Asymptotic behavior of a viscoelastic fluid in a closed loop thermosyphon: physical derivation, asymptotic analysis, and numerical experiments

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Volume 2013, Article ID 748683,20pages http://dx.doi.org/10.1155/2013/748683

Research Article

Asymptotic Behavior of a Viscoelastic Fluid in a Closed Loop

Thermosyphon: Physical Derivation, Asymptotic Analysis, and

Numerical Experiments

Justine Yasappan,

1

Ángela Jiménez-Casas,

1

and Mario Castro

2

1Grupo de Din´amica No Lineal (DNL), Escuela T´ecnica Superior de Ingenier´ıa (ICAI), Universidad Pontificia Comillas,

28015 Madrid, Spain

2ICAI, Grupo Interdisciplinar de Sistemas Complejos (GISC) and DNL, Universidad Pontificia Comillas, 28015 Madrid, Spain

Correspondence should be addressed to Justine Yasappan; justemmasj@gmail.com

Received 11 February 2013; Accepted 19 April 2013

Academic Editor: Douglas Anderson

Copyright © 2013 Justine Yasappan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Fluids subject to thermal gradients produce complex behaviors that arise from the competition with gravitational effects. Although such sort of systems have been widely studied in the literature for simple (Newtonian) fluids, the behavior of viscoelastic fluids has not been explored thus far. We present a theoretical study of the dynamics of a Maxwell viscoelastic fluid in a closed-loop thermosyphon. This sort of fluid presents elastic-like behavior and memory effects. We study the asymptotic properties of the fluid inside the thermosyphon and the exact equations of motion in the inertial manifold that characterizes the asymptotic behavior. We derive, for the first time, the mathematical derivations of the motion of a viscoelastic fluid in the interior of a closed-loop thermosyphon under the effects of natural convection and a given external temperature gradient.

1. Introduction

Instabilities and chaos in fluids subject to temperature gradi-ents have been the subject of intense work for its applications in engineering and in atmospheric sciences. In such sort of systems, the fluid displays nontrivial behaviors (as turbulence or the formation of convective rolls) when subject to a heating that competes with buoyancy effects. A traditional approach that goes back to the pioneering work by Lorenz consists of the study of the system under some simplifications. Another approach is to study the controlled setups that capture the underlying complexity of the full system, being a thermosyphon one of those simpler cases [1].

In the engineering literature, a thermosyphon is a device composed of a closed-looppipecontaining a fluid where some soluble substance has been dissolved [2,3]. The motion of the fluid is driven by the action of several forces such as gravity and natural convection. The flow inside the loop is driven by an energetic balance between thermal energy and mechan-ical energy. The interest on this system comes both from engineering and as atoy model of natural convection (for

instance, to understand the origin of chaos in atmospheric systems).

Thus far, all the works on thermosyphons analyze the behavior of a Newtonian fluid inside the loop, consequently neglecting elastic effects in the system coming from either the fluid itself or the elastic walls of the loop. However, many interesting fluids are known to behave slightly different from the common (Newtonian) fluids in terms of their response to an applied stress and are commonly referred to as viscoelastic. Among them, it is worth emphasizing volcanic lavas, snow avalanches, flowing paint, or biological mucosas membranes. Here, we consider a thermosyphon model in which the confined fluid is viscoelastic. This has some a priori

interesting peculiarities that could affect the dynamics with respect to the case of a Newtonian fluid. On the one hand, the dynamics has memory and so its behavior depends on the whole past history, and, on the other hand, at small perturbations the fluid behaves like an elastic solid and a characteristic resonance frequency could, eventually, be relevant (e.g., consider the behavior of jelly or toothpaste).

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The simplest approach to viscoelasticity comes from the so-called Maxwell constitutive equation [4]. In this model, both Newton’s law of viscosity and Hooke’s law of elasticity are generalized and complemented through an evolution equation for the stress tensor,𝜎.

The stress tensor comes into play in the equation for the conservation of momentum:

𝜌 (𝜕k𝜕𝑡 +k⋅ ∇k) = −∇𝑝 + ∇ ⋅ 𝜎. (1)

For a Maxwellian fluid, the stress tensor takes the form

𝜇 𝐸

𝜕𝜎

𝜕𝑡 + 𝜎 = 𝜇 ̇𝛾, (2)

where 𝜇is the fluid viscosity, 𝐸the Young’s modulus, and

̇𝛾the shear strain rate (or rate at which the fluid deforms). Under stationary flow, (2) reduces to Newton’s law, and consequently, (1) reduces to the celebrated Navier-Stokes equation. On the contrary, for short times (where impulsive behavior from rest can be expected) equation (2) reduces to Hooke’s law of elasticity.

Memory effects can be well understood from (2) after performing a separation of variables and integrating. Thus, we can rewrite

𝜎 (𝑡) = ∫𝑡 0𝑒

𝐸/𝜇(𝑡−𝑠)𝜇 ̇𝛾 (𝑠) 𝑑𝑠, (3)

where it is clear that the local state of stress𝜎(𝑡)is calculated from the present and values of ̇𝛾(𝑡)with amemorytime scale of order𝜇/𝐸.

In a thermosyphon, the equations of motion can be greatly simplified because of the quasi-one-dimensional geometry of the loop. Thus, we assume that the section of the loop is constant and small compared with the dimensions of the physical device, so that the arc length coordinate along the loop (𝑥) gives the position in the circuit. The velocity of the fluid is assumed to be independent of the position in the circuit; that is, it is assumed to be a scalar quantity depending only on time. This approximation comes from the fact that the fluid is assumed to be incompressible, and so

∇ ⋅k= 0, (4)

besides the quasi-one-dimensional assumption. On the con-trary, temperature is assumed to depend both on time and position along the loop.

The derivation of the thermosyphon equations of motion is similar to that in [2,3]. The simplest way to incorporate (2) is by differentiating (1) with respect to time and replacing the resulting time derivative of𝜎with (2). This way to incorporate the constitutive equation allows to reduce the number of unknowns (we remove𝜎from the system of equations) at the cost of increasing the order of the time derivatives to second order.

The resulting second-order equation is then averaged along the loop section (as in [2]). Hence, after nondimen-sionalizing the variables (to reduce the number of free parameters) we arrive at our main system of equations

𝜀𝑑2V

𝑑𝑡2 +𝑑V𝑑𝑡 + 𝐺 (V)V= ∮ 𝑇𝑓, V(0) =V0, 𝑑V𝑑𝑡 (0) = 𝑤0, 𝜕𝑇

𝜕𝑡 +V 𝜕𝑇

𝜕𝑥 = ℎ (𝑥,V, 𝑇) +] 𝜕2𝑇

𝜕𝑥2, 𝑇 (0, 𝑥) = 𝑇0(𝑥) ,

(5) whereℎ(𝑥,V, 𝑇) = 𝑙(V)(𝑇𝑎− 𝑇); that is, we consider Newton’s linear cooling law as in [3,5–10] also in this paper we consider the diffusion of temperature given by the term](𝜕2𝑇/𝜕𝑥2). The parameter𝜀in (5) is the nondimensional version of𝜇/𝐸 which has dimensions of time. Roughly speaking, it gives the (nondimensional) time scale in which the transition from elastic to fluid-like occurs in the fluid.

The model we will take into consideration forms an ODE/PDE system for the velocityV(𝑡), the distribution of the temperature𝑇(𝑡, 𝑥)of the fluid into the loop, (5) with]≥ 0, where∮ = ∫01𝑑𝑥denotes integration along the closed path of the circuit. The function𝑓describes the geometry of the loop and the distribution of gravitational forces [2,3]. Note that∮ 𝑓 = 0.

The system of (5) is not a trivial extension of the New-tonian model in [11] due to the first term in the differential equation for the velocity. Specifically, the addition of a term proportional to the second derivative ofVissingular, in the sense that it changes qualitatively the character of the equa-tions. The implications of this singular perturbation cannot be ascertained using a standard boundary layer analysis (see the Appendix for details). So a complementary theoretical and numerical approach is mandatory.

We assume that𝐺(V)which specifies the friction law at the inner wall of the loop is positive and bounded away from zero. This function has been usually taken to be𝐺(V) = 𝐺, a positive constant for the linear friction case [2] (Stokes flow), or𝐺(V) = |V|for the quadratic law [12,13], or even a rather general function given by𝐺(V) = ̃𝑔(Re)|V|, where Re is the Reynolds number, Re = 𝜌V𝐿/𝜇. Here we will consider a general function of the velocity assumed to be large [9,14]. The functions 𝐺, 𝑓, 𝑙, and ℎ incorporate relevant physical constants of the model, such as the cross-sectional area,𝐷, the length of the loop,𝐿, and the Prandtl, Rayleigh or Reynolds number; see [9]. Here, we consider Newton’s linear cooling law 𝑙(V)(𝑇𝑎 − 𝑇) where 𝑙 represents the heat transfer law across the loop wall, and is a positive quantity depending on the velocity, and 𝑇𝑎 is the (given) ambient temperature distribution, see [3,5,9,10].

Hereafter, we consider𝐺and𝑙as continuous functions, such that𝐺(V) ≥ G0 > 0, and𝑙(V) ≥ 𝑙0 > 0, for𝐺0 and𝑙0 positive constants.

Our contributions in this paper are the following. (i) To obtain the system of (5) governing a closed loop

thermosyphon model with a viscoelastic fluid and to study the asymptotic behavior of this system which

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is a generalization of the previous model [11]. Thus, although the dynamics of viscoelastic fluids has been extensively studied in the engineering literature, it has scarcely been considered in the applied mathematics community.

(ii) To present an analysis beginning with the well-posedness and boundedness of solution. The exis-tence of an attractor and an inertial manifold is shown and an explicit reduction to low-dimensional systems is obtained. It is noteworthy that we are able to obtain an exact finite-dimensional reduction (87) that may have a much lower number of degrees of freedom. (iii) To provide a detailed numerical analysis of the

behav-ior of acceleration, velocity, and temperature which includes a thorough study of the various behaviors of the system for different values of viscoelastic fluid and ambient temperature distribution.

(iv) The numerical analysis will show that viscoelasticity induces a chaotic behavior that is not captured by a boundary layer analysis (that would predict the same qualitative behaviors as in the original model in [11] see the Appendix for details) being the new (nontriv-ial) emergent behaviors induced by the viscoelasticity worth characterizing.

The structure of the paper is as follows. The first section provides a general introduction to the system, explaining briefly the dynamics of a thermosyphon, viscoelastic fluids, and the objectives of this work. InSection 2, we prove the existence, uniqueness, and boundedness of the solutions. In

Section 3, we provide a detailed derivation of the dynamics of the system in the inertial manifold as a reduced dimen-sionality version of the full system of (5). InSection 4, we present the numerical integration of the reduced system of equations valid in the manifold to understand the role of the main parameters of the physical system. Finally, inSection 5, we conclude with a proposal for future works.

2. Well-Posedness and Boundedness:

Global Attractor

2.1. Existence and Uniqueness of Solutions. In this section we prove the existence and uniqueness of solutions of the thermosyphon model (5).

