Exponential stability of an incompressible non-Newtonian fluid with delay

Manuscript submitted to doi:10.3934/xx.xx.xx.xx AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX EXPONENTIAL STABILITY OF AN INCOMPRESSIBLE NON-NEWTONIAN FLUID WITH DELAY Linfang Liu1, Tomás Caraballo2,∗, and Xianlong Fu1 1Department of Mathematics, Shanghai Key Laboratory of PMMP East China Normal University, Shanghai 200241, P. R. of China 2Dpto. Ecuaciones Diferenciales y Análisis Numérico Fac. Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain (Communicated by the associate editor name) Abstract. The existence and uniqueness of stationary solutions to an incompressible non-Newtonian fluid are first established. The exponential stability of steady-state solutions is then analyzed by means of four different approaches. The first is the classical Lyapunov function method, while the second one is based on a Razumikhin type argument. Then, a method relying on the construction of Lyapunov functionals and another one using a Gronwall-like lemma are also exploited to study the stability, respectively. Some comments concerning several open research directions about this model are also included. 1. Introduction. The investigation of non-Newtonian fluids has been receiving much attention over the recent years, mainly due to their importance for the understanding of fluid materials motion in real life which cannot be characterized by Newtonian fluids (such as the classic Navier-Stokes equations). As examples of these fluids we can cite ketchup, toothpaste, saliva, and synovial fluid, Bingham plastics (like clay suspensions, drilling mud and mustard), latex paint, blood plasma, etc (see [2,3,16,18]). On the other hand, delay effects have been proved to be useful in the modeling of physical and biological phenomena, as well as in other real world applications. For instance, when we want to use some types of external forcing terms to control a system in control engineering, it seems natural to assume that these forces should take into account not only the current state of the system, but also some part of its history, sometimes even the whole history. The existence and uniqueness of solution, the existence of maximal compact attractor and global (or pullback) attractor for non-Newtonian equations have been studied in [1,2,3,13,17,23,24,25], while Guo et al. analyzed in [12] the martingale stationary solutions for some stochastic non-Newtonian fluids without delay. However, to the best of our knowledge, there are no available works concerning the 2010 Mathematics Subject Classification. Primary: 34D20; 34K20; 34K21; 37L15; 76A05. Key words and phrases. Exponential stability, stationary solution, non-Newtonian fluids, delay. This research was partially supported by the projects MTM2015-63723-P (MINECO/ FEDER, EU) and P12-FQM-1492 (Junta de Andalucía), and by NSF of China (Nos. 11671142 and11371087), Science and Technology Commission of Shanghai Municipality (No. 13dz2260400) and Shanghai Leading Academic Discipline Project (No. B407), respectively. ∗Corresponding author: [email protected]. 1 2 L. LIU, T. CARABALLO AND X. FU local stability analysis of incompressible non-Newtonian fluids containing hereditary characteristics (constant, distributed or variable delay, memory, etc). Enlightened by the analysis carried out in [6] in the case of 2D–Navier-Stokes equations, in the current paper we focus on the asymptotic behavior of the stationary solution of the following incompressible non-Newtonian fluid with delay in a 2D bounded domain Ω⊂R2whose existence and uniqueness of solutions have already been analyzed in [17] (see Theorem 3.1 in Section 3for more details), ∂u ∂t + (u·∇)u+∇p=∇·µ(e(u)) + g(t, ut) + f(x, t),in (τ, +∞)×Ω,(1) ∇·u= 0,in (τ, +∞)×Ω,(2) u(τ+θ, x) = φ(θ, x), θ ∈[−h, 0], x ∈Ω.