Navier-Stokes Equations with Delays

Navier-Stokes Equations with Delays By Tom´ as Caraballo, & Jos´ e Real† 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 external force contains some hereditary characteristics are proved. Keywords: Navier-Stokes equations, variable and distributed delays. 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). However, up to date, we have not found in the literature any work which takes into account the possibility of appearing some kind of delay in these equations. The main aim of this work is to consider several situations in which the external force contains some hereditary features and prove existence of solutions. These situations may appear when we want to control the system (in certain sense) by applying a force which takes into account not only the present state of the system but the history of the solution. It is worth pointing out that a similar analysis was carried out by Artola (1969) for general linear partial differential equations with delays. Let Ω ⊂RN(N= 2 or 3) be an open and bounded set with regular boundary Γ, T > 0 given, and 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−∇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 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. †E-mails: [email protected] ; [email protected] Article submitted to Royal Society T EX Paper 2T. Caraballo, & J. Real 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 k·k,and associated scalar 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 compact. 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). Finally, we will use k·k∗for the norm in V0and h·,·i for the duality hV0, V i. In order to state the problem in the correct framework, let us firstly establish the 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, (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) Article submitted to Royal Society Navier-Stokes equations with delays 3 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), g1: [0, T ]×C0([−h, 0]; V)→(L2(Ω))Nsatisfies hypotheses (I)-(IV) with X=V, Y= (L2(Ω))N,Lg1=L1and Cg1=C1, and g2: [0, T]×C0([−h, 0]; V)→V0 satisfies 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), (1.1) where the equation in (1.1) must be understood in the sense of D0(0, T). Remark 1.1. Observe that the terms in (1.1) 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 (1.1), 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 (1.1), it then belongs to the space C0([0, T ]; H). In the next section, we shall prove existence of solutions to (1.1) and the uniqueness of solution to the problem in the case N= 2.In Section 3, we show several general situations containing delayed terms including, in particular, those with variable and distributed delays, and we conclude the work by proving, in the Appendix, a finite-dimensional result needed for the proof of the existence of solutions to (1.1). 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. Theorem 2.1. Let us consider u0∈H,φ∈L2(−h, 0; V),f∈L2(0, T ;V0), and assume that g1: [0, T ]×C0([−h, 0]; V)→(L2(Ω))Nsatisfies hypotheses (I)- (IV) with X=V,Y= (L2(Ω))N,Lg1=L1and Cg1=C1, and g2: [0, T ]× C0([−h, 0]; V)→V0satisfies hypotheses (I)-(IV) with X=V,Y=V0,Lg2=L2 and Cg2=C2. Then: a) If N= 2 and ν2> C2,there exists at most one solution to problem (1.1). b) If N∈ {2,3}and ν2> C2, there exists a solution to (1.1) if, in addition, the following assumption (C) holds: (C) If vmconverges weakly to vin L2(−h, T ;V)and strongly in L2(−h, T;H), then gi(·, vm ·)converges weakly to gi(·, v·)in L2(0, T ;V0)for