof For the existence, we split the proof into several steps.
Step : A Galerkin scheme. Since the injection ofV⊂Yis compact. Let {(ωi,ψi),i=
(u,φ)t)) is a local Lipschitz function in (u,φ), it follows from the theory of ordinary
differ-ential equations that this equation has a local solution (um,φm).
Thus
Etmδ≤max
sup θ∈(–∞,τ–t]
eδθEmϑ(t+θ–τ),
sup θ∈[τ–t,]
eδθE(τ) +eδθ t+θ
τ
ν–g(s)∗+ LgEsmδ
ds
≤max
sup θ∈(–∞,τ–t]
eδθEmϑ(t+θ–τ),
E(τ) + t
τ
ν–g(s)∗+ LgEmsδ
ds
and
sup θ∈(–∞,τ–t]
eδθEmϑ(t+θ–τ)=sup θ≤
eδ(θ–(t–τ))Eϑ(θ)
=e–δ(t–τ)E(ϑ)δ≤E(ϑ)δ
andE(τ) =E(ϑ())≤ E(ϑ)δ, so we can obtain
Etmδ≤E(ϑ)δ+ t
τ
ν–g(s)∗+ LgEmsδ
ds.
Thus by Gronwall’s lemma, we have
Etmδ≤eLg(t–τ)E(ϑ) δ+
t
τ
ν–g(s)∗ds
.
Using this inequality, we also see that there exists a constantC, depending on some con-stants of the problem (namely,ν,Lg andg) and onτ,T andR> , such that
Etmδ≤C(τ,T,R), Em(t)≤C(τ,T,R). ()
As|(um,φm)|
Y≤Em(t), this implies that (um,φm) is bounded inL∞(τ,T;Y)∩L(τ,T,V) ∀T>τ. Noting that
μm= Aγφm+αfγφm
we get []
Aγφm
L≤cμmL+Qφm, ()
A/γ φmL≤cμmL+Qφm, ()
whereQis a monotone nondecreasing function independent on time, the initial condition
andm. Then we see thatA/γ φmis bounded inL(τ,T;D(A/γ )). With ()-(), () and () we get
d dt
Step : Approximation inCδ(Y) of the initial datum. For the initial datumϑ∈Cδ(Y) we
have used the projections in the Galerkin scheme in Step . Let us check that
Pmϑ→ϑ inCδ(Y).
Indeed assume there do not exist > and a subsequence such that
eδθmPmϑ(θ
m) –ϑ(θm)> .
One can assume thatθm→–∞. Otherwise, ifθm→θ thenPmϑ(θm)→ϑ(θ). We have
|Pmϑ(θm) –ϑ(θ)| ≤ |Pmϑ(θm) –Pmϑ(θ)|+|Pmϑ(θ) –ϑ(θ)| → asm→+∞. But with
θm→–∞asm→+∞if we denotex=limθ→–∞eδθϑ(θ), we obtain
eδθmPmϑ(θm) –ϑ(θm) = Pm
eδθmϑ(θm)
–eδθmϑ(θm)
≤ Pmeδθmϑ(θ
m)
–Pmx+Pmx–x+x–eδθmϑ(θ
m)
→.
Thus there is a contradiction.
Step : Energy method and compactness results. Now using the standard methods as in [], we can pass to the limit in () asm→ ∞and we see that (u,φ) is a weak solution of ().
From Step we get a subsequence (still) denoted by (um,φm) and using the compactness theorem
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(um,φm)∗ (u,φ) weakly star inL∞(τ,T;Y),
(um,φm)(u,φ) weakly inL(τ,T;V),
d
dt(um,φm) d
dt(u,φ) weakly inL
(τ,T;V∗
)×L(τ,T;V∗),
(um,φm)→(u,φ) strongly inL(τ,T;Y),
G(·, (um,φ)) ζ weakly inL(τ,T; (L())),
()
for allT>τ. Furthermore we can also assume that
um,φm(t)→(u,φ)(t) inYa.e.t∈(τ,T), ()
which nevertheless is not enough.
Since the injection ofVintoYis compact, the injection ofYintoV∗is compact too. So by the Ascoli-Arzela theorem we have
um,φm(t)→(u,φ)(t) inC[τ,T],V∗. ()
Then, for any{tm} ⊂[τ,T], withtm→t, we have
um,φm(tm)
and
Em(tm)E(t) weakly inY. ()
Next we will prove that
um,φm(tm)→
um,φm(t) inC[τ,T];Y.
