Navier-Stokes equations with delays on unbounded domains

Navier-Stokes equations with delays on unbounded domains By Mar´ ıa Jos´ e Garrido-Atienza, & Pedro Mar´ ın-Rubio † Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla. Spain Some results on the existence and uniqueness of solutions to Navier-Stokes equations when the domain is unbounded and the external force contains some hereditary characteristics are proved for both the evolutionary and the stationary cases. Exponential stability of the stationary solution is also established in dimension two. Keywords: Navier-Stokes equations, variable and distributed delays, unbounded domains. 1. Introduction and statement of the problem The Navier-Stokes equations govern the motion of usual fluids like water, air, oil, etc. These equations have been the object of numerous works since the first paper of Leray was published in 1933 (see Constantin & Foias 1988; Lions 1969; Temam 1979, and the references therein), even with unbounded domains, allowing the possibility of channel and multi-channel flows among other variations (see for instance Rosa 1998; Temam 1979). On other hand, delay effects have been proved to be useful in many physical and biological situations. These situations may appear when we want to control the system (in a certain sense) by applying a force which takes into account not only the present state of the system but the history of the solution. To our knowledge, this has been rarely treated in the context of Navier-Stokes equations (cf. Caraballo & Real 2001, 2003, 2004). To date, we have not found in the literature any work that considers the combination of delay terms and unbounded domains. The aim of the paper is two-fold: firstly we consider several situations in which the external force contains some hereditary features and the domain is not bounded, and prove existence (for dimension N= 2 and 3) and uniqueness (N= 2) of solutions. In a second part of the paper the existence and uniqueness of a stationary solution are established in dimensions 2 and 3,and, in the case N= 2 exponential stability of the solution under an additional assumption. Let Ω ⊂RN(N= 2 or 3) be an open set with boundary Γ that is not necessarily bounded but satisfies a Poincar´e inequality: There exists λ1>0 such that ZΩ|φ|2dx≤1 λ1ZΩ|∇φ|2dx, ∀φ∈H1 0(Ω).(1.1) †E-mails: [email protected] ; [email protected] Article submitted to Nonlinear Analysis TMA T EX Paper 2M. J. Garrido-Atienza, & P. Mar´ın-Rubio Consider the following functional Navier-Stokes problem (for further details and notations see Lions 1969 and Temam 1979):                ∂u ∂t −ν∆u+PN i=1 ui ∂u ∂xi =f(t)−∇p+g(t, ut) in (0, T)×Ω, div u= 0 in (0, T)×Ω, u= 0 on (0, T)×Γ, u(0, x) = u0(x), x ∈Ω, u(t, x) = φ(t, x), t ∈(−h, 0) x∈Ω, where we assume that T > 0 is given, ν > 0 is the kinematic viscosity, uis the velocity field of the fluid, pthe pressure, u0the initial velocity field, fa nondelayed external force field, ganother external force containing some hereditary characteristic