Access the full text.
Sign up today, get DeepDyve free for 14 days.
DM Anderson, GB McFadden, AA Wheeler (1998)
Diffuse-interface methods in fluid mechanicsAnnu. Rev. Fluid Mech., 30
T Caraballo, J Real (2004)
Attractors for 2D-Navier-Stokes models with delaysJ. Differ. Equ., 205
TT Medjo (2016)
A two-phase flow model with delaysDiscrete Contin. Dyn. Syst., 21
CG Gal, M Grasselli (2010)
Longtime behavior for a model of homogeneous incompressible two-phase flowsDiscrete Contin. Dyn. Syst., 28
TT Medjo (2012)
Pullback attractors for a non-autonomous homogeneous two-phase flow modelJ. Differ. Equ., 253
T Blesgen (1999)
A generalization of the Navier-Stokes equation to two-phase flowsJ. Appl. Phys., 32
J García-Luengo, P Marín-Rubio, J Real (2012)
Pullback attractors in V for non-autonomous 2D-Navier-Stokes equations and their tempered behaviourJ. Differ. Equ., 252
P Marín-Rubio, J Real, J Valero (2011)
Pullback attractors for a two-dimensional Navier-Stokes model in an infinte delay caseNonlinear Anal., 74
PE Kloeden, B Schmalfuss (1997)
Nonautonomous systems, cocycle attractors and variable time-step discretizationNumer. Algorithms, 14
CG Gal, M Grasselli (2010)
Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2DAnn. Inst. Henri Poincaré, Anal. Non Linéaire, 27
PE Kloeden, DJ Stonier (1998)
Cocycle attractors in nonautonomously perturbed differential equationsDyn. Contin. Discrete Impuls. Syst., 4
TT Medjo (2016)
Attractors for a two-phase flow model with delaysDiffer. Integral Equ., 29
R Temam (2000)
Infinite dimensional dynamical systems in mechanics and physicsAppl. Math. Sci., 68
JJ Feng, C Liu, J Shen, P Yue (2005)
Advantages and Challenges
J García-Luengo, P Marín-Rubio, J Real (2014)
Regularity of pullback attractors and attraction in H1 in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delayDiscrete Contin. Dyn. Syst., 34
E Feireisl, H Petzeltova, E Rocca, G Schimperna (2010)
Analysis of a phase-field model for two-phase compressible fluidsMath. Models Methods Appl. Sci., 20
College of Aerospace Engineering, In this paper we study the existence of solutions for a coupled Nanjing University Of Aeronautics and Astronautics, Nanjing, 210016, Allen-Cahn-Navier-Stokes model in two dimensions with an external force containing China infinite delay effects in the weighted space C (Y). We prove the existence of pullback Department of Mathematics, attractors for the dynamical system associated to the problem under more general College of Sciences, Hohai University, Nanjing, 210098, China 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 resulting 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 velocity u and the order parame- ter φ. 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 attractor A and establish the existence of an exponential attractor E which entails that A 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 Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Yang Advances in Difference Equations (2017) 2017:238 Page 2 of 20 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) values τ < T in R.Thenwecon- sider the following system: ∂u – νu +(u ·∇)u + ∇p – Kμ∇φ = g(t)+ G(t,(u, φ) ), ⎪ t ∂t div u =, () ∂φ ⎪ + u ·∇φ + μ =, ∂t μ =– φ + αf (φ), in × (τ, T). In (), the unknown functions are the velocity u =(u , u ) of the fluid, its pressure p and the order parameter φ. The quantity μ is the variational derivative of the following free energy functional: F(φ)= |∇φ| + αF(φ) ds, where F(r)= f (ζ ) dζ,the constants ν > and K > 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 number τ ∈ R is the initial time. We endow ()with the boundary condition ∂φ u =, = on ∂ × (τ, T), ∂η where ∂ is the boundary of and η is its outward normal. The initial condition is given by (u, φ)(t + τ)= ϑ(t)=(ϑ , ϑ )(t), t ∈ (–∞, ]. () The terms g(t) is a non-delayed external force field, G(t,(u, φ) ) is another external force containing some hereditary characteristics and we denote by (u, φ) the function defined on (–∞,] by the relation (u, φ) =(u, φ)(t + s), s ∈ (–∞,]. t Yang Advances in Difference Equations (2017) 2017:238 Page 3 of 20 We assume that f ∈ C (R)satisfies lim f (r)>, f (r) ≤ c + |r| , ∀r ∈ R,() |r|→+∞ where c is some positive constant and k ∈ [, +∞) is fixed. So we get k+ f (r) ≤ c + |r| , ∀r ∈ R.