That is, we will prove that
Em(tm)→E(t) inC
[τ,T];Y.
If this were not so we take into account that (u,φ)∈C([τ,T];Y), there would exist > andt∈[τ,T] and subsequences (relabeled the same){(um,φm)}and{tm} ⊂[τ,T] with
limm→+∞tm=tsuch that
Em(tm) –E(t)≥ ∀m.
To prove that this is absurd, we will use the energy method. Observe that the following energy inequality holds for all (um,φm):
E
m(t) +
t
s
νum(r)+μmL
dr
≤ t
s
g(r),um(r)dr+ E
m(s) +C(t–s) ∀s,t∈[τ,T], ()
whereC= D
νλandDcorresponds to the upper bound t
s
G
r,um,φmrdr≤D(t–s), τ≤s≤t≤T,
by (g), (g) and ().
On the other hand, by (), passing to the limit in (), we see that (u,φ)∈C([τ,T];Y) is a solution of a similar problem to (),
dE
dt + νu
+ |μ|
L=
g(t),u+ (ζ,u) ()
with the initial data (u,φ)(τ) =ϑ(). Therefore, it satisfies the energy equality
E(t) + s
t
νu+ |μ|L
dr
=E(s) + t
s
g(r),u(r)+Gr, (u,φ)r
,u(r)dr ∀s,t∈[τ,T].
Furthermore, from the last convergence in (), we deduce that
t
s
ζ(r)dr≤ lim
m→+∞sup
t
s
Now consider the functionsJm,J: [τ,T]→Rdefined by
Jm(t) =
E
m(t) –
t
τ
g(r),um(r)dr–Ct,
J(t) = E(t) –
t
τ
g(r),u(r)dr–Ct.
Obviously,Jm,Jare non-increasing functions. By (), we get
Jm(t)→J(t) a.e.t∈[τ,T]. ()
On the one hand, from ()
E(tm)E(t) weakly inY ()
and we get
E(t)≤ lim
m→+∞infE(tm).
On the other hand, ift=τ we get from Step and () withs=τ
lim
m→+∞supE(tm)≤E(t),
so we assume thatt≥τ. This is important, since we will approach this valuetfrom the
left by a sequence{˜tk}, i.e.,limk→+∞˜tk=twith{˜tk}being values where () holds. Since
E(t) is continuous att, for any there isk such that
J(˜tk) –J(t)≤ / ∀k≥k .
Then takingtm≥tk˜ , asJmis non-increasing and for all˜tkthe convergence () holds, and
one has
Jm(tm) –J(t)≤Jm(˜tk) –J(˜tk )+J(˜tk) –J(t)
and with tm
τ
g(r),um(r)dr→ t
τ
g(r),u(r)dr
we get
lim
m→+∞supE(tm)≤E(t)
and
and
um,φm(tm)→(u,φ)(t) inY,
um,φm→(u,φ) C[τ,T],Y.
Thus, we can finally pass to the limit in ().
The uniqueness of the solution can be obtained in the following way. Consider two weak solutions, (u,φ), (u,φ) of () with the same initial data, and denoteω=u–u,ψ= φ–φ, (u,φ)t= (u,φ)(t+s), (u,φ)t= (u,φ)(t+s). We derive as Lemma . in []
that
dy
dt≤ϒ(t)y(t) +cG
t, (u,φ)t
–Gt, (u,φ)t
L≤ϒ(t)y(t) +cLg(ω,ψ)t
δ,
wherec=cis a constant that depends only onand
y(t) =(ω,ψ)Y,
ϒ(t) =cu+
+φ
|Aγφ|L+|u|Lu
+Q
|φ|H,|φ|H
.
As (ω,ψ)(θ) = , ifθ≤τ
(ω,ψ)t
δ =sup θ≤
eδθ(ω,ψ)(t+θ)Y
≤ sup θ∈[τ–t,]
(ω,ψ)(t+θ)Y
≤ sup
r∈[τ,t]
(ω,ψ)(r)Y, ∀τ≤t≤T,
so
dy
dt≤ϒ(t)y(t) +cL
g sup r∈[τ,t]
y(r);
we have
y(t)≤y() + t
τ
ϒ(s) +cLg sup
r∈[τ,s]
y(r)ds.