and φthe initial datum in the interval of time (−h, 0),where his a positive fixed number. To start, we consider the following usual abstract spaces: V=nu∈(C∞ 0(Ω))N: divu= 0o, H= the closure of Vin (L2(Ω))Nwith the norm |·|,and inner product (·,·) where for u, v ∈(L2(Ω))N, (u, v) = N X j=1 ZΩ uj(x)vj(x)dx, V= the closure of Vin (H1 0(Ω))Nwith the norm [thanks to (1.1)] k·k associated to the inner product ((·,·)),where for u, v ∈(H1 0(Ω))N, ((u, v)) = N X i,j=1 ZΩ ∂uj ∂xi ∂vj ∂xi dx. It follows that V⊂H≡H0⊂V0,where the injections are dense and continuous. We will use k·k∗for the norm in V0and h·,·i for the duality hV0, V i.Now we denote a(u, v) = ((u, v)), and define the trilinear form bon V×V×Vby b(u, v, w) = N X i,j=1 ZΩ ui ∂vj ∂xi wjdx∀u, v, w ∈V. Let Xbe a Banach space. Given a function u: (−h, T)→X, for each t∈(0, T) we denote by utthe function defined on (−h, 0) by the relation ut(s) = u(t+s), s ∈ (−h, 0). In order to state the problem in the correct framework, let us first establish suitable assumptions on the term in which the delay is present. In a general way, let Xand Ybe two separable Banach spaces, and g: [0, T ]× C0([−h, 0]; X)→Ysuch that (I) for all ξ∈C0([−h, 0]; X), the mapping t∈[0, T]→g(t, ξ)∈Yis measurable, Article submitted to Nonlinear Analysis TMA Navier-Stokes equations with delays on unbounded domains 3 (II) for each t∈[0, T], g(t, 0) = 0, (III) there exists Lg>0 such that ∀t∈[0, T],∀ξ, η ∈C0([−h, 0]; X) kg(t, ξ)−g(t, η)kY≤Lgkξ−ηkC0([−h,0];X), (IV) there exists Cg>0 such that ∀t∈[0, T],∀u, v ∈C0([−h, T]; X) Zt 0kg(s, us)−g(s, vs)k2 Yds≤CgZt −hku(s)−v(s)k2 Xds. Observe that (I)-(III) imply that given u∈C0([−h, T]; X), the function gu:t∈ [0, T]→Ydefined by gu(t) = g(t, ut)∀t∈[0, T], is measurable (see Bensoussan et al. 1992) and, in fact, belongs to L∞(0, T;Y). Then, thanks to (IV), the mapping G:u∈C0([−h, T]; X)→gu∈L2(0, T;Y) has a unique extension to a mapping e Gwhich is uniformly continuous from L2(−h, T;X) into L2(0, T;Y). From now on, we will denote g(t, ut) = e G(u)(t) for each u∈ L2(−h, T;X), and thus, ∀t∈[0, T],∀u, v ∈L2(−h, T;X),we will have Zt 0kg(s, us)−g(s, vs)k2 Yds≤CgZt −hku(s)−v(s)k2 Xds. With the convention above, assume that u0∈H,φ∈L2(−h, 0; V) , f∈L2(0, T ;V0), and the following delay operators: g1: [0, T]×C0([−h, 0]; V)→(L2(Ω))N satisfying hypotheses (I)-(IV) with X=V,Y= (L2(Ω))N,Lg1=L1and Cg1=C1, and g2: [0, T]×C0([−h, 0]; V)→V0 satisfying hypotheses (I)-(IV) with X=V,Y=V0,Lg2=L2and Cg2=C2. We are interested in the following problem:          To find u∈L2(−h, T;V)∩L∞(0, T;H) such that, for all v∈V, d dt(u(t), v) + νa(u(t), v) + b(u(t), u(t), v) = hf(t), vi+ (g1(t, ut), v) +hg2(t, ut), vi, u(0) = u0, u(t) = φ(t), t ∈(−h, 0), (1.2) where the equation in (1.2) must be understood in the sense of D0(0, T ). Remark 1.1. Observe that the terms in (1.2) are well defined. In particular, by hypotheses (I)-(IV), if u∈L2(−h, T;V)the term g1(t, ut)defines a function in L2(0, T; (L2(Ω))N), and the term g2(t, ut)defines a function in L2(0, T;V0). Thus (see Lions 1969), if u∈L2(−h, T;V)∩L∞(0, T;H)satisfies the equation in (1.2),u is weakly continuous from [0, T ]into H, and therefore the initial condition u(0) = u0 makes sense. Of course, for N= 2, if there exists a solution uto the problem (1.2), it then belongs to the space C0([0, T ]; H). In Section 2 we shall prove existence of solutions to (1.2) and the uniqueness of solution to the problem in the case N= 2.In Section 3, general situations containing delayed terms –variable and distributed– are considered. We finish with the study of existence and uniqueness of a stationary solution and its exponential stability (for N= 2) in Section 4. Article submitted to Nonlinear Analysis TMA 4M. J. Garrido-Atienza, & P. Mar´ın-Rubio 2. Existence of solutions In this section we will prove a general theorem on the existence of solutions when N= 2 or 3, and uniqueness if N= 2. In the proof of existence we will need the following two results: Theorem 2.1. (cf. [Caraballo & Real 2001, Theorem A.1]) Let u0∈Rm,φ∈ L2(−h, 0; Rm), k ∈L2(0, T ;Rm),g: [0, T]×C0([−h, 0]; Rm)→Rmsatisfying hypotheses (I)-(IV) with X=Y=Rm, and f: [0, T]×Rm→Rma continuous function such that f(t, 0) = 0 and for all n > 0there exists Ln>0such that |f(t, u)−f(t, v)|Rm≤Ln|u−v|Rm,∀|u|Rm≤n, |v|Rm≤n, ∀t∈[0, T ]. Then: a) For each t∗∈(0, T ]there exists at most one solution to the problem        To find u∈L2(−h, t∗;Rm)∩C0([0, t∗]; Rm)such that u(t) = φ(t), t ∈(−h, 0), u(t) = u0+Zt 0 f(s, u(s)) ds+Zt 0 g(s, us) ds+Zt 0 k(s) ds∀t∈[0, t∗]. (2.1) b) There exists t∗∈(0, T]such that there exists one (and only one)solution to the problem (2.1). c) Suppose that there exists a constant C > 0such that if t∗∈(0, T ]is such that there is a solution uof (2.1), then maxt∈[0,t∗]|u(t)|Rm≤C. Then, under this additional assumption, there exists a solution to problem (2.1) with t∗=T. Theorem 2.2. (cf. [Simon 2003, Corollary 2.34]) Let Θbe a bounded open set of Rd,and X⊂EBanach spaces with compact injection. Consider 1≤r < q ≤ ∞. Suppose F⊂Lr(Θ; E)satisfies (i) ∀ω⊂⊂ Θ,supf∈Fkτhf−fkLr(ω;E)→0when h→0 [where τhfis the translation: (τhf)(x) = f(x+h)], (ii) Fis bounded in Lq(Θ; E)∩L1(Θ; X). Then Fis precompact in Lr(Θ; E). The main result in this section needs extra notation for an additional condition (V), which will be discussed in some detail below in Remark 2.5. Denote V(O) the same space as Vbut with an open set Oinstead of Ω,and analogously define V(O) the closure of V(O) in (H1 0(Ω))N. Theorem 2.3. Let u0∈H,φ∈L2(−h, 0; V),f∈L2(0, T;V0), and assume that g1: [0, T]×C0([−h, 0]; V)→(L2(Ω))Nand g2: [0, T]×C0([−h, 0]; V)→V0satisfy hypotheses (I)-(IV) in their corresponding spaces. Then: a) If N= 2 and ν2> C2,there exists at most one solution to problem (1.2). b) If N∈ {2,3}and ν2> C2, there exists a solution to (1.2) if, in addition, the following assumption (V) holds: (V) If vmconverges weakly to vin L2(−h, T;V),weakly-star in L∞(0, T;H), and strongly in L2(−h, T; (L2(O))N)for a bounded open set O ⊂ Ωwith smooth boundary, then gi(·, vm ·)converges