() If X is a real Hilbert space with inner product (·, ·) , we will denote the induced norm ∗ ∗ by |·| while X will indicate its dual with ·, · for the duality between X and X and the norm by · .Weset V = u ∈ C (): div u = in . We denote by H and V the closure of V in (L ()) and (H ()) ,respectively. Thescalar product in H is denoted by (·, ·) and the associated norm by |·| .Moreover, thespace L L V is endowed with the scalar product / (u, v) = (∂ u, ∂ v) , u = (u, u) . x x L i i i= We define the operator A by A u = Pu, ∀u ∈ D(A )= H () ∩ V , where P is the Leray-Helmholtz projector from L ()onto H .Then A is a self-adjoint positive unbounded operator in H which is associated with the scalar product defined – above. Furthermore, A is a compact linear operator on H and |A ·| is a norm on L D(A ) that is equivalent to the H -norm. From (), we can find γ > such that lim f (r)>γ >. () |r|→+∞ Then we can define the linear positive unbounded operator A on L ()by A φ =–φ + γφ, ∀φ ∈ D(A ), γ γ where ∂ρ D(A )= ρ ∈ H (); = on ∂ . ∂η – Note that A is a compact linear operator on L ()and |A ·| is a norm on D(A ) γ L γ that is equivalent to the H -norm. We set V = H (). Furthermore we denote by λ > a positive constant satisfying / λ |ω| ≤ ω ∀ω ∈ V ; λ A ψ ≤|A ψ | ∀ψ ∈ H (). () γ L L L Yang Advances in Difference Equations (2017) 2017:238 Page 4 of 20 We also introduce the bilinear operators B , B and their associated trilinear forms b , b as well as the coupling mapping R ,which aredefinedfrom D(A ) × D(A )into H , / D(A ) × D(A )into L ()and L () × D(A )into H , respectively. More precisely, we γ set B (u, v), w = (u ·∇)v · w dx = b (u, v, w), ∀u, v, w ∈ D(A ), B (u, φ), ρ = (u ·∇)φ ρ dx = b (u, φ, ρ), ∀u ∈ D(A ), φ, ρ ∈ D(A ), γ R (μ, φ), w / = μ[∇φ · w] dx = b (w, φ, μ), ∀w ∈ D(A ), (μ, φ) ∈ L () × D A . Note that R (μ, φ)= P μ∇φ and B , B and R satisfy the following estimates: / / / / B (u, v) ≤ c|u| u |v| v , ∀u, v ∈ V,() ∗ V L L / / / / B (u, φ) ≤ c|u| u |φ| φ , ∀u ∈ V , φ ∈ V,() ∗ V L L / / R (A φ, ρ) ≤ c ρ |A ρ| |A φ| , ∀φ, ρ ∈ D(A ). () γ γ γ L γ V L Nowwedefine theHilbert spaces Y and V by Y = H × H (), V = V × D(A )endowed γ with the scalar products whose associated norms are – – / (u, φ) = K |u| + |∇φ| + γ |φ| = K |u| + A φ ; Y L L L L L (u, φ) = u + |A φ| . γ V L – Let f (r)= f (r)– α γ r and observe that f still satisfies ()with γ in place of γ since γ γ ≤ α,and F (r)= f (ζ ) dζ is bounded from below. γ γ There are several phase spaces which allow us to deal with infinite delays. For instance, for a given δ > , and a given Banach space X, we may consider the space δs C (X)= ϕ ∈ C (–∞,]; X : ∃ lim e ϕ(s) ∈ X s→–∞ with the norm δs ϕ := sup e ϕ(s) . s∈(–∞,] In order to state the problem in the correct framework, we assume that G :[τ, T] × C (Y) → (L ()) satisfies (g) for any (u, φ) ∈ C (Y), t ∈ [τ, T] → G(t,(u, φ)) is measurable; (g) for any t ∈ R, G(t,) = ; (g) there exists a constant L > such that, for any t ∈ [τ, T], (u , φ ), (u , φ ) ∈ C (Y), g δ |G(t,(u , φ )) – G(t,(u , φ ))| ≤ L (u , φ )–(u , φ ) . g δ An example of an operator satisfying assumptions (g)-(g) is given here. Yang Advances in Difference Equations (2017) 2017:238 Page 5 of 20 We will assume that the distributed delay term on the Banach space C (Y)is given by G t,(u, φ) := G t, s,(u, φ)(t + s) ds, t –∞ where the function G :[τ, T] × (–∞,] × Y → (L ()) satisfies G (t, s,) = , G t, s,(u, φ ) – G t, s,(u , φ ) ≤ L (s) (u , φ )–(u , φ ) , R Y –(δ+θ)· for the function L >. We assume that L (·)e ∈ L (–∞, ) for certain θ >. 