Now we deduce that
sup
r∈[τ,t]
y(r)≤y() + t
τ
ϒ(s) +cLg sup
r∈[τ,s]
y(r)ds.
By the Gronwall lemma we finish the proof of uniqueness.
Proposition Assume thatg∈L(τ,T;V∗
),G: [τ,T]×Cδ(Y)→(L()) satisfies
corresponding to the initial dataϑandϑ.Then the following continuity properties hold:
Proof Arguing as in the proof of Theorem , we have
Substitutingtbyr∈[τ,t] and considering the maximum when varying thisrwe can con-clude that
Then by the Gronwall lemma, we get the result:
max
Proposition Assume thatg∈L(τ,T;V∗
),G: [τ,T]×Cδ(Y)→(L()) satisfies
as-sumptions(g)-(g).Let us denote(u,φ)(·;s,ϑ)the solution of()with initial time s.Then, for each t∈[τ,T]andϑ∈Cδ(Y)fixed,the mapping s→(u,φ)t(·;s,ϑ)∈Cδ(Y),s∈[t,τ]is
continuous.
The method of proof is similar to Proposition in [].
4 Existence of pullback attractors
In this section we will prove the existence of a pullback attractor for the problem () with the distributed delay under additional assumptions. We firstly recall some basic defini-tions and main results that we will use later about properties required of a process for a non-autonomous dynamical system in order to have a pullback attractor. These results can be found in [] and [] and here we only reproduce the statements for the sake of completeness.
Definition LetXbe a complete metric space. A family of mappings{U(t,τ),t,τ∈R,t≥
τ} ⊂C(X,X) is said to be a process inXifU(t,τ)U(τ,r) =U(t,r) for anyτ≤r≤t, and U(τ,τ) =Idfor allτ. The processU(·,·) is said to be continuous if the mapping (t,τ)→ U(t,τ)xis continuous for allx∈X.
Corollary Assume that g∈Lloc(R;V∗),G:R×Cδ(Y)→(L())satisfies assumptions (g)-(g)for anyτ<T.Then the bi-parametric family of mappings U(t,τ) :Cδ(Y)→Cδ(Y) with t≥τ,defined by
U(t,τ)ϑ= (u,φ)t,
where(u,φ)(·;τ,ϑ)is the unique weak solution of(),defines a semi-process on Cδ(Y).
Proof The proof is a consequence of Theorem and Proposition .
The following result can be obtained analogously to [], Propositions ., . with the natural changes in the delay norms.
Proposition Assume that g∈Lloc(R;V∗),G:R×Cδ(Y)→(L())satisfies assump-tions(g)-(g)for anyτ<T.Then for any bounded set B⊂Cδ(Y):
() The set weak of weak solutions{(u,φ)(·;τ,ϑ) :ϑ∈B}is bounded inL∞(τ+ ,T;V) for any > and anyT>τ+ .
() Moreover,if{ϑ() :ϑ∈B}is bounded inV,then{(u,φ)(·;τ,ϑ) :ϑ∈B}is bounded in L∞(τ,T;Y)for allT>τ.
Definition A processU onXis said to be closed if for any τ ≤t, and any sequence {xn} ⊂Xwithxn→x∈XandU(t,τ)xn→y∈X, thenU(t,τ)x=y.
Let us denote byP(X) the family of all nonempty subsets ofXand consider a family of nonempty setsDˆ={D(t) :t∈R} ⊂P(X).
Definition We say that a processUonXis pullbackDˆ-asymptotically compact if for
anyt∈Rand any sequences{τn} ⊂(–∞,t] and{xn} ⊂Xsatisfyingτn→–∞andxn∈
Denote
(Dˆ,t) =
s≤t
τ≤s
U(t,τ)D(τ)
X
∀t∈R,
where{· · ·}Xis the closure inX.
GivenA,B⊂Xwe denote bydist(A,B) the Hausdorff semi-distance inXbetween them, defined as
dist(A,B) =sup
a∈A
inf
b∈Bd(a,b).
LetDbe a nonempty class of families parameterized in timeDˆ ={D(t) :t∈R} ⊂P(X). The classDwill be called a universe inP(X).
Definition A processUonXis said to be pullbackD-asymptotically compact if it is pullbackD-asymptotically compact for anyˆ Dˆ ∈D.
It is said thatDˆ={D(t) :t∈R} ⊂P(X) is pullbackD-absorbing for processUonXif
for anyt∈Rand anyDˆ ∈D, there exists aτ(D,ˆ t)≤tsuch that
U(t,τ)D(τ)⊂D(t) ∀τ≤τ(D,ˆ t).