First, we observe that for]≥ 0, if we integrate the equa-tion for the temperature along the loop taking into account the periodicity of𝑇, that is,∮(𝜕𝑇/𝜕𝑥) = ∮(𝜕2𝑇/𝜕𝑥2) = 0, we have(𝑑/𝑑𝑡)(∮ 𝑇) = 𝑙(V)(∮ 𝑇𝑎− ∮ 𝑇). Therefore,∮ 𝑇 → ∮ 𝑇𝑎 exponentially as time goes to infinity for every∮ 𝑇0.

Moreover, if we consider𝜏 = 𝑇 − ∮ 𝑇, then from the second equation of system (5), we obtain that𝜏verifies the equation

𝜕𝜏 𝜕𝑡 +V

𝜕𝜏 𝜕𝑥 =]

𝜕2𝜏

𝜕𝑥2 + 𝑙 (V) (𝜏𝑎− 𝜏) ,

𝜏 (0, 𝑥) = 𝜏0(𝑥) = 𝑇0− ∮ 𝑇0,

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where𝜏𝑎= 𝑇𝑎− ∮ 𝑇𝑎.

Finally, since∮ 𝑓 = 0, we have∮ 𝑇𝑓 = ∮ 𝜏𝑓and the equation forVreads

𝜀𝑑𝑑𝑡2V2 +𝑑V𝑑𝑡 + 𝐺 (V)V= ∮ 𝜏𝑓, V(0) =V0, 𝑑V𝑑𝑡(0) = 𝑤0.

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Thus, with𝑤 = 𝑑V/𝑑𝑡, we get(𝑤,V, 𝜏)verifying the system (5) with𝜏𝑎, 𝜏0 replacing𝑇𝑎, 𝑇0, respectively, and now∮ 𝜏 =

∮ 𝜏𝑎 = ∮ 𝜏0 = ∮ 𝑓 = 0. Therefore, hereafter we consider the system (5) where all functions have zero average.

Also, if] > 0, the operator]𝐴 = −](𝜕2/𝜕𝑥2), together with periodic boundary conditions, is an unbounded, self-adjoint operator with compact resolvent in𝐿2per(0, 1)that is

positive when restricted to the space of zero average functions

̇𝐿2

per(0, 1). Hence, the equation for the temperature𝑇in (5) is

of parabolic type for]> 0.

2.1.1. The Case with Diffusion:]> 0. We consider the accel-eration 𝑤 = 𝑑V/𝑑𝑡 and write the system (5) as the following evolution system for the acceleration, velocity, and temperature:

𝑑𝑤 𝑑𝑡 +

1 𝜀𝑤 = −

1

𝜀𝐺 (V)V+ 1

𝜀∮ 𝑇𝑓, 𝑤 (0) = 𝑤0, 𝑑V

𝑑𝑡 = 𝑤, V(0) =V0, 𝜕𝑇

𝜕𝑡 +V 𝜕𝑇 𝜕𝑥−]

𝜕2𝑇

𝜕𝑥2 = 𝑙 (V) (𝑇𝑎− 𝑇) , 𝑇 (0, 𝑥) = 𝑇0(𝑥) ;

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that is,

𝑑 𝑑𝑡(

𝑤 V 𝑇) + (

1

𝜀 0 0

0 0 0

0 0 −]𝜕𝑥𝜕22 ) (𝑤V

𝑇) = (

𝐹1(𝑤,V, 𝑇) 𝐹2(𝑤,V, 𝑇) 𝐹3(𝑤,V, 𝑇)) ,

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with𝐹1(𝑤,V, 𝑇) = −(1/𝜀)𝐺(V)V+ (1/𝜀) ∮ 𝑇𝑓, 𝐹2(𝑤,V, 𝑇) = 𝑤 and𝐹3(𝑤,V, 𝑇) = −V(𝜕𝑇/𝜕𝑥)+𝑙(V)(𝑇𝑎−𝑇)and the initial data

(𝑤V

𝑇) (0) = ( 𝑤0

V0

𝑇0).

The operator𝐵 = (1/𝜀 00 0 00

0 0 −](𝜕2/𝜕𝑥2))is a sectorial operator

inY=R2× ̇𝐻per1 (0, 1)with domain𝐷(𝐵) =R2× ̇𝐻per3 (0, 1) and has compact resolvent, where

̇𝐿 2

per(0, 1)={𝑢∈𝐿2loc(R) , 𝑢 (𝑥+1)=𝑢 (𝑥)a.e., ∮ 𝑢=0} ,

̇

𝐻per𝑚 (0, 1) = 𝐻loc𝑚 (R) ∩ ̇𝐿2per(0, 1) .

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Using the results and techniques of sectorial operator of [15], we obtainTheorem 1.

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Theorem 1. We assume that𝐻(𝑟) = 𝑟𝐺(𝑟)and𝑙(V)are locally Lipschitz,𝑓 ∈ ̇𝐿2per(0, 1),𝑇𝑎 ∈ ̇𝐻per1 (0, 1), and𝑙(V) ≥ 𝑙0 > 0. Then, given(𝑤0,V0, 𝑇0) ∈Y=R2× ̇𝐻1per(0, 1), there exists a

unique solution of(5)satisfying

(𝑤,V, 𝑇) ∈ 𝐶 ([0, ∞) ,R2× ̇𝐻per1 (0, 1))

∩ 𝐶 (0, ∞,R2× ̇𝐻3

per(0, 1)) ,

( ̇𝑤, 𝑤,𝜕𝑇𝜕𝑡) ∈ 𝐶 (0, ∞,R2× ̇𝐻per3−𝛿(0, 1)) ,

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where𝑤 = ̇V = 𝑑V/𝑑𝑡and ̇𝑤 = 𝑑2V/𝑑𝑡2 for every𝛿 > 0. In particular,(5)defines a nonlinear semigroup,𝑆(𝑡)inY =

R2× ̇𝐻per1 (0, 1), with𝑆(𝑡)(𝑤0,V0, 𝑇0) = (𝑤(𝑡),V(𝑡), 𝑇(𝑡)).

Proof. We cover several steps.

Step 1. We prove the local existence and regularity. This follows easily from the variation of constants formula of [15]. In order to prove this we write the system as (9), and we have

𝑈𝑡+ 𝐵𝑈 = 𝐹 (𝑈) , with𝑈 = ( 𝑤

V 𝑇) ,

𝐵 = ( 1

𝜀 0 0

0 0 0

0 0 −]𝜕𝑥𝜕22

) , 𝐹 = (𝐹𝐹12 𝐹3) ,

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where the operator𝐵is a sectorial operator inY = R2 ×

̇

𝐻1

per(0, 1) with domain 𝐷(𝐵) = R2 × ̇𝐻per3 (0, 1) and has

compact resolvent. In this context, the operator𝐴 = −𝜕2/𝜕𝑥2 must be understood in the variational sense; that is, for every

𝑇, 𝜑 ∈ ̇𝐻1

per(0, 1),

⟨𝐴 (𝑇) , 𝜑⟩ = ∮𝜕𝑇𝜕𝑥𝜕𝜑𝜕𝑥, (13)

and ̇𝐿2per(0, 1)coincides with the fractional space of exponent

1/2[15]. Hereafter we denote by‖ ⋅ ‖the norm on the space

̇𝐿2

per(0, 1). Now, if we prove that the nonlinearity𝐹 : Y =

R2× ̇𝐻per1 (0, 1) 󳨃→ Y−1/2 = R2× ̇𝐿2per(0, 1)is well defined

and is Lipschitz and bounded on bounded sets, we obtain the local existence for the initial data inY=R2× ̇𝐻per1 (0, 1).

Using 𝐻(V) = 𝐺(V)V and 𝑙(V) being locally Lipschitz together with𝑓 ∈ ̇𝐿2per(0, 1)and𝑇𝑎 ∈ ̇𝐻per1 (0, 1), we will

prove the nonlinear terms, 𝐹1(𝑤,V, 𝑇) = −(1/𝜀)𝐺(V)V +

(1/𝜀) ∮ 𝑇𝑓, 𝐹2(𝑤,V, 𝑇) = 𝑤, and𝐹3(𝑤,V, 𝑇) = −V(𝜕𝑇/𝜕𝑥) +

𝑙(V)(𝑇𝑎− 𝑇)and satisfy𝐹1 : R2× ̇𝐻per1 (0, 1) 󳨃→R,𝐹2 : R2×

̇

𝐻1

per(0, 1) 󳨃→ R, and𝐹3 : R2× ̇𝐻per1 (0, 1) 󳨃→ ̇𝐿2per(0, 1); that

is,𝐹 : Y 󳨃→ Y−1/2is well defined, Lipschitz, and bounded on bounded sets. It is possible to prove this by considering

𝑇𝑎 ∈ ̇𝐿2

per(0, 1). In order to prove these properties of the

nonlinearity𝐹, let𝑈𝑖= (𝑤𝑖,V𝑖, 𝑇𝑖)𝑡, and we note that

󵄩󵄩󵄩󵄩𝐹3(𝑈1) − 𝐹3(𝑈2)󵄩󵄩󵄩󵄩 ≤󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩−V1𝜕𝑇1

𝜕𝑥 + 𝑙 (V1) (𝑇𝑎− 𝑇1) +V2 𝜕𝑇2

𝜕𝑥

−𝑙 (V2) (𝑇𝑎− 𝑇2)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

≤ 󵄨󵄨󵄨󵄨𝑙 (V1) − 𝑙 (V2)󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩 + (1) + (2),

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where

(1) ≡󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩−V1𝜕𝑇1 𝜕𝑥 +V2

𝜕𝑇2

𝜕𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩, (2) ≡ 󵄩󵄩󵄩󵄩𝑙 (V2) 𝑇2− 𝑙 (V1) 𝑇1󵄩󵄩󵄩󵄩,

(15) and adding ±V1(𝜕𝑇2/𝜕𝑥),∓V2(𝜕𝑇1/𝜕𝑥)and ∓V1(𝜕𝑇1/𝜕𝑥) in

(1), we have

(1) ≤ (󵄨󵄨󵄨󵄨V1󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨V2󵄨󵄨󵄨󵄨)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜕𝑇2 𝜕𝑥 −

𝜕𝑇1

𝜕𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 +󵄨󵄨󵄨󵄨V2−V1󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜕𝑇𝜕𝑥1󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

+ 󵄨󵄨󵄨󵄨V1󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜕𝑇𝜕𝑥1 −𝜕𝑇𝜕𝑥2󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩,

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and adding±𝑙(V2)𝑇1in(2), we get

(2) ≡ 󵄩󵄩󵄩󵄩𝑙 (V2) 𝑇2− 𝑙 (V1) 𝑇1󵄩󵄩󵄩󵄩 ≤ 󵄨󵄨󵄨󵄨𝑙(V1) − 𝑙 (V2)󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩𝑇1󵄩󵄩󵄩󵄩 + 󵄨󵄨󵄨󵄨𝑙 (V2)󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩𝑇2− 𝑇1󵄩󵄩󵄩󵄩, (17)

and from the previous hypothesis on function 𝑙(V), there exists 𝑀 > 0 such that‖𝐹3(𝑈1) − 𝐹3(𝑈2)‖ ≤ 𝑀(|V1

V2| + ‖𝑇1− 𝑇2‖𝐻̇1

per(0,1)) ≤ 𝐶‖𝑈1− 𝑈2‖R2× ̇𝐻per1 and the rest is

obvious.