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 ()inthe form du + νA u + B (u, u)– KR ( A φ, φ)= g(t)+ G(t,(u, φ) ), ⎪ γ t dt dφ + B (u, φ)+ μ =, dt () ⎪ μ = A φ + αf (φ), γ γ (u, φ)(t + τ)= ϑ(t)=(ϑ , ϑ )(t), t ∈ (–∞,]. Remark In the weak formulation (), the term μ∇φ is replaced by A ∇φ.Thisisjus- tified since f (φ) is the gradient F (φ) and can be incorporated into the pressure gradient; see []. To simplify the notation, we set α = K =. Remark Set ∀(ω, ψ) ∈ Y E(t)= ε(ω, ψ)= (ω, ψ) + F (ψ), + α , γ Y L where α is a constant large enough and independent on (ω, ψ)such that E(t) is nonneg- ative. Definition Apair (u, φ) is called a weak solution to ()if du ∗ (u, φ) ∈ C((–∞, T]; Y) ∩ L [τ, T]; V , ∈ L [τ, T]; V , dt dφ ∗ , μ ∈ L [τ, T]; V , dt ∗ ∗ and (u, φ)satisfies () and () in V and V ,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)+ ν u +|μ| dr = g(r), u(r) + G r,(u, φ) , u(r) dr ∀s, t ∈ [τ, T], where E(t)= ε(u, φ). Yang Advances in Difference Equations (2017) 2017:238 Page 6 of 20 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 that g ∈ L (τ, T; V ), G :[τ, T] ×C (Y) → (L ()) satisfies (g)-(g). Then, for ϑ ∈ C (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 of V ⊂ Y is compact. Let {(ω , ψ ), i = i i ,,,...}⊂ V be an orthonormal basis of Y,where {ω , i = ,,...}, {ψ , i = ,,...} are i i eigenvectors of A and A ,respectively. We set V = Y = span{(ω , ψ ), (ω , ψ ),..., γ m m m m (ω , ψ )}.Welookfor (u , φ ) ∈ Y , the solution to the ordinary differential equations m m m du m m m m m m + P (νA u + B (u , u )– R ( A φ , φ )) γ ⎪ dt ⎪ m m m = P (g(t)+ G(t,(u , φ ) )), dφ m m m m () + P (B (u , φ )+ μ )=, dt m m m m ⎪ μ = P ( A φ + f (φ )), γ γ ⎪ m m m (u , φ )(t + τ)= P ϑ(t), t ∈ (–∞,], m m m m m where P =(P , P ): H × L () → V is the orthogonal projection. Since P (, G(t, (u, φ) )) is a local Lipschitz function in (u, φ), it follows from the theory of ordinary differ- m m ential equations that this equation has a local solution (u , φ ). Next we will deduce apriori estimates that ensure that the solutions do exist for all time. Step : Apriori estimates. By taking the scalar product in H of () with u ,thentaking m the scalar product in L ()of () with μ ,weget dE m m m m m m +ν u + μ = g(t), u + G t, u , φ , u,() L t dt m m m where E (t)= ε(u (t), φ (t)). Then by (g), we have dE m m m m m m +ν u + μ ≤ g(t) u +L u , φ u g L ∗ t δ L dt m – m m ≤ ν u + ν g(t) +L u , φ ; ∗ t δ then dE m m – m m + ν u + μ ≤ ν g(t) +L u , φ g L ∗ t δ dt – m ≤ ν g(t) +L E , ∗ δ where δs E = sup e E(s) s∈(–∞,] and therefore t t m m m – m E (t)+ ν u + μ ds ≤ E(τ)+ ν g(s) +L E ds. () g L ∗ δ τ τ Yang Advances in Difference Equations (2017) 2017:238 Page 7 of 20 Thus m δθ m E ≤ max sup e E ϑ(t + θ – τ) , θ ∈(–∞,τ–t] t+θ δθ δθ – m sup e E(τ)+ e ν g(s) +L E ds ∗ δ θ ∈[τ–t,] τ δθ m ≤ max sup e E ϑ(t + θ – τ) , θ ∈(–∞,τ–t] – m E(τ)+ ν g(s) +L E ds ∗ δ and δθ m δ(θ–(t–τ)) sup e E ϑ(t + θ – τ) = sup e E ϑ(θ) θ ∈(–∞,τ–t] θ ≤ –δ(t–τ) = e E(ϑ) ≤ E(ϑ) δ δ and E(τ)= E(ϑ()) ≤ E(ϑ) , so we can obtain m – m E ≤ E(ϑ) + ν g(s) +L E ds. t s δ δ ∗ δ Thus by Gronwall’s lemma, we have m L (t–τ) – E ≤ e E(ϑ) + ν g(s) ds . δ δ ∗ Using this inequality, we also see that there exists a constant C, depending on some con- stants of the problem (namely, ν, L and g)and on τ , T and R >, such that m m E ≤ C(τ, T, R), E (t) ≤ C(τ, T, R). () m m m m m ∞ As |(u , φ )| ≤ E (t), this implies that (u , φ )is bounded in L (τ, T; Y) ∩ L (τ, T, V) ∀T > τ.Notingthat m m m μ = A φ + αf φ γ γ we get [] m m m A φ ≤ c μ + Q φ , () γ L L / m m m A φ ≤ c μ + Q φ,() L L where Q is a monotone nondecreasing function independent on time, the initial condition / m / and m.Thenwesee that A φ is bounded in L (τ, T; D(A )). With ()-(), ()and () γ γ we get m m ∗ ∗ u , φ is bounded in L τ, T; V × L τ, T; V . dt Yang Advances in Difference Equations (2017) 2017:238 Page 8 of 20 Step : Approximation in C (Y) of the initial datum. For the initial datum ϑ ∈ C (Y)we δ δ have used theprojections in theGalerkinschemeinStep.Let us checkthat P ϑ → ϑ in C (Y). Indeed assume there do not exist > and a subsequence such that δθ m e P ϑ(θ )– ϑ(θ ) > . m m One can assume that θ → –∞.Otherwise,if θ → θ then P ϑ(θ ) → ϑ(θ). We have m m m m m m m |P ϑ(θ )– ϑ(θ)|≤|P ϑ(θ )– P ϑ(θ)| + |P ϑ(θ)– ϑ(θ)|→ as m → +∞.But with m m δθ θ → –∞ as m → +∞ if we denote x = lim e ϑ(θ), we obtain m θ →–∞ δθ m m δθ δθ m m e P ϑ(θ )– ϑ(θ ) = P e ϑ(θ ) – e ϑ(θ ) m m m m m δθ m m δθ ≤ P e ϑ(θ ) – P x + P x – x + x – e ϑ(θ ) m 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 ()as m →∞ and we see that (u, φ) is a weak solution of (). m m From Step we get a subsequence (still) denoted by (u , φ ) and using the compactness theorem m m ∞ (u , φ ) (u, φ)weaklystarin L (τ, T; Y), m m (u , φ ) (u, φ)weaklyin L (τ, T; V), d d m m ∗ ∗ () (u , φ ) (u, φ)weaklyin L (τ, T; V ) × L (τ, T; V ), dt dt ⎪ m m ⎪ (u , φ ) → (u, φ)stronglyin L (τ, T; Y), m G(·,(u , φ)) ζ weakly in L (τ, T;(L ()) ), for all T > τ . Furthermore we can also assume that m m u , φ (t) → (u, φ)(t)in Y a.e. t ∈ (τ, T), () whichneverthelessisnot enough. Since the injection of V into Y is compact, the injection of Y into V is compact too. So by the Ascoli-Arzela theorem we have m m ∗ u , φ (t) → (u, φ)(t)in C [τ, T], V.() Then, for any {t }⊂ [τ, T], with t → t,wehave m m m m m m u , φ (t ) u , φ (t)weaklyin Y () m Yang Advances in Difference Equations (2017) 2017:238 Page 9 of 20 and E (t ) E(t)weaklyin Y.() Next we will prove that m m m m u , φ (t ) → u , φ (t)in C [τ, T]; Y . That is, we will prove that E (t ) → E(t)in C [τ, T]; Y . If this were not so we take into account that (u, φ) ∈ C([τ, T]; Y), there would exist > m m and t ∈ [τ, T] and subsequences (relabeled the same) {(u , φ )} and {t }⊂ [τ, T]with m lim t = t such that m→+∞ m E (t )– E(t ) ≥ ∀m. m To prove that this is absurd, we will use the energy method. Observe that the following m m energy inequality holds for all (u , φ ): m m m E (t)+ ν u (r) + μ dr m m ≤ g(r), u (r) dr + E (s)+ C(t – s) ∀s, t ∈ [τ, T], () where C = and D corresponds to the upper bound νλ m m G r, u , φ dr ≤ 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 asolutionofasimilarproblem to (), dE +ν u +|μ| = g(t), u +(ζ , u)() dt with the initial data (u, φ)(τ)= ϑ(). Therefore, it satisfies the energy equality E(t)+ ν u +|μ| dr = E(s)+ g(r), u(r) + G r,(u, φ) , u(r) dr ∀s, t ∈ [τ, T]. Furthermore, from the last convergence in (), we deduce that t t ζ (r) dr ≤ lim sup G r,(u, φ) dr ≤ D(t – s) ∀τ ≤ s ≤ t ≤ T. m→+∞ s s Yang Advances in Difference Equations (2017) 2017:238 Page 10 of 20 Now consider the functions J , J :[τ, T] → R defined by m m J (t)= E (t)– g(r), u (r) dr – Ct, J(t)= E(t)– g(r), u(r) dr – Ct. Obviously, J , J are non-increasing functions. By (), we get J (t) → J(t)a.e. t ∈ [τ, T]. () On the one hand, from () E(t ) E(t)weaklyin Y () and we get E(t ) ≤ lim inf E(t ). m m→+∞ On the other hand, if t = τ we get from Step and ()with s = τ lim sup E(t ) ≤ E(t ), m m→+∞ so we assume that t ≥ τ.Thisisimportant,since we will approach this value t from the ˜ ˜ ˜ left by a sequence {t }, i.e., lim t = t with {t } being values where ()holds.Since k k→+∞ k k E(t) is continuous at t , for any there is k such that J(t )– J(t ) ≤ / ∀k ≥ k . k Then taking t ≥ t ,as J is non-increasing and for all t the convergence ()holds,and m ˜ m k one has ˜ ˜ ˜ J (t )– J(t ) ≤ J (t )– J(t ) + J(t )– J(t ) m m m k k k and with t t m g(r), u (r) dr → g(r), u(r) dr τ τ we get lim sup E(t ) ≤ E(t ) m m→+∞ and E(t ) → E(t )in Y m Yang Advances in Difference Equations (2017) 2017:238 Page 11 of 20 and m m u , φ (t ) → (u, φ)(t )in Y, m m m u , φ → (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 denote ω = u – u , ψ = φ – φ ,(u , φ ) =(u , φ )(t + s), (u , φ ) =(u , φ )(t + s). We derive as Lemma . in [] t t that dy ≤ ϒ(t)y(t)+ c G t,(u , φ ) – G t,(u , φ ) ≤ ϒ(t)y(t)+ cL (ω, ψ) , t t t L δ dt where c = c is a constant that depends only on and y(t)= (ω, ψ) , ϒ(t)= c u + + φ |A φ | + |u | u γ L L + Q |φ | , |φ | . H H As (ω, ψ)(θ)=, if θ ≤ τ δθ (ω, ψ) = sup e (ω, ψ)(t + θ) δ Y θ ≤ ≤ sup (ω, ψ)(t + θ) θ ∈[τ–t,] ≤ sup (ω, ψ)(r) , ∀τ ≤ t ≤ T, r∈[τ,t] so dy ≤ ϒ(t)y(t)+ cL sup y(r); dt r∈[τ,t] we have y(t) ≤ y() + ϒ(s)+ cL sup y(r) ds. τ r∈[τ,s] Now we deduce that sup y(r) ≤ y() + ϒ(s)+ cL sup y(r) ds. r∈[τ,t] τ r∈[τ,s] By the Gronwall lemma we finish the proof of uniqueness. ∗ Proposition Assume that g ∈ L (τ, T; V ), G :[τ, T] × C (Y) → (L ()) satisfies as- sumptions (g)-(g). Let us denote by (u , φ )(·; τ, ϑ ), (u , φ )(·; τ, ϑ ) the weak solutions Yang Advances in Difference Equations (2017) 2017:238 Page 12 of 20 corresponding to the initial data ϑ and ϑ . Then the following continuity properties hold: L t g (L +ϒ(s)) ds max (u , φ )–(u , φ ) ≤ ϑ () – ϑ () + ϑ – ϑ e , r∈[τ,t] δ L t g (L +ϒ(s)) ds (u , φ ) –(u , φ ) ≤ + ϑ – ϑ e . t t δ Proof Arguing as in the proof of Theorem ,wehave dy ≤ ϒ(t)y(t)+ G t,(u , φ ) – G t,(u , φ ) , ω , t t dt where y(t)= (u – u , φ – φ ) = (ω, ψ) , Y Y ϒ(t)= c u + + φ |A φ | + |u | u γ L L + Q |φ | , |φ | . H H As δθ (ω, ψ) = sup e (ω, ψ)(t + θ) δ Y θ ≤ δθ = max sup e ϑ (t – τ + θ)– ϑ (t – τ + θ) , θ ∈(–∞,τ–t] δθ sup e (ω, ψ)(t + θ) θ ∈[τ–t,] δ(τ–t) ≤ max e ϑ – ϑ , max (ω, ψ)(θ),() θ ∈[τ,t] we conclude that δ(τ–s) y ≤ y() + L ϑ – ϑ e ω(s) ds g δ t t + L ω(s) max (ω, ψ)(θ) ds + ϒ(s)y(s) ds. θ ∈[τ,t] τ τ Substituting t by r ∈ [τ, t] and considering the maximum when varying this r we can con- clude that max y ≤ y() + ϑ – ϑ + L + ϒ(s) max y(r) ds. g r∈[τ,t] δ r∈[τ,s] Then by the Gronwall lemma, we get the result: L t g (L +ϒ(s)) ds max (u , φ )–(u , φ ) ≤ ϑ () – ϑ () + ϑ – ϑ e .() r∈[τ,t] δ With equations ()and (), we get the last inequality. Yang Advances in Difference Equations (2017) 2017:238 Page 13 of 20 ∗ Proposition Assume that g ∈ L (τ, T; V ), G :[τ, T] × C (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] and ϑ ∈ C (Y) fixed, the mapping s → (u, φ) (·; s, ϑ) ∈ C (Y), s ∈ [t, τ] is δ t δ continuous. Themethodofproof is similartoProposition 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 Let X be 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 in X if U(t, τ)U(τ, r)= U(t, r) for any τ ≤ r ≤ t,and U(τ, τ)= Id for all τ.The process U(·, ·) is said to be continuous if the mapping (t, τ) → U(t, τ)x is continuous for all x ∈ X. ∗ Corollary Assume that g ∈ L (R; V ), G : R × C (Y) → (L ()) satisfies assumptions loc (g)-(g) for any τ < T. Then the bi-parametric family of mappings U(t, τ): C (Y) → C (Y) δ δ with t ≥ τ , defined by U(t, τ)ϑ =(u, φ) , where (u, φ)(·; τ, ϑ) is theuniqueweaksolutionof (), defines a semi-process on C (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 ∈ L (R; V ), G : R × C (Y) → (L ()) satisfies assump- loc tions (g)-(g) for any τ < T. Then for any bounded set B ⊂ C (Y): () The set weak of weak solutions {(u, φ)(·; τ, ϑ): ϑ ∈ B} is bounded in L (τ + , T; V) for any > and any T > τ + . () Moreover, if {ϑ() : ϑ ∈ B} is bounded in V , then {(u, φ)(·; τ, ϑ): ϑ ∈ B} is bounded in L (τ, T; Y) for all T > τ . Definition Aprocess U on X is said to be closed if for any τ ≤ t, and any sequence {x }⊂ X with x → x ∈ X and U(t, τ)x → y ∈ X,then U(t, τ)x = y. n n n Let us denote by P(X) the family of all nonempty subsets of X and consider a family of nonempty sets D = {D (t): t ∈ R}⊂ P(X). Definition