With the above definitions, we have the main result which is given in [].
Theorem Consider a closed process U,a universe D in P(X),and a familyDˆ={D(t) :
t∈R} ⊂P(X)which is pullback D-absorbing for U and assume that U is pullbackDˆ
-asymptotically compact.Then the family A(t)defined by A(t) =Dˆ∈D(D,ˆ t) X
has the fol-lowing properties:
() For anyt∈Rthe setA(t)is a nonempty compact subset ofXandA(t)⊂(Dˆ,t). () Ais pullbackD-attracting,i.e.limr→–∞dist(U(t,τ)D(τ),A(t)) = for allDˆ ∈Dand
anyt∈R.
() Ais invariant,i.e.U(t,τ)A(τ) =A(t)for all(t,τ)∈R. () IfDˆ∈DthenA(t) =(Dˆ,t)⊂D(t)
X
for allt∈R.
The familyA(t) is minimal in the sense that ifCˆ ={C(t) :t∈R} ⊂P(X) is a family of closed sets such that, for anyDˆ ={D(t) :t∈R},limr→–∞dist(U(t,τ)D(τ),C(t)) = , then
A(t)⊂C(t).
Remark Under the assumptions of Theorem , the familyA(t) is called the minimal pullbackD-attractor for the processU.
Theorem Assume thatg∈L
loc(R;V∗),G:R×Cδ(Y)→(L())satisfies(g)-(g)for
anyτ<T, (u,φ)(·;τ,ϑ)is the unique weak solution to(),then the following estimates hold for all t>τ,and anyσ∈(,α)such that(α–σ)λ≤δ:
(u,φ)t
δ≤e
–((α–σ)λ–Lg)(t–τ)E(ϑ)
δ
+ t
τ
e–((α–σ)λ–Lg)(t–s)σ–g(s) ∗+c
ds, ()
σ
t
τ
(u,φ)ds≤eLg(t–τ)E(ϑ) δ
+eLgt–(α–σ)λτ t
τ
e–((α–σ)λ–Lg)sσ–g(s) ∗+c
ds, ()
where cis a positive constant.
Proof Multiplying ()withu, ()withμ, ()with φand adding the resulting
equa-tions, we derive as the proof of Theorem that
dE
dt + νu
+ |μ|
L=
u,g(t) +Gt, (u,φ)t
, ()
where
E(t) =(u,φ)(t)Y+ Fγφ(t), L+α
andαis a constant large enough to ensure theE(t) is nonnegative.
We can conclude that
dE
dt + νu
+ |μ|
L≤σ–g∗+σu+ Lg(u,φ)tδ. ()
Thus,
dE
dt + (ν–σ)u
+ |μ|
L≤σ–g∗+ Lg(u,φ)t
δ.
So
dE
dt + (ν–σ)λ|u|
L+ |μ|L≤σ–g∗+ Lg(u,φ)t
δ.
Setα= νand as we can chooseσ> , such thatα>σandcis a positive constant, then
dE
dt ≤–(α–σ)λE+σ
–g
∗+ Lg(u,φ)t
δ+c.
So
E(t)≤e–(α–σ)λ(t–τ)E(τ)
+ t
τ
e–(α–σ)λ(s–τ)σ–g
∗+ Lg(u,φ)sδ+c
Consequently,
Etδ≤max
sup θ∈(–∞,τ–t]
eδθEϑ(t+θ–τ), sup θ∈[τ–t,]
eδθ–(α–σ)λ(t+θ–τ)E(τ)
+eδθ t+θ
τ
e–(α–σ)λ(s+θ–τ)σ–g(s)
∗+ LgEsδ+c
ds
.
We assume that, moreover,σsatisfies (α–σ)λ≤δ.
On the one hand
sup θ∈(–∞,τ–t]
eδθEϑ(t+θ–τ)=sup θ≤
eδ(θ–(t–τ))Eϑ(θ)
=e–δ(t–τ)E(ϑ)δ≤e–(α–σ)λ(t–τ)E(ϑ) δ
and on the other hand
sup θ∈[τ–t,]
eδθ–(α–σ)λ(t+θ–τ)E(τ)≤e–(α–σ)λ(t–τ)E(τ)
and
sup θ∈[τ–t,]
eδθ t+θ
τ
e–(α–σ)λ(s+θ–τ)σ–g(s)
∗+ LgEsδ+c
ds
≤ t τ
e–(α–σ)λ(s–τ)σ–g(s)
∗+ LgEsδ+c
ds
so we get
Etδ≤e–(α–σ)λ(t–τ)E(ϑ)δ
+ t
τ
e–(α–σ)λ(t–τ)σ–g(s)
∗+ LgEsδ+c
ds ∀t≥τ.