(3) The above system (1)-(3) is equipped with below boundary conditions u= 0, τijνjνl= 0, i, j, l = 1,2,on ∂Ω×(τ, +∞),(4) where the velocity of the fluid u=u(x, t) = (u(1), u(2))is an unknown vector function, the external function f(x, t) = (f(1), f(2))is time-dependent, pis the pressure and h > 0is the delay time. Besides, g(t, ut)represents the influence of an external force with some kind of delay, memory or hereditary characteristics. In (4), u= 0 is the no-slip condition, namely, the fluid has zero velocity relative to the boundary, and ν= (ν1, ν2)is the outer unit normal to ∂Ω, while τij νjνl= 0 with τije= 2µ1∂eij ∂xl,i, j, l = 1,2, expresses that the first moments of the traction vanishes on ∂Ω, which is an immediate consequence of the principle of practical work. Problem (1)-(4) describes the motion of an isothermal incompressible viscous fluid with the extra stress tensor µ(e(u)) = (µij(e(u)))2×2, and which is a matrix of order 2×2with µij(e(u)) = 2µ0(+|e|2)−α 2eij −2µ1∆eij, i, j = 1,2, eij =eij(u) = 1 2(∂ui ∂xj +∂uj ∂xi ),|e|2= 2 X i,j=1 |eij|2,(5) where µ0,µ1,and α(0 < α < 1) are temperature and pressure-depended positive constants. Our primary objective is to show the applicability of various methods developed in [6,9,14,15] to analyze the exponential stability of steady-state solutions of our problem. More specifically, the classical Lyapunov theory is used to show the exponential stability of stationary solutions in the cases that the delay terms are continuously differentiable. Fortunately, this assumption, which may be somehow restrictive, can be weaken by an appropriate application of the Razumikhin technique, where only the continuity on the operators of the model is needed but more general delay terms are allowed, in fact, continuity is the only requirement for delay terms. Furthermore, our third way to study the asymptotic behavior of problem (1)-(4) is to exploit the construction of Lyapunov functionals. It is worth pointing out that when we are able to construct suitable Lyapunov functionals, better stability results can be achieved. The fourth alternative is based on a Gronwall-like lemma, which only requires measurability of the delay functions but still provides us with exponential stability of the steady-state solutions under appropriate sufficient conditions. INCOMPRESSIBLE NON-NEWTONIAN FLUID WITH DELAY 3 Nevertheless, we first need to prove the existence and eventual uniqueness of stationary solutions, which is not a trivial task due to the difficulties in handling the nonlinear term N(u). Indeed, the proof of the existence of stationary solutions is much more complicated and involved when we compare with other models, for example, Navier-Stokes. In other words, many more technicalities are required to deal with the nonlinear term N(u)and to obtain the existence of stationary solution, which represents one of the main difficulties of this work. In this respect, it is worth mentioning that Guo and Lin studied in [11] the existence and uniqueness of stationary solutions of non-Newtonian viscous incompressible fluids without delay, but this reference does not contain a complete proof for the existence of such stationary solution, a gap which is solved in our current paper since the result in [11] can be obtained as a particular case of the analysis we are doing in this paper by just taking h= 0. We would also like to recall that the existence and uniqueness of solutions, and the existence of pullback attractors of our delay model have been investigated in our previous work [17]. The contents of this paper are the following. In Section 2, we recall some abstract phase spaces and operators that will be used in this work, and we present two typical examples of delay terms. Section 3is devoted to proving existence and eventual uniqueness of stationary solutions for our problem (1)-(4). Four different methods are applied to study the exponential stability of the steady-state of Eq. (1) in Section 4. Finally, in Section 5, we include some remarks about possible generalizations and variants as well as some future open directions to continue investigating this challenging field of non-Newtonian fluids. 