weakly to gi(·, v·)in L2(0, T ;V(O)0)for i= 1,2. Article submitted to Nonlinear Analysis TMA Navier-Stokes equations with delays on unbounded domains 5 Proof. a) Uniqueness for N= 2 follows as the case with bounded domain given in Theorem 2.1 in [Caraballo & Real 2001] and it is reproduced here for the sake of completeness. If ν2> C2, let u, v be two solutions to (1.2) and set w=u−v. Then, from the energy equality, and the bound for the trilinear form (see Ladyzhenskaya 1992), it follows that for all t∈(0, T) |w(t)|2+ 2νZt 0kw(s)k2ds=−2Zt 0 b(w(s), u(s), w(s)) ds +2 Zt 0 (g1(s, us)−g1(s, vs), w(s)) ds +2 Zt 0hg2(s, us)−g2(s, vs), w(s)ids ≤21/2Zt 0|w(s)|kw(s)kku(s)kds +2 Zt 0|g1(s, us)−g1(s, vs)||w(s)|ds +2 Zt 0kg2(s, us)−g2(s, vs)k∗||w(s)|| ds. Then, from assumption (IV), taking into account that w(s) = 0 for s∈(−h, 0), and denoting 2ε=ν−√C2>0, we have for all t∈(0, T ) |w(t)|2+ 2νZt 0kw(s)k2ds≤1 2εZt 0|w(s)|2ku(s)k2ds+εZt 0kw(s)k2ds +C1 εZt 0|w(s)|2ds+εZt 0kw(s)k2ds +2pC2Zt 0kw(s)k2ds, and so, |w(t)|2+ 2εZt 0kw(s)k2ds≤1 2εZt 0|w(s)|2ku(s)k2ds+C1 εZt 0|w(s)|2ds, from which uniqueness follows thanks to the Gronwall lemma: indeed, denoting C= max(2−1, C1)/ε, we have that d dtµ|w(t)|2exp ½−CZt 0 (ku(s)k2+ 1)ds¾¶≤0. b) Now for the existence, we assume N∈ {2,3},ν2> C2and that condition (V) holds. We will develop a proof based on the unbounded case without delays (see for instance Temam 1979), and on the case with delays on bounded domains (Caraballo & Real 2001), but with both difficulties treated jointly. Article submitted to Nonlinear Analysis TMA 6M. J. Garrido-Atienza, & P. Mar´ın-Rubio Consider an orthonormal basis B={w1, . . . , wn, . . .} ⊂ V of Hsuch that linear combinations of elements of Bare dense in V.†[Notice one usually takes a special basis, using the Stokes operator. This is not valid here since compactness is lost, and it will have influence on the way one can obtain a convergent subsequence, because standard estimates on the derivatives of the velocity fields are not valid neither.] Let us denote Vm=span[w1, . . . , wm], PH Vm:H→Vmthe projector given by PH Vmu=Pm j=1(u, wj)wj.We will also denote PV Vm:V→Vmthe projector given by PV Vmu=Pm j=1((v, ˜wj)) ˜wj(where the sequence {˜w1, . . . , ˜wn}comes from the Gram-Schmidt orthonormalization process in V; this will be useful since the initial convergence in (−h, 0) must hold in V). Finally, define um(t) = Pm j=1 γmj(t)wj, where            um∈L2(−h, T;Vm)∩C0([0, T]; Vm) d dt(um(t), wj) + νa(um(t), wj) + b(um(t), um(t), wj) = hf(t), wji+ + (g1(t, um t), wj) + hg2(t, um t), wjiin D0(0, T),1≤j≤m, um(0) = PH Vmu0, um(t) = PV Vmφ(t), t ∈(−h, 0). (2.2) The preceding is a system of ordinary functional differential equations in the unknown γm(t) = (γm1(t), ..., γmm(t)). Existence and uniqueness of solution is obtained by applying Theorem 2.1 stated above. Observe that problem (2.2) has one solution defined in an interval [0, t∗] with 0< t∗≤T. However, as usual, it can be deduced by the a priori estimates below, we can set t∗=T. In fact, multiplying in (2.2) by γmj(t) and summing in j, we get for all t∈[0, t∗] |um(t)|2+ 2νZt 0kum(s)k2ds≤ |u0|2+ 2 Zt 0hf(s), um(s)ids +2 Zt 0 (g1(s, um s), um(s)) ds +2 Zt 0hg2(s, um s), um(s)ids, and arguing in a similar manner as in the proof of uniqueness in the 2-dimensional case, we easily get two constants (depending on φ, ν, f, g1, g2, h, T, but not on m nor t∗)K1and K2such that sup t∈[0,t∗]|um(t)|2≤K1,Zt∗ 0kum(s)k2ds≤K2.