Therefore, using the techniques of variations of con-stants formula of [15], we obtain the unique local solutions

(𝑤,V, 𝑇) ∈ 𝐶([0, 𝜏],Y)of (8) which are given by

𝑤 (𝑡) = 𝑤0𝑒−(1/𝜀)𝑡−1𝜀∫ 𝑡

0𝑒

−(1/𝜀)(𝑡−𝑟)𝐻 (𝑟) 𝑑𝑟

+1 𝜀∫

𝑡

0𝑒

−(1/𝜀)(𝑡−𝑟)(∮ 𝑇 (𝑟) 𝑓) 𝑑𝑟,

(18)

with𝐻(𝑟) = 𝐻(V(𝑟)),

V(𝑡) =V0+ ∫𝑡

0𝑤 (𝑟) 𝑑𝑟, (19)

𝑇 (𝑡, 𝑥) = 𝑒−]𝐴𝑡𝑇0(𝑥)

+ ∫𝑡 0𝑒

−]𝐴(𝑡−𝑟)𝑙 (V(𝑟)) [𝑇

𝑎(𝑟, 𝑥) − 𝑇 (𝑟, 𝑥)] 𝑑𝑟

− ∫𝑡 0𝑒

−]𝐴(𝑡−𝑟)V(𝑟)𝜕𝑇 (𝑟, 𝑥) 𝜕𝑥 𝑑𝑟,

(20) and using again the results of [15], we get the regularity of solutions. In fact, from the smoothing effect of the equations,

(5)

we have(𝑤,V, 𝑇) ∈ 𝐶([0, 𝜏],Y=R2× ̇𝐻per1 (0, 1)) ∩ 𝐶((0, 𝜏),

R2 × ̇𝐻per2 (0, 1))and( ̇𝑤, 𝑤, (𝜕𝑇/𝜕𝑡)) ∈ 𝐶((0, 𝜏),R2× ̇𝐻per2−𝛿

(0, 1)), for some positive𝜏and any𝛿 > 0. Now, for𝜖 > 0we have(𝑤(𝜖),V(𝜖), 𝑇(𝜖)) ∈ R2× ̇𝐻per2 (0, 1), and since𝐹 :R2×

̇

𝐻per2 (0, 1) 󳨃→ R2× ̇𝐻per1 (0, 1)is well defined, Lipschitz, and

bounded on bounded sets, we have(𝑤,V, 𝑇) ∈ 𝐶([𝜖, 𝜏],R2×

̇

𝐻2

per(0, 1)) ∩ 𝐶((𝜖, 𝜏),R2× ̇𝐻per3 (0, 1))and( ̇𝑤, 𝑤, (𝜕𝑇/𝜕𝑡)) ∈

𝐶((𝜖, 𝜏),R2× ̇𝐻per3−𝛿(0, 1)). Since𝜖is arbitrary, we obtain the

regularity of the local solution.

Step 2.Now, we prove the solutions of (8) for every time𝑡 ≥ 0. To prove the global existence, we must show that the solutions are bounded inY=R2× ̇𝐻per1 (0, 1)norm on finite time intervals. First, to obtain that the norm of𝑇is bounded in finite time, we note that multiplying the equations for the temperature by𝑇in ̇𝐿2per(0, 1)and integrating by parts, we have:

1 2

𝑑

𝑑𝑡‖𝑇‖2+]󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 𝜕𝑇 𝜕𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

2

= ∮ 𝑙 (V) (𝑇𝑎− 𝑇) 𝑇 𝑑𝑥, (21)

since∮ 𝑇(𝜕𝑇/𝜕𝑥) = (1/2) ∮(𝜕/𝜕𝑥)(𝑇2) = 0.

Using Cauchy-Schwartz and the Young inequality and then the Poincar´e inequality, since∮ 𝑇 = 0together with𝜋2 is the first nonzero eigenvalue of𝐴 = −𝜕2/𝜕𝑥2in ̇𝐿2per(0, 1),

we obtain

1 2

𝑑

𝑑𝑡‖𝑇‖2+ (]𝜋2+ 𝑙 (V)) ‖𝑇‖2≤ 𝑙 (V)2 󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩2+𝑙 (V)2 ‖𝑇‖2,

(22) and using𝑙(V) ≥ 𝑙0> 0, we get

𝑑

𝑑𝑡‖𝑇‖2+ (2]𝜋2+ 𝑙0) ‖𝑇‖2≤ 𝑙 (V) 󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩

2, (23)

and we conclude that the norm of𝑇in ̇𝐿2per(0, 1)remains bounded in finite time.

Now, we note that differentiating the second equation of (5) with respect to 𝑥, we obtain the same equations for

‖𝜕𝑇/𝜕𝑥‖, and considering now‖𝜕𝑇𝑎/𝜕𝑥‖, we obtain

𝑑 𝑑𝑡󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

𝜕𝑇 𝜕𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

2

+ (2]𝜋2+ 𝑙0)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜕𝑇 𝜕𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

2

≤ 𝑙 (V)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜕𝑇𝑎 𝜕𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

2

. (24)

Thus, we show that the norm of𝑇in𝐻̇per1 (0, 1)remains bounded in finite time. Then, using‖𝑇‖bounded for finite time, we prove that|𝑤(𝑡)|and|V(𝑡)|remain bounded in finite time and we conclude.

2.1.2. The Case with No Diffusion:]=0. The system now reads

𝜀𝑑𝑑𝑡2V2 +𝑑V𝑑𝑡 + 𝐺 (V)V= ∮ 𝑇𝑓, V(0) =V0, 𝑑V𝑑𝑡 (0) = 𝑤0,

𝜕𝑇 𝜕𝑡 +V

𝜕𝑇

𝜕𝑥 = ℎ (𝑥,V, 𝑇) , 𝑇 (0, 𝑥) = 𝑇0(𝑥) ,

(25)

whereℎ(𝑥,V, 𝑇) = 𝑙(V)(𝑇𝑎− 𝑇); that is, we consider Newton’s linear cooling law as in [6–8, 10], and it is no longer of a parabolic type system and is given by

𝑑𝑤 𝑑𝑡 +

1 𝜀𝑤 = −

1

𝜀𝐺 (V)V+ 1

𝜀∮ 𝑇𝑓, 𝑤 (0) = 𝑤0, 𝑑V

𝑑𝑡 = 𝑤, V(0) =V0, 𝜕𝑇

𝜕𝑡 +V 𝜕𝑇

𝜕𝑥 = 𝑙 (V) (𝑇𝑎− 𝑇) , 𝑇 (0, 𝑥) = 𝑇0(𝑥) .

(26)

To prove the system is well posed, we use the techniques from [16] considering the same transport equation for tem-perature in different thermosyphon models as [6–8,10,16].

We note that ifV(𝑡)is a given continuous function, then the equation for the temperature can be integrated along characteristics to obtain

𝑇V(𝑡, 𝑥) = 𝑇0(𝑥 − ∫ 𝑡

0V) 𝑒 − ∫0𝑡𝑙(V)

+ ∫𝑡

0[𝑙 (V(𝑟)) 𝑒 − ∫𝑟𝑡𝑙(V)𝑇

𝑎(𝑥 − ∫ 𝑡

𝑟V)] ,

(27)

and plugging this into the nonlocal differential equation for the acceleration and into the equation for the velocity yields

𝑤V(𝑡) = 𝑤0𝑒−(1/𝜀)𝑡−1𝜀∫ 𝑡

0𝑒

−(1/𝜀)(𝑡−𝑟)𝐺 (V(𝑟))V(𝑟) 𝑑𝑟

+1𝜀∫𝑡 0𝑒

−(1/𝜀)(𝑡−𝑟)(∮ 𝑇V(𝑟) 𝑓) 𝑑𝑟,

V(𝑡) =V0+ ∫𝑡 0𝑤

V(𝑟) 𝑑𝑟.

(28)

We note that for 𝑇0, 𝑇𝑎 ∈ ̇𝐿2per(0, 1) and since in this space the translations are continuous isometries, (27) defines a continuous function of time with values in this space. Although we restrict ourselves to ̇𝐿2per(0, 1), many other choices of space are possible for solving problem (26). In fact any Banach space of 1-periodic functions of𝑥 having zero mean and in which translations are continuous isometries can be used as an “admissible space”𝑋; see [16]. In particular

𝑊𝑚,𝑝

per(0, 1), 𝐶𝑘per(0, 1)are admissible spaces between others.

Then, we can proveLemma 2.

Lemma 2. Let𝜏 > 0, fixV∈ 𝐶[0, 𝜏], and assume that𝑇0, 𝑇𝑎∈ 𝑋 where 𝑋 is an “admissible space”; see [16], in particular 𝑇0, 𝑇𝑎 ∈ 𝐻̇1

per(0, 1). Then, the function given in (27), 𝑇V

∈ 𝐶([0, 𝜏],X), is an integral solution of the PDE which is satisfied only if 𝑇0 and 𝑇𝑎 are differentiable. In particular, if𝑇0, 𝑇𝑎 ∈ ̇𝐻per1 (0, 1), then𝑇V is continuous with values in

̇

𝐻per1 (0, 1)and satisfies the PDE as an equality in ̇𝐿2per(0, 1), a.e.

in time. Moreover,(27)satisfies the following properties:

(i)

󵄩󵄩󵄩󵄩𝑇V󵄩󵄩󵄩󵄩

(6)

(ii)if there exist positive constants𝑐𝑑, 𝑑 = 𝑎and𝑑 = 0such that𝑇0, 𝑇𝑎satisfy‖𝑇𝑑(⋅ + ℎ) − 𝑇𝑑(⋅)‖𝑋≤ 𝑐𝑑|𝑘|for all𝑘, then𝑇Vsatisfies

󵄩󵄩󵄩󵄩𝑇V(𝑡 + 𝑘) − 𝑇V

(𝑡)󵄩󵄩󵄩󵄩𝑋≤ 𝐶 |𝑘| , 𝐶 = 𝐶 (‖V‖∞, 󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩𝑋) (30)

positive constant independent on time;

(iii)we assume that𝑋 ⊂ ̇𝐿2per(0, 1)and there exist positive constants𝑐𝑑, 𝑑 = 𝑎and𝑑 = 0such that𝑇0, 𝑇𝑎satisfy ‖𝑇d(⋅ + ℎ) − 𝑇𝑑(⋅)‖̇𝐿2

per ≤ 𝑐𝑑|𝑘| for all 𝑘. If we also

assume thatV𝑖, 𝑖 = 1, 2are continuous in𝑡 ∈ [0, 𝜏], then

sup

𝑟∈[0,𝜏]󵄩󵄩󵄩󵄩𝑇

V

1(𝑟) − 𝑇2V(𝑟)󵄩󵄩󵄩󵄩 ≤ 𝐾𝜏󵄩󵄩󵄩󵄩V1−V2󵄩󵄩󵄩󵄩∞, (31)

𝐾is a positive constant, and‖V1−V2 = sup𝑟∈[0,𝜏]

|V1(𝑟) −V2(𝑟)|.

