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This work is concerned with limit behavior of the Navier–Stokes–Voigt equation with degenerate white noise in a 3D thin domain. Although the individual solutions may be chaotic in fluid dynamics, the stationary statistical solutions are essential to capture complex dynamical behaviors in the view of statistic. We therefore prove that the stationary statistical solution of the system converges weakly to the counterpart of the 2D stochastic Euler equation as the viscosity, the elasticity parameter and the thickness of the thin domain tend to zero. [ABSTRACT FROM AUTHOR]

In this paper, we derive the effective model describing a thin-domain flow with permeable boundary through which the fluid is injected into the domain. We start with incompressible Stokes system and perform the rigorous asymptotic analysis. Choosing the appropriate scaling for the injection leads to a compressible effective model. In this paper, we derive the effective model describing a thin-domain flow with permeable boundary through which the fluid is injected into the domain. We start with incompressible Stokes system and perform the rigorous asymptotic analysis. Choosing the appropriate scaling for the injection leads to a compressible effective model. [ABSTRACT FROM AUTHOR]

This study deals with the limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays and multiplicative noise on unbounded thin domains. We first prove the existence and uniqueness of pullback tempered random attractors for the systems and then establish the upper semicontinuity of these attractors when the thin domains collapse onto R{\mathbb{R}}.

This paper deals with the limiting behavior of invariant measures of stochastic reaction–diffusion equations on thin domains. We first show the existence of invariant measures of the stochastic equations in a bounded domain in ℝ n + 1 which can be viewed as a perturbation of a bounded domain in ℝ n . We then prove that the set of all invariant measures of the perturbed equations is tight and any limit of invariant measures of the perturbed systems must be an invariant measure of the limiting system when the (n + 1) -dimensional thin domains collapses onto an n -dimensional domain. [ABSTRACT FROM AUTHOR]

This paper is concerned with pullback measure attractors of the non-autonomous stochastic reaction-diffusion equations defined in thin domains. We first prove the existence and uniqueness of pullback measure attractors for the inhomogeneous Markov process associated with the stochastic equations. We also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback measure attractors. Then we establish the upper semi-continuity of these attractors as a family of thin domains collapses into a lower-dimensional domain. [ABSTRACT FROM AUTHOR]

This work is concerned with the modified stochastic Swift–Hohenberg equation in a 3D thin domain. Although the diffusion motion of molecules is irregular with the interference of the film-fluid fluctuation, the invariant measure on the trajectory space reveals delicate transition of the dynamical behavior when the interior forces change. We therefore prove that the invariant measure of the system converges weakly to the unique counterpart of the stochastic Swift–Hohenberg equation in a 2D bounded domain with a concrete convergence rate, as the modified parameter and the thickness of the thin domain tend to zero. Furthermore, we address that the smooth density of the limit invariant measure fulfills a Fokker–Planck equation. [ABSTRACT FROM AUTHOR]

We prove the existence of global in time strong solutions of incompressible flow equations through isotropic granular (unconsolidated) porous media in thin domains of the form $ \Omega _\varepsilon := \mathbb {R}^{2} \times (0, \varepsilon) $ Ω ϵ := R 2 × (0 , ϵ) with $ \varepsilon \in (0,1] $ ϵ ∈ (0 , 1 ]. Also, we demonstrate that as $ \varepsilon \to 0^+ $ ϵ → 0 + the fluid velocity tends to vanish away from the initial time. [ABSTRACT FROM AUTHOR]

