Three Dimensional System of Globally Modified Navier-Stokes Equations with Infinite Delays

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume 00, Number 0, Xxxx XXXX pp. 000–000 THREE DIMENSIONAL SYSTEM OF GLOBALLY MODIFIED NAVIER-STOKES EQUATIONS WITH INFINITE DELAYS P. Mar´ ın-Rubio, A. M. M´ arquez-Dur´ an & J. Real Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080–Sevilla, Spain (Communicated by Aim Sciences) Abstract. Existence and uniqueness of solution for a globally modified version of Navier-Stokes equations containing infinite delay terms are established. Moreover, we also analyze the stationary problem and, under suitable additional conditions, we obtain global exponential decay of the solutions of the evolutionary problem to the stationary solution. Keywords: Globally Modified Navier-Stokes Equations; infinite delays. Mathematics Subject Classifications (2000): 35K55, 35Q30, 34D23, 34K20 1. Introduction and statement of the problem. Let Ω ⊂R3be an open bounded set with regular boundary Γ, and let N∈(0,+∞) be fixed. Let us define FN: [0,+∞)→(0,1] by FN(r) := min 1,N r, r ∈[0,+∞), and consider the following system of globally modified Navier-Stokes equations on Ω with homogeneous Dirichlet boundary condition                ∂u ∂t −ν∆u+FN(kuk) [(u· ∇)u] + ∇p=f(t) in (τ, T∗)×Ω, ∇ · u= 0 in (τ, T∗)×Ω, u= 0 on (τ, T∗)×Γ, u(τ, x) = u0(x), x ∈Ω, (1) where ν > 0 is the kinematic viscosity, uthe velocity field of the fluid, pthe pressure, τ∈Ran initial time, u0the initial velocity field, f(t) a given external force field, and T∗∈(τ, +∞] a given final time. The system (1) is indeed a globally modified version of the Navier-Stokes system – the modifying factor FN(kuk) depends on the norm kuk=k∇uk(L2(Ω))3×3, 2000 Mathematics Subject Classification. Primary: 35K55; 35Q30; 34D23; 34K20. Key words and phrases. Globally Modified Navier-Stokes Equations; infinite delays. Partially supported by Ministerio de Ciencia e Innovaci´on (Spain), project MTM2008-00088 and Junta de Andaluc´ıa, grant P07-FQM-02468. 1 2 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL which in turn depends on ∇uover the whole domain Ω and not just at or near the point x∈Ω under consideration. Essentially, it prevents large gradients dominating the dynamics and leading to explosions. It violates the basic laws of mechanics, but mathematically the system (1) is a well defined system of equations, just like the modified versions of the Navier-Stokes equations of Leray and others with other mollifications of the nonlinear term, see the review paper [8]. It is worth mentioning that a global cut off function involving the D(A1/4) norm for the two dimensional stochastic Navier-Stokes equations is used in [9], and a cut-off function similar to the one we will use here was considered in [23]. The system (1) was introduced and studied in [1] (see also [2,14,15,16,3] and the review paper [13]). However, there are situations in which the model is better described if