Proof. See [6–8,10,16]. Then, we haveTheorem 3.

Theorem 3. Assume that 𝐺(V)V is locally Lipschitz, 𝑓 ∈ ̇𝐿2

per(0, 1),𝑇0, 𝑇𝑎 ∈ ̇𝐻per1 (0, 1), and𝑤0,V0 ∈ R2. Then there

exists a unique solution of(26)satisfying

(𝑤,V, 𝑇) ∈ 𝐶 ((0, ∞) ,R2× ̇𝐻per1 (0, 1)) (32)

and 𝑇 satisfies the PDE in the sense of (27). Moreover ( ̇𝑤, ̇V, (𝜕𝑇/𝜕𝑡)) ∈ 𝐶([0, ∞),R2× ̇𝐿2

per(0, 1)).

Proof. As noted earlier, we need to solve the fixed point problem

V(𝑡) = F(V) (𝑡) =V0

+∫𝑡 0(𝑤0𝑒

−(1/𝜀)𝑠1 𝜀∫

𝑠

0𝑒

−(1/𝜀)(𝑠−𝑟)𝐺 (V(𝑟))V(𝑟) 𝑑𝑟) 𝑑𝑠

+1𝜀∫𝑡 0(∫

𝑠

0𝑒

−(1/𝜀)(𝑠−𝑟)(∮ 𝑇V(𝑟) 𝑓) 𝑑𝑟) 𝑑𝑠,

(33) on a space of continuous functions. More precisely, we take

𝑊 = {V∈ 𝐶[0, 𝐿],V(0) =V0, |V(𝑡) −V0| ≤ 𝑀}, endowed with

the sup norm, with𝐿and𝑀to be chosen and prove thatF is a contraction on𝑊.

From (29) in Lemma 2 for 𝑇V, we have ‖𝑇V‖ ≤ max{‖𝑇0‖‖𝑇𝑎‖}, and this, together with the local Lipschitz property of𝐺(V)Vshows that for fixed𝑀,F(M) ⊂ Mif𝐿 is sufficiently small.

To show thatFis a contraction, it is clear that we must prove some Lipschitz dependence on∮ 𝑇V𝑓with respect to

V∈ 𝑊.

First, we note that from𝑇0, 𝑇𝑎 ∈ ̇𝐻per1 (0, 1), then verify

(30), that is,‖𝑇𝑑(⋅ + ℎ) − 𝑇𝑑(⋅)‖̇𝐿2

per ≤ 𝑐𝑑|𝑘|for all𝑘with𝑑 =

0, 𝑎, and givenV𝑖∈ 𝑊, again from (31) inLemma 2, we have sup

𝑟∈[0,𝜏]󵄩󵄩󵄩󵄩𝑇

V

1(𝑟) − 𝑇2V(𝑟)󵄩󵄩󵄩󵄩 ≤ 𝐿𝑀󵄩󵄩󵄩󵄩V1−V2󵄩󵄩󵄩󵄩∞. (34)

With these we find thatFis Lipschitz on𝑊with a Lipschitz constant depending on𝐿and𝑀that tends to zero as𝐿 → 0 and thenFis a contraction for small enough𝐿. Therefore, local well-posedness follows.

To prove the global existence, it is sufficient to prove that

(𝑤(𝑡),V(𝑡))is bounded on finite time intervals, since from

𝑤V(𝑡1) − 𝑤V(𝑡2)

= 𝑤0𝑒−(1/𝜀)(𝑡1−𝑡2)

−1 𝜀∫

𝑡2

𝑡1

𝑒−(1/𝜀)(𝑡−𝑟)𝐺 (V(𝑟))V(𝑟) 𝑑𝑟

+1𝜀∫𝑡2 𝑡1

𝑒−(1/𝜀)(𝑡−𝑟)(∮ 𝑇V(𝑟) 𝑓) 𝑑𝑟,

V(𝑡1) −V(𝑡2)

= − ∫𝑡2 𝑡1

(𝑤0𝑒−𝑠/𝜀−1 𝜀∫

𝑠

0𝑒

−(1/𝜀)(𝑠−𝑟)𝐺 (V(𝑟))V(𝑟) 𝑑𝑟

+1𝜀∫𝑠 0𝑒

−(1/𝜀)(𝑠−𝑟)∮ 𝑇V(𝑟) 𝑓 𝑑𝑟 𝑑𝑠) ,

(35)

we find that(𝑤(𝑡),V(𝑡))is of Cauchy type as𝑡 → 𝑡0for finite

𝑡0. Consequently, the limit of(𝑤(𝑡),V(𝑡), 𝑇(𝑡))exists inR2×

̇𝐿2

per(0, 1)and the solution can be prolonged.

But again from (29) together with (18) and (19), we obtain boundedness on finite time intervals and global existence follows.

As noted earlier, if𝑇0, 𝑇𝑎 ∈ ̇𝐻per1 (0, 1), then𝑇satisfies the PDE equation as an equality in ̇𝐿2per(0, 1)a.e. in time. In

par-ticular, we have(𝜕𝑇/𝜕𝑡) ∈ 𝐶((0, ∞), ̇𝐿2per(0, 1)).

2.2. Boundedness of the Solutions and Global Attractor. In order to obtain asymptotic bounds on the solutions as

𝑡 → ∞, we consider the friction function𝐺satisfying the hypotheses from the previous section and we also assume that there exists a constantℎ0≥ 0such that

lim sup

𝑡 → ∞ 󵄨󵄨󵄨󵄨 󵄨𝐺󸀠(𝑡)󵄨󵄨󵄨󵄨󵄨

𝐺 (𝑡) = 0, lim sup𝑡 → ∞ 󵄨󵄨󵄨󵄨 󵄨𝑡𝐺󸀠(𝑡)󵄨󵄨󵄨󵄨󵄨

𝐺 (𝑡) ≤ ℎ0. (36)

We make use of L’Hopital’s lemma proved in [11] to prove several results in this section.

Lemma 4 (L’Hopital’s lemma). Assume that𝑓and𝑔are real differentiable functions on(𝑎, 𝑏), 𝑏 ≤ ∞, 𝑔󸀠(𝑥) ̸= 0on(𝑎, 𝑏) andlim𝑥 → 𝑏𝑔(𝑥) = ∞.

(i)If lim sup𝑥 → 𝑏(𝑓󸀠(𝑥)/𝑔󸀠(𝑥)) = 𝐿, then lim sup𝑥 → 𝑏

(𝑓(𝑥)/𝑔(𝑥)) ≤ 𝐿.

(ii)If lim inf𝑥 → 𝑏(𝑓󸀠(𝑥)/𝑔󸀠(𝑥)) = 𝐿, then lim inf𝑥 → 𝑏

(𝑓(𝑥)/𝑔(𝑥)) ≥ 𝐿.

(7)

Lemma 5. If one assumes that𝐺(𝑟)and𝐻(𝑟) = 𝑟𝐺(𝑟)satisfy the hypothesis fromTheorem 1orTheorem 3together with(36), then

lim sup

𝑡 → ∞ 󵄨󵄨󵄨󵄨

󵄨󵄨𝐻 (𝑡) − (1/𝜀) ∫0𝑡𝑒−(1/𝜀)(𝑡−𝑟)𝐻 (𝑟) 𝑑𝑟󵄨󵄨󵄨󵄨󵄨󵄨

𝐺 (𝑡) ≤ 𝐻0, (37)

with𝐻0 = (1 + ℎ0)𝜀a positive constant such that𝐻0 → 0if 𝜀 → 0.

Proof. Integrating by parts we have

𝐻 (𝑡) −1𝜀∫𝑡 0𝑒

−(1/𝜀)(𝑡−𝑟)𝐻 (𝑟) 𝑑𝑟 = ∫𝑡 0𝑒

−(1/𝜀)(𝑡−𝑟)𝐻󸀠(𝑟) 𝑑𝑟,

(38) and usingLemma 4we obtain

lim sup

𝑡 → ∞

0𝑡𝑒(1/𝜀)𝑟󵄨󵄨󵄨󵄨

󵄨𝐻󸀠(𝑟)󵄨󵄨󵄨󵄨󵄨 𝑑𝑟

𝑒(1/𝜀)𝑡𝐺 (𝑡) ≤ 𝜀lim sup𝑡 → ∞ 󵄨󵄨󵄨󵄨 󵄨𝐻󸀠(𝑡)󵄨󵄨󵄨󵄨󵄨 𝐺 (𝑡) + 𝜀𝐺󸀠(𝑡)

≤ 𝜀lim sup

𝑡 → ∞ 󵄨󵄨󵄨󵄨

󵄨𝐺 (𝑡) + 𝑡𝐺󸀠(𝑡)󵄨󵄨󵄨󵄨󵄨 𝐺 (𝑡) + 𝜀𝐺󸀠(𝑡),

(39) and from (36) we conclude.

Remark 6. We note that the conditions (36) are satisfied for all friction functions𝐺considered in the previous works, that is, the thermosyphon models, where𝐺is constant or linear or quadratic law. Moreover, the conditions (36) are also true for

𝐺(𝑠) ≈ 𝐴|𝑠|𝑛, as𝑠 → ∞.

Now, we use the asymptotic bounded for temperature to obtain the asymptotic bounded for the velocity and the acceleration functions.

Theorem 7. Under the previous notations and hypothesis of

Theorem 1or Theorem 3, if one assumes also that𝐺 satisfies

(37)for some constant𝐻0≥ 0, then one has the following. Part (I). General case:

(i) lim sup

𝑡 → ∞ |V(𝑡)| ≤ 1

𝐺0lim sup𝑡 → ∞ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨∮𝑇(𝑡,⋅)𝑓(⋅)󵄨󵄨󵄨󵄨󵄨󵄨󵄨 + 𝐻0. (40)

In particular: Iflim sup𝑡 → ∞‖𝑇‖ ∈R, then

lim sup

𝑡 → ∞ |V(𝑡)| ≤ 1

𝐺0󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩lim sup𝑡 → ∞ ‖𝑇‖ + 𝐻0∈R. (41)

(ii)Iflim sup𝑡 → ∞‖𝑇‖ ∈Rand𝐺∗0 =lim sup𝑡 → ∞𝐺(V(𝑡)), then

lim sup

𝑡 → ∞ |𝑤 (𝑡)| ≤ 𝐺 ∗

0𝐻0+ (1 +𝐺 ∗ 0 𝐺0) 𝐼

with𝐼 =lim sup

𝑡 → ∞ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨∮𝑇(𝑡,⋅)𝑓(⋅)󵄨󵄨󵄨󵄨󵄨󵄨󵄨,

(42)

lim sup

𝑡 → ∞ |𝑤 (𝑡)| ≤ 𝐺 ∗

0𝐻0+ (1 +𝐺 ∗ 0

𝐺0) 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩lim sup𝑡 → ∞ ‖𝑇‖ ∈R.