In this work, we present a lower-dimensional model for flow and transport problems in thin domains with rough walls. The full-order model is given for a fully resolved geometry, wherein we consider Stokes flow and a time-dependent diffusion–convection equation with inlet and outlet boundary conditions and zero-flux boundary conditions for both the flow and transport problems on domain walls. Generally, discretizations of a full-order model by classical numerical schemes result in very large discrete problems, which are computationally expensive given that sufficiently fine grids are needed for the approximation. To construct a computationally efficient numerical method, we propose a model-order-reduction numerical technique to reduce the full-order model to a lower-dimensional model. The construction of the lower-dimensional model for the flow and the transport problem is based on the finite volume method and the concept of numerical averaging. Numerical results are presented for three test geometries with varying roughness of walls and thickness of the two-dimensional domain to show the accuracy and applicability of the proposed scheme. In our numerical simulations, we use solutions obtained from the finite element method on a fine grid that can resolve the complex geometry at the grid level as the reference solution to the problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the flow of a non-Newtonian fluid through a thin three-dimensional domain with an oscillating boundary. Starting from a nonlinear coupled system describing a non-isothermal regime of the flow, we rigorously derive three different effective models, depending on the period of the boundary roughness. We employ a homogenization technique based on the adaption of the unfolding method and deduce the roughness-induced effects on the fluid flow. [ABSTRACT FROM AUTHOR]

In this paper, we perform the fast rotation limit ε → 0 + of the density-dependent incompressible Navier–Stokes–Coriolis system in a thin strip Ω ε : = R 2 × - ℓ ε , ℓ ε , where ε ∈ 0 , 1 is the size of the Rossby number and ℓ ε > 0 for any ε > 0 . By letting ℓ ε ⟶ 0 + for ε → 0 + and considering Navier-slip boundary conditions at the boundary of Ω ε , we give a rigorous justification of the phenomenon of the Ekman pumping in the context of non-homogeneous fluids. With respect to previous studies (performed for flows of contant density and for compressible fluids), our approach has the advantage of circumventing the complicated analysis of boundary layers. To the best of our knowledge, this is the first study dealing with the asymptotic analysis of fast rotating incompressible fluids with variable density in a 3-D setting. In this respect, we remark that the case ℓ ε ⩾ ℓ > 0 for all ε > 0 remains largely open at present. [ABSTRACT FROM AUTHOR]

This paper is concerned with the limiting behavior of a stochastic integro-differential equation driven by additive noise defined on thin domains. We prove the existence and uniqueness of random attractors for the equation in an (n + 1) -dimensional narrow domain. We also establish the upper-semicontinuity of these attractors when a family of (n + 1) -dimensional thin domains collapses onto an n-dimensional domain. The main difficulty of this paper is the non-compactness of the generated RDS based on the fact that the memory term includes the whole past history of the phenomenon. To solve this, a splitting method is employed to prove the asymptotic compactness. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the thermomicropolar fluid flow through a thin channel with rough boundary. The flow is governed by the prescribed pressure drop between the channel's ends and the heat exchange through the rough wall is allowed. Depending on the limit of the ratio between channel's thickness and the wavelength of the roughness, we rigorously derive different asymptotic models clearly showing the roughness‐induced effects on the average velocity and microrotation. To accomplish that, we employ the adaptation of the unfolding method to a thin‐domain setting. [ABSTRACT FROM AUTHOR]

This paper is concerned with upper semicontinuity of the family of random attractors associated with non-autonomous stochastic parabolic problems dominated by p-Laplacian operator on thin domains in which the domain collapses onto lower dimensional domain. The existence and uniqueness of random attractors for the equations are strictly proved on an (n + 1)-dimensional thin domain, and the upper semicontinuity of these attractors are established when an (n + 1)-dimensional thin domain collapses onto an n-dimensional domain. [ABSTRACT FROM AUTHOR]

In this work, we present a lower-dimensional model for flow and transport problems in thin domains with rough walls. The full-order model is given for a fully resolved geometry, wherein we consider Stokes flow and a time-dependent diffusion–convection equation with inlet and outlet boundary conditions and zero-flux boundary conditions for both the flow and transport problems on domain walls. Generally, discretizations of a full-order model by classical numerical schemes result in very large discrete problems, which are computationally expensive given that sufficiently fine grids are needed for the approximation. To construct a computationally efficient numerical method, we propose a model-order-reduction numerical technique to reduce the full-order model to a lower-dimensional model. The construction of the lower-dimensional model for the flow and the transport problem is based on the finite volume method and the concept of numerical averaging. Numerical results are presented for three test geometries with varying roughness of walls and thickness of the two-dimensional domain to show the accuracy and applicability of the proposed scheme. In our numerical simulations, we use solutions obtained from the finite element method on a fine grid that can resolve the complex geometry at the grid level as the reference solution to the problem.

