On The Convergence of Solutions of Globally Modified Navier-Stokes Equations with Delays To Solutions of Navier-Stokes Equations with Delays

Advanced Nonlinear Studies 11 (2011), xxx–xxx On the Convergence of Solutions of Globally Modified Navier-Stokes Equations with Delays to Solutions of Navier-Stokes Equations with Delays Pedro Mar´ ın-Rubio, Jos´ e Real∗ Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla,Apdo. de Correos 1160, 41080-Sevilla, Spain e-mail: [email protected], jr[email protected] Antonio M. M´ arquez-Dur´ an Departamento de Econom´ıa, M´etodos Cuantitativos e Historia Econ´omica Universidad Pablo de Olavide,Ctra. de Utrera, Km. 1, 41013-Sevilla, Spain e-mail: ammar[email protected] Received 5 December 2010 Communicated by Peter E. Kloeden Abstract We prove that under suitable assumptions, from a sequence of solutions of Globally Modified Navier-Stokes equations with delays we can extract a subsequence which converges in an adequate sense to a weak solution of a three-dimensional Navier-Stokes equation with delays. An additional case with a family of different delays involved in the approximating problems is also discussed. 2010 Mathematics Subject Classification.35Q30, 35K55, 35R10. Key words.Globally Modified Navier-Stokes equations; three-dimensional Navier-Stokes equations; Navier-Stokes equations with delays. ∗This work was partially supported by the Spanish Ministerio de Ciencia e Innovaci´ on, Project MTM2008-00088, and the Consejer´ ıa de Innovaci´ on, Ciencia y Empresa (Junta de Andaluc´ ıa), Proyecto de Excelencia P07-FQM-02468. 1 2P. Mar´ ın-Rubio, J. Real, A. M. M´ arquez-Dur´ an 1 Introduction Let Ω⊂R3be an open bounded set with regular boundary Γ. For each N∈(0,+∞) 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 in Ω, with delays and homogeneous Dirichlet boundary condition:                                ∂u ∂t−ν∆u+FN(kuk)[(u· ∇)u]+∇p=G(t,u(t−ρ(t))) in (0,T)×Ω, ∇ · 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),in (−h,0) ×Ω, (1.1) where uis the unknown velocity field of the fluid, pis the unknown pressure, and we have given ν > 0, the kinematic viscosity, T>0 a final time, the external force term G(t,u(t−ρ(t))), which depends on the value u(t−ρ(t)), where 0 ≤ρ(t)≤h,with h>0,is a delay function, u0an initial velocity field, and φa velocity field defined in (−h,0). The globally modified Navier-Stokes equations, in the case without delays, were introduced and studied in [1] (see also [2, 3, 10, 11, 12, 21] and the review paper [9]). However, there are situations in which the model is better described if some terms containing delays appear in the equations. The system (1.1) has been studied in [4], where existence, uniqueness and asymptotic behaviour of a solution u(N)=u(N)(·;u0, φ) were analyzed (see also [16, 17, 18] for the case of infinite delays with and without modification). This system is a modification of the the following system of Navier-Stokes equations, with delays and homogeneous Dirichlet boundary condition:                                ∂u ∂t−ν∆u+(u· ∇)u+∇p=G(t,u(t−ρ(t))) in (0,T)×Ω, ∇ · 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),in (−h,0) ×Ω, (1.2) which has been studied in [5] and [8] (see also [6, 7, 14, 23, 19, 15] for the two-dimensional case). In this note we prove, in particular, that under suitable assumptions, from the family of solutions {u(N)(·;u0, φ) : N>0}, we can extract a sequence {u(Nj)(·;u0, φ) : j=1,2, ...