2. Preliminaries. Although the content of this section with preliminary results and notations can be shortened and, instead, we could suggest the reader to read some already published literature (e.g. [1,3,13,24]), we prefer to include them here for the reader convenience and to make the paper as much self-contained as possible. Unless otherwise is stated, the letters ci,i= 1,2,3,··· denote positive constants, and for short we will write k·kinstead of k·kL2(Ω). Let Vdenote the set {u∈C∞ 0(Ω) ×C∞ 0(Ω) : u= (u1, u2),∇·u= 0}, then His the closure of Vin L2(Ω) with norm k·k, and H0is the dual space of H,Wdenotes the closure of Vin H2(Ω) with norm k·kW, and W0is the dual space of W, while (·,·)−the inner product in H, and h·,·i the dual pairing between Wand W0. And distM(X, Y )is the Hausdorff semi-distance between X, Y ⊂M, where M is a normed space, defined by distM(X, Y ) = sup x∈X inf y∈Ykx−ykM. Set a(u, v) = 2 X i,j,k=1 ∂eij(u) ∂xk ,∂eij (v) ∂xk= 2 X i,j,k=1 ZΩ ∂eij (u) ∂xk·∂eij(v) ∂xk dx, u, v ∈W, (6) which defines a positive definite symmetric bilinear form on W(see [24]). It follows from Lax-Milgram theorem that we can define an isometric operator A∈ L(W, W0) by hAu, vi=a(u, v), u, v ∈W. 4 L. LIU, T. CARABALLO AND X. FU Notice that D(A) = {u∈W:Au ∈H}, it turns out that D(A)is a Hilbert space and Ais also an isometry from D(A)to H. In fact, A=P∆2, where Pis the Leray projector from L2(Ω) to H. We also define a continuous trilinear form on H1 0(Ω) ×H1 0(Ω) ×H1 0(Ω) by b(u, v, w) = 2 X i,j=1 ZΩ ui ∂vj ∂xi wjdx, u, v, w ∈H1 0(Ω). Since W⊂H1 0(Ω),b(·,·,·)is continuous on W×W×Wand it is easy to check (see [22]) that b(u, v, w) = −b(u, w, v), b(u, v, v)=0,∀u, v, w ∈W. (7) Now we can define a continuous functional B(u) := B(u, u)from W×Wto W0, for any u∈W, hB(u), wi=b(u, u, w),∀w∈W. (8) To finish, we set µ(u) = 2µ0(+|e(u)|2)−α/2, for u∈W, and define a nonlinear form on W×W×Wby n(u, v, w) = 2 X i,j=1 ZΩ µ(u)eij(v)eij(w)dx, u, v, w ∈W. Similarly, we define a nonlinear functional N(u) hN(u), vi= 2 X i,j=1 ZΩ µ(u)eij(u)eij(v)dx, ∀v∈W. (9) Then the functional N(u)is continuous from Wto W0. When u∈D(A), we can extend N(u)to Hby setting hN(u), vi=−ZΩ{∇·[µ(u)e(u)] ·v}dx, ∀v∈H. (10) From a physical point of view, problem (1)-(4) can be formulated as ∂u ∂t + (u·∇)u+∇p=∇·2µ0(+|e|2)−α 2e−2µ1∆e +g(t, ut) + f(x, t),in (τ, +∞)×Ω,(11) ∇·u= 0,in (τ, +∞)×Ω,(12) u= 0, τijνjνl= 0,on ∂Ω×(τ, ∞),(13) u(τ+θ, x) = φ(θ, x), θ ∈[−h, 0], x ∈Ω.(14) As usual, with an abstract formulation we can ignore the pressure and rewrite our problem (11)-(14) as (see [3,23]) ∂u ∂t + 2µ1Au +B(u) + N(u) = g(t, ut) + f(x, t),in (τ, +∞)×Ω,(15) u(τ+θ, x) = φ(θ, x), θ ∈[−h, 0], x ∈Ω.(16) Here, we define the mapping ut: [−h, 0] ×Ω→R2by ut(θ, x) = u(t+θ, x), θ ∈[−h, 0], x ∈Ω. It is worth mentioning that this abstract formulation includes several types of delay terms in a unified way. Readers are referred to [4,7,8] for more details. INCOMPRESSIBLE NON-NEWTONIAN FLUID WITH DELAY 5 Let Xbe a Banach space. Denote by CXthe Banach space C([−h, 0]; X)endowed with the norm kφkCX= sup θ∈[−h,0] kφ(θ)kX. We assume that g: [τ, T ]×CH7→ (L2(Ω))2satisfies (g1) For any ξ∈CH, the mapping [τ, T ]3t7→ g(t, ξ)∈(L2(Ω))2is measurable, (g2) g(·,0) = 0, (g3) ∃Lg>0such that for any t∈[τ, T ]and all ξ, η ∈CH, kg(t, ξ)−g(t, η)k ≤ Lgkξ−ηkCH, Remark 1. As it is pointed out in [7,10,19], (g2) is not really a restriction, and condition (g2) and (g3) imply that kg(t, ξ)k ≤ LgkξkCH, so that kg(·, ξ)k ∈ L∞(τ, T). Two typical examples of delay terms satisfying (g1)−(g3) are presented below. The proofs can be found, e.g., in the references [6,10]. Example 1. Forcing term with variable delay Let G: [τ, T]×R2→R2be a measurable function satisfying G(t, 0) = 0 for all t∈[τ, T], and assume that there exists LG>0such that |G(t, u)−G(t, v)|R2≤LG|u−v|R2,∀u, v ∈R2. Consider a function ρ(t), which plays the role of the delay. Suppose that ρ(·)is measurable and define g(t, ξ)(x) = G(t, ξ(−ρ(t))(x)) for each ξ∈CH,x∈Ωand t∈[τ, T]. Notice that, in this case, the delay term gin our problem becomes g(t, ξ) = G(t, ξ(−ρ(t))), and conditions (g1)-(g3) are fulfilled. Example 2. Forcing term with distributed delay Let G: [τ, T]×[−h, 0] ×R2→R2be a measurable function satisfying G(t, s, 0) = 0 for all (t, s)∈[τ, T]×[−h, 0], and there exists a function α(·)∈L2(−h, 0) such that |G(t, s, u)−G(t, s, v)|R2≤α(s)|u−v|R2,∀u, v ∈R2,∀(t, s)∈[τ, T ]×[−h, 0]. Define g(t, ξ)(x) = R0 −hG(t, s, ξ(s)(x))ds for each ξ∈CH,t∈[τ, T], and x∈Ω. Then the delay term gin our problem becomes g(t, ξ) = Z0 −h G(t, s, ξ(s))ds, and conditions (g1)-(g3) are satisfied. 3. Existence and uniqueness of stationary solutions. In this paragraph, we first recall a result on the existence and uniqueness of solutions for our model, complemented with a statement about the regularity of solutions. Next we prove a result ensuring the existence and uniqueness of stationary solutions to our problem by exploiting the techniques of Galerkin’s approximations, Lax-Milgram’s theorem as well as Schauder’s fixed pointed theorem. The presence of the nonlinear term N(·) requires of a more involved and technical analysis compared with the Newtonian case, which implies the nontrivial character of this proof. In the sequel, we will use the following inequalities. kAuk2≥λ1kuk2 W,kuk2 W≥ kuk2,(17) 6 L. LIU, T. CARABALLO AND X. FU where constant λ1>0denotes the first eigenvalue of the operator A. Let us recall a result ensuring existence and uniqueness of solution to our problem which was stated and proved in [17]. Theorem 3.1. ([17]) Assume that (g1) −(g3) hold. Let f∈L2 loc(R, W 0)and φ∈CH. Then, for any τ∈R, (a) there exists a unique weak solution uto problem (15)satisfying u∈C([τ−h, T]; H)∩L∞(τ, T ;H)∩L2(τ, T;W),∀T > τ. (b) If φ(0) ∈CW, and f∈L2 loc(R, H), then there exists a unique strong solution uto problem (15)satisfying u∈C([τ−h, T]; W)∩L∞(τ, T ;W)∩L2(τ, T;D(A)),∀T > τ. Although our interest in this paper is to analyze the stability properties of solutions in the case of variable delays, we can consider the existence of steady-state solutions in a much more general case which is described below. Indeed, to start our analysis, we suppose that there exists a function G:R2→R2such that for any constant function ξ(·) : [−h, 0] →W, i.e. ξ(θ) = ξ∗∈Wfor all θ∈[−h, 0], it holds g(t, ξ∗)(x) = G(ξ∗(x)),for all t∈R, x ∈Ω(18) where Gsatisfies G(0) = 0 (19) and that there exits LG>0for which |G(u)−G(v)|R2≤LG|u−v|R2,∀u, v ∈R2.(20) Now, we can study existence and uniqueness of steady-state solutions to the equation du dt + 2µ1Au +B(u) + N(u) = g(t, ut) + f, (21) with f∈W0independent of t. Recall that such a stationary (or steady-state) solution to (21) is a u∗∈Wsuch that 2µ1Au∗+B(u∗) + N(u∗) = g(t, u∗) + f for all t≥0, which can be written, according to our assumption, as 2µ1Au∗+B(u∗) + N(u∗) = f+G(u∗).(22) Theorem 3.2. Suppose that Gsatisfies (19)-(20)and 2λ1µ1> LG. Then, (a) for all f∈W0, there exists a stationary solution to (21); (b) if f∈(L2(Ω))2, the stationary solutions belong to D(A); (c) there exists a constant C0(Ω) >0, such that if (2λ1µ1−LG)2> C0(Ω)kfk∗, then the stationary solution to (21)is unique. Proof. (a) Denote Wm=span{w1, w2,··· , wm}, where {wn}∞ n=1 ⊂W∩D(A)form a basis of Wand are orthonormal in H. Now we claim that for fixed zm∈Wm, there exists umsatisfying, for every vm∈Wm, 2µ1(Aum, vm) + b(zm, um, vm) + n(zm, um, vm)=(G(zm), vm) + hf, vmi.