Asymptotic Behaviour of 2D-Navier-Stokes Equations with Delays

Asymptotic behaviour of 2D−Navier-Stokes Equations with Delays By Tom´ as Caraballo and 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 asymptotic behaviour of solutions to Navier-Stokes equations when the external force contains some hereditary characteristics are proved. We show two different approaches to prove the convergence of solutions to the stationary one, when this is unique. The first is a direct method while the second is based in the Razumikhin type one. Keywords: Stability, long-time behaviour, 2DNavier-Stokes equations, variable delays. 1. Introduction The Navier-Stokes equations govern the motion of usual fluids like water, air, oil, etc. These equations have been the subject 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). In our recent work Caraballo & Real (2001) we consider a Navier-Stokes model in which the external force contains some hereditary features and prove the existence of weak solutions. These situations containing delays 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. Another interesting problem concerns the asymptotic behaviour of the systems, since this analysis can provide useful information on the future evolution of the system. This will be the main aim of this paper. To this end, let Ω ⊂R2be an open bounded set with regular boundary Γ, and consider the following functional 2D−Navier-Stokes problem (for further details and notations see Lions 1969 and Temam 1979):                ∂u ∂t −ν∆u+P2 i=1 ui ∂u ∂xi =f− ∇p+g(t, ut) in (0,+∞)×Ω, div u= 0 in (0,+∞)×Ω, u= 0 on (0,+∞)×Γ, u(0, x) = u0(x), x ∈Ω, u(t, x) = φ(t, x), t ∈(−h, 0) x∈Ω, †E-mail: [email protected] ; [email protected] Article submitted to Royal Society T EX Paper 2T. Caraballo & J. Real 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 with some hereditary characteristics and φthe initial datum in the interval of time (−h, 0),where his a fixed positive number. In Section 2, we will recall some preliminary results on the existence and uniqueness of weak solutions of our model that were proved in Caraballo & Real (2001). Then, we will complete these by proving a regularity result concerning the existence of strong solutions; such solutions will be needed in our stability analysis. In Section 3, we analyze the exponential convergence towards a unique stationary solution; such a solution exists if, for instance, the viscosity is large enough. Our main aim is to exhibit two different approaches ensuring this asymptotic behaviour: a direct method, and a Razumikhin type one. To apply the former, we need only the existence of weak solutions but must impose a restriction on the delay term, while the latter needs strong solutions but the extra assumption on the delay forcing term can be removed. A surprising fact, in principle, is that we can establish a sufficient condition on the parameters and functions appearing in the model, such that both results hold true in some particular cases. 