i= 1,2. Article submitted to Royal Society 4T. Caraballo, & J. Real Proof. a) If N= 2 and ν2> C2, let u, v be two solutions to (1) and set w=u−v. Then, from the energy equality, and the bounds for the trilinear form (see Lions 1969), it follows that for all t∈(0, T) |w(t)|2+ 2νRt 0||w(s)||2ds =−2Rt 0b(w(s), u(s), w(s)) ds +2 Rt 0(g1(s, us)−g1(s, vs), w(s)) ds +2 Rt 0hg2(s, us)−g2(s, vs), w(s)ids ≤2k1Rt 0|w(s)|(||w(s)||)||u(s)||ds +2 Rt 0|g1(s, us)−g1(s, vs)||w(s)|ds +2 Rt 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+ 2νRt 0||w(s)||2ds ≤k2 1 εRt 0|w(s)|2||v(s)||2ds +εRt 0||w(s)||2ds +C1 εRt 0|w(s)|2ds +εRt 0||w(s)||2ds +2√C2Rt 0||w(s)||2ds, and so, |w(t)|2+ 2εZt 0||w(s)||2ds ≤k2 1 εZt 0|w(s)|2||v(s)||2ds +C1 εZt 0|w(s)|2ds, from which uniqueness follows thanks to the Gronwall lemma. b) Now, we assume N∈ {2,3},ν2> C2and that condition (C) holds. For the proof of existence, we will follow a Galerkin scheme similar to the one in Constantin & Foias (1988), so we only emphasize the details involving the new terms gi. Let us consider {wj} ⊂ V∩¡H2(Ω)¢Nthe orthonormal basis of Hof all the eigenfunctions of the Stokes problem in Ω with homogeneous Dirichlet conditions. The subspace of Vspanned by w1, ..., wmwill be denoted Vm. Consider the projector Pm:H→Vmgiven by Pmu=Pm j=1(u, wj)wj,and 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) = Pmu0, um(t) = Pmφ(t), t ∈(−h, 0). (2.1) The preceding is a system of ordinary functional differential equations in the unknown γm(t) = (γm1(t), ..., γmm(t)). We can get existence and uniqueness of solution by applying Theorem 3.1 in the Appendix. Observe that, according to Theorem 3.1, we can ensure that problem (2.1) has one solution defined in an interval [0, t∗] with 0 < t∗≤T. However, as can be deduced by the a priori estimates below, we can set t∗=T. Article submitted to Royal Society Navier-Stokes equations with delays 5 In fact, multiplying in (2.1) by γmj(t) and summing in j, we get for all t∈[0, t∗] |um(t)|2+ 2νRt 0kum(s)k2ds ≤ |u0|2+ 2 Rt 0hf(s), um(s)i +2 Rt 0(g1(s, um s), um(s)) ds +2 Rt 0hg2(s, um s), um(s)ids, and arguing in a similar manner as we did in the proof of uniqueness in the 2dimensional case, we easily get two constants (depending on φ, ν, f, g1, g2, h, T, but not on mnor t∗)K1and K2such that sup t∈[0,t∗]|um(t)|2≤K1,Zt∗ 0||um(s)||2ds ≤K2. So we can take t∗=T, and obtain that {um}is bounded in L2(0, T;V)∩L∞(0, T ;H). Moreover, observe that um=Pmφin (−h, 0) and, by the choice of the basis {wj}, the sequence umconverges to φin L2(−h, 0; V), and, in particular, g1(·, um ·) + g2(·, um ·) is bounded in L2(0, T;V0). Now, it is a standard matter to bound the nonlinear term b(um, um,·), and using the same reasoning that in Constantin & Foias (1988) (see page 67), one can obtain that ©dum dt ªis bounded in L4/3(0, T;V0) (in fact, if N= 2, ©dum dt ªis bounded in L2(0, T;V0)). Using the compactness of the injection of the space W={u∈L2(0, T;V) : du dt ∈L4/3(0, T;V0)}into L2(0, T;H), from the preceding analysis and the assumptions on g1and g2, we can deduce that there exist a subsequence (denoted again um) and u∈L2(−h, T ;V) such that: um→uweakly in L2(−h, T;V), um→uweakly star in L∞(0, T ;H), um→uin L2(−h, T;H), gi(·, um ·)→gi(·, u·) weakly in L2(0, T ;V0), i = 1,2. Now, as in the non-delay case, we can take limits in (2.1) after integrating over the interval (0, t) (for t∈(0, T)), getting that uis a solution to our problem (1.1) (see once again, e.g., Constantin & Foias 1988 for the complete details). Remark 2.2. Observe that if g1satisfies (I)-(IV) with X=Hand Y=¡L2(Ω)¢N, then, as a direct consequence of (IV), g1satisfies assumption (C). 