some terms containing delays appear in the equations. These delays may appear, for instance, when one wants to control the system by applying a force which takes into account not only the present state but the complete history of the solutions. Therefore, in this paper we are interested in the case in which terms containing infinite delays appear. We consider the following version:              ∂u ∂t −ν∆u+FN(kuk) [(u· ∇)u] + ∇p=f(t) + g(t, ut) in (τ, T∗)×Ω, ∇ · u= 0 in (τ, T∗)×Ω, u= 0 on (τ, T∗)×Γ, u(τ+s, x) = φ(s, x), s ∈(−∞,0], x ∈Ω, (2) where gis another external force containing some hereditary characteristic and φ is a given function defined in the interval (−∞,0]. Our goal is to establish the existence and uniqueness of solution for the above problem and to study its asymptotic behaviour (for a similar goal in Navier-Stokes models with finite delay cf. [5,6,7], and for ODEs and PDEs with unbounded delay terms cf. [4,18]). The structure of the paper is the following: in Section 2we recall some spaces useful for the abstract framework and some properties and estimates related to the operators involved in the model. In Section 3the existence and uniqueness of solution for the (evolutionary) problem is given. The stationary problem is treated in Section 4, where we also prove that under adequate assumptions, any solution of the evolutionary problem has an exponential decay toward the stationary solution. 2. Preliminaries. To set our problem in the abstract framework, we consider the following usual abstract spaces (see [17] and [21,22]): V=nu∈(C∞ 0(Ω))3: div u= 0o, H= the closure of Vin (L2(Ω))3with inner product (·,·) and associate norm |·| , where for u, v ∈(L2(Ω))3, (u, v) = 3 X j=1 ZΩ uj(x)vj(x)dx, 3D-GMNS EQUATIONS WITH INFINITE DELAYS 3 V= the closure of Vin (H1 0(Ω))3with scalar product ((·,·)) and associate norm k·k ,where for u, v ∈(H1 0(Ω))3, ((u, v)) = 3 X i,j=1 ZΩ ∂uj ∂xi ∂vj ∂xi dx. We will use k·k∗for the norm in V0and h·,·i for the duality pairing between V0 and V. Finally, we will identify every u∈Hwith the element fu∈V0given by hfu, vi= (u, v)∀v∈V. It follows that V⊂H⊂V0,where the injections are dense and compact. We consider the linear continuous operator A:V→V0defined by hAu, vi= ((u, v)) ∀u, v ∈V. (3) Denoting D(A) = {u∈V:Au ∈H},with inner product (u, v)D(A)= (Au, Av), then, by the regularity of Γ, D(A) = (H2(Ω))3∩V, and Au =−P∆u, ∀u∈D(A), is the Stokes operator (Pis the ortho-projector from (L2(Ω))3onto H). Let us denote λ1= inf v∈V\{0} kvk2 |v|2>0, the first eigenvalue of the Stokes operator. Now we define the trilinear form bon V×V×Vby b(u, v, w) = 3 X i,j=1 ZΩ ui ∂vj ∂xi wjdx, ∀u, v, w ∈V, and we denote bN(u, v, w) = FN(kvk)b(u, v, w),∀u, v, w ∈V. The form bNis linear in uand w, but it is nonlinear in v. Evidently we have bN(u, v, v) = 0,for all u, v ∈V. We will also make use of the following inequality (see [21] and [10]) |b(u, v, w)| ≤ 2−1|u|1/4kuk3/4kvk|w|1/4kwk3/4,∀u, v, w ∈V. (4) In particular, this implies