2. Preliminaries and statement of the problem To start, we consider the following usual abstract spaces: V=nu∈(C∞ 0(Ω))2: div u= 0o, H= the closure of Vin (L2(Ω))2with norm |·| ,and inner product (·,·) where for u, v ∈(L2(Ω))2, (u, v) = 2 X j=1 ZΩ uj(x)vj(x)dx, V= the closure of Vin (H1 0(Ω))2with norm k·k ,and associated scalar product ((·,·)),where for u, v ∈(H1 0(Ω))2, ((u, v)) = 2 X i,j=1 ZΩ ∂uj ∂xi ∂vj ∂xi dx. It follows that V⊂H≡H0⊂V0,where the injections are dense and compact. Finally, we will use k·k∗for the norm in V0and h·,·i for the duality pairing between Vand V0. Now we define the trilinear form bon V×V×Vby b(u, v, w) = 2 X i,j=1 ZΩ ui ∂vj ∂xi wjdx∀u, v, w ∈V. The trilinear form satisfies b(u, v, w) = −b(u, w, v),∀u, v, w ∈V, and consequently b(u, v, v) = 0,∀u, v ∈V, Article submitted to Royal Society Asymptotic behaviour of 2D−Navier-Stokes equations with delays 3 b(u, u, v −u)−b(v, v, v −u) = −b(v−u, u, v −u),∀u, v ∈V. We recall that (H1 0(Ω))2(and, as a consequence, Vtoo) is embedded in (L4(Ω))2, this injection being compact. Also, there exists C1(Ω) >0 such that |b(u, v, w)| ≤ C1(Ω)|u|(L4(Ω))2kvkkwk,∀u, v, w ∈V. Now, consider a fixed T > 0. 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, (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 0 kg(s, us)−g(s, vs)k2 Yds≤CgZt −h ku(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 0 kg(s, us)−g(s, vs)k2 Yds≤CgZt −h ku(s)−v(s)k2 Xds. With the convention above, assume that u0∈H,φ∈L2(−h, 0; V), f∈L2(0, T;V0), g: [0, T]×C0([−h, 0]; V)→(L2(Ω))2satisfies hypotheses (I)-(IV) with X=V, Y= (L2(Ω))2,Lg=L1and Cg1=C1. For example, when the function gis defined by g(t, φ) = G(φ(−ρ(t)) for a suitable differentiable delay function ρand a Lipschitz continuous G, the assumptions above hold (see Caraballo and Real (2001) for more details and further examples). Thus, 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) + ν((u(t), v)) + b(u(t), u(t), v) = hf(t), vi+ (g(t, ut), v) u(0) = u0, u(t) = φ(t), t ∈(−h, 0), (2.1) where the equation in (2.1) must be understood in the sense of D0(0, T). Article submitted to Royal Society 4T. Caraballo & J. Real Remark 2.1. Observe that the terms in (2.1) are well defined. In particular, by hypotheses (I)-(IV), if u∈L2(−h, T ;V)the term g(t, ut)defines a function in L2(0, T; (L2(Ω)2). Moreover, if there exists a solution to this problem, then it belongs to the space C0([0, T ]; H). Remark 2.2. Define the operator A:V7→ V0by hAu, vi= ((u, v)),∀u, v ∈V, consider the operator B:V×V7→ V0defined by hB(u, v), wi=b(u, v, w),∀u, v, w ∈V, and set B(u) = B(u, u). Then, (2.1) can be rewritten as      To find u∈L2(−h, T ;V)∩L∞(0, T;H)such that, , d dtu(t) + νAu(t) + B(u(t)) = f(t) + g(t, ut)in V0 u(0) = u0, u(t) = φ(t), t ∈(−h, 0), (2.2) Observe that if we denote D(A) = (H2(Ω))2∩V, then Au =−P∆u, ∀u∈D(A), where Pis the orthogonal projector from (L2(Ω))2onto H, and there exists a constant C2(Ω) >0such that |u|2≤C2(Ω)|Au|,∀u∈D(A),(2.3) where |u|2denotes the norm of uin (H2(Ω))2(see Constantin & Foias 1988). In the following theorem we recall an existence and uniqueness result concerning our 2D-Navier-Stokes model with delays, completed with a statement about the existence of strong solutions. Theorem 2.3. Let us consider u0∈H,φ∈L2(−h, 0; V),f∈L2(0, T;V0), and assume that g: [0, T ]×C0([−h, 0]; V)→(L2(Ω))2satisfies hypotheses (I)-(IV) with X=V,Y= (L2(Ω))2,Lg=L1and Cg=C1. Also, we assume that the following condition (V) holds: (V) If vmconverges weakly to vin L2(−h, T ;V)and strongly in L2(−h, T ;H), then g(·, vm ·)converges weakly to g(·, v·)in L2(0, T ;V0). Then, a) There exists a unique weak solution to (2.1) which, in addition, belongs to the space C0([0, T ]; H). b) If f∈L2(0, T; (L2(Ω))2)and u0∈V, then, the solution uto (2.1) is a strong solution, that is, u∈L2(0, T;D(A)) ∩C0([0, T]; V)and u0∈L2(0, T;H).