(23) Indeed, notice that for each zm∈Wm, the functional (u, v)7→ 2µ1(Au, v) + b(zm, u, v) + n(zm, u, v)is bilinear, continuous and coercive in Wm×Wm. On the other hand, the functional v7→ (G(zm), v) + hf, viis obviously linear and continuous. Thanks to Lax-Milgram’s theorem, for each zm∈Wm, there exists a unique um∈Wmsuch that (23) holds true. INCOMPRESSIBLE NON-NEWTONIAN FLUID WITH DELAY 7 Define the mapping Tm:Wm7→ Wmgiven by T(zm) = um. We will see that for each mwe can apply Schauder’s fixed point theorem to the map Tm(restricted to a suitable subset Km⊂Wm) and ensure that we obtain um∈Wm such that 2µ1(Aum, vm) + b(um, um, vm) + n(um, um, vm) = (G(um), vm) + hf, vmi,∀vm∈Wm.(24) Indeed, setting vm=umin (23) yields 2µ1(Aum, um) + n(zm, um, um)=(G(zm), um) + hf, umi.(25) By (17), 2µ1(Aum, um)≥2λ1µ1kumk2 W, and (G(zm), um) + hf, umi ≤ LGkzmkkumk+kfk∗kumkW ≤LGkzmkkumkW+kfk∗kumkW. Since n(zm, um, um)≥0, the previous inequalities imply 2λ1µ1kumkW≤LGkzmk+kfk∗. Because 2λ1µ1> LG, one may take k > 0such that k(2λ1µ1−LG)≥ kfk∗and, consequently, 2λ1µ1kumkW≤LGkzmk+k(2λ1µ1−LG). Define Km={z∈Wm:kzkW≤k}, which is a convex set of W, and also compact since the inclusion W⊂H1 0(Ω) is compact as well. Obviously, Tm:Km→ Kmis well defined due to the choice of the constant k. Now we will use Schauder’s fixed point theorem to establish the existence of stationary solutions. To do this, we still need to verify the continuity of Tm. Actually, take zm 1, zm 2∈Wm, and denote um i=T(zm i), i = 1,2, the respective solutions of (23). For any vm∈Wmwe deduce 2µ1(A(um 1−um 2), vm) + b(zm 1, um 1, vm)−b(zm 2, um 2, vm) +n(zm 1, um 1, vm)−n(zm 2, um 2, vm)=(G(zm 1)−G(zm 2), vm).(26) In particular, we set vm=um 1−um 2in (26), and then using (17) once more, 2λ1µ1kum 1−um 2k2 W≤b(zm 2, um 2, vm)−b(zm 1, um 1, vm) +n(zm 2, um 2, vm)−n(zm 1, um 1, vm) +(G(zm 1)−G(zm 2), vm).(27) As for the trilinear term, b(zm 2, um 2, vm)−b(zm 1, um 1, vm) =b(zm 2−zm 1, um 1, um 1−um 2) ≤ kzm 2−zm 1k(L4(Ω))2k∇um 1k(L2(Ω))2kum 2−um 1k(L4(Ω))2 ≤c0kzm 2−zm 1kWkum 1kWkum 2−um 1kW ≤c1kzm 2−zm 1kWkum 2−um 1kW.(28) Then we estimate the nonlinear term, n(zm 2, um 2, vm)−n(zm 1, um 1, vm) = 2 X i,j=1 ZΩ [µ(zm 2)eij(um 2)−µ(zm 1)eij(um 1)]eij(vm)dx 8 L. LIU, T. CARABALLO AND X. FU = 2 X i,j=1 ZΩ [µ(zm 2)−µ(zm 1)]eij(um 2)eij(vm)dx − 2 X i,j=1 ZΩ µ(zm 1)|eij(um 2−um 1)|2dx ≤ 2 X i,j=1 ZΩ [µ(zm 2)−µ(zm 1)]eij(um 2)eij(vm)dx. (29) Using the mean value theorem to µ(zm 2)−µ(zm 1), there exists a constant swith |e(zm 1)|<s<|e(zm 2)|, such that µ(zm 2)−µ(zm 1) = 2µ0(+|e(zm 2)|2)−α 2−2µ0(+|e(zm 1)|2)−α 2 = 2µ0(−α 2)(+s2)−α+2 2(|e(zm 2)|2−|e(zm 1)|2) =−αµ0(+s2)−α+2 2(|e(zm 2)|+|e(zm 1)|)(|e(zm 2)|−|e(zm 1)|). (30) Hence, n(zm 2, um 2, vm)−n(zm 1, um 1, vm) ≤2αµ0 2 X i,j=1 ZΩ (+|e(zm 1)|2)−α+2 2|e(zm 2)||e(zm 2−zm 1)||eij(um 2)||eij(vm)|dx ≤2αµ0−α+2 2ke(zm 2)k(L4(Ω))2ke(zm 2−zm 1)k(L4(Ω))2keij(um 2)k(L4(Ω))2× ×keij (um 1−um 2)k(L4(Ω))2 ≤2αµ0−α+2 2c2kzm 2kWkzm 1−zm 2kWkum 2kWkum 1−um 2kW ≤2αµ0−α+2 2c3kzm 1−zm 2kWkum 1−um 2kW. (31) On the other hand, (G(zm 1)−G(zm 2), um 1−um 2)≤LGkzm 1−zm 2kkum 1−um 2k ≤LGkzm 1−zm 2kWkum 1−um 2kW. Thanks to all above inequalities we obtain 2λ1µ1kum 1−um 2k2 W≤(c1+ 2αµ0−α+2 2c3+LG)kzm 1−zm 2kWkum 1−um 2kW.