that there exists a constant C1>0 only dependent on Ω (namely, C1= (2λ1/4 1)−1) such that |b(u, v, w)| ≤ C1kukkvkkwk,∀u, v, w ∈V. Thus by the definition of FN,if we denote hBN(u, v), wi=bN(u, v, w),∀u, v, w ∈V, we have kBN(u, v)k∗≤NC1kuk,∀u, v ∈V. (5) We recall (see [21]) that there exists a constant C2>0 depending only on Ω such that |b(u, v, w)| ≤ C2kuk1/2|Au|1/2kvk|w|,(6) for all u∈D(A), v ∈V, w ∈H, and |b(u, v, w)| ≤ C2kukkvk|w|1/2kwk1/2,(7) for all u, v, w ∈V. (See [20] for the proof of (7)). 4 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL Let Xbe a Banach space. The notation BX(a, r) will be used to denote the open ball of center aand radius rin the space X. Given a function u: (−∞, T∗)→X, for each t<T∗we denote by utthe function defined on (−∞,0) by the relation ut(s) = u(t+s), s ∈(−∞,0). One possibility to deal with infinite delays, and which we will use here (cf. [18,11,12]), is to consider, for any γ > 0, the space : Cγ(H) = ϕ∈C((−∞,0]; H) : ∃lim s→−∞ eγsϕ(s)∈H, which is a Banach space with the norm kϕkγ:= sup s∈(−∞,0] eγs|ϕ(s)|. In order to state the problem in the correct framework, let us first establish some initial assumptions on some terms in the equation: We will assume that f∈L2(τ, T; (L2(Ω))3) for all T∈(τ, T∗]∩R. For the term g, in which the delay is present, we assume that g: (τ, T∗)× Cγ(H)→(L2(Ω))3satisfies (g1) For any ξ∈Cγ(H) the mapping (τ, T∗)3t7→ g(t, ξ) is measurable, (g2) g(t, 0) = 0 for all t∈(τ, T∗), (g3) there exists a constant Lg>0 such that for any t∈(τ, T∗) and all ξ, η ∈ Cγ(H), |g(t, ξ)−g(t, η)| ≤ Lgkξ−ηkγ. Remark 1. (i) Condition (g2) is not really a restriction, since otherwise, if |g(·,0)| ∈ L2(τ, T ) for all T∈(τ, T∗]∩R,we could redefine ˆ f(t) = f(t) + g(t, 0) and ˆg(t, ·) = g(t, ·)−g(t, 0).In this way the problem is exactly the same and ˆ fand ˆgsatisfy the required assumptions. (ii) Conditions (g2) and (g3) imply that |g(t, ξ)| ≤ Lgkξkγ, so that |g(·, ξ)| ∈ L∞(τ, T∗). We will denote Pmthe orthogonal projector of Honto the vector space generated by the first meigenfunctions of the Stokes problem in Ω with homogeneous Dirichlet boundary conditions. An example of operator satisfying assumptions (g1)-(g3) is given here. Example 1. We consider the operator g: (τ, T∗)×Cγ(H)→(L2(Ω))3defined as follows: g(t, ξ) := Z0 −∞ G(t, s, ξ(s))ds ∀t∈(τ, T∗),∀ξ∈Cγ(H), where the function G: (τ, T∗)×(−∞,0) ×R3→R3satisfies the following assumptions: (a) G(t, s, 0) = 0 for all (t, s)∈(τ, T∗)×(−∞,0). (b) There exists a function κ: (−∞,0) →(0,+∞) such that kG(t, s, u)−G(t, s, v)kR3≤κ(s)ku−vkR3 ∀u, v ∈R3,∀(t, s)∈(τ, T∗),×(−∞,0), 3D-GMNS EQUATIONS WITH INFINITE DELAYS 5 (c) and the function κsatisfies that κ(·)e−(γ+)·∈L2((−∞,0)) for some  > 0. Namely, the operator gdefines an element of (L2(Ω))3in the following way: g(t, ξ)(x) = Z0 −∞ G(t, s, ξ(s)(x))ds ∀x∈Ω. We check now that gsatisfies the assumption (g3), and using (a) above, we obtain that it is well defined as a map with values in (L2(Ω))3: ZΩZ0 −∞ κ(s)kξ(s)(x)−η(s)(x)kR3ds2 dx =ZΩZ0 −∞ κ(s)e−(γ+)se(γ+)skξ(s)(x)−η(s)(x)kR3ds2 dx ≤ZΩZ0 −∞ κ2(s)e−2(γ+)sdsZ0 −∞ e2(γ+)skξ(s)(x)−η(s)(x)k2 R3dsdx =Z0 −∞ κ2(s)e−2(γ+)sdsZΩZ0 −∞ e2(γ+)skξ(s)(x)−η(s)(x)k2 R3dsdx =CκZ0 −∞ ZΩ e2(γ+)skξ(s)(x)−η(s)(x)k2 R3dxds ≤Cκ"sup s∈(−∞,0] e2γs ZΩ kξ(s)(x)−η(s)(x)k2 R3dx#Z0 −∞ e2sds =Cκkξ−ηk2 γ 1 2 =Lgkξ−ηk2 γ, where we have denoted Cκ=kκ(·)e−(γ+)·k2 L2(−∞,0) and Lg=Cκ/(2). 