(2.4) In particular, if φ∈C0([−h, 0]; V)and u0=φ(0),then u∈C0([−h, T ]; V). Article submitted to Royal Society Asymptotic behaviour of 2D−Navier-Stokes equations with delays 5 Proof. For the proof of a), see Caraballo & Real (2001). To prove b), observe that if f∈L2(0, T ; (L2(Ω))2) and we denote ˜ f(t) = P(f(t) + g(t, ut)), then, for all v∈V,usatisfies      u∈L2(0, T;V)∩L∞(0, T;H), d dt(u(t), v) + ν((u(t), v)) + b(u(t), u(t), v) = ( ˜ f(t), v) in D0(0, T ), u(0) = u0, (2.5) with ˜ f∈L2(0, T;H),and consequently, if u0∈V,usatisfies (2.4) (see Temam 1979, 1995). 3. Stability of stationary solutions First of all, we will prove a result ensuring the existence of stationary solutions to our Navier-Stokes model when the delay term has a special form, provided the viscosity is large enough. Then, we will prove that when this stationary solution is unique all the solutions to our problem converge to it exponentially fast. This result requires a strong assumption on the delay function, which we will relax later on by using a different approach, namely, a Razumikhin type argument (see Razumikhin (1956), (1960) and Hale and Lunel (1995) for a modern and nice presentation of the method in the finite-dimensional case), but for this it will be necessary to deal with strong solutions. In the sequel, λ1will denote the first eigenvalue of A. (a)Existence and uniqueness of stationary solutions Let us consider the following equation du dt +νAu +B(u) = f+g(t, ut),(3.1) with f∈V0independent of t. A stationary solution to (3.1) is a u?such that νAu?+B(u?) = f+g(t, u?) for all t≥0. Now we will prove a result ensuring existence and uniqueness of stationary solutions when the delay term has a particular form, namely, we will assume that the term gis given by g(t, ut) = G(u(t−ρ(t))), with G:R2→R2a function satisfying G(0) = 0 and such that there exists L1>0 for which |G(u)−G(v)|R2≤L1|u−v|R2,∀u, v ∈R2, and ρ∈C1([0,+∞)), ρ(t)≥0 for all t≥0, h= supt≥0ρ(t)∈(0,+∞) and ρ∗= supt≥0ρ0(t)<1. Observe that this situation is within our framework and satisfies our assumptions (Conditions (I)-(V)) ensuring the existence and uniqueness of solutions (see Caraballo & Real 2001). Now we are interested in proving that there exist stationary solutions to our equation (3.1), i.e., there exists u?∈Vsuch that νAu?+B(u?) = f+G(u?). Article submitted to Royal Society 6T. Caraballo & J. Real We can prove the following result. Theorem 3.1. Suppose that Gsatisfies the conditions above and ν > λ−1 1L1. Then, (a)for all f∈V0there exists a stationary solution to (3.1); (b)if f∈(L2(Ω))2, the stationary solutions belong to D(A); (c)there exists a constant C3(Ω) >0such that if (ν−λ−1 1L1)2> C3(Ω)kfkV0, then the stationary solution to (3.1) is unique. Proof. (a) Let f∈V0be fixed. By the Lax-Milgram Theorem, it is easy to see that for each z∈Vgiven, there exists a unique u∈Vsuch that ν((u, v)) + b(z, u, v) = hf, vi+ (G(z), v),∀v∈V. (3.2) Moreover, taking v=uin (3.2), it follows that νkuk ≤ kfkV0+λ−1 1L1kzk.(3.3) Take k > 0 such that k(ν−λ−1 1L1)≥ kfkV0, and denote C={z∈V;kzk ≤ k}. Then, Cis a convex and compact subset of (L4(Ω))2, and moreover, by (3.3), the mapping z7→ u, defined by means of (3.2), maps Cinto C. If we see that this mapping is continuous in Cwith the topology induced by (L4(Ω))2, then, thanks to Schauder’s Theorem, we will obtain the existence of a fixed point in Cfor the mapping, and obviously this fixed point is a stationary solution to (3.1). The continuity of the mapping z7→ ucan be seen as follows. Let zi∈ C and ui∈ C be such that ν((ui, v)) + b(zi, ui, v) = hf, vi+ (G(zi), v),∀v∈V, i = 1,2. Then, it is easy to obtain that νku1−u2k2=b(z2−z1, u1, u1−u2)+(G(z1)−G(z2), u1−u2) ≤kC1(Ω)|z1−z2|L4(Ω))2ku1−u2k +L1λ−1/2 1|z1−z2|ku1−u2k.