(32) The continuity of the mapping T:z7→ uin Kmfollows from (32). Therefore, by Schauder’s fixed point theorem, there exists zm∈Kmsuch that T(zm) = zm, which means that (24) holds true for every m. Next, we pass to the limit on the solutions and conclude the existence of a stationary solution uto (21). Choosing vm=umin (24), we deduce 2µ1(Aum, um) + n(um, um, um)=(G(um), um) + hf, umi. Due to some standard computations, (2λ1µ1−LG)kumkW≤ kfk∗, which provides a uniform bound of umin W(namely, kumkW≤(2λ1µ1−LG)−1kfk∗). We can extract a weakly convergent subsequence (relabeled the same) um* u in W, by the compact injections ((H2(Ω))2⊂(H1 0(Ω))2⊂(L2(Ω))2), we have kum−uk(H1 0(Ω))2→0and kum−uk(L2(Ω))2→0. INCOMPRESSIBLE NON-NEWTONIAN FLUID WITH DELAY 9 To proceed, we fix any wj∈Wm. Since we have a subsequence of equations (24) for every mgreater than j, it is clear that we can pass to the limit on every term to obtain 2µ1(Au, wj) + b(u, u, wj) + n(u, u, wj)=(G(u), wj) + hf, wji.(33) The first term is obtained by the weak convergence um* u in W. Indeed, 2µ1(Aum, wj)=2µ1(∂eij(um) ∂xk ,∂eij (wj) ∂xk )*2µ1(∂eij (u) ∂xk ,∂eij (wj) ∂xk ) = 2µ1(Au, wj), as m→ ∞. The trilinear term b(um, um, wj)−b(u, u, wj) = −b(um−u, wj, um)−b(u, wj, um−u) ≤c4kum−uk(L4(Ω))2kwjk(H1 0(Ω))2kumk(L4(Ω))2 +c5kuk(L4(Ω))2kwjk(H1 0(Ω))2kum−uk(L4(Ω))2 ≤c6kum−uk1/2 (L2(Ω))2kum−uk1/2 (H1 0(Ω))2kwjk(H1 0(Ω))2kumk(L2(Ω))2kumk1/2 (H1 0(Ω))2 +c7kuk(L2(Ω))2kuk1/2 (H1 0(Ω))2kwjk(H1 0(Ω))2kum−uk1/2 (L2(Ω))2kum−uk1/2 (H1 0(Ω))2→0. The nonlinear term n(um, um, wj)−n(u, u, wj) = hN(um)−N(u), wji ≤ |hN(um)−N(u), wji| ≤c8kum−uk(H1 0(Ω))2kwjk(H1 0(Ω))2→0. And the delay term (G(um)−G(u), wj)≤LGkum−uk(L2(Ω))2kwjk(L2(Ω))2→0. Thus, (33) holds true for every wj∈Wm. Since the set of linear combinations of w1, w2,··· , wm,··· is dense in W, we deduce that (21) is satisfied at least by u∗=u. (b) Regularity. From (a) we know that 2µ1Au +B(u) + N(u) = G(u) + f, (34) which must be understood in the sense of D0. Now taking the inner product of (34) with ugives 2µ1(Au, u)+(N(u), u)=(G(u), u)+(f, u). By standard calculations, kukW≤(2λ1µ1−LG)−1kfk.(35) From (34), we have 2µ1kAuk≤kB(u, u)k+kN(u)k+kG(u)k+kfk. Notice that kB(u, u)k ≤ c9kukkuk(H1 0(Ω))2≤c10kuk2 W, and kN(u)k= 2µ0(ZΩ (+|∇u|2)−α|∆u|2dx)1/2 ≤2µ0−α/2k∆uk ≤2µ0−α/2c11kukW. 16 L. LIU, T. CARABALLO AND X. FU and we obtain d dtV(t, ut) = d dt eλtku(t)k2+εLG 1−ρ∗Zt t−ρ(t) eλ(s+h)ku(s)k2 Wds! =λeλtku(t)k2+ 2eλt(−2µ1Au(t)−B(u(t)) −N(u(t)), u(t)) + 2eλt(G(u(t−ρ(t))), u(t)) +εLG 1−ρ∗ eλ(t+h)ku(t)k2 W−εLG 1−ρ∗ (1 −ρ0)eλ(t−ρ(t)+h)ku(t−ρ(t))k2 W ≤(λ−4λ1µ1+LG ε)eλtku(t)k2 W+εLGeλtku(t−ρ(t))k2 W +εLGeλh 1−ρ∗ eλtku(t)k2 W−εLGeλtku(t−ρ(t))k2 W =−(4λ1µ1−LG ε−λ−εLGeλh 1−ρ∗ )eλtku(t)k2 W Choosing ε=√1−ρ∗, we have d dtV(t, ut)≤ −(4λ1µ1−LG √1−ρ∗−λ−LGeλh √1−ρ∗ )eλtku(t)k2 W =−(4λ1µ1−2LG √1−ρ∗−λ−LG(eλh −1) √1−ρ∗ )eλtku(t)k2 W. (47) Writing h(λ) = λ+LG(eλh −1) √1−ρ∗ , h(0) = 0, since the function h(λ)is continuous respect to λ, there exists λ > 0small enough such that 2(2λ1µ1−LG √1−ρ∗ )≥h(λ). Then it follows directly from (47) that d dt V(t, ut)≤0, and the Lyapunov functional V(t, ut) = eλtku(t)k2+εLG 1−ρ∗Rt t−ρ(t)eλ(s+h)ku(s)k2 Wds satisfies the conditions in Theorem 4.3, which implies that the trivial solution of Eq. (21) is exponentially stable. Remark 2. (a) Here G:W→W0is a Lipschitz continuous operator with Lipschitz constant LG>0and G(0) = 0. If G:H→Hwith Lipschitz constant Lgwith Lg≥LG, then G:W→W0is Lipschitz, and from 2λ1µ1>Lg √1−ρ∗ , we obtain that 2λ1µ1>LG √1−ρ∗ . (b) Although applying this method, we also need the differentiability of variable delay function, the stability result that we obtained is better than the first case, in which 2λ1µ1>(2−ρ∗)LG 2(1−ρ∗)is required, but here we only need 2λ1µ1>LG √1−ρ∗ , which means we have more choices for µ1. 