3. Existence of solutions. In this section we establish existence of solution for (2) by a compactness method using a Faedo-Galerkin scheme. Definition 1. A weak solution of (2) is a function u∈C((−∞, T]; H)∩L2(τ, T ;V) for all T∈(τ, T ∗]∩R, with uτ=φand such that for all v∈V, d dt(u(t), v) + ν((u(t), v)) + bN(u(t), u(t), v)=(f(t), v)+(g(t, ut), v), in the sense of D0(τ, T ∗). Remark 2. If uis a solution of (2) in the sense given above, then usatisfies an energy equality, namely: |u(t)|2+ 2νZt s ku(r)k2dr =|u(s)|2+ 2 Zt s [(f(r), u(r)) + (g(r, ur), u(r))] dr ∀s, t ∈[τ, T∗]∩R. First, we will prove the uniqueness of weak solutions for our model in a similar way as done in [20] for the model without delay. We will only include the detailed estimates which involve the delay term. 6 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL Theorem 1. Under the above assumptions, there exists at most a weak solution u of (2). The proof is similar to, but a bit more complicated than for the 2D-Navier-Stokes equations and depends on the following Lemma 1. ([20]) For every u, v ∈V, and each N > 0, 1. 0 ≤ kukFN(kuk)≤N, 2. |FN(kuk)−FN(kvk)| ≤ 1 NFN(kuk)FN(kvk)ku−vk. Proof of Theorem 1:Let u, v be two weak solutions with the same initial conditions and set w=v−u. Then, using the energy equality, we obtain 1 2 d dt|w|2+νkwk2+hNL(u, v), wi= (g(t, vt)−g(t, ut), w), t ∈(τ, T∗), where we have set hNL(u, v), wi=FN(kuk)b(u, u, w)−FN(kvk)b(v, v, w).From the properties of the trilinear form bit easily follows that hNL(u, v), wi=FN(kuk)b(w, u, w)+(FN(kuk)−FN(kvk))b(v, u, w) +FN(kvk)b(v, w, w). Now using Lemma 1, formula (7) and Young’s inequality (see [20] for the details) there exists a constant C3>0,which depends on C2and ν, such that, | hNL(u, v), wi | ≤ νkwk2+C3N4|w|2. Consequently, taking (g3) into account, we obtain d dt|w(t)|2≤2C3N4|w(t)|2+ 2Lgkwtkγ|w(t)|, t ∈(τ, T∗). Observe that w(s) = 0 if s≤τ. Therefore, for t∈(τ, T∗) : kwtkγ= sup θ≤0 eγθ|w(t+θ)| ≤ sup θ∈[τ−t,0] |w(t+θ)|. Thus we obtain |w(t)|2≤2C3N4Zt τ |w(s)|2ds + 2LgZt τ sup r∈[τ,s] |w(r)||w(s)|ds ≤(2C3N4+ 2Lg)Zt τ sup r∈[τ,s] |w(r)|2ds, for any t∈[τ, T∗). Now we deduce that sup r∈[τ,t] |w(r)|2≤(2C3N4+ 2Lg)Zt τ sup r∈[τ,s] |w(r)|2ds, for any t∈[τ, T∗),whence the Gronwall lemma finishes the proof. Our main result is the following Theorem 2. Suppose that f∈L2(τ, T; (L2(Ω))3)for all T∈(τ, T∗]∩R, g : (τ, T∗)×Cγ(H)→(L2(Ω))3satisfying the assumptions (g1)–(g3), and φ∈Cγ(H) are given, and that 2γ > νλ1.Then, there exists a unique weak solution uof (2), which in fact is a strong solution in the sense that u∈C((τ, T ]; V)∩L2(τ+ε, T;D(A)), 3D-GMNS EQUATIONS WITH INFINITE DELAYS 7 for all 0< ε < T∗−τand any T∈(τ+ε, T∗]∩R. Moreover, if φ(0) ∈V, then usatisfies u∈C([τ, T ]; V)∩L2(τ, T;D(A)), for all T∈(τ, T∗]∩R. Proof. We split the proof in several steps. Step 1: A Galerkin scheme. Let us consider {vj} ⊂ V, the orthonormal basis of Hof all the eigenfunctions of the Stokes operator. Denote Vm= span[v1, . . . , vm] and consider the projector Pmu=Pm j=1(u, vj)vj. Define also um(t) = m X j=1 αm,j(t)vj where the upper script mwill be used instead of (m) for short since no confusion is possible with powers of u, and where the coefficients αm,j are required to satisfy the following system: d dt(um(t), vj) + ν((um(t), vj)) + bN(um(t), um(t), vj) = (f(t), vj)+(g(t, um t), vj),1≤j≤m, (8) where the equations are understood in the sense of D0(τ, T∗),and the initial conditions is um(τ+s) = Pmφ(s) for s∈(−∞,0]. The above system of ordinary functional differential equations with infinite delay fulfills the conditions for existence and uniqueness of local solution of [12, Th.1.1, p.36]. Next, we will deduce a priori estimates that assure that the solutions do exist for all time t∈[τ, T∗]∩R. Step 2: A priori estimates. Multiplying (8) by umwe obtain 1 2 d dt|um(t)|2+νλ1 2|um(t)|2+ν 2kum(t)k2≤(f(t), um(t)) + (g(t, um t), um(t)) ≤ |f(t)||um(t)|+Lgkum tkγ|um(t)| ≤ν 4kum(t)k2+|f(t)|2 νλ1 +Lgkum tk2 γ. Hence |um(t)|2+ν 2Zt τ e−νλ1(t−s)kum(s)k2ds ≤e−νλ1(t−τ)|u(τ)|2+ 2 Zt τ e−νλ1(t−s)|f(s)|2 νλ1 +Lgkum sk2 γds. (9) Further kum tk2 γ≤max (sup θ∈(−∞,τ−t] e2γθ|φ(θ+t−τ)|2,sup θ∈[τ−t,0] e2γθ−νλ1(t−τ+θ)|u(τ)|2 +2e2γθ Zt+θ τ e−νλ1(t+θ−s)|f(s)|2 νλ1 +Lgkum sk2 γds). 8 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL On the one hand sup θ∈(−∞,τ−t] eγθ|φ(θ+t−τ)|= sup θ≤0 eγ(θ−(t−τ))|φ(θ)| =e−γ(t−τ)kφkγ. On the other hand, as we are assuming that 2γ > νλ1, sup θ∈[τ−t,0] e2γθ−νλ1(t−τ+θ)|u(τ)|2≤e−νλ1(t−τ)|u(τ)|2 and sup θ∈[τ−t,0] e2γθ Zt+θ τ e−νλ1(t+θ−s)|f(s)|2 νλ1 +Lgkum sk2 γds ≤Zt τ e−νλ1(t−s)|f(s)|2 νλ1 +Lgkum sk2 γds. Collecting these inequalities we deduce kum tk2 γ≤e−νλ1(t−τ)kφk2 γ+ 2 Zt τ e−νλ1(t−s)|f(t)|2 νλ1 +Lgkum sk2 γds. By the Gronwall lemma we have kum tk2 γ≤e−(νλ1−2Lg)(t−τ)kφk2 γ+2 νλ1Zt τ e−(λ1ν−2Lg)(t−s)|f(s)|2ds. Then we obtain the following estimates: for any R > 0 and T∈(τ, T∗]∩R,there exists a constant C=C(τ, T, R),depending on some constants of the problem (namely, λ1, ν, Lgand f), and on τ, T and R, such that kum tk2 γ≤C(τ, T, R)∀t∈[τ, T],∀ kφkγ≤R, ∀m≥1.(10) In particular, this implies that {um}is bounded in L∞(τ, T;H)∀T∈(τ, T∗]∩R.(11) Now, it follows from (9) and (10) that ν 2e−νλ1(T−τ)ZT τ kum(s)k2ds ≤ν 2ZT τ e−νλ1(T−s)kum(s)k2ds ≤ |u(τ)|2+ 2 ZT τ e−νλ1(T−s)|f(s)|2 νλ1 +Lgkum sk2 γds ≤R2+ 2 ZT τ e−νλ1(T−s)|f(s)|2 νλ1 +LgC(τ, T, R)ds, so that we conclude the existence of another constant (relabelled the same) C(τ, T, R) such that kumk2 L2(τ,T ;V)≤C(τ, T, R)∀ kφkγ≤R∀m≥1,∀T∈(τ, T∗]∩R.(12) Now, observe that (8) is equivalent to dum dt =−νAum−PmBN(um, um) + Pmf(t) + Pmg(t, um t).