},with Nj→+∞, which converges in an adequate sense to a weak solution of (1.2). We thus extend the result obtained in [1] for the case without delays. The structure of the paper is the following. In Section 2 we give some preliminaries on the abstract framework to deal with the problems, and the basic assumptions under which well-possedness of the approximating problems hold, and so that additional estimates can be obtained. In Section On solutions of Globally Modified Navier-Stokes equations with delays 3 3 we establish our main result on the convergence of a subsequence of solutions of approximating problems to one weak solution of the problem without modification. Finally, a result about the case when different delays may appear in the approximating problems is also discussed. 2 Preliminaries To set our problem in the abstract framework, we consider the following usual abstract spaces (see [13] and [24, 25]): V=u∈C∞ 0(Ω)3: div u=0, 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=1ZΩ uj(x)vj(x)dx, 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=1ZΩ ∂uj ∂xi ∂vj ∂xi dx. We we will use k·k∗for the norm in V0and h·,·ifor the duality pairing between Vand V0.Finally, we will identify every u∈Hwith the element fu∈V0given by hfu,vi=(u,v) for all v∈V. It follows that V⊂H⊂V0,where the injections are dense and compact. Now we define b(u,v,w)= 3 X i,j=1ZΩ ui ∂vj ∂xi wjdx, for all measurable functions u,v,wdefined on Ωwith values in R3for which the integrals in the right-hand member of the above equality are finite. In particular, bis a trilinear continuous form on V×V×V. 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.Moreover, from the properties of b(e.g. see [20] or [24]), and the definition of FN,one easily obtains the existence of a constant C1>0 only dependent on Ωsuch that |bN(u,v,w)| ≤ C1|u|1/4kuk3/4|v|1/4kvk3/4kwk,∀u,v,w∈V, |bN(u,v,w)| ≤ NC1kukkwk,∀u,v,w∈V. Thus, if we denote hBN(u,v),wi=bN(u,v,w),∀u,v,w∈V, 4P. Mar´ ın-Rubio, J. Real, A. M. M´ arquez-Dur´ an we have kBN(u,v)k∗≤C1|u|1/4kuk3/4|v|1/4kvk3/4,∀u,v∈V,(2.3) kBN(u,v)k∗≤NC1kuk,∀u,v∈V.(2.4) We also consider A:V→V0defined by hAu,vi=((u,v)).Denoting D(A)=(H2(Ω))3∩V,then Au =−P∆u,∀u∈D(A),is the Stokes operator (Pis the ortho-projector from (L2(Ω))3onto H). We recall (see [21]) that there exists a constant C2>0 depending only on Ωsuch that |b(u,v,w)| ≤ C2kukkvk|w|1/2kwk1/2,∀u,v,w∈V, and |b(u,v,w)| ≤ C2|u|kvk|Aw|,∀u∈H,v∈V,w∈D(A). Moreover, we assume given a mapping G: (0,T)×H→Hsuch that c1) G(·,u) : (0,T)→His measurable, ∀u∈H, c2) there exists a nonnegative function g∈Lp(0,T) for some 1 ≤p≤+∞,and a nondecreasing function L: (0,+∞)→(0,+∞), such that for all R>0 if |u|,|v|≤R,then |G(t,u)−G(t,v)|≤L(R)g1/2(t)|u−v|, a.e. t∈(0,T), and c3) there exists a nonnegative function f∈L1(0,T), such that for any u∈H, |G(t,u)|2≤g(t)|u|2+f(t),a.e. t∈(0,T). Finally, we suppose φ∈L2p0(−h,0; H) and u0∈H, where 1/p+1/p0=1. In this situation, we consider a delay function ρ∈C1([0,T]) such that 0 ≤ρ(t)≤hfor all t∈[0,T], and there exists a constant ρ∗satisfying ρ0(t)≤ρ∗<1∀t∈[0,T]. Definition 2.1 Let u0∈H and