3. Some general situations In this section, we are going to show some situations where our theory can be applied. The cases considered include situations such as distributed delay, variable delay, and delay in gradient or second order derivatives terms. (a)Case 1 Let G: [0, T]×RN→RNbe a measurable function satisfying G(t, 0) = 0 for all t∈[0, T],and assume that there exists L1>0 such that |G(t, u)−G(t, v)|RN≤L1|u−v|RN,∀u, v ∈RN. Article submitted to Royal Society 6T. Caraballo, & J. Real Consider a function ω(t), which is going to play the role of the delay function. We suppose that ω∈C1([0, T]), ω(t)≥0 for all t∈[0, T], h= maxt∈[0,T ]ω(t)>0 and ω∗= maxt∈[0,T ]ω0(t)<1. Then, we define g1(t, ξ)(x) = G(t, ξ(−ω(t))(x)) for each ξ∈C0([0, T]; H), x∈Ω and t∈[0, T]. Notice that, in this case, the delayed term g1in our problem turns to g1(t, ut) = G(t, u(t−ω(t))).Then, g1satisfies the hypotheses in Theorem 2.1 with X=Hand Y=L2(Ω)N. Indeed, (I)-(III) follow immediately. On the other hand, if u, v ∈L2(−h, T;H), using the change of variable τ=s−ω(s) it is easy to see that Zt 0|g1(s, us)−g1(s, vs)|2ds ≤Zt −h|u(τ)−v(τ)|2dτ ∀t∈[0, T], and, consequently, (IV) and (C) are fulfilled. (b)Case 2 Let now G: [0, T]×[−h, 0] ×RN→RNbe a measurable function satisfying G(t, s, 0) = 0 for all (t, s)∈[0, T ]×[−h, 0] and such that there exists a function γ∈L2(−h, 0) such that |G(t, s, u)−G(t, s, v)|RN≤γ(s)|u−v|RN,∀u, v ∈RN∀(t, s)∈[0, T ]×[−h, 0]. Then, we define g1(t, ξ)(x) = R0 −hG(t, s, ξ(s)(x)) ds for each ξ∈C0([0, T]; H), x∈Ω and t∈[0, T]. In this case, the delayed term g1in our problem becomes g1(t, ut) = Z0 −h G(t, s, u(t+s)) ds. As in Case 1, g1satisfies the hypotheses in Theorem 2.1 with X=Hand Y= ¡L2(Ω)¢N. Indeed, (I) and (II) can be deduced immediately. On the other hand, if ξ, η ∈ C0([0, T]; H), for each t∈[0, T] we obtain |g1(t, ξ)−g1(t, η)|2≤RΩ³R0 −h|G(t, s, ξ(s)(x)) −G(t, s, η(s)(x))|RNds´2 dx ≤RΩ³R0 −hγ(s)|ξ(s)(x)−η(s)(x)|RNds´2 dx ≤RΩkγk2 L2(−h,0) ³R0 −h|ξ(s)(x)−η(s)(x)|2 RNds´dx ≤hkγk2 L2(−h,0)kξ−ηk2 C0([0,T ];H). Finally, if u, v ∈L2(−h, T ;H) then, for each t∈[0, T ] it follows Zt 0|g1(τ, uτ)−g1(τ, vτ)|2dτ ≤hkγk2 L2(−h,0) Zt 0µZ0 −h|u(s+τ)−v(s+τ)|2ds¶dτ, and, with the change r=s+τ, Rt 0|g1(τ, uτ)−g1(τ, vτ)|2dτ ≤hkγk2 L2(−h,0) Rt 0³Rτ τ−h|u(r)−v(r)|2dr´dτ ≤hTkγk2 L2(−h,0) Rt −h|u(r)−v(r)|2dr. Article submitted to Royal Society Navier-Stokes equations with delays 7 (c)Case 3 Now, we shall exhibit a situation where certain delay can appear in terms containing partial derivatives with respect to the spatial variables. Let B(·)∈L∞(0, T;L(V;¡L2(Ω)¢N)) and ω∈C1([0, T]), such that ω(t)≥0 for all t∈[0, T ], h= maxt∈[0,T ]ω(t)>0 and ω∗= maxt∈[0,T ]ω0(t)<1. We now define g1(t, ξ) = B(t)ξ(−ω(t)) for each ξ∈C0([0, T]; V), and t∈[0, T ]. Thus, in this case the delayed term g1in problem (1.1) turns to g1(t, ut) = B(t)u(t−ω(t)). It is easy to see that g1satisfies the hypotheses in Theorem 2.1 with X=Vand Y=¡L2(Ω)¢N. Indeed, (I)-(IV) obviously hold. On the other hand, if vmconverges to zero weakly in L2(−h, T;V) and ψ∈L2(0, T ;V) is given, we have ZT 0hg1(t, vm t), ψ(t)idt =ZT 0hB∗(t)ψ(t), vm(t−ω(t))idt, with B∗(·)∈L∞(0, T;L(L2(Ω)N;V0)) ⊂L∞(0, T;L(V;V0)) the adjoint of B(·). Using the change of variables τ=t−ω(t) = ρ(t), we obtain RT 0hg1(t, vm t), ψ(t)idt =Rρ(T) ρ(0) B∗(ρ−1(τ))ψ(ρ−1(τ)), vm(τ)®1 ρ0(ρ−1(τ)) dτ =RT −hhΨ(τ), vm(τ)idτ, with Ψ(τ) =      1 ρ0(ρ−1(τ))B∗(ρ−1(τ))ψ(ρ−1(τ)) if τ∈[ρ(0), ρ(T)], 0 if τ∈[−h, T]\[ρ(0), ρ(T)]. For this function Ψ it