We say that a process U on X is pullback D -asymptotically compact if for any t ∈ R and any sequences {τ }⊂ (–∞, t]and {x }⊂ X satisfying τ → –∞ and x ∈ n n n n D (τ ) for all n,the sequence {U(t, τ )x } is relatively compact in X. n n n Yang Advances in Difference Equations (2017) 2017:238 Page 14 of 20 Denote (D , t)= U(t, τ)D (τ) ∀t ∈ R, s≤t τ ≤s where {···} is the closure in X. Given A, B ⊂ X we denote by dist(A, B)the Hausdorff semi-distance in X between them, defined as dist(A, B)= sup inf d(a, b). b∈B a∈A Let D be a nonempty class of families parameterized in time D = {D(t): t ∈ R}⊂ P(X). The class D will be called a universe in P(X). Definition Aprocess U on X is said to be pullback D-asymptotically compact if it is ˆ ˆ pullback D-asymptotically compact for any D ∈ D. It is said that D = {D (t): t ∈ R}⊂ P(X) is pullback D-absorbing for process U on X if ˆ ˆ for any t ∈ R and any D ∈ D,there exists a τ (D, t) ≤ t such that U(t, τ)D(τ) ⊂ D (t) ∀τ ≤ τ (D, t). With the above definitions, we have the main result which is given in []. Theorem Consider a closed process U, auniverse D in P(X), and a family D = {D (t): t ∈ R}⊂ P(X) which is pullback D-absorbing for U and assume that U is pullback D - asymptotically compact. Then the family A(t) defined by A(t)= (D, t) has the fol- D∈D lowing properties: () For any t ∈ R the set A(t) is a nonempty compact subset of X and A(t) ⊂ (D , t). () A is pullback D-attracting, i.e. lim dist(U(t, τ)D(τ), A(t)) = for all D ∈ D and r→–∞ any t ∈ R. () A is invariant, i.e. U(t, τ)A(τ)= A(t) for all (t, τ) ∈ R . ˆ ˆ () If D ∈ D then A(t)= (D , t) ⊂ D (t) for all t ∈ R. The family A(t) is minimal in the sense that if C = {C(t): t ∈ R}⊂ P(X)is a family of closed sets such that, for any D = {D(t): t ∈ R}, lim dist(U(t, τ)D(τ), C(t)) = , then r→–∞ A(t) ⊂ C(t). Remark Under the assumptions of Theorem ,the family A(t) is called the minimal pullback D-attractor for the process U. Proposition Let X be a connected metric space. Assume that the semi-process U satisfies additionally the requirement that for every t and x ∈ Xthe map τ → U(t, τ)x, τ ∈ (–∞, t] is continuous. If U possesses a pullback attractor A, then A(t) is connected for every t ∈ R. Yang Advances in Difference Equations (2017) 2017:238 Page 15 of 20 ∗ Theorem Assume that g ∈ L (R; V ), G : R × C (Y) → (L ()) satisfies (g)-(g) for loc any τ < T,(u, φ)(·; τ, ϑ) is the unique weak solution to (), then the following estimates hold for all t > τ , and any σ ∈ (, α ) such that (α – σ )λ ≤ δ: –((α –σ )λ –L )(t–τ) g (u, φ) ≤ e E(ϑ) δ δ –((α –σ )λ –L )(t–s) – g + e σ g(s) + c ds,() L (t–τ) σ (u, φ) ds ≤ e E(ϑ) L t–(α –σ )λ τ –((α –σ )λ –L )s – g g + e e σ g(s) + c ds,() where c is a positive constant. Proof Multiplying () with u,() with μ,() with φ and adding the resulting equa- tions, we derive as the proof of Theorem that dE +ν u +|μ| = u, g(t)+ G t,(u, φ),() dt where E(t)= (u, φ)(t) + F φ(t) , + α γ Y L and α is a constant large enough to ensure the E(t) is nonnegative. We can conclude that dE – +ν u +|μ| ≤ σ g + σ u +L (u, φ).() g t L δ dt Thus, dE – +(ν – σ ) u +|μ| ≤ σ g +L (u, φ) . ∗ g t L δ dt So dE – +(ν – σ )λ |u| +|μ| ≤ σ g +L (u, φ) . ∗ g t L L δ dt Set α =ν and as we can choose σ >,suchthat α > σ and c is a positive constant, then dE – ≤ –(α – σ )λ E + σ g +L (u, φ) + c . ∗ g t dt So –(α –σ )λ (t–τ) E(t) ≤ e E(τ) –(α –σ )λ (s–τ) – + e σ g +L (u, φ) + c ds.