(3.4) As V⊂(L4(Ω))2and (L4(Ω))2⊂(L2(Ω))2with continuous injections, the continuity of the mapping z7→ uin C, with respect to the topology induced by (L4(Ω))2, is a direct consequence of (3.4). (b) Notice that if f∈(L2(Ω))2, then every stationary solution u∗to (3.1) is also a solution to (2.1), but with initial data u0=φ(t) = u∗for t∈[−h, 0), and forcing term ˜ f=P(f+G(u∗)) ∈H⊂L2(0, T;H).Thus, we can apply the standard regularity results from the theory of the Navier-Stokes equations without delays. (c) Let f∈Vbe given, and let u1and u2be two stationary solutions to (3.1). Then, arguing as we did for the inequality (3.4), we obtain νku1−u2k2≤C1(Ω)|u1−u2|L4(Ω))2ku1kku1−u2k+L1λ−1/2 1|u1−u2|ku1−u2k.(3.5) But νku1k2=hf, u1i+ (G(u1), u1) ≤ kfkV0ku1k+λ−1 1L1ku1k2, Article submitted to Royal Society Asymptotic behaviour of 2D−Navier-Stokes equations with delays 7 Thus, (ν−λ−1 1L1)ku1k ≤ kfkV0. Using this last inequality, and the continuous injection of Vinto (L4(Ω))2, we obtain easily from (3.5) that there exists C3(Ω) >0 such that (ν−λ−1 1L1)2ku1−u2k2≤C3(Ω)kfkV0ku1−u2k2. This completes the proof of the theorem. Remark 3.2. a) Although we have proved the existence of a unique stationary solution in this particular situation, this may well happen in more general cases. b) Also, it is worth pointing out that the regularity result in part (b) of the previous theorem does not depend on the particular form of the term gwe have chosen; this is clear from the argument used in the proof. (b)Exponential convergence of solutions: a direct approach Now we will prove that, under appropriate assumptions, our model has a unique stationary solution, u∞, and every weak solution approaches u∞exponentially fast as tgoes to +∞. Theorem 3.3. Assume that the forcing term g(t, ut)is given by g(t, ut) = G(u(t− ρ(t)) with ρ∈C1(R+; [0, h]) such that ρ0(t)≤ρ∗<1for all t≥0. Then, there exist two constants ki>0,i= 1,2, depending only on Ω, such that if f∈(L2(Ω))2 and ν > λ−1 1L1satisfy in addition 2νλ1>(2 −ρ∗)L1 1−ρ∗ +k1|f| ν−λ−1 1L1 +k2|f|3 ν2(ν−λ−1 1L1)3,(3.6) then there is a unique stationary solution u∞of (3.1) and every solution of (2.1) converges to u∞exponentially fast as t→+∞. More exactly, there exist two positive constants Cand λ, such that for all u0∈Hand φ∈L2(−h, 0; V), the solution u of (2.1) with f(t)≡fsatisfies |u(t)−u∞|2≤Ce−λt ³|u0−u∞|2+kφ−u∞k2 L2(−h,0;V)´,(3.7) for all t≥0. Proof. Let f∈(L2(Ω))2be fixed. Consider u, the solution of (2.2) for f(t)≡f, and let u∞∈D(A) be a stationary solution to (3.1). We set w(t) = u(t)−u∞, and observe that d dtw(t) + νAw(t) + B(u(t)) −B(u∞) = G(u(t−ρ(t)) −G(u∞). Now fix a positive λto be determined later on. By standard computations we get d dt(eλt|w(t)|2) = λeλt|w(t)|2+ eλt d dt|w(t)|2 ≤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+L1−2νλ1)kw(t)k2 +2eλt|b(w(t), w(t), u∞)|+L1eλt|w(t−ρ(t))|2.(3.8) Article submitted to Royal Society 8T. Caraballo & J. Real Obviously |b(w(t), w(t), u∞)| ≤ c|w(t)|kw(t)k|u∞|∞,(3.9) where we denote by |u∞|∞the norm of u∞in (L∞(Ω))2. Observe that H2(Ω) ⊂ L∞(Ω) with continuous injection, thus, using (2.3), we obtain the existence of a constant c1>0 depending only on Ω such that |b(w(t), w(t), u∞)| ≤ c1λ−1/2 1kw(t)k2|Au∞|.