φ∈L2p0(−h,0; H)be given. A weak solution of (1.1) is a function u∈L2p0(−h,T;H)∩L2(0,T;V)∩L∞(0,T;H)such that (u(t),w)+νZt 0 ((u(s),w)) ds +Zt 0 bN(u(s),u(s),w)ds =(u0,w)+Zt 0 (G(s,u(s−ρ(s))),w)ds, for all t ∈[0,T]and all w ∈V, and coincides with φ(t)a.e. in (−h,0). Remark 2.1 The definition of a weak solution of (1.2) is the analogous of Definition 2.1, but with binstead of bN. On solutions of Globally Modified Navier-Stokes equations with delays 5 Remark 2.2 If uis a weak solution of (1.1) and we define ˜g(t)=g(θ−1(t)), where θ: [0,T]→ [−ρ(0),T−ρ(T)] is the differentiable and strictly increasing function given by θ(s)=s−ρ(s), then, by c3), we obtain ZT 0 |G(t,u(t−ρ(t)))|2dt ≤1 1−ρ∗ZT−ρ(T) −ρ(0) ˜g(t)|u(t)|2dt +ZT 0 f(t)dt, and therefore, taking into account the facts that ˜g∈Lp(−ρ(0),T−ρ(T)) and u∈L2p0(−h,T;H)∩ L∞(0,T;H), we have that G(·,u(·−ρ(·))) belongs to L2(0,T;H).Thus, as u∈L2(0,T;V) and satisfies the equation u0(t)+νAu(t)+BN(u(t),u(t)) =G(t,u(t−ρ(t))), in D0(0,T;V0), as a consequence of (2.4), u0∈L2(0,T;V0),and consequently (see [25]) u∈ C([0,T]; H) and satisfies the energy equality |u(t)|2+2νZt s ku(r)k2dr =|u(s)|2+2Zt s (G(r,u(r−ρ(r))),u(r)) dr, for all 0 ≤s,t≤T. In [4], the following theorem of existence and uniqueness of solutions of (1.1) was proved. Theorem 2.1 Assume that conditions c1)-c3) are satisfied, and that u0∈H and φ∈L2p0(−h,0; H) are given. Then, there exists a unique weak solution u =u(·;u0, φ)of (1.1) which is, in fact, a strong solution in the sense that u∈C([ε, T]; V)∩L2(ε, T;D(A)), for all 0<ε<T.Moreover, if u0∈V, then u∈C([0,T]; V)∩L2(0,T;D(A)). 3 Convergence to weak solutions of Navier-Stokes Equations The main result of this paper is the following. Theorem 3.1 Assume that conditions c1)-c3) are satisfied, and let {u(Nk)=u(Nk)(·;u(Nk) 0, φ(Nk)) : k= 1,2, ...}, where Nk→+∞as k →+∞, be a sequence of weak solutions of (1.1) with N =Nk,and with initial data such that u(Nk) 0*u0weakly in H, φ(Nk)→φstrongly in L 2p 2p−1(−h,0; H)as k →+∞, and the sequence {φ(Nk):k=1,2, ...}is bounded in L2p0(−h,0; H). Then, there exists a subsequence {u(Nj):j=1,2, ...}⊂{u(Nk):k=1,2, ...},with Nj→+∞, which converges weakly-star in L∞(0,T;H), weakly in L2(0,T;V),weakly-star in L2p0 (−h,T;H), strongly in L 2p 2p−1(−h,T;H),and strongly in L2(0,T;H), to a weak solution u of (1.2). 6P. Mar´ ın-Rubio, J. Real, A. M. M´ arquez-Dur´ an Proof. By Remark 2.2, we know that each of these solutions u(Nk)satisfies the energy equality 1 2 d dt |u(Nk)(t)|2+νku(Nk)(t)k2=(G(t,u(Nk)(t−ρ(t))),u(Nk)(t)),a.e. t∈(0,T), and therefore, taking into account c3) and that λ1|u(Nk)(t)|2≤ ku(Nk)(t)k2,where λ1is the first eigenvalue of the Stokes operator, we obtain d dt |u(Nk)(t)|2+νku(Nk)(t)k2≤1 νλ1 (g(t)|u(Nk)(t−ρ(t))|2+f(t)) a.e. t∈(0,T). Consequently, integrating between 0 and t, using the function ˜gdefined in Remark 2.2, and denoting ˜g(s)=0 for all s>T−ρ(T),we obtain |u(Nk)(t)|2+νZt 0 ku(Nk)(s)k2ds ≤ |uNk 0|2+1 νλ1(1 −ρ∗)Zt 0 ˜g(s)|u(Nk)(s)|2ds +1 νλ1(1 −ρ∗)Z0 −ρ(0) ˜g(s)|φ(Nk)(s)|2ds +1 νλ1ZT 0 f(s)ds,(3.5) for all t∈[0,T]. From the assumptions there exists a constant C>0 such that |uNk 0| ≤ Cand kφ(Nk)kL2p0(−h,0;H)≤C,for all k. (3.6) Therefore, from (3.5) we obtain |u(Nk)(t)|2+νZt 0 ku(Nk)(s)k2ds ≤KT+1 νλ1(1 −ρ∗)Zt 0 ˜g(s)|u(Nk)(s)|2ds,(3.7) for all t∈[0,T],where KT=C2+C2 νλ1(1 −ρ∗)k˜gkLp(−ρ(0),0) +1 νλ1ZT 0 f(s)ds. Thus, by the Gronwall lemma, |u(Nk)(t)|2≤KTexp 1 νλ1(1 −ρ∗)ZT 0 ˜g(s)ds!