() ∗ g s τ Yang Advances in Difference Equations (2017) 2017:238 Page 16 of 20 Consequently, δθ δθ–(α –σ )λ (t+θ–τ) E ≤ max sup e E ϑ(t + θ – τ) , sup e E(τ) t δ θ ∈(–∞,τ–t] θ ∈[τ–t,] t+θ δθ –(α –σ )λ (s+θ–τ) – + e e σ g(s) +L E + c ds . g s δ We assume that, moreover, σ satisfies (α – σ )λ ≤ δ. On the one hand δθ δ(θ–(t–τ)) sup e E ϑ(t + θ – τ) = sup e E ϑ(θ) θ ∈(–∞,τ–t] θ ≤ –δ(t–τ) –(α –σ )λ (t–τ) = e E(ϑ) ≤ e E(ϑ) δ δ and on the other hand δθ–(α –σ )λ (t+θ–τ) –(α –σ )λ (t–τ) sup e E(τ) ≤ e E(τ) θ ∈[τ–t,] and t+θ δθ –(α –σ )λ (s+θ–τ) – sup e e σ g(s) +L E + c ds g s δ θ ∈[τ–t,] τ –(α –σ )λ (s–τ) – ≤ e σ g(s) +L E + c ds g s δ so we get –(α –σ )λ (t–τ) E ≤ e E(ϑ) t δ –(α –σ )λ (t–τ) – + e σ g(s) +L E + c ds ∀t ≥ τ. g s δ Then by Gronwall’s lemma, we have (u, φ) ≤ E t t δ –((α –σ )λ –L )(t–τ) g ≤ e E(ϑ) –((α –σ )λ –L )(t–s) – g + e 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 ≤ α such that L ≤ (α – σ )λ ≤ δ g and βs – e σ g(s) + c ds <+∞,() –∞ Yang Advances in Difference Equations (2017) 2017:238 Page 17 of 20 where β = (α – σ )λ – L . g ∗ Remark If we assume that g ∈ L (R; V ), then equation ()isequivalentto loc –β(t–s) – e σ g(s) + c ds <+∞, ∀t ∈ R. –∞ ∗ Corollary Assume that g ∈ L (R; V ), G : R × C (Y) → (L ()) satisfies assumptions loc (g)-(g) for any τ < T and condition () is satisfied, then the family D = {D (t): t ∈ R}, with D (t)= D (, ρ(t)), where C (Y) –β(t–s) – ρ (t)= + e 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 D - asymptotically compact. m m Proof Let (u , φ )be a sequence in Theorem , we will prove this sequence is relatively compact in C (Y). Let τ (D, t, h)< t – h – be such that –β(t–h–) βτ e e E(ϑ) ≤ ∀τ ≤ τ (D, t, h), ϑ ∈ D(τ). Consider fixed τ ≤ τ (D, t, h)and ϑ ∈ D(τ). Firstly, from the result ()ofTheorem and using the definition of the norm · ,we can deduce that –β(t–h–) βs – (u, φ) ≤ + e e (σλ ) g(s) + c ds t –∞ ρ (t) ∀r ∈ [t – h –, t]. Secondly, we derive from Theorem by integrating ()between r – and r r r m m m m E (r)– E (r –)+(α – σ ) u + μ (t) dt r– r– – m m ≤ σ g +L u , φ ds, s δ r– then from ()and u(t), φ(t) ≤ E(t) ≤ Q u(t), φ(t) , Y Y Yang Advances in Difference Equations (2017) 2017:238 Page 18 of 20 where Q is a monotone non-decreasing function independent of the time and the initial data, we can deduce that r r m m (α – σ ) u + μ (t) dt r– r– – m m m ≤ σ g +L u , φ ds + E (r –) s δ r– r s – –β(s–τ) –β(s–ξ) – ≤ σ g +L e E(ϑ) + e (σλ ) g(ξ) dξ + c ds g r– τ m m + Q u (t), φ (t) . As α – σ >, m m u + μ dt ≤ C . r– From μ = A φ + αf φ we have γ γ r r m m m A φ dt ≤ c μ + α f φ dt ≤ C γ γ L L L r– r– so m m u + A φ dt ≤ C,() γ r– where C , C and C are constants. For the rest of the estimates we take the inner product in H of () with A u, the inner product in L (M)of () with A φ and add the resulting equalities; we get (see []for the details) the inner product by integrating between r – and r, dY m / m + ν A u + A φ ≤ (t)Y (t)+ (t), L L dt where m m Y (t)= u (t) + A φ (t) , γ m m m m (t)= c A φ (t) φ (t) + u u , L H L m ϒ(t)= c f (φ)∇φ + |g| , L L m m (t)= ϒ(t)+ G t, u , φ . t L By integrating between r – and r and from () we can obtain r r r r m m Y (r) ≤ Y (s) ds + ϒ(s) ds + L u , φ ds × exp (s) ds s δ r– r– r– r– r r r –β(s–τ) ≤ Y (s) ds + ϒ(s) ds + L e E(ϑ) r– r– r– s r –β(s–ξ) – + e (σλ ) g(ξ) dξ + c ds × exp (s) ds . τ r– Yang Advances in Difference Equations (2017) 2017:238 Page 19 of 20 Recalling (), ()and (), and employing standard Sobolev inequalities, it is easy to check that there exist positive constants a , i = ,,, such that r r (s) ds ≡ a < ∞; ϒ(s) ds ≡ a < ∞, r– r– Y (s) ds ≡ a < ∞. r– Then we get m m Y (r)= u (r) + A φ (r) is bounded ∀r ∈ [t – h –, t]. γ Letting (u, φ)=(·; t – s, ϑ)we get m m u , φ is bounded ∀r ∈ [t – h –, t]. t V Thirdly, we have dY m / m ν A u + A φ ≤ – + (t)Y (t)+ (t). L L dt By integrating between r – and r,weobtain m / m ν A u + A φ ds L L r– ≤ Y (r –)+ (s)Y (s)+ (s) ds r– m m ≤ Y (r –)+ (s)Y (s)+ ϒ(s)+ L u , φ ds. s δ r– With ()wehave m / m ν A u + A φ ds is bounded ∀r ∈ [t – h, t]. L L r– Similartothe proofofTheorem ,wealsoconcludethat m m u , φ ds is bounded ∀r ∈ [t – h, t] dt r– m m ∞ d m m so (u (t), φ (t)) is bounded in L (t – h –, t; Y) ∩ L (t – h –, t; V)with (u , φ )is dt ∗ ∗ bounded in L (t – h –, t; V ) × L (t – h –, t; V ) and for a subsequence (relabeled the same) the following convergences hold: m m ∞ (u , φ ) (u, φ)weaklystarin L (t – h –, t; Y), m m (u , φ ) (u, φ)weaklyin L (t – h –, t; V), d d m m ∗ ∗ (u , φ ) (u, φ)weaklyin L (t – h –, t; V ) × L (t – h –, t; V ), dt dt ⎪ m m ⎪ (u , φ ) → (u, φ)stronglyin L (t – h –, t; Y), m m (u (t), φ (t)) → (u(t), φ(t)) a.e., t ∈ (t – h –, t). Yang Advances in Difference Equations (2017) 2017:238 Page 20 of 20 Then the following proof can be obtained analogously to Step of Theorem .Toavoid unnecessary repetition, we skip the proof. ∗ Theorem Assume that g ∈ L (R; V ), G : R × C (Y) → (L ()) satisfies assumptions loc (g)-(g) for any τ < T. Also, suppose that L ≤ (α – σ )λ . Then the semi-process U defined g in C (Y) associated to () has a pullback attractor A = {A(t)}. Moreover, every A(t) is connected in C (Y). Proof The existence of the pullback attractor is a direct consequence of Theorem , Corol- lary , Corollary and Proposition . The connectedness follows from Propositions and and the fact that the space C (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 Theauthorwrote thepaper andapproved thefinalmanuscript. Authors’ information Min Yang (1978-), female, lecturer, engaged in the research of numerical solution of partial differential equations. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 16 February 2017 Accepted: 3 August 2017 References 1. Anderson, DM, McFadden, GB, Wheeler, AA: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30(1), 139-165 (1998) 2. Feng, JJ, Liu, C, Shen, J, Yue, P: An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids. In: Advantages and Challenges, vol. 141, pp. 1-26. Springer, New York (2005) 3. Blesgen, T: A generalization of the Navier-Stokes equation to two-phase flows. J. Appl. Phys. 32(10), 1119-1123 (1999) 4. Feireisl, E, Petzeltova, H, Rocca, E, Schimperna, G: Analysis of a phase-field model for two-phase compressible fluids. Math. Models Methods Appl. Sci. 20(7), 1129-1160 (2010) 5. Gal, CG, Grasselli, M: Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete Contin. Dyn. Syst. 28(1), 1-39 (2010) 6. Gal, CG, Grasselli, M: Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27(1), 401-436 (2010) 7. Medjo, TT: Pullback attractors for a non-autonomous homogeneous two-phase flow model. J. Differ. Equ. 253(6), 1779-1806 (2012) 8. Caraballo, T, Real, J: Attractors for 2D-Navier-Stokes models with delays. J. Differ. Equ. 205(2), 271-297 (2004) 9. Marín-Rubio, P, Real, J, Valero, J: Pullback attractors for a two-dimensional Navier-Stokes model in an infinte delay case. Nonlinear Anal. 74(5), 2012-2030 (2011) 10. García-Luengo, J, Marín-Rubio, P, Real, J: Regularity of pullback attractors and attraction in H1 in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay. Discrete Contin. Dyn. Syst. 34(1), 181-201 (2014) 11. Medjo, TT: A two-phase flow model with delays. Discrete Contin. Dyn. Syst. 21, 2263-2285 (2016) 12. Medjo, TT: Attractors for a two-phase flow model with delays. Differ. Integral Equ. 29, 1071-1092 (2016) 13. Temam, R: Infinite dimensional dynamical systems in mechanics and physics. Appl. Math. Sci. 68(5), 2135-2143 (2000) 14. Kloeden, PE, Schmalfuss, B: Nonautonomous systems, cocycle attractors and variable time-step discretization. Numer. Algorithms 14(1), 141-152 (1997) 15. Kloeden, PE, Stonier, DJ: Cocycle attractors in nonautonomously perturbed differential equations. Dyn. Contin. Discrete Impuls. Syst. 4, 211-226 (1998) 16. García-Luengo, J, Marín-Rubio, P, Real, J: Pullback attractors in V for non-autonomous 2D-Navier-Stokes equations and their tempered behaviour. J. Differ. Equ. 252(8), 4333-4356 (2012)
Advances in Difference Equations – Springer Journals
Published: Aug 15, 2017
You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.