(43)

Part (II). If] ̸= 0and also assume that there exists𝐿0a positive constant such that𝐿0 ≥ 𝑙(V) ≥ 𝑙0, then for any solution of (5)

in the spaceY=R2× ̇𝐻per1 (0, 1), one has

(i)

lim sup

𝑡 → ∞ ‖𝑇 (𝑡)‖ ≤ ( 𝐿0 2]𝜋2+ 𝑙

0) 1/2

󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩,

lim sup

𝑡 → ∞ 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 𝜕𝑇

𝜕𝑥(𝑡)󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 ≤ ( 𝐿0 2]𝜋2+ 𝑙

0) 1/2󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩𝜕𝑇𝜕𝑥𝑎󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩;

(44)

(ii)

lim sup

𝑡 → ∞ |V(𝑡)| ≤ 1 𝐺0(

𝐿0 2]𝜋2+ 𝑙

0) 1/2

󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩 + 𝐻0; (45)

(iii)if𝐺∗0 =lim sup𝑡 → ∞𝐺(V(𝑡)),

lim sup

𝑡 → ∞ |𝑤 (𝑡)| ≤ 𝐺 ∗

0𝐻0+ 𝐺2 󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩 √2]𝜋2+ 𝑙 0

with𝐺2= (1 + 𝐺

∗ 0 𝐺0) √𝐿0.

(46)

In particular,(5)has a global compact and connected attractor, A, inY=R2× ̇𝐻per1 (0, 1).

Proof.

Part (I).General case. (i) From (8) we have that

𝑑𝑤 𝑑𝑡 +

1 𝜀𝑤 = −

1

𝜀𝐺 (V)V+ 1

𝜀∮ 𝑇 ⋅ 𝑓, (47)

and𝑤(𝑡) = 𝑑V/𝑑𝑡satisfies

𝑑V

𝑑𝑠 = 𝑤 (0) 𝑒−(1/𝜀)𝑠− 1 𝜀∫

𝑠

0 𝑒

−(1/𝜀)(𝑠−𝑟)𝐻 (𝑟) 𝑑𝑟

+1 𝜀∫

𝑠

0(∮ 𝑇 (𝑟) ⋅ 𝑓) 𝑒

−(1/𝜀)(𝑠−𝑟)𝑑𝑟,

(48)

where𝐻(𝑟) = 𝐻(V(𝑟)) = V(𝑟)𝐺(V(𝑟)). First, we rewrite (48) as

𝑑V

𝑑𝑠 + 𝐺 (𝑠)V= 𝑤 (0) 𝑒−(1/𝜀)𝑠+ 𝐼1(𝑠) + 𝐼2(𝑠) , (49)

with

𝐼1(𝑠) = 1 𝜀∫

𝑠

0(∮ 𝑇 (𝑟) ⋅ 𝑓) 𝑒

−(1/𝜀)(𝑠−𝑟)𝑑𝑟,

𝐼2(𝑠) = 𝐻 (𝑠) −1 𝜀∫

𝑠

0𝑒

−(1/𝜀)(𝑠−𝑟)𝐻 (𝑟) .

(50)

Next, for any𝛿 > 0 there exists𝑡0 > 0such that 𝛿(𝑠) =

𝑤(0)𝑒−1/𝜀 < 𝛿 for any𝑠 ≥ 𝑡

0 and integrating with𝑡 ≥ 𝑡0,

we obtain

|V(𝑡)| ≤ 󵄨󵄨󵄨󵄨V(𝑡0)󵄨󵄨󵄨󵄨 𝑒− ∫𝑡0𝑡𝐺(𝑠)𝑑𝑠

+ 𝑒− ∫𝑡0𝑡𝐺(𝑠)𝑑𝑠∫

𝑡

𝑡0

𝑒∫𝑡0𝑠𝐺(𝑟)𝑑𝑟(𝛿 + 󵄨󵄨󵄨󵄨𝐼1(𝑠)󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨𝐼2(𝑠)󵄨󵄨󵄨󵄨) .

(8)

Using L’Hopital’sLemma 4proved in [11], we get

lim sup

𝑡 → ∞ 𝑒

− ∫𝑡0𝑡𝐺(𝑠)𝑑𝑠𝑡 𝑡0

𝑒∫𝑡0𝑠𝐺(𝑟)𝑑𝑟(󵄨󵄨󵄨󵄨𝐼1(𝑠)󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨𝐼2(𝑠)󵄨󵄨󵄨󵄨 + 𝛿)

=lim sup

𝑡 → ∞ ∫𝑡𝑡

0𝑒

𝑡0𝑠𝐺(𝑟)𝑑𝑟

(󵄨󵄨󵄨󵄨𝐼1(𝑠)󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨𝐼2(𝑠)󵄨󵄨󵄨󵄨 + 𝛿) 𝑑𝑠 𝑒∫𝑡0𝑡𝐺(𝑠)𝑑𝑠

≤lim sup

𝑡 → ∞

󵄨󵄨󵄨󵄨𝐼1(𝑡)󵄨󵄨󵄨󵄨 +󵄨󵄨󵄨󵄨𝐼2(𝑡)󵄨󵄨󵄨󵄨 + 𝛿

𝐺 (𝑡) for any 𝛿 > 0.

(52)

Moreover, using again the L’Hopital’sLemma 4proved in [11], we get

lim sup

𝑡 → ∞ 󵄨󵄨󵄨󵄨𝐼1(𝑡)󵄨󵄨󵄨󵄨 ≤lim sup𝑡 → ∞ ∫0𝑡𝑒𝑟/𝜀󵄨󵄨󵄨󵄨

󵄨∮ 𝑇 (𝑡) ⋅ 𝑓󵄨󵄨󵄨󵄨󵄨 𝜀𝑒𝑡/𝜀

≤lim sup

𝑡 → ∞ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨∮𝑇(𝑡) ⋅ 𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨,

(53)

and from (51) together with (37), we conclude that

lim sup

𝑡 → ∞ |V(𝑡)| ≤lim sup𝑡 → ∞

lim sup𝑡 → ∞󵄨󵄨󵄨󵄨󵄨∮ 𝑇 (𝑡) ⋅ 𝑓󵄨󵄨󵄨󵄨󵄨

𝐺0 + 𝐻0+ 𝛿,

(54)

for any𝛿.

(ii) From (47) together with Gronwall’s lemma, we get

|𝑤 (𝑡)| ≤ 󵄨󵄨󵄨󵄨𝑤 (𝑡0)󵄨󵄨󵄨󵄨 𝑒−(1/𝜀)𝑡

+1𝜀∫𝑡 𝑡0

𝑒−(1/𝜀)(𝑡−𝑟)[𝐺 (𝑟) |V(𝑟)| +󵄨󵄨󵄨󵄨󵄨󵄨󵄨∮𝑇(𝑟) ⋅ 𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨]𝑑𝑟,

(55)

where𝐺(𝑟) = 𝐺(V(𝑟)). Consequently, for any𝛿 > 0, there exists𝑡0such that for any𝑡 ≥ 𝑡0,

1 𝜀∫

𝑡

𝑡0

𝑒−(1/𝜀)(𝑡−𝑟)[𝐺 (V(𝑟)) |V(𝑟)| +󵄨󵄨󵄨󵄨󵄨󵄨󵄨∮𝑇(𝑟) ⋅ 𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨]𝑑𝑟

≤ [𝛿 +lim sup

𝑡 → ∞ (𝐺 (V(𝑡)) |V(𝑡)| +󵄨󵄨󵄨󵄨󵄨󵄨󵄨∮𝑇(𝑡) ⋅ 𝑓󵄨󵄨󵄨󵄨󵄨󵄨󵄨)]

× (1 − 𝑒−(1/𝜀)(𝑡−𝑡0)) ,

(56)

that is,

lim sup

𝑡 → ∞ |𝑤 (𝑡)|

≤lim sup

𝑡 → ∞ (𝐺 (V(𝑡)) |V(𝑡)| +󵄨󵄨󵄨󵄨󵄨∮ 𝑇 (𝑡) ⋅ 𝑓󵄨󵄨󵄨󵄨󵄨 + 𝛿) ,

(57)

for any𝛿 > 0, and using the previous results (i) we get (42).

Part (II).(i) From (23) together with (24), we get

‖𝑇‖2≤ 𝐿0 2]𝜋2+ 𝑙

0󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩 2

+ (󵄩󵄩󵄩󵄩𝑇0󵄩󵄩󵄩󵄩2−2]𝜋𝐿20+ 𝑙 0󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩

2) +

× 𝑒−(2𝜋2]+𝑙 0)𝑡,

󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝜕𝑇𝜕𝑥󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

2

≤ 𝐿0

2]𝜋2+ 𝑙 0

󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝜕𝑇𝜕𝑥𝑎󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

2

+ (󵄩󵄩󵄩󵄩𝑇0󵄩󵄩󵄩󵄩2−2]𝜋𝐿20+ 𝑙 0󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩

2) +

×𝑒−(2𝜋2]+𝑙0)𝑡 ,

(58) and by elementary integration we obtain (44). Using Part (I), the rest (ii) and (iii) are obvious. Since the sectorial operator𝐵 defined inSection 2.1.1has compact resolvent, the rest follows from [17, Theorems4.2.2and3.4.8].

Remark 8. First, we note that the hypothesis about the function𝑙(V)inTheorem 7, 𝑙(V) ≤ 𝐿0 is satisfied when we consider Newton’s linear cooling lawℎ = 𝑘(𝑇𝑎− 𝑇), where𝑘 is a positive quantity; that is,𝑙(V) = 𝑘 = 𝐿0, as [18]. Moreover, this condition is also satisfied if we considerℎ = 𝑙(V)(𝑇𝑎− 𝑇) where𝑙(V)is a positive upper bounded function.

Second, it is important to note that we prove in the next section the existence of the global compact and connected attractor and the inertial manifold for the system (8), when we consider the general Newton’s linear cooling law without the additional previous hypothesis on𝑙(V); but we assume that the friction function𝐺(V)always satisfies (36).

In order to get this, we consider the Fourier expansions and observe the dynamics of each coefficient of Fourier expansions to improve the asymptotic bounded of temper-ature. In particular, we will prove lim sup𝑡 → ∞‖𝑇(𝑡)‖ ≤ ‖𝑇𝑎‖ for every locally Lipschitz and positive function𝑙(V)and also for every ] ≥ 0(see (72) in Proposition 9) and for every friction function𝐺(V)satisfying (36).

3. Asymptotic Behavior: Reduction to

Finite-Dimensional Systems

We take a close look at the dynamics of (5) by considering the Fourier expansions of each function and observing the dynamics of each Fourier mode. Assume that𝑇𝑎∈ ̇𝐻per1 (0, 1) and 𝑓 ∈ ̇𝐿2per(0, 1) are given by the following Fourier expansions:

𝑇𝑎(𝑥) = ∑ 𝑘∈Z∗

𝑏𝑘𝑒2𝜋𝑘𝑖𝑥,

𝑓 (𝑥) = ∑ 𝑘∈Z∗𝑐𝑘𝑒

2𝜋𝑘𝑖𝑥 withZ, (59)

while the initial data 𝑇0 ∈ 𝐻̇per1 (0, 1)is given by𝑇0(𝑥) =

𝑘∈Z∗𝑎𝑘0𝑒2𝜋𝑘𝑖𝑥.