=:CT,(3.8) for all t∈[0,T]. From (3.6)–(3.8), we obtain that the sequence {u(Nk)}is bounded in the space L2p0(−h,T;H)∩ L∞(0,T;H)∩L2(0,T;V). Moreover, taking into account (2.3) and the equation satisfied by u(Nk), we deduce that the sequence of time derivatives d dt u(Nk)is bounded in L4/3(0,T;V∗).Thus, using in particular the Aubin-Lions compactness theorem (see [13]), there exists a subsequence u(Nj)of the On solutions of Globally Modified Navier-Stokes equations with delays 7 u(Nk)which converges to a function uweakly-star in L∞(0,T;H), weakly in L2(0,T;V),weakly-star in L2p0(−h,T;H), strongly in L 2p 2p−1(−h,T;H) and strongly in L2(0,T;H), and such that |u(Nj)(t)− u(t)| → 0 a.e. in (−h,T), u(Nj)→ua.e. in (−h,T)×Ω, and d dt u(Nk)converges to d dt uweakly in L4/3(0,T;V∗).Moreover, we can assume, without loss of generality, that there exists a function z∈L 2p 2p−1(−h,T) such that |u(Nj)(t)| ≤ z(t) a.e. t∈(−h,T). Now, we are going to prove that this limiting function uis a weak solution of (1.2). Evidently, as φ(Nj)→φstrongly in L 2p 2p−1(−h,0; H),we have that u(t)=φ(t) a.e. t∈(−h,0). Thus, by the density of D(A) into V, to finish the proof it is enough to see that for all w∈D(A), the function usatisfies (u(t),w)+νZt 0 ((u(s),w)) ds +Zt 0 b(u(s),u(s),w)ds =(u0,w)+Zt 0 (G(s,u(s−ρ(s))),w)ds t ∈[0,T].(3.9) Let us fix w∈D(A). We know that for all jone has (u(Nj)(t),w)+νZt 0 ((u(Nj)(s),w)) ds +Zt 0 bNj(u(Nj)(s),u(Nj)(s),w)ds =(u(Nj) 0,w)+Zt 0 (G(s,u(Nj)(s−ρ(s))),w)ds t ∈[0,T].(3.10) Evidently, by the assumptions, (u(Nj) 0,w)→(u0,w).(3.11) Also, as u(Nj)converges to uweakly in L2(0,T;V), we have that ((u(Nj),w)) →((u,w)) in L2(0,T).(3.12) Now, observe that by c2), a.e. t∈(0,T) the function G(t,·) : H→His continuous; thus, as |u(Nj)(s)−u(s)| → 0 a.e. in (−h,T), we have that G(t,u(Nj)(t−ρ(t))) →G(t,u(t−ρ(t))) in Ha.e. in (0,T). On the other hand, by c3) and (3), |G(t,u(Nj)(t−ρ(t)))| ≤ g1/2(t)z(t−ρ(t)) +f1/2(t) a.e. in (0,T). Thus, by dominated convergence, G(·,u(Nj)(· − ρ(·))) →G(·,u(· − ρ(·))) in L1(0,T;H).(3.13) We claim that bNju(Nj)(·),u(Nj)(·),w*b(u(·),u(·),w)weakly in L1(0,T).(3.14) 8P. Mar´ ın-Rubio, J. Real, A. M. M´ arquez-Dur´ an This can be seen as follows. First, we observe that by the boundedness of the sequence {u(Nj)}in L2(0,T;V), we have (see [1, Lem.12]) that FNj(ku(Nj)(·)k)→1 in L2(0,T). On the other hand, reasoning as in [13, pp.76–77], one obtains that bu(Nj)(·),u(Nj)(·),w*b(u(·),u(·),w)weakly in L2(0,T). Moreover, from the boundedness of the sequence {u(Nj)}in L2(0,T;V)∩L∞(0,T;H) and (2), we see that the sequence {bu(Nj)(·),u(Nj)(·),w}is bounded in L2(0,T).Then, from (3) and (3), we can deduce (3.14). Finally, as u(Nj)→uin L2(0,T;H), we have that (u(Nj),w)→(u,w) in L2(0,T).(3.15) From (3.10)–(3.14) and (3.15), we deduce that (3.9) holds a.e. t∈(0,T).Now observe that u∈ L∞(0,T;H), and also, due to the fact that u0∈L4/3(0,T;V0), we have that u∈C([0,T]; V0).Thus, uis weakly continuous as a function from [0,T] into H(see [22]), and therefore (3.9) holds for all t∈[0,T]. With a slight modification of the proof of Theorem 3.1, it is possible to prove the following result. Theorem 3.2 Assume that G(·,·):(−h,T+h)×H→H is continuous, and conditions c2) and c3) hold with p =1. Let {ρ(Nk):k=1,2, ...