(13) 3D-GMNS EQUATIONS WITH INFINITE DELAYS 9 From (5), (11), (12) and (13), by the choice of the basis one also deduces that k(um)0k2 L2(τ,T ;V0)≤C(τ, T, R)∀ kφkγ≤R∀m≥1,∀T∈(τ, T∗]∩R.(14) So, this implies the existence of a u∈L∞(τ, T ;H)∩L2(τ, T;V) with u0∈L2(τ, T;V0),∀T∈(τ, T∗]∩R, and a subsequence of {um}which converges weak-star to uin L∞(τ, T;H),weakly to uin L2(τ, T ;V),with {(um)0}converging weakly to u0in L2(τ, T;V0) for all T∈(τ, T∗]∩R. Observe in particular that u∈C([τ, T]; H) for all T∈(τ, T∗]∩R. By a compactness result (cf. [17, Ch.1,Th.5.1]), one can then deduce that a subsequence in fact converges strongly to uin L2(τ, T;H) and a.e. in (τ, T) with values in Hand a.e. in (τ, T)×Ω for all T∈(τ, T∗]∩R. Step 3: Some more a priori estimates. The estimates obtained above are not enough to pass to the limit and deduce that uis a solution of (2). Namely, we have two main difficulties. On the one hand, we need to pass to the limit in g(um), this will be done in Step 4, proving that actually um t→utin Cγ(H).On other hand, the weak convergence in L2(τ, T;V) is not enough to ensure that kum(t)k→ku(t)k or at least FN(kum(t)k)→FN(ku(t)k) for a.a. t, which is needed to manage the nonlinear term BN(um, um). In order to sort out this last trouble, we need to find a stronger estimate. We proceed now with that. Take the inner product of the Galerkin ODE (8) with Aum and obtain 1 2 d dtkum(t)k2+ν|Aum(t)|2+bN(um(t), um(t), Aum(t)) = (f(t), Aum(t)) + (g(t, um t), Aum(t)).(15) Obviously, (f(t), Aum(t)) ≤ |f(t)||Aum(t)| ≤ ν 8|Aum(t)|+2 ν|f(t)|2 and |(g(t, um t), Aum(t))| ≤ ν 8|Aum(t)|2+2 ν|g(t, um t)|2. By (6), Lemma 1and Young’s inequality, it follows |bN(um(t), um(t), Aum(t))| ≤ N kum(t)kC2kum(t)k3/2|Aum(t)|3/2 =NC2kum(t)k1/2|Aum(t)|3/2 ≤ν 4|Aum(t)|2+CNkum(t)k2, with CN=27(NC2)4 4ν3. Thus (15) simplifies to d dtkum(t)k2+ν|Aum(t)|2≤4 ν|f(t)|2+4 ν|g(t, um t)|2+ 2CNkum(t)k2.(16) 16 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL Thus, if we take β=λ−1/2 1|f| ν−Lgλ−1 1 , we obtain ((Rmu, u)) ≥0∀u∈Vmsuch that kuk=β. Consequently by a corollary of the Brouwer’s fixed point theorem (see [17, p.53]), for each m≥1 there exist um∈Vmsuch that Rm(um)=0,with kumk ≤ β. Observe moreover that Aum∈Vm,and therefore ν|Aum|2=−hBN(um, um), Aumi+ (f, Aum)+(g(um), Aum) (40) ≤ν 2|Aum|2− hBN(um, um), Aumi+|f|2 ν+L2 gβ2 νλ1 . Moreover, by (6) and Young’s inequality, |hBN(um, um), Aumi| ≤ ν 4|Aum|2+CNkumk2(41) ≤ν 4|Aum|2+CNβ2, with CN=27(NC2)4 4ν3. From (40) and (41), we deduce that the sequence {um}is bounded in D(A), and consequently, by the compact injection of D(A) in V, we can extract a subsequence {um0} ⊂ {um}, that converges weakly in D(A) and strongly in Vto an element u∗∈D(A). It is now standard to take limits in (38) and to obtain that u∗is an stationary solution. In order to prove the final regularity remark, it is enough to take into account that every stationary solution u∗to (34) is also a solution to (2), but with initial data φ(t) = u∗for t∈(−∞,0], and forcing term f+g(u∗). Thus, we can apply Theorem 2. Uniqueness Let us suppose that u∗and eu∗are two stationary solutions of (34). Then, νhAu∗−Aeu∗, vi+hBN(u∗, u∗)−BN(eu∗,eu∗), vi = (g(u∗)−g(eu∗), v),∀v∈V. (42) Taking v=u∗−eu∗and proceeding as in (39) we obtain from (42) νku∗−eu∗k2≤ |FN(ku∗k)b(u∗−eu∗, u∗, u∗−eu∗)| +|(FN(ku∗k)−FN(keu∗k))b(eu∗, u∗, u∗−eu∗)| +λ−1 1Lgku∗−eu∗k2.