Assume that𝑇(𝑡, 𝑥) ∈ ̇𝐻per1 (0, 1)is given by

𝑇 (𝑡, 𝑥) = ∑ 𝑘∈Z∗

(9)

Then, we find that the coefficient𝑎𝑘(𝑡)in (60) is a solution of

̇𝑎

𝑘(𝑡) + (2𝜋𝑘V𝑖 + 4]𝜋2𝑘2+ 𝑙 (V)) 𝑎𝑘(𝑡) = 𝑙 (V) 𝑏𝑘,

𝑎𝑘(0) = 𝑎𝑘0, 𝑘 ∈Z∗. (61)

Since all the functions involved are real, we have 𝑎𝑘 =

𝑎−𝑘, 𝑏𝑘 = 𝑏−𝑘, and𝑐𝑘 = 𝑐−𝑘. Therefore, (5) is equivalent to

the infinite system of ODEs consisting of (61) coupled with

𝜀𝑑2V 𝑑𝑡 +

𝑑V

𝑑𝑡 + 𝐺 (V)V= ∑𝑘∈Z

𝑎𝑘(𝑡) 𝑐−𝑘. (62)

The system of (5) reflects two of the main features: (i) the coupling between the modes enter only through the velocity, while diffusion acts as a linear damping term, and (ii) it is important to note in this model that we have also the nonlinear term given by Newton’s linear cooling law.

We note that the system (5) is equivalent to the system (8) for acceleration, velocity and temperature. It is equivalent to the following infinite system of ODEs:

𝑑𝑤 𝑑𝑡 +

1 𝜀𝑤 = −

1

𝜀𝐺 (V)V+ 1 𝜀𝑘∈Z

𝑎𝑘(𝑡) 𝑐−𝑘, 𝑤 (0) = 𝑤0,

𝑑V

𝑑𝑡 = 𝑤, V(0) =V0,

̇𝑎

𝑘(𝑡) + (2𝜋𝑘V𝑖 + 4]𝜋2𝑘2+ 𝑙 (V)) 𝑎𝑘(𝑡) = 𝑙 (V) 𝑏𝑘,

𝑎𝑘(0) = 𝑎𝑘0, 𝑘 ∈Z∗.

(63) In what follows, we will exploit this explicit equation for the temperature modes to analyze the asymptotic behavior of the system and to obtain the explicit low-dimensional models.

3.1. Attractors and Inertial Manifolds. The existence of an inertial manifold does not rely, in this case, on the existence of large gaps in the spectrum of the elliptic operator but on the invariance of certain sets of Fourier modes.

A similar explicit construction was given by Bloch and Titi in [19] for a nonlinear beam equation where the nonlin-earity occurs only through the appearance of the𝐿2norm of the unknown. A related construction was given by Stuart in [20] for a nonlocal reaction-diffusion equation.

In order to get the inertial manifold for this system, we first improve the bounds on acceleration, velocity, and temperature of the previous section for all situations with]≥

0and𝑙(V) ≥ 𝑙0 > 0general locally Lipschitz function under the hypotheses ofLemma 5for some𝐻0 ≥ 0, in particular

𝐺(V)satisfying (36).

We will prove in Proposition 9that we have always an upper bounded for the temperature in ̇𝐿2(0, 1)independent of the velocity and the function𝑙(V), considered in Newton’s linear cooling law and also independent of the diffusion coefficient. That is,

lim sup

𝑡 → ∞ ‖𝑇 (𝑡, ⋅)‖ ≤ 󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩. (64)

Proposition 9. Under the previous notations, for every solu-tion of the system(5),(𝑤,V, 𝑇), and for every𝑘 ∈Z∗, one has

(i) lim sup

𝑡 → ∞ 󵄨󵄨󵄨󵄨𝑎𝑘(𝑡)󵄨󵄨󵄨󵄨≤󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨, in particularlim sup𝑡 → ∞ ‖𝑇 (𝑡, ⋅)‖≤󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩,

(65)

(ii)

lim sup

𝑡 → ∞ |V(𝑡)| ≤ 𝐼0

𝐺0 + 𝐻0, with 𝐼0= ∑𝑘∈Z∗󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑐𝑘󵄨󵄨󵄨󵄨 (66)

and𝐺0a positive constant such that𝐺(V) ≥ 𝐺0;

(iii)

lim sup

𝑡 → ∞ |𝑤 (𝑡)| ≤ 𝐺 ∗

0𝐻0+ (1 +𝐺 ∗ 0 𝐺0) 𝐼0,

with𝐼0= ∑

𝑘∈Z∗󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑐𝑘󵄨󵄨󵄨󵄨, 𝐺

0 =lim sup

𝑡 → ∞ 𝐺 (V(𝑡)) .

(67)

In particular, one has a global compact and connected attractor A⊂ [−𝑀, 𝑀]×[−𝑁, 𝑁]×Cwhere𝑀, 𝑁are the upper bounds for acceleration and velocity as given in(67)and(66)and𝑇0∈ C= {𝑅(𝑥) = ∑𝑘∈Z∗𝑟𝑘𝑒2𝜋𝑘𝑖𝑥, |𝑟𝑘| ≤ |𝑏𝑘|}.

Proof. From (61), we have

𝑎𝑘(𝑡) = 𝑎𝑘0𝑒−4]𝜋

2𝑘2𝑡

𝑒− ∫0𝑡[2𝜋𝑘V𝑖+𝑙(V)]

+ 𝑏𝑘∫ 𝑡

0𝑒

−4]𝜋2𝑘2(𝑡−𝑠)

𝑙 (V(𝑠)) 𝑒− ∫𝑠𝑡[2𝜋𝑘V𝑖+𝑙(V)]𝑑𝑠

(68)

with

󵄨󵄨󵄨󵄨

󵄨󵄨󵄨𝑒− ∫0𝑡2𝜋𝑘V𝑖󵄨󵄨󵄨󵄨

󵄨󵄨󵄨 =󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑒− ∫𝑠𝑡2𝜋𝑘V𝑖󵄨󵄨󵄨󵄨

󵄨󵄨󵄨 = 1, 𝑒−4]𝜋2𝑘2(𝑡−𝑠)≤ 1, ∫𝑡

0𝑙 (V(𝑠)) 𝑒

− ∫𝑠𝑡𝑙(V)𝑑𝑠 = 1 − 𝑒− ∫0𝑡𝑙(V).

(69)

Thus, we obtain

󵄨󵄨󵄨󵄨𝑎𝑘(𝑡)󵄨󵄨󵄨󵄨 ≤󵄨󵄨󵄨󵄨𝑎𝑘0󵄨󵄨󵄨󵄨𝑒−4]𝜋

2𝑘2𝑡

𝑒− ∫0𝑡𝑙(V)+ 󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨(1 − 𝑒− ∫ 𝑡

0𝑙(V)) , (70)

and we get lim sup𝑡 → ∞|𝑎𝑘(𝑡)| ≤ |𝑏𝑘|. Using Theorem 7

together with∮ 𝑇𝑓 = ∑𝑘∈Z∗𝑎𝑘(𝑡)𝑐𝑘, we get

lim sup

𝑡 → ∞ |V(𝑡)| ≤ 1

𝐺0lim sup𝑡 → ∞ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨∮𝑇(𝑡,⋅)𝑓(⋅)󵄨󵄨󵄨󵄨󵄨󵄨󵄨 + 𝐻0,

lim sup

𝑡 → ∞ |𝑤 (𝑡)| ≤ 𝐺 ∗

0𝐻0+ (1 +𝐺 ∗ 0

𝐺0) 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩lim sup𝑡 → ∞ ‖𝑇‖ ∈R.

(71) From this upper bounded for the velocity, we also have𝐿0the upper bound for the continuous positive function𝑙(V), and using Part (II) fromTheorem 7we conclude.

(10)

We note from the previous result that we have always the upper bound for ‖𝑇‖ and from Theorem 7 for the velocity. Therefore, we can consider𝐿0the upper bound for the continuous positive function 𝑙(V); we note that 𝐿0 = lim sup𝑡 → ∞𝑙(V)and we prove inProposition 10the bound of solutions to show the influence of diffusion coefficient].

Proposition 10. Under the previous notations, for every solu-tion of the system(5),(𝑤,V, 𝑇), and for every𝑘 ∈Z∗, one has

lim sup

𝑡 → ∞ 󵄨󵄨󵄨󵄨𝑎𝑘(𝑡)󵄨󵄨󵄨󵄨 ≤ 𝐿 𝑘

]󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨, with𝐿𝑘]=4]𝜋2𝐿𝑘02+ 𝑙 0

in particular lim sup

𝑡 → ∞ ‖𝑇 (𝑡, ⋅)‖ ≤ 𝐿0 4]𝜋2+ 𝑙

0󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩.

(72)

Moreover, one has that

‖𝑇‖2𝐻̇𝑚+2

per

𝐿0

4]𝜋2󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩2𝐻̇𝑚

per,

lim sup

𝑡 → ∞ |V(𝑡)| ≤ 𝐼0

𝐺0 + 𝐻0, with𝐼0= 4]𝜋𝐿20+ 𝑙 0

× ∑

𝑘∈Z∗󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑐𝑘󵄨󵄨󵄨󵄨,

(73)

and𝐺0positive constant such that𝐺(V) ≥ 𝐺0;

lim sup

𝑡 → ∞ |𝑤 (𝑡)| ≤ 𝐺 ∗

0𝐻0+ 2𝐼0,

with𝐼0= 𝐿0 4]𝜋2+ 𝑙

0𝑘∈∑Z∗󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑐𝑘󵄨󵄨󵄨󵄨, 𝐺

0 =lim sup𝑡 → ∞ 𝐺 (V(𝑡)) .

(74)

Proof. Using again

𝑎𝑘(𝑡) = 𝑎𝑘0𝑒−4]𝜋2𝑘2𝑡𝑒− ∫0𝑡[2𝜋𝑘V𝑖+𝑙(V)]

+𝑏𝑘∫𝑡 0𝑒

−4]𝜋2𝑘2(𝑡−𝑠)

𝑙 (V(𝑠)) 𝑒− ∫𝑠𝑡[2𝜋𝑘V𝑖+𝑙(V)]𝑑𝑠,

(75)

if we assume that0 < 𝑙0≤ 𝑙(V) ≤ 𝐿0, then we obtain

󵄨󵄨󵄨󵄨𝑎𝑘(𝑡)󵄨󵄨󵄨󵄨 ≤󵄨󵄨󵄨󵄨𝑎𝑘0󵄨󵄨󵄨󵄨𝑒−(4]𝜋

2𝑘2+𝑙 0)𝑡+ 󵄨󵄨󵄨󵄨𝑏

𝑘󵄨󵄨󵄨󵄨𝐿0∫ 𝑡

0𝑒 −4(]𝜋2𝑘2+𝑙

0)(𝑡−𝑠), (76)

and we get

󵄨󵄨󵄨󵄨𝑎𝑘(𝑡)󵄨󵄨󵄨󵄨 ≤ (󵄨󵄨󵄨󵄨𝑎𝑘0󵄨󵄨󵄨󵄨 − 𝐿4]𝜋2𝑘02+ 𝑙

0󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨)+𝑒 −(4]𝜋2𝑘2+𝑙

0)𝑡

+4]𝜋2𝐿𝑘02+ 𝑙 0 󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨;

(77)

that is, if |𝑎𝑘0| ≤ (𝐿0/(4]𝜋2𝑘2 + 𝑙0))|𝑏𝑘|, then |𝑎𝑘(𝑡)| ≤

(𝐿0/(4]𝜋2𝑘2+ 𝑙 0))|𝑏𝑘|.