(2.3) So we can take t∗=T, and obtain that {um}is bounded in L2(0, T;V)∩L∞(0, T;H), so there exists a subsequence, relabelled the same, such that um* u in L2(0, T;V) weakly and in L∞(0, T;H) weak-star as m→ ∞.(2.4) †This can be obtained as follows: Vis separable (since it is a subset of (H1 0(Ω))N), and by definition Vis dense in Vand H, which implies that Vis also dense in H. Thus, given a sequence {vi}i≥1⊂Vdense in V, we may take a sequence {wi n}i,n≥1⊂ V which accumulates to every point vi,and therefore, linear combinations of these elements are dense in Vand H. Since V⊂Hare vectorial subspaces of (L2(Ω))N,linear (in)dependence is equivalent considered in any of them, whence Bis obtained applying the Gram-Schmidt orthonormalization process. Article submitted to Nonlinear Analysis TMA Navier-Stokes equations with delays on unbounded domains 7 Moreover, observe that um=PV Vmφin (−h, 0) converges to φin L2(−h, 0; V), and, in particular, thanks to (IV), g1(·, um) + g2(·, um) is bounded in L2(0, T;V0). As we noticed before, when it is possible to choose a special basis (in bounded domains it is so), standard estimates on kd dtumkL2(0,T ;V0)(for N= 2,and L4/3(0, T;V0) for N= 3) allow us to obtain a compactness result: a subsequence umconverges to uin L2(0, T;H). Here we will have a similar result but not in a straightforward way, nor on the whole domain Ω.Actually, what holds in this case is the following: For any bounded open set O ⊂ Ω there exists a subsequence (depending on Owhich we relabel) satisfying um|O→u|Oin L2(0, T; (L2(O))N).(2.5) For the sake of clarity, we postpone the proof (we will use Theorem 2.2) to Lemma 2.4 below. Now, let ψbe a continuously differentiable function on [0, T ] with ψ(T) = 0.Consider equation (2.2) and a fixed element wjof B. Since (um(·), wj)ψ(·)∈ W1,1(0, T) (actually in H1(0, T) for N= 2,and W1,4/3(0, T) for N= 3) we have −ZT 0 (um(t), ψ0(t)wj)dt+νZT 0 ((um(t), wjψ(t)))dt +ZT 0 b(um(t), um(t), wjψ(t))dt= (um(0), wj)ψ(0) + ZT 0hf(t), wjψ(t)idt +ZT 0 (g1(t, um t), wjψ(t))dt+ZT 0hg2(t, um t), wjψ(t)idt. Taking a diagonal subsequence, denoted again um,that satisfies (2.5) for a sequence of regular bounded open sets Oj⊂Ω that contain all supports of functions wjof the basis, we may now pass to the limit, thanks to the weak convergence in (2.4) and condition (V) too. Thus, we obtain (first for any w∈ {w1, w2, . . .