(43) From this inequality, taking into account (4), that b(eu∗, u∗, u∗−eu∗) = 0,and the fact that FN(ku∗k)≤1,we obtain νku∗−eu∗k2≤λ−1/4 1ku∗k ku∗−eu∗k2+λ−1 1Lgku∗−eu∗k2, and therefore, by the estimate (36), νku∗−eu∗k2≤ λ−1/4 1 λ−1/2 1|f| ν−λ−1 1Lg +λ−1 1Lg!ku∗−eu∗k2.(44) 3D-GMNS EQUATIONS WITH INFINITE DELAYS 17 On the other hand, if in (43) we use (4), and Lemma 1, i.e. ku∗kFN(ku∗k)≤N and keu∗kFN(keu∗k)≤N, we obtain νku∗−eu∗k2≤Nλ−1/4 1ku∗−eu∗k2+λ−1 1Lgku∗−eu∗k2.(45) From (44) and (45) we deduce that if (37) holds, then u∗=eu∗. Theorem 4. Assume that the assumptions in Theorem 2with fand gindependent of time and (37) hold. Then there exists a value 0< λ < 2γsuch that for the solution u(·,0, φ)of (2) with τ= 0,T∗= +∞and φ∈Cγ(H),the following estimates hold for all t≥0: |u(t, 0, φ)−u∗|2≤e−λt |φ(0) −u∗|2+Lg 2γ−λkφ−u∗k2 γ,(46) kut(·,0, φ)−u∗k2 γ ≤max e−2γtkφ−u∗k2 γ, e−λt |φ(0) −u∗|2+Lg 2γ−λkφ−u∗k2 γ,(47) where u∗is the unique stationary solution of (34) given by Theorem 3. Proof. For short denote u(t) = u(·,0, φ).Let us also denote w(t) = u(t)−u∗. Considering equations (34) for u(t) and (35) for u∗,one has d dt(w(t), v) + ν((w(t), v)) + bN(u(t), u(t), v)−bN(u∗, u∗, v)=(g(ut)−g(u∗), v), for t > 0, for any v∈V. From energy equality and the Lipschitz condition on g, and introducing an exponential term eλt with a positive value λto be fixed later on, we obtain d dt(eλt|w(t)|2)≤eλt λ|w(t)|2−2νkw(t)k2 +2|bN(u(t), u(t), w(t))−bN(u∗, u∗, w(t))|+ 2Lgkwtkγ|w(t)|), for t > 0. Reasoning as for (44) and (45), we have |bN(u(t), u(t), w)−bN(u∗, u∗, w)| ≤ µku(t)−u∗k2, where µ:= min (Nλ−1/4 1,λ−3/4 1|f| ν−λ−1 1Lg). Hence, using a Young inequality with δ > 0 to be fixed later on, we conclude that d dt(eλt|w(t)|2)≤eλt(−2ν+λλ−1 1+ 2µ+δλ−1 1Lg)kw(t)k2+Lg δeλtkwtk2 γ. Therefore, integrating from 0 to t, we have eλt|w(t)|2≤ |w(0)|2+Lg δZt 0 eλskwsk2 γds +(−2ν+λλ−1 1+ 2µ+δλ−1 1Lg)Zt 0 eλskw(s)k2ds. (48) 18 P. MAR´ IN-RUBIO, A. M. M´ ARQUEZ-DUR´ AN & J. REAL In order to control the term Rt 0eλskwsk2 γds, we proceed as follows. Zt 0 eλs sup θ≤0 e2γθ|w(s+θ)|2ds =Zt 0 eλs max{sup θ≤−s e2γθ|w(s+θ)|2,sup θ∈[−s,0] e2γθ|w(s+θ)|2}ds =Zt 0 max{e−(2γ−λ)skφ−u∗k2 γ,sup θ∈[−s,0] e(2γ−λ)θeλ(s+θ)|w(s+θ)|2}ds. So, if λ≤2γ, using the above equality in (48), we obtain eλt|w(t)|2≤ |w(0)|2+Lg δkφ−u∗k2 γZt 0 e(λ−2γ)sds +−2ν+λλ−1 1+2µ+δλ−1 1Lg+Lg(λ1δ)−1Zt 0 max r∈[0,s]eλrkw(r)k2ds. Observe that the (optimal) choice of δ= 1 makes that δλ−1 1Lg+Lg(λ1δ)−1is minimal and the coefficient of the last integral is negative with a suitable choice of λ∈(0,2γ) by (37). 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