In particular |𝑎𝑘(𝑡)| ≤ (𝐿0/(4]𝜋2𝑘2 + 𝑙0))|𝑏𝑘| ≤ (𝐿0/

(4]𝜋2 + 𝑙0))|𝑏𝑘| for every 𝑘 ∈ Z∗ and also |𝑎𝑘(𝑡)| ≤

(𝐿0/(4]𝜋2𝑘2+ 𝑙

0))|𝑏𝑘| ≤ (𝐿0/(4]𝜋2𝑘2))|𝑏𝑘|; that is,|𝑎𝑘(𝑡)| ≤ (𝐿0/4]𝜋2)|𝑏

𝑘|.

Then, we also have that

‖𝑇‖2𝐻̇𝑚+2

per ≤

∞ ∑ |𝑘|=1

|𝑘|2𝑚+4󵄨󵄨󵄨󵄨𝑎𝑘(𝑡)󵄨󵄨󵄨󵄨 ≤ 4]𝜋𝐿02 ∞ ∑ |𝑘|=1

|𝑘|2𝑚󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨

4]𝜋𝐿02󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩2𝐻̇𝑚

per.

(78)

Finally, we note that if 𝐿0 = lim sup𝑡 → ∞𝑙(V(𝑡)), then given 𝛿 > 0, there exists 𝑡0 such that𝐿(V(𝑡)) ≤ 𝐿0 + 𝛿 for every 𝑡 ≥ 𝑡0, and integrating in 𝑡 ≥ 𝑡0, we obtain lim sup𝑡 → ∞|𝑎𝑘(𝑡)| ≤ (𝐿0/(4]𝜋2𝑘2+ 𝑙0))|𝑏𝑘| + 𝛿, for any𝛿 > 0, and

‖𝑇‖2𝐻̇𝑚

per ≤ ‖𝑇‖

2

̇

𝐻𝑚+2

per ≤

𝐿0

4]𝜋2󵄩󵄩󵄩󵄩𝑇𝑎󵄩󵄩󵄩󵄩2𝐻̇𝑚

per. (79)

Using againTheorem 7, we conclude.

We note that if] = 0, then|𝑎𝑘| ≤ |𝑏𝑘| ≤ (𝐿0/𝑙0)|𝑏𝑘|. If

]> 0we observe in the numerical experiments, as]is bigger, that the solution becomes stable or periodic.

As a consequence, we have the following result on the smoothness of the attractor of (5).

Corollary 11. (i) If |𝑎𝑘0| ≤ (𝐿0/(4]𝜋2𝑘2 + 𝑙0))|𝑏𝑘|, then |𝑎𝑘(𝑡)| ≤ (𝐿0/(4]𝜋2𝑘2+ 𝑙

0))|𝑏𝑘|for every𝑡 ≥ 0.

(ii) IfAis the global attractor in the spaceY = R2 ×

̇

𝐻1

per(0, 1), then for every (𝑤0,V0, 𝑇0) ∈ A, with 𝑇0(𝑥) =

𝑘∈Z∗𝑎𝑘𝑒2𝜋𝑘𝑖𝑥, one gets

󵄨󵄨󵄨󵄨𝑎𝑘󵄨󵄨󵄨󵄨 ≤ 𝐿4]𝜋2𝑘02+ 𝑙

0 󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨, 𝑘 ∈Z

. (80)

In particular, if𝑇𝑎 ∈ ̇𝐻per𝑚 (0, 1)with𝑚 ≥ 1, the global attractor

A󳨅→R2× ̇𝐻𝑚+2

per (0, 1)and is compact in this space.

Proof. (i) From (77) we have|𝑎𝑘(𝑡)| ≤ (𝐿0/(4]𝜋2𝑘2+𝑙0))|𝑏𝑘|+

(|𝑎𝑘0| − (𝐿0/(4]𝜋2𝑘2 + 𝑙

0))|𝑏𝑘|)+𝑒− ∫

𝑡

0𝑙(V), therefore, if|𝑎𝑘0| ≤ (𝐿0/(4]𝜋2𝑘2+ 𝑙

0))|𝑏𝑘|, then|𝑎𝑘(𝑡)| ≤ (𝐿0/(4]𝜋2𝑘2+ 𝑙0))|𝑏𝑘|

for every𝑡 ≥ 0and𝑘 ∈Z∗.

(ii) We note that from (i), if 𝑇𝑎(𝑥) = ∑𝑘∈Z∗𝑏𝑘𝑒2𝜋𝑘𝑖𝑥 ∈

̇

𝐻𝑚

per, then∑𝑘∈Z∗𝑘2𝑚|𝑏𝑘|2 < ∞, and therefore𝑇0 ∈ C =

{𝑅(𝑥) = ∑𝑘∈Z∗𝑟𝑘𝑒2𝜋𝑘𝑖𝑥 ∈ ̇𝐻per𝑚+2, 4]𝜋2𝑘2|𝑟𝑘| ≤ |𝑏𝑘|}; that is, A ⊂ [−𝑀, 𝑀] × [−𝑁, 𝑁] × Cwhere𝑀, 𝑁are the upper bounds for acceleration and velocity as given in (74) and (73). But, the setCis compact in𝐻̇per𝑚+2since for any sequence{𝑇𝑛} inCwe can extract a subsequence that we still denote{𝑇𝑛} such that it converges weakly to a function𝑇and such that for any𝑘 ∈ Z∗, the Fourier coefficients verify𝑎𝑘𝑛 → 𝑎𝑘

(11)

as𝑛 → ∞, where 𝑎𝑘 is the 𝑘th Fourier coefficient of 𝑇. Therefore,4]𝜋2𝑘2|𝑎𝑘| ≤ |𝑏𝑘|and for every integer𝐸,

󵄩󵄩󵄩󵄩𝑇𝑛

− 𝑇󵄩󵄩󵄩󵄩2𝑚+2≤ 𝐸 ∑ |𝑘|=1

|𝑘|2𝑚+4󵄨󵄨󵄨󵄨𝑎𝑘𝑛− 𝑎𝑘󵄨󵄨󵄨󵄨2

+ 𝐶0 ∑∞ |𝑘|=𝐸+1

|𝑘|2𝑚+4|𝑘|2𝑚󵄨󵄨󵄨󵄨𝑏𝑘󵄨󵄨󵄨󵄨2,

(81)

where ‖ ⋅ ‖𝑚+2 denotes the norm in 𝐻̇per𝑚+2. Hence, the first

term goes to zero as𝑛 → ∞and the second term can be made arbitrarily small as𝐸 → ∞. Consequently,𝑇 ∈Cand

𝑇𝑛 → 𝑇in𝐻̇𝑚+2

per , and the result follows.

Note that this result reveals in particular the asymptotic smoothing of (5). In the next result we will prove that the dynamical system induced by (8) in the phase space Y =

R2× ̇𝐻per𝑚 (0, 1),𝑚 ≥ 1, has an inertial manifold. According

to [21], we have the following definition.

Definition 12. Let𝑆(𝑡), 𝑡 ≥ 0, be a nonlinear semigroup in a Banach space inYthat has a global attractorA. Then, a smooth manifoldM⊂Yis called an inertial manifold if

(i)Mis positively invariant, that is, 𝑆(𝑡)M ⊂ M for every𝑡 ≥ 0;

(ii)Mcontains the attractor, that is,A⊂M;

(iii)Mis exponentially attracting in the sense that there exists a constant𝛿 > 0such that for every bounded set𝐵 ⊂Ythere exists𝐶 = 𝐶(𝐵) ≥ 0such that

dist(𝑆 (𝑡) 𝐵,M) ≤ 𝐶𝑒−𝛿𝑡, (82)

for every𝑡 ≥ 0.

See, for example, [21,22].

Assume the ambient temperature given by

𝑇𝑎(𝑥) = ∑ 𝑘∈𝐾

𝑏𝑘𝑒2𝜋𝑘𝑖𝑥∈ ̇𝐻per𝑚 (0, 1) , (83)

where𝐾 ⊂ Z, that is, with𝑏𝑘 ̸= 0for every𝑘 ∈ 𝐾 ⊂Zwith

0 ∉ 𝐾, since∮ 𝑇𝑎= 0.

Then, we denote by 𝑉𝑚 the closed linear subspace of

̇

𝐻𝑚

per(0, 1) spanned by {𝑒2𝜋𝑘𝑖𝑥, 𝑘 ∈ 𝐾} and consider the

following spectral decomposition in𝐻̇per𝑚 (0, 1) : 𝑇 = 𝑇1+ 𝑇2, where 𝑇1 denotes the projection of 𝑇onto 𝑉and 𝑇2 the

projection onto the space generated by{𝑒2𝜋𝑘𝑖𝑥, 𝑘 ∉ 𝐾}. Note that (8) is equivalent to

𝜀𝑑𝑑𝑡2V2 +𝑑V𝑑𝑡 + 𝐺 (V)V= ∮ (𝑇1+ 𝑇2) 𝑓,

V(0) =V0, 𝑑V

𝑑𝑡(0) = 𝑤0, 𝜕𝑇1

𝜕𝑡 +V 𝜕𝑇1

𝜕𝑥 =𝑙 (V) (𝑇𝑎− 𝑇1)+] 𝜕2𝑇1

𝜕𝑥2 , 𝑇1(0, 𝑥)=𝑇01(𝑥) , 𝜕𝑇2

𝜕𝑡 +V 𝜕𝑇2

𝜕𝑥 = −𝑙 (V) 𝑇2+] 𝜕2𝑇2

𝜕𝑥2 , 𝑇2(0, 𝑥) = 𝑇02(𝑥) .

(84)

Note that from (61), if 𝑏𝑘 = 0, then the 𝑘th mode for the temperature is damped out exponentially, and therefore the space𝑉attracts the dynamics for the temperature. This is precisely stated in the following result.

Theorem 13. Assume that𝑇𝑎 ∈ ̇𝐻per𝑚 (0, 1)and𝑓 ∈ ̇𝐿2per(0, 1). Then, the setM=R2× 𝑉𝑚is an inertial manifold for the flow of𝑆(𝑡)(𝑤0,V0, 𝑇0) = (𝑤(𝑡),V(𝑡), 𝑇(𝑡))in the spaceY =R2×

̇

𝐻𝑚

per(0, 1). Moreover, if𝑓 ∈ 𝑉𝑚the inertial manifoldMhas the

exponential tracking property; that is, for every(𝑤0,V0, 𝑇0) ∈

R2 × ̇𝐻𝑚

per(0, 1), there exists (𝑤1,V1, 𝑇1) ∈ M such that if

(𝑤𝑖(𝑡),V𝑖(𝑡), 𝑇𝑖(𝑡)), 𝑖 = 0, 1, are the corresponding solutions of

(8), then(𝑤0(𝑡),V0(𝑡), 𝑇0(𝑡)) − (𝑤1(𝑡),V1(𝑡), 𝑇1(𝑡)) → 0in

R2× ̇𝐻𝑚

per(0, 1). In particular if𝐾is a finite set, the dimension

ofMis|𝐾| + 2, where|𝐾|is the number of elements in𝐾.