(3.10) On the other hand, ν|Au∞|≤|f|+|G(u∞)|+|B(u∞)| ≤ |f|+L1|u∞|+c0ku∞k|u∞|∞, and consequently, from the continuous injection of H2(Ω) into L∞(Ω), the inequality (2.3), and the Gagliardo-Nirenberg interpolation inequality, we obtain ν|Au∞| ≤ |f|+L1|u∞|+c00ku∞k|u∞|1/2|Au∞|1/2.(3.11) But c00ku∞k|u∞|1/2|Au∞|1/2≤(c00)2λ−1/2 1 2νku∞k3+ν 2|Au∞|, and thus, from (3.11) we deduce |Au∞| ≤ 2 ν|f|+2L1λ−1/2 1 νku∞k+(c00)2λ−1/2 1 ν2ku∞k3.(3.12) Now, as νku∞k2= (f, u∞)+(G(u∞), u∞) ≤ |f|λ−1/2 1ku∞k+L1λ−1 1ku∞k2, we obtain from (3.12) |Au∞| ≤ 2 ν|f|+2L1λ−1 1 ν(ν−L1λ−1 1)|f|+(c00)2λ−2 1 ν2(ν−L1λ−1 1)3|f|3 =2 (ν−L1λ−1 1)|f|+(c00)2λ−2 1 ν2(ν−L1λ−1 1)3|f|3.(3.13) From (3.8), (3.10), (3.13), and denoting k1= 4c1λ1/2 1, k2= 2c1λ−3/2 1(c00)2, it follows that d dt(eλt|w(t)|2) ≤λ−1 1eλt µλ+L1−2νλ1+k1|f| (ν−L1λ−1 1)+k2|f|3 ν2(ν−L1λ−1 1)3¶kw(t)k2 +L1eλt|w(t−ρ(t))|2.(3.14) Article submitted to Royal Society Asymptotic behaviour of 2D−Navier-Stokes equations with delays 9 Now, taking into account the properties of the function ρ, we deduce that if we denote τ(t) = t−ρ(t), the function τis strictly increasing in [0,+∞), and that there exists a µ > 0 such that τ−1(t)≤t+µfor all t≥ −ρ(0). Thus, by the change of variable η=s−ρ(s) = τ(s), we have Zt 0 eλs|w(s−ρ(s))|2ds=Zt−ρ(t) −ρ(0) eλτ−1(η)|w(η)|1 τ0(τ−1(η)dη ≤eλµ 1−ρ∗Zt −h eλη|w(η)|2dη. (3.15) If (3.6) is satisfied, then there exists λ > 0 small enough such that λ+L1−2νλ1+k1|f| (ν−L1λ−1 1)+k2|f|3 ν2(ν−L1λ−1 1)3+L1eλµ 1−ρ∗ ≥0, integrating (3.14) over the interval [0, t], and taking into account (3.15), we deduce that for this λ > 0 eλt|w(t)|2≤ |w(0)|2+L1eλµ 1−ρ∗Z0 −h eλη|w(η)|2dη, and thus (3.7) is satisfied. The uniqueness of u∞follows from the fact that if ˆu∞ is another stationary solution of (3.1), then u(t)≡ˆu∞is a solution of (2.1) with u0= ˆu∞and φ= ˆu∞, and consequently, applying (3.7) and making t→+∞, one has |ˆu∞−u∞|2≤0. Remark 3.4. Notice that the result holds true for more general delay terms. For instance, if gsatisfies Zt 0 e²s|g(s, us)−g(s, vs)|2ds≤C(h)Zt −h e²s|u(s)−v(s)|2ds, (3.16) for sufficiently small ² > 0, and the existence of a unique stationary solution is known. (c)Exponential convergence of solutions: a Razumikhin approach In the previous section we proved a result on the exponential convergence of weak solutions to the unique stationary solution when the delay term ghas a particular form, but we needed to impose a restriction on the delay function, so that our proof worked appropriately or that condition (3.16) holds. However, it is possible to prove a result for more general forcing terms and relax this restriction by using a different method which is also much used in dealing with the stability properties of delay differential equations. This method was firstly developed by Razumikhin (see Razumikhin (1956), (1960)) in the context of ordinary differential functional equations, and has already been applied to some stochastic ODEs and PDEs (e.g. Caraballo et al. 2000). However, one interesting point to be noted is that this method requires some kind of continuity concerning the operators in the model and the solutions. This will allow us to prove a result that weaken the assumptions on the delay function or condition (3.16), but concerns only the strong solutions to (2.1). Article submitted to Royal Society