4.4. Exponential stability: A Gronwall argument. Now we investigate the stability of stationary solutions to Eq. (21) via a Gronwall-like lemma. For convenience, we will consider Eq. (21) with f(t)≡0and g(t, φ) = G(φ(−ρ(t))), for φ∈CH, where G:H→His Lipschitz continuous with Lipschitz constant Lg>0and G(0) = 0. For the delay term ρwe only assume that it is measurable and bounded, i.e., ρ: [0,+∞)→[0, h]. Compared with the ones required in the INCOMPRESSIBLE NON-NEWTONIAN FLUID WITH DELAY 17 three previous approaches, this is the weakest assumption. But we still can prove exponential stability of the steady-state solution. Lemma 4.5. ([9]) Let y(·):[−h, +∞)→[0,+∞)be a function. Assume that there exist positive numbers γ, α1, α2such that the following inequality holds: y(t)≤   α1e−γt +α2Rt 0e−γ(t−s)sup θ∈[−h,0] y(s+θ)ds, t ≥0, α1e−γt, t ∈[−h, 0]. Then, y(t)≤α1e−µt,for all t≥ −h, where µ∈(0, γ)is given by the unique root of the equation α2 γ−µeµh = 1 in this interval. Theorem 4.6. Suppose that f(t)≡0and g(t, ut) = G(u(t−ρ(t))), where G:H→ His Lipschitz constant Lg>0and satisfies G(0) = 0. Assume that ρ: [0,+∞)7→ [0, h]is a measurable function. Then the zero solution of (1)is exponentially stable provided 4λ1µ1> Lg. Proof. Let us choose a positive constant λsuch that λ−4λ1µ1+Lg>0. Notice that the weak solution u(·)to model (1) corresponding to the initial datum φsatisfies eλtku(t)k2=kφ(0)k2+λZt 0 eλsku(s)k2ds −4µ1Zt 0 eλs(Au(s), u(s))ds −2Zt 0 eλshN(u(s)), u(s)ids + 2 Zt 0 eλs(G(u(s−ρ(s))), u(s))ds ≤ kφ(0)k2+λZt 0 eλsku(s)k2ds −4λ1µ1Zt 0 eλsku(s)k2ds + 2LgZt 0 eλsku(s−ρ(s))kku(s)kds ≤ kφ(0)k2+ (λ−4λ1µ1+Lg)Zt 0 eλsku(s)k2ds +LgZt 0 eλsku(s−ρ(s))k2ds ≤ kφk2 C([−h,0];H)+ (λ−4λ1µ1+ 2Lg)Zt 0 eλs sup θ∈[−h,0] ku(s+θ)k2ds. Hence, from the Lemma 4.5, we know that the unique zero solution to Eq. (1) is exponentially stable. 5. Conclusions and comments. In this work we have exhibited several methods to analyze the exponential stability of incompressible non-Newtonian fluids when some hereditary properties are taken into account in the forcing term of the model, and our analysis has been carried out when the delays are bounded. 18 L. LIU, T. CARABALLO AND X. FU In the case of constant delays, the autonomous theory of global attractor may provide an appropriate framework to study the problem. But for more general delay terms, such as variable or distributed delays, the problem becomes non-autonomous and it is necessary to consider a non-autonomous framework for the global asymptotic behavior of the model. Several options, for instance, the theories of skewproduct and uniform attractor are available, but we would like to emphasize that the theory of pullback attractors may allow more general non-autonomous terms in the models. In this respect, the existence of pullback attractor of an incompressible non-Newtonian fluids with bounded delay has been established in [17]. Although many other aspects on this model have already been investigated (see [1,2,3,11,12,13,17,23,24,25] and the references therein), there are still many interesting problems related to incompressible non-Newtonian fluids that need to be studied in future. For instance, what are the effects that some environmental noise may produce in the phenomenon, which will then become a stochastic non- Newtonian fluid. Amongst the many topics that we could analyze within the field of stochastic non-Newtonian fluids with delay (bounded or unbounded), we could wonder about the existence and uniqueness of solutions, in particular the stationary one, their stability properties, and the existence and structure of random attractors as well. We plan to work on these problems in some forthcoming papers. REFERENCES [1] H.-O. Bae. Existence, regularity, and decay rate of solutions of non-Newtonian flow. J. Math. Anal. Appl., 231(2):467–491, 1999. [2] H. Bellout, F. Bloom, and J. Nečas. Young measure-valued solutions for non-Newtonian incompressible fluids. Comm. Partial Differential Equations, 19(11-12):1763–1803, 1994. [3] F. Bloom and W. Hao. Regularization of a non-Newtonian system in an unbounded channel: existence and uniqueness of solutions. Nonlinear Anal., 44(3):281–309, 2001. [4] T. Caraballo and A. M. Márquez-Durán. Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay. Dyn. Partial Differ. Equ., 10(3):267– 281, 2013. [5] T. Caraballo, J. Real, and L. Shaikhet. Method of Lyapunov functionals construction in stability of delay evolution equations. J. Math. Anal. Appl., 334(2):1130–1145, 2007. [6] T. Caraballo and X. Han. A survey on Navier-Stokes models with delays: existence, uniqueness and asymptotic behavior of solutions. Discrete Contin. Dyn. Syst. Ser. S, 8(6):1079–1101, 2015. [7] T. Caraballo, J. A. Langa, and J. C. Robinson. Attractors for differential equations with variable delays. J. Math. Anal. Appl., 260(2):421–438, 2001. [8] T. Caraballo, A. M. Márquez-Durán, and F. Rivero. Well-posedness and asymptotic behavior of a nonclassical nonautonomous diffusion equation with delay. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25(14):1540021, 11, 2015. [9] H. Chen. Asymptotic behavior of stochastic two-dimensional Navier-Stokes equations with delays. Proc. Indian Acad. Sci. Math. Sci., 122(2):283–295, 2012. [10] J. García-Luengo, P. Marín-Rubio, and J. Real. Pullback attractors for 2D Navier-Stokes equations with delays and their regularity. Adv. Nonlinear Stud., 13(2):331–357, 2013. [11] B. Guo, G. Lin, and Y. Shang. Non-Newtonian Fluids Dynamical Systems. National Defense Industry Press, in Chinese, 2006. [12] B. Guo, C. Guo, and J. Zhang. Martingale and stationary solutions for stochastic non- Newtonian fluids. Differential Integral Equations, 23(3-4):303–326, 2010. [13] J. U. Jeong and J. Park. Pullback attractors for a 2D-non-autonomous incompressible non- Newtonian fluid with variable delays. Discrete Contin. Dyn. Syst. Ser. B, 21(8):2687–2702, 2016. INCOMPRESSIBLE NON-NEWTONIAN FLUID WITH DELAY 19 [14] V. Kolmanovskii and L. Shaikhet. Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results. Math. Comput. Modelling, 36(6):691–716, 2002. Lyapunov’s methods in stability and control. [15] V. Kolmanovskii and L. Shaikhet. General method of Lyapunov functionals construction for stability investigation of stochastic difference equations. In Dynamical systems and applications, volume 4 of World Sci. Ser. Appl. Anal., pages 397–439. World Sci. Publ., River Edge, NJ, 1995. [16] O. Ladyzhenskaya. New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them, in: Boundary Value Problem of Mathematical Physics. American Mathematical Society, Providence, 1970. [17] L. Liu and T. Caraballo. Dynamics of a non-autonomous incompressible non-newtonian fluid with delay. Dynamics of PDE (to appear). [18] J. Málek, J. Nečas, M. Rokyta, and M. R užička. Weak and measure-valued solutions to evolutionary PDEs, volume 13 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London, 1996. [19] P. Marín-Rubio, J. Real, and J. Valero. Pullback attractors for a two-dimensional Navier- Stokes model in an infinite delay case. Nonlinear Anal., 74(5):2012–2030, 2011. [20] L. Shaikhet. Lyapunov functionals and stability of stochastic difference equations. Springer, London, 2011. [21] L. Shaikhet. Lyapunov functionals and stability of stochastic functional differential equations. Springer, Cham, 2013. [22] R. Temam. Infinite-dimensional dynamical systems in mechanics and physics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997. [23] C. Zhao and Y. Li. H2-compact attractor for a non-Newtonian system in two-dimensional unbounded domains. Nonlinear Anal., 56(7):1091–1103, 2004. [24] C. Zhao and S. Zhou. Pullback attractors for a non-autonomous incompressible non- Newtonian fluid. J. Differential Equations, 238(2):394–425, 2007. [25] C. Zhao, S. Zhou, and Y. Li. Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid. J. Math. Anal. Appl., 325(2):1350–1362, 2007. Received xxxx 20xx; revised xxxx 20xx. E-mail address:[email protected] E-mail address:[email protected] E-mail address:[email protected]