Proof. In order to prove thatMis invariant, we note if𝑘 ∉ 𝐾, then𝑏𝑘= 0, and therefore𝑎𝑘0= 0, (𝑇02= 0), from (70), we get that𝑎𝑘(𝑡) = 0, for every𝑡; that is,𝑇(𝑡, 𝑥) = ∑𝑘∈𝐾𝑎𝑘(𝑡)𝑒2𝜋𝑘𝑖𝑥=

𝑇1. Thus, if(𝑤

0,V0, 𝑇0) ∈ M, then(𝑤(𝑡),V(𝑡), 𝑇(𝑡)) ∈Mfor

every𝑡, that is, invariant manifold.

We consider the decomposition in𝐻̇per𝑚 ,𝑇 = 𝑇1 + 𝑇2,

where𝑇1is the projection of𝑇on𝑉𝑚and𝑇2is the projection of𝑇on the subspace generated by{𝑒2𝜋𝑘𝑖𝑥, 𝑘 ∈ Z∗\ 𝐾}; that is,𝑇1= ∑𝑘∈𝐾𝑎𝑘𝑒2𝜋𝑘𝑖𝑥and𝑇2= ∑𝑘∈Z\𝐾𝑎𝑘𝑒2𝜋𝑘𝑖𝑥= 𝑇 − 𝑇1.

From (77) taking into account that 𝑏𝑘 = 0 for 𝑘 ∈

Z∗ \ 𝐾, we have that|𝑎

𝑘(𝑡)| ≤ |𝑎𝑘0|𝑒−(4]𝜋

2𝑘2+𝑙

0)𝑡 for every 𝑘 ∈ Z∗ implies that there exist positive constants𝐶

𝑖, such

that‖𝑇2(𝑡)‖𝐻̇2𝑚

per ≤ 𝐶1‖𝑇

2(𝑡)‖

̇

𝐻𝑚

per ≤ 𝐶2‖𝑇

2 0‖𝐻̇𝑚

per𝑒

−(4]𝜋2+𝑙 0)𝑡, that

is,𝑇2(𝑡) → 0in𝐻̇per2𝑚(0, 1)if𝑡 → ∞.

Therefore, we have in particular that ‖𝑇2(𝑡)‖𝐻̇𝑚

per → 0

as 𝑡 → ∞with exponential decay rate 𝑒−(4]𝜋2+𝑙0)𝑡. Thus, Malso attracts(𝑤(𝑡),V(𝑡), 𝑇(𝑡))with exponential rate𝑒−4]𝜋2𝑡, since distY((𝑤(𝑡),V(𝑡), 𝑇(𝑡)),M) = dist𝐻̇𝑚

per(𝑇(𝑡), 𝑉) =

‖𝑇2(𝑡)‖

̇

𝐻𝑚

per ≤ 𝐶2‖𝑇

2 0‖𝐻̇𝑚

per𝑒

−(4]𝜋2+𝑙 0)𝑡.

(12)

To prove the exponential tracking property just note that the flow insideMis given by setting𝑇2= 0where∮ 𝑇(𝑥)⋅𝑓 =

𝑘∈𝐾𝑎𝑘(𝑡) ⋅ 𝑐−𝑘; that is,

̇𝑤 + 1𝜀𝑤 +1

𝜀𝐺 (V)V= 1

𝜀𝑘∈𝐾∑𝑎𝑘(𝑡) ⋅ 𝑐−𝑘,

̇

V= 𝑤,

̇𝑎

𝑘(𝑡) + (2𝜋𝑘V𝑖 + 4]𝜋2𝑘2+ 𝑙 (V)) 𝑎𝑘(𝑡) = 𝑙 (V) 𝑏𝑘, 𝑘 ∈ 𝐾,

𝑎𝑘= 0, 𝑘 ∉ 𝐾.

(85) Therefore, if𝑓 ∈ 𝑉, then ∮ 𝑇𝑓 = ∮ 𝑇1𝑓, and given(𝑤0,

V0, 𝑇0) ∈ Yand(𝑤(𝑡),V(𝑡), 𝑇(𝑡)), the solution of (84), we decompose 𝑇0 = 𝑇01 + 𝑇02 and 𝑇(𝑡) = 𝑇1(𝑡) + 𝑇2(𝑡). Then, we consider(𝑤(𝑡),V(𝑡), 𝑇1(𝑡)) ∈ M and it is still a solution of (85). Hence,(𝑤(𝑡),V(𝑡), 𝑇(𝑡))−(𝑤(𝑡),V(𝑡), 𝑇1(𝑡)) =

(0, 0, 𝑇2(𝑡))and the right-hand side is of order𝑒−(4]𝜋2+𝑙 0)𝑡. In

particular, if the set𝐾is finite, then the inertial manifoldM is of finite dimension and the flow inside is equivalent to the finite system of ODEs given by (85). Thus, the theorem is proved.

3.2. The Reduced Subsystem. Under the hypotheses and notations ofTheorem 13, we suppose that

𝑓 (𝑥) = ∑ 𝑘∈𝐽

𝑐𝑘𝑒2𝜋𝑘𝑖𝑥, (86)

with 𝑐𝑘 ̸= 0 for every 𝑘 ∈ 𝐽 ⊂ Z. Then, ∮(𝑇 ⋅ 𝑓) =

𝑘∈𝐾∩𝐽𝑎𝑘(𝑡)𝑐−𝑘. So, the evolution of velocity Vand acceler-ation𝑤depend only on the coefficients of𝑇which belong to the set𝐾 ∩ 𝐽. Note that in (85) the set of equations for𝑎𝑘with

𝑘 ∈ 𝐾∩𝐽together with the equation forVand𝑤is a subsystem of coupled equations denoted by the reduced subsystem.

Thus, we will reduce the asymptotic behavior of the initial system (5) to the dynamics of the reduced explicit system (87) when we consider the relevant modes of temperature𝑎𝑘, 𝑘 ∈

𝐾 ∩ 𝐽. Consider

𝑑𝑤 𝑑𝑡 +

1 𝜀𝑤 +

1

𝜀𝐺 (V)V= 1

𝜀𝑘∈(𝑘∩𝐽)∑ 𝑎𝑘(𝑡) 𝑐−𝑘, 𝑤 (0) = 𝑤0,

𝑑V

𝑑𝑡 = 𝑤, V(0) =V0,

̇𝑎

𝑘(𝑡) + (2𝜋𝑘V𝑖 + 4]𝜋2𝑘2+ 𝑙 (V)) 𝑎𝑘(𝑡) = 𝑙 (V) 𝑏𝑘,

𝑎𝑘(0) = 𝑎𝑘0, 𝑘 ∈ 𝐾 ∩ 𝐽,

(87) where𝑎−𝑘= 𝑎𝑘,𝑏−𝑘= 𝑏𝑘, and𝑐−𝑘= 𝑐𝑘as we consider only the real functions. After solving this, we must solve the equations for𝑘 ∉ 𝐾 ∩ 𝐽which are linear autonomous equations.

Now, we will show the modes in𝑘 ∈ 𝐾∩𝐽that will play an essential role in the dynamics. With the previous notations, we further decompose𝑇1as follows:

𝑇1= 𝜏 + 𝜃, (88)

where 𝜏 is the projection onto the space generated by

{𝑒2𝜋𝑘𝑖𝑥, 𝑘 ∈ 𝐾 ∩ 𝐽}and 𝜃 is the projection onto the space generated by{𝑒2𝜋𝑘𝑥, 𝑘 ∈ 𝐾\𝐽}. We denote by𝑃the projection

𝑃(𝑤,V, 𝑇) = (𝑤,V, 𝜏)and𝑄 = 𝐼 − 𝑃. With these notations and decomposing𝑇𝑎 as𝑇𝑎 = 𝜏𝑇𝑎 + 𝜃𝑇𝑎, (8) and (84) can be decomposed as a system of the form

𝑑𝑤 𝑑𝑡 +

1 𝜀𝑤 = −

1

𝜀𝐺 (V)V+ 1

𝜀∮ (𝜏 + 𝑇2) 𝑓, 𝑤 (0) = 𝑤0, 𝑑V

𝑑𝑡 = 𝑤, V(0) =V0, 𝜕𝜏

𝜕𝑡 +V 𝜕𝜏

𝜕𝑥= 𝑙 (V) (𝜏𝑇𝑎− 𝜏) +] 𝜕2𝜏 𝜕𝑥2, 𝜏 (0, 𝑥) = (𝑇0)𝜏(𝑥) , 𝜕𝜃

𝜕𝑡 +V 𝜕𝜃

𝜕𝑥 = 𝑙 (V) (𝜃𝑇𝑎− 𝜃) +] 𝜕2𝜃 𝜕𝑥2, 𝜃 (0, 𝑥) = (𝑇0)𝜃(𝑥) , 𝜕𝑇2

𝜕𝑡 +V 𝜕𝑇2

𝜕𝑥 = −𝑙 (V) 𝑇2+] 𝜕2𝑇2

𝜕𝑥2 , 𝑇2(0, 𝑥) = 𝑇02(𝑥) .

(89)

Since ∮ 𝜃𝑓 = 0 and setting 𝑇2 = 0, the first four equa-tions give the flow inside the inertial manifold M; that is, they are equivalent to (84) while the first three are the only nonlinearity coupled equations. Therefore, once this subsystem is solved, the other unknowns are determined through linear nonhomogeneous equations.

To make this idea more precise in terms of semigroup and attractors, we proceed as in [14]. We denote by𝑆(𝑡)the semigroup generated by (8) onY:= R2× ̇𝐻per1 (0, 1)and by

𝑆𝑀(𝑡)its restriction to the inertial manifoldM, that is, the semigroup generated by (90). Consider

𝑑𝑤 𝑑𝑡 +

1 𝜀𝑤 = −

1

𝜀𝐺 (V)V+ 1

𝜀∮ 𝑇1𝑓, 𝑤 (0) = 𝑤0, 𝑑V

𝑑𝑡 = 𝑤, V(0) =V0, 𝜕𝑇1

𝜕𝑡 +V 𝜕𝑇1

𝜕𝑥 = 𝑙 (V) (𝑇𝑎− 𝑇1) +]𝜕 2𝑇1 𝜕𝑥2 , 𝑇1(0, 𝑥) = 𝑇01(𝑥) ,

𝜕𝑇2 𝜕𝑡 +V

𝜕𝑇2

𝜕𝑥 = −𝑙 (V) 𝑇2+] 𝜕2𝑇2

𝜕𝑥2 , 𝑇2(0, 𝑥) = 𝑇2

0(𝑥) ,

𝑇2= 0.

(90)

We will find a reduced semigroup on the reduced space

YR := 𝑃(Y), denoted as𝑆𝑅(𝑡), that, in a sense, determines

Figure

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