},and by density for every w∈V): −ZT 0 (u(t), ψ0(t)w)dt+νZT 0 ((u(t), wψ(t)))dt +ZT 0 b(u(t), u(t), wψ(t))dt= (u0, w)ψ(0) + ZT 0hf(t), wψ(t)idt +ZT 0 (g1(t, ut), wψ(t))dt+ZT 0hg2(t, ut), wψ(t)idt. (2.6) Writing (2.6) for ψ∈ D(0, T ), u satisfies (2.2) in the distribution sense. By Remark 1.1 it makes sense to wonder about the value at time t= 0.Now, since (u(t), wj)ψ(t)∈W1,4/3(0, T),for both N= 2 or 3,arguing as before, we obtain an analogous expression to (2.6) with (u(0), w) instead of (u0, w).This implies (u(0) −u0, w) = 0 for all w∈V, so u(0) = u0. Article submitted to Nonlinear Analysis TMA 8M. J. Garrido-Atienza, & P. Mar´ın-Rubio For the following result, let us observe that the constants appearing in standard estimates of the trilinear form b–for a proof see for instance Temam 1979, lemmas 3.3, 3.4, 3.5 and Theorem 3.3, pp. 291 and forward– may be improved (cf. Ladyzhenskaya 1992 for the case N= 2). Although it is not essential to make the most of them for existence and uniqueness, since we will establish a stability result later, we use them in order to ensure it under the minimal conditions. Lemma 2.4. Under the assumptions of Theorem 2.3, the sequence umgiven in (2.2) is precompact in the following sense: suppose a bounded open set O ⊂ Ωis given, then there exists a subsequence depending on O,which we relabel, such that um|O→u|Oin L2(0, T; (L2(O))N), where uis the weak limit given in (2.4). Proof. We will adapt the proof of Theorem 9.4 [cf. Simon 2003] to check our situation fits to the Theorem 2.2. More exactly, we claim it can be applied taking r= 2, q= +∞,Θ = (0, T). For the set O ⊂ Ω let us make precise a technical detail: if O ⊂⊂ Ω one may obtain a finite recovering of balls, denoted ˜ O ⊂ Ω,which is bounded and open, and then X= (H1(˜ O))N⊂E= (L2(˜ O))Nwith compact injection. However, for a general O ⊂ Ω the above comment may not be true since O and Ω can share part of their boundaries. The compact injection from H1may not hold for lack of regularity on the boundary (it is not imposed for Γ), however it does in H1 0.One may then use a truncation argument (see for instance Rosa 1998): fix χ∈C1(R+) with χ(s) = 1 for s∈[0,1] and χ(s) = 0 for s≥4.Consider Oas in the statement, let R > 0 be such that O ⊂ B(0, R) and denote ˜ O= Ω ∩B(0,2R),and um,R(x) = um(x)χ(|x|2/R2).Again the compactness holds for X= (H1 0(˜ O))N⊂E= (L2(˜ O))Nwith compact injection, and we conserve the original functions umon Ω ∩B(0, R). For the sake of clarity, we continue the proof directly with uminstead of um,R. Since condition (ii) in Theorem 2.2 is obviously satisfied by (2.3), we concentrate on (i). Actually, we will prove that for the whole domain Ω the following property holds: sup m∈Nkτhum−umkL2(0,T −h;(L2(Ω))N)→0 when h→0. Consider h > 0 arbitrarily small. From (2.2) we deduce for (t, t +h)⊂(0, T) that ZΩ (um(t+h)−u(t))wjdx+νZt+h tZΩ∇um(s)·∇wjdxds+Zt+h t b(um(s), um(s), wj)ds =Zt+h thf(s), wjids+Zt+h tZΩ g1(s, um s)wjdxds+Zt+h thg2(s, um s), wjids. Article submitted to Nonlinear Analysis TMA Navier-Stokes equations with delays on unbounded domains 9 Multiplying by γmj(t+h)−γmj(t) and summing in jwe obtain ZΩ|um(t+h)−u(t)|2dx=−νZt+h tZΩ∇um(s)·(∇um(t+h)−∇um(t))dxds +Zt+h t b(um(s), um(s), um(t+h)−um(t))ds +Zt+h tZΩ g1(s, um s)(um(t+h)−um(t))dxds +Zt+h thf(s) + g2(s, um s), um(t+h)−um(t)ids. The right hand side may be bounded by ν|∇um(t+h)−∇um(t)|Zt+h t|∇um(s)|ds +Zt+h t GN(|um(s)|,kum(s)k,kum(t+h)−um(t)k) ds +Zt+h t|g1(s, um s)||um(t+h)−um(t)|ds +Zt+h t (kf(s)k∗+kg2(s, um s)k∗)kum(t+h)−um(t)kds where the trilinear form bis bounded (depending on the dimension) by the function GN:R3→Rdefined as GN(x, y, z) = ½2−1/2xyz if N= 2, 2−1x1/2y3/2zif N= 3.(2.7) Thus, using (1.1) and (2.3), we have proved that ZΩ|um(t+h)−um(t)|2dx≤ kum(t+h)−um(t)kZt+h t Gm(s)ds where the function Gm:R→Ris defined (recall definition given in (2.7)) as Gm(s)=(νkum(s)k+(2−1K1)1/2kum(s)k+kf(s)k∗+kg2(s, um s)k∗+λ−1/2 1|g1(s, um s)|if N= 2, νkum(s)k+2−1K1/4 1kum(s)k3/2+kf(s)k∗+kg2(s, um s)k∗+λ−1/2 1|g1(s, um s)|if N= 3. To finish the proof, we will estimate kτhum−umk2 L2(0,T −h;(L2(Ω))N)=ZT−h 0ZΩ|τhum−um|2dxdt ≤ZT−h 0kum(t+h)−um(t)kZt+h t Gm(s)dsdt. For the right hand side, the Fubini theorem yields, using the function ¯s=   0 if s≤0, sif 0 < s ≤T−h, T−hif s > T −h, Article submitted to Nonlinear Analysis TMA 16 M. J. Garrido-Atienza, & P. Mar´ın-Rubio Then there is a unique stationary solution u∗of (4.2) and every solution of (1.2) converges to u∗exponentially fast as t→+∞, that is, there exist two positive constants Cand λ, such that for all u0∈Hand φ∈L2(−h, 0; V), the solution u of (1.2) with f(t)≡fsatisfies for all t≥0 : |u(t)−u∗|2≤Ce−λt ³|u0−u∗|2+kφ−u∗k2 L2(−h,0;V)´.(4.7) Proof. Consider uthe solution of (1.2) for f(t)≡f, and denote u∗∈Vthe stationary solution to (4.2), which existence and uniqueness is ensured by Theorem 4.1. Set w(t) = u(t)−u∗, and observe that d dt(w(t), v)+ν((w(t), v))+b(u(t), u(t), v)−b(u∗, u∗, v) = (G(u(t−ρ(t))), v)−(G(u∗), v). Since b(u(t), u(t), w(t)) −b(u∗, u∗, w(t)) =b(u∗, w(t), u∗)−b(u(t), w(t), u(t)) =b(u∗, w(t), u∗)∓b(u(t), w(t), u∗)−b(u(t), w(t), u(t)) =−b(w(t), w(t), u∗)−b(u(t), w(t), w(t)) =−b(w(t), w(t), u∗), we can obtain the following estimation (here λand δare fixed positive values to be determined later on): d dt(eλt|w(t)|2) = λeλt|w(t)|2+ eλt d dt|w(t)|2 =λeλt|w(t)|2+ 2eλt(−νkw(t)k2−b(u(t), u(t), w(t)) +b(u∗, u∗, w(t)) + (G(u(t−ρ(t))) −G(u∗), w(t))) ≤eλt ¡λ|w(t)|2−2νkw(t)k2+ 2b(w(t), w(t), u∗) +2L1|w(t−ρ(t))||w(t)|) ≤λ−1 1eλt(λ+δL1−2νλ1)kw(t)k2+ 2eλt|b(w(t), w(t), u∗)| +(δλ1)−1L1eλtkw(t−ρ(t))k2.(4.8) Using again that |b(w(t), w(t), u∗)| ≤ (2λ1)−1/2kw(t)k2ku∗kand taking into account the estimate we proved in Step 3 of Theorem 4.1 for the stationary solution, ku∗k ≤ kfk∗/(ν−λ−1 1L1),we have |b(w(t), w(t), u∗)| ≤ (2λ1)−1/2kfk∗ ν−λ−1 1L1kw(t)k2. Substituting this last inequality into (4.8) it follows that d dt(eλt|w(t)|2)≤λ−1 1eλt µλ+δL1−2νλ1+(2λ1)1/2kfk∗ ν−λ−1 1L1¶kw(t)k2 +(δλ1)−1L1eλtkw(t−ρ(t))k2, Article submitted to Nonlinear Analysis TMA Navier-Stokes equations with delays on unbounded domains 17 and so for all t∈[0, T] eλt|w(t)|2≤ |w(0)|2+ (δλ1)−1L1Zt 0 eλskw(s−ρ(s))k2ds +λ−1 1µλ+δL1−2νλ1+(2λ1)1/2kfk∗ ν−λ−1 1L1¶Zt 0 eλskw(s)k2ds. We concentrate momentarily in