Then by Gronwall’s lemma, we have
(u,φ)t
δ≤ Etδ
≤e–((α–σ)λ–Lg)(t–τ)E(ϑ)
δ
+ t
τ
e–((α–σ)λ–Lg)(t–s)σ–g(s) ∗+c
ds ∀t≥τ.
With () and the above equality we get the result ().
From now on, we will assume that there exist ≤σ≤αsuch thatLg≤(α–σ)λ≤δ
and
–∞
eβsσ–g(s)∗+c
where
β=(α–σ)λ–Lg
.
Remark If we assume thatg∈Lloc(R;V∗), then equation () is equivalent to
t
–∞e
–β(t–s)σ–g(s) ∗+c
ds< +∞, ∀t∈R.
Corollary Assume thatg∈L
loc(R;V∗),G:R×Cδ(Y)→(L())satisfies assumptions
(g)-(g)for anyτ <T and condition()is satisfied,then the familyDˆ={D(t) :t∈R},
with D(t) =DCδ(Y)(,ρ(t)),where
ρ(t) = + t
–∞e
–β(t–s)σ–g(s) ∗+c
ds
is pullback absorbing for bounded sets for the semi-process U.
Proof The proof follows immediately from Theorem .
Proposition Under the assumptions of Corollary , the semi-process U is Dˆ
-asymptotically compact.
Proof Let (um,φm) be a sequence in Theorem , we will prove this sequence is relatively compact inCδ(Y).
Letτ(D,ˆ t,h) <t–h– be such that
e–β(t–h–)eβτE(ϑ)
δ≤ ∀τ≤τ(D,ˆ t,h),ϑ∈D(τ).
Consider fixedτ≤τ(D,ˆ t,h) andϑ∈D(τ).
Firstly, from the result () of Theorem and using the definition of the norm · δ, we can deduce that
(u,φ)tY≤ +e–β(t–h–)
t
–∞e
βs(σ λ
)–g(s)+c
ds
ρ(t) ∀r∈[t–h– ,t].
Secondly, we derive from Theorem by integrating () betweenr– andr
Em(r) –Em(r– ) + (α–σ)
r
r–
um
+ r
r–
μm(t)Ldt
≤σ–g∗+ Lg
r
r–
um,φm s
δds,
then from () and
u(t),φ(t)Y≤E(t)≤Qu(t),φ(t)
Y
whereQis a monotone non-decreasing function independent of the time and the initial
data, we can deduce that
(α–σ) the details) the inner product by integrating betweenr– andr,
dY
By integrating betweenr– andrand from () we can obtain
Recalling (), () and (), and employing standard Sobolev inequalities, it is easy to check
By integrating betweenr– andr, we obtain r
Similar to the proof of Theorem , we also conclude that
r
Then the following proof can be obtained analogously to Step of Theorem . To avoid
unnecessary repetition, we skip the proof.
Theorem Assume thatg∈L
loc(R;V∗),G:R×Cδ(Y)→(L())satisfies assumptions
(g)-(g)for anyτ<T.Also,suppose that Lg≤(α–σ)λ.Then the semi-process U defined
in Cδ(Y)associated to() has a pullback attractor A={A(t)}.Moreover,every A(t)is connected in Cδ(Y).
Proof The existence of the pullback attractor is a direct consequence of Theorem , Corol-lary , CorolCorol-lary and Proposition . The connectedness follows from Propositions and and the fact that the spaceCδ(Y) is connected.
Acknowledgements
The paper was supported by the Fundamental Research Funds for the Central Universities (2013B07314). The author also wants to thank the anonymous referees for their valuable comments on the paper.
Competing interests
The author declares that there is no competition of interests regarding the publication of this paper.
Authors’ contributions
The author wrote the paper and approved the final manuscript.
Authors’ information
Min Yang (1978-), female, lecturer, engaged in the research of numerical solution of partial differential equations.
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Received: 16 February 2017 Accepted: 3 August 2017
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