} ⊂ C1([−h,T+h]),where Nk→+∞as k →+∞, be a sequence of functions such that 0≤ρ(Nk)(t)≤h for all t ∈[−h,T+h]and any k ≥1,there exist constants ρ∗<1and ρ∗≥0such that −ρ∗≤(ρ(Nk))0(t)≤ρ∗for all t ∈[−h,T+h]and any k ≥1, and ρ(Nk)→¯ρ(t)uniformly in C1([−h,T+h]) as k →+∞. For each k ≥1,let us denote u(Nk), the corresponding weak solution of (1.1) with N =Nk, ρ=ρ(Nk), and with initial data (u(Nk) 0, φ(Nk))such that u(Nk) 0*u0weakly in H, φ(Nk)→φstrongly in L2(−h,0; H)as k →+∞, and the sequence {φ(Nk):k=1,2, ...}is bounded in L∞(−h,0; H). Then, there exists a subsequence {u(Nj):j=1,2, ...}⊂{u(Nk):k=1,2, ...},with Nj→+∞, which converges weakly-star in L∞(−h,T;H), weakly in L2(0,T;V),and strongly in L2(−h,T;H), to a weak solution u of (1.2) with ρ=¯ρ. Proof. The beginning of the proof follows the same steps as in Theorem 3.1. Indeed it only requires to be adapted slightly to deal with the different delays, but the uniform estimates can be obtained similarly. However, although it is possible to obtain a uniform estimate for |G(t,u(Nj)(t−ρ(Nj)(t)))|,since now z∈L∞(−h,T),the pass to the limit from G(t,u(Nj)(t−ρ(Nj)(t))) to G(t,u(t−¯ρ(t))) is not so clear, since the time arguments of u(Nj)and uare different, and the convergence only seems to hold in V0 but not in H. On solutions of Globally Modified Navier-Stokes equations with delays 9 This can be circumvented in the following way. To conclude that uis a solution for (1.2) it only remains to prove that Zt 0 (G(s,u(Nj)(s−ρ(Nj)(s))),w)ds →Zt 0 (G(s,u(s−¯ρ(s))),w)ds as j→+∞,(3.16) since the rest of the convergences in the approximating equations toward the expected limit follow analogously as in Theorem 3.1. Using the change of variable θj(s) :=s−ρ(Nj)(s)=r,(3.16) is equivalent to prove that Zt−ρ(Nj)(t) −ρ(Nj)(0) 1 1−(ρ(Nj))0(θ−1 j(r))(G(θ−1 j(r),u(Nj)(r)),w)dr →Zt−¯ρ(t) −¯ρ(0) 1 1−¯ρ0(¯ θ−1(r))(G(¯ θ−1(r),u(r)),w)dr as j→+∞,(3.17) where ¯ θ(s) :=s−¯ρ(s). Indeed, the proof of the above convergence again will follows from the dominated convergence theorem (observe that now the functions u(Nj)and uare written in the same time values), thanks to the continuity of G,and if we finally check that θ−1 j(r)→¯ θ−1(r) as j→+∞,for almost all r∈(−h,T).Actually, we claim that θ−1 j(r)→¯ θ−1(r) as j→+∞,for all r∈(−h,T).(3.18) Indeed, if it is not so, there exist values ˆr∈(−h,T) and ε > 0,and a subsequence relabelled {θ−1 j}, such that |θ−1 j(ˆr)−¯ θ−1(ˆr)| ≥ εfor all j. Then, there are two possibilities. Case 1: For a subsequence, again relabelled {θ−1 j}, θ−1 j(ˆr)≥¯ θ−1(ˆr)+εfor all j. (3.19) Due to the strict monotonicity of ¯ θand ¯ θ−1and the fact that ˆr<T,then ¯ θ−1(ˆr)<T+h.Taking if necessary a smaller value ε > 0,we may assume that ¯ θ−1(ˆr)+ε∈[−h,T+h].We now make the composition of the expressions in (3.19) with θj.This leads to ˆr=θj(θ−1 j(ˆr)) ≥θj(¯ θ−1(ˆr)+ε). Passing to the limit when j→+∞and using that ¯ θis strictly increasing, we obtain that ˆr≥¯ θ(¯ θ−1(ˆr)+ε)>¯ θ(¯ θ−1(ˆr)) =ˆr, which is a contradiction. Case 2: For a subsequence, again relabelled {θ−1 j}, ¯ θ−1(ˆr)≥θ−1 j(ˆr)+εfor all j. (3.20) We claim that θ−1 j(ˆr)+ε≤T+hfor all j, (3.21) if ε > 0 is small enough.