the delay term on the right hand side. Observing that the function φ(t) := t−ρ(t) is strictly increasing, that ρtakes values on [0, h], and so φ−1(η)≤η+h, we can apply the change of variable η=s−ρ(s) = φ(s) : Zt 0 eλskw(s−ρ(s))k2ds=Zt−ρ(t) −ρ(0) eλτ−1(η)kw(η)k21 1−ρ0(φ−1(η)) dη ≤eλh 1−ρ∗Zt −h eληkw(η)k2dη. Combining the above two inequalities we obtain eλt|w(t)|2≤ |w(0)|2+ (δλ1)−1L1 eλh 1−ρ∗Zt −h eλskw(s)k2ds +λ−1 1µλ+δL1−2νλ1+(2λ1)1/2kfk∗ ν−λ−1 1L1¶Zt 0 eλskw(s)k2ds. Observe the coefficients of the integral Rt 0eλskw(s)k2ds. Let us note that δ∗= (1 −ρ∗)−1/2is the minimum of the map δ7→ δ+ 1/(δ(1 −ρ∗)).Then, thanks to (4.6) there exists λ > 0 small enough such that λ+δ∗L1−2νλ1+(2λ1)1/2kfk∗ ν−λ−1 1L1 +L1eλh δ∗(1 −ρ∗)≤0. Thus, we deduce that eλt|u(t)−u∗|2≤ |u0−u∗|2+λ−1 1L1eλh 1−ρ∗Z0 −h eληkw(η)k2dη, whence (4.7) is satisfied with C= max ½1,λ−1 1L1eλh 1−ρ∗¾. Remark 4.3. The above result has been given with g1as in Case 1 of Section 3to simplify the notation. Other types of delay could be used, for example one could consider an autonomous case with distributed delay by removing in Case 2 the dependence of Gon its first variable. Conclusions and final comments Existence, uniqueness and stability results have been established under different conditions –essentially viscosity is asked to be large enough–. One may wonder about results under weaker assumptions, where uniqueness or stability may not be Article submitted to Nonlinear Analysis TMA 18 M. J. Garrido-Atienza, & P. Mar´ın-Rubio ensured. This leads us to consider additional concepts from the theory of dynamical systems, namely attractors, both the classical (forward) one, and the ’pullback’ definition that is well-suited to non-autonomous systems (see Caraballo & Real 2004 and Kloeden & Schmalfuß 1997). Acknowledgement This work was partially supported by Ministerio de Ciencia y Tecnolog´ıa (Spain) and FEDER Proyecto BFM2002-03068. The authors thanks Professor Jos´e Real for his advice on the subject of this paper. References Artola, M. 1969 Sur les perturbations des ´equations d’evolution, application `a des probl`emes de retard, Ann. Scient. Ec. Norm. Sup. 4s´erie, t. 2, 137-253. Bensoussan, A., Da Prato, G., Delfour, M.C., & Mitter, S.K. 1992 Representation and Control of Infinite Dimensional Systems, Vol. I, Birkh¨auser, Boston-Basel-Berlin. Caraballo, T. & Real, J. 2001 Navier-Stokes equations with delays, Proc. R. Soc. Lond. A(2001) 457, 2441-2453. 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Lions, J.L. 1969 Quelques m´ethodes de r´esolution des probl`emes aux limites non lin´eaires, Dunod, Gauthier- Villars, Paris. Rosa, R. 1998 The global attractor for the 2D Navier-Stokes flow on some unbounded domains., Nonlinear Anal. 32, 71-85. Simon, J. 2003 ´ Equations de Navier-Stokes, Cours de DEA 2002-2003, Universit´e Blaise Pascal. http://wwwlma.univ-bpclermont.fr/∼simon/ Temam, R. 1979 Navier-Stokes equations, Theory and Numerical Analysis, 2nd. ed., North Holland, Amsterdam. Article submitted to Nonlinear Analysis TMA