follows ZT −hkΨ(τ)k2 ∗dτ =Zρ(T) ρ(0) 1 (ρ0(ρ−1(τ)))2kB∗(ρ−1(τ))ψ(ρ−1(τ))k2 ∗dτ, and thus, by means of the change τ=ρ(t) = t−ω(t), ZT −hkΨ(τ)k2 ∗dτ =ZT 0 1 1−ω0(t)kB∗(t)ψ(t)k2 ∗dt ≤b2 0 1−ω∗ZT 0kψ(t)k2dt, where b0=kB∗(·)kL∞(0,T ;L(V;V0)) .Consequently, Ψ ∈L2(−h, T ;V0) and lim m→∞ ZT 0hg1(t, vm t), ψ(t)idt = lim m→∞ ZT −hhΨ(τ), vm(τ)idτ = 0. Therefore, hypothesis (C) is satisfied and once again we can apply our theory to this situation. Article submitted to Royal Society 8T. Caraballo, & J. Real (d)Case 4 Let K∈L∞(−h, T;L(V;V0)) and consider in problem (1.1) a term of the form g2(t, ut) = R0 −hK(t+s)u(t+s)ds, defined for all u∈L2(−h, T ;V). This term corresponds to the situation g2(t, ξ) = R0 −hK(t+s)ξ(s)ds for each t∈[0, T] and ξ∈ C0([0, T]; V). In this case, it is easy to see that g2is well defined and satisfies (I)-(IV) with X=Vand Y=V0. In particular, if we denote k=kK(·)kL∞(−h,T ;L(V;V0)), we can see that, for each t∈[0, T] and each u∈L2(−h, T;V), we have Zt 0kg2(s, us)k2 ∗ds ≤k2hmin(h, T)Zt −hku(s)k2ds, and thus, (IV) holds by setting C2=k2hmin(h, T). On the other hand, let vmbe weakly converging to zero in L2(−h, T;V), and fix ψ∈L2(0, T;V). Then ZT 0hg2(t, vm t), ψ(t)idt =ZT 0¿Zt t−h K(τ)vm(τ)dτ, ψ(t)Àdt, and, by Fubini’s theorem, it is easy to see that ZT 0hg2(t, vm t), ψ(t)idt =ZT −hhΣ(τ), vm(τ)idτ, with Σ(τ) = K∗(τ)Ψ(τ) and Ψ(τ) =          Rτ+h 0ψ(t)dt if −h≤τ < 0, Rτ+h τψ(t)dt if 0 ≤τ < T −h, RT τψ(t)dt if T−h≤τ≤T, in the case h≤T, and Ψ(τ) =          Rτ+h 0ψ(t)dt if −h≤τ < T −h, RT 0ψ(t)dt if T−h≤τ < 0, RT τψ(t)dt if 0 ≤τ≤T, in the case h > T . In both cases Ψ ∈C0([0, T ]; V), and in particular Σ ∈L2(0, T ;V0). Consequently, if vmconverges weakly to zero in L2(−h, T;V), then g2(·, vm ·) converges weakly to zero in L2(−h, T;V0) and thus, g2satisfies hypothesis (C). Appendix In this section we will prove a theorem on the existence of solutions for a finitedimensional problem. This result has been used in the proof of Theorem 2.1, and is a variant of Theorem 3.2 Chapter 4 (p. 213) in Bensoussan et al. (1992). We present the proof for the sake of completeness. However, it is worth mentioning that a similar result is proved in Hale & Lunel (1995) in the case of continuous initial datum, although this result cannot be applied to our situation. Article submitted to Royal Society Navier-Stokes equations with delays 9 Theorem 3.1. 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 > 0 there 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+Rt 0f(s, u(s)) ds +Rt 0g(s, us)ds +Rt 0k(s)ds ∀t∈[0, t∗]. (3.1) b) There exists t∗∈(0, T ]such that there exists one (and only one)solution to the problem (3.1). c) Suppose that there exists a constant C > 0such that if t∗∈(0, T ]is such that there is a solution uof (3.1), then maxt∈[0,t∗]|u(t)|Rm≤C. Then, under this additional assumption, there exists a solution to problem (3) with t∗=T. Proof. a) If uand vare two solutions of (3.1) then, denoting w=u−v, we obtain w= 0 in (−h, 0), and for all t∈[0, t∗] |w(t)|Rm≤LnZt 0|w(s)|Rmds +µCgt∗Zt 0|w(s)|2 Rmds¶1/2 , with n= max ¡maxt∈[0,t∗]|u(t)|Rm,maxt∈[0,t∗]|v(t)|Rm¢. Consequently, |w(t)|2 Rm≤(Ln+C1/2 g)2t∗Zt 0|w(s)|2 Rmds, ∀t∈[0, t∗], and thus, w= 0 on [0, t∗]. b) Take any C > 0 such that |u0|Rm≤C. Denote M= 1 + C+C1/2 gkφkL2(−h,0;Rm)+C(Cgh)1/2, and fix t∗∈(0, T] such that 2t∗≤³1 + M(LM+C1/2 g) + kkkL2(0,T ;Rm)´−2 , and T t∗being an integer. Let X={u∈L2(−h, t∗;Rm)∩C0([0, t∗]; Rm); u=φin (−h, 0),|u(t)|Rm≤M∀t∈[0, t∗]}, with the metric d(·,·) given by d(u, v) = max t∈[0,t∗]|u(t)−v(t)|Rm. Article submitted to Royal Society