We prove the global existence of strong solutions to the 2D compressible isothermal Navier–Stokes equations with relatively large initial data and the initial density bounded away from zero on periodic thin domains Ω δ = T × δ T , δ > 0 , when the thickness δ is sufficiently small. [ABSTRACT FROM AUTHOR]

We study the homogenization of a double porosity diffusion problem in a thin highly heterogeneous composite medium formed by two materials separated by an imperfect interface, where the solution and its flux exhibit jumps. By applying homogenization techniques specific to the thin periodic domain under study, one derives the limit problem. [ABSTRACT FROM AUTHOR]

We investigate the limiting behavior of dynamics of non-autonomous stochastic FitzHugh–Nagumo equations driven by a nonlinear multiplicative colored noise on unbounded thin domains. We first establish the existence and uniqueness of random attractors for the equations on the thin domains and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain. [ABSTRACT FROM AUTHOR]

In this paper we analyze the asymptotic behavior of a control problem set by a convection-reaction-diffusion equation with mixed boundary conditions and defined in a tubular thin domain with rough boundary. The control term acts on a subset of the rough boundary where a Robin-type boundary condition and a catalyzed reaction mechanism are set. The reaction mechanism depends on a parameter α ∈ R. Such parameter establishes different regimes which also depend on the profile and geometry of the tube defined by a periodic function g : R 2 ↦ R. We see that, if ∂ 2 g is not null (that is, when g really depends on the second variable), then three regimes with respect to α are established: α < 2 , α = 2 (the critical value) and α > 2. On the other hand, if ∂ 2 g ≡ 0 , similar regimes are obtained but now with a different critical value. Indeed, if ∂ 2 g ≡ 0 , then the critical value is achieved at α = 1. For each one of these six regimes, we obtain the asymptotic behavior of the control problem when the cylindrical thin domain degenerates to the unit interval. We show that the problem is asymptotically controllable just when α assumes the critical values. Our analysis is mainly performed using the periodic unfolding method adapted to cylindrical coordinates in R 3. [ABSTRACT FROM AUTHOR]

In this paper, we study stochastic partial differential equations driven by nonlinear multiplicative colored noise in varying phase spaces. We establish the C 1 , ν -convergence of center manifolds of these equations. Finally, we apply the results to the stochastic reaction-diffusion equations driven by nonlinear multiplicative colored noise on thin domain. [ABSTRACT FROM AUTHOR]

In this article, we prove global existence of classical solutions to the incompressible isotropic Hookean elastodynamics in three-dimensional thin domain Ωδ = R² × [0, δ] with periodic boundary condition. This system essentially consists of two-dimensional quasilinear wave-type equations. Following the classical vector field theory and the generalized energy method of Klainerman, we introduce the anisotropic weighted Sobolev-type inequalities, the anisotropic generalized energy, and the anisotropic weighted L² norm adapted to the thin domain. The main issue in the anisotropic generalized energy estimate is that the pressure gradient brings an extra singular δ-1 factor arising from the δ²3 derivative. Based on the inherent cancellation structure, we introduce an auxiliary anisotropic generalized energy to overcome this difficulty. The ghost weight technique introduced by Alinhac [Invent. Math., 145 (2001), pp. 597--618] and the strong null condition introduced by Lei [Comm. Pure Appl. Math., 69 (2016), pp. 2072--2106] play important roles for temporal decay estimates. [ABSTRACT FROM AUTHOR]

We consider the compressible Euler system describing the motion of an ideal fluid confined to a straight layer Ω δ = (0 , δ) × R 2 , δ > 0 . In the framework of dissipative measure-valued solutions, we show the convergence to the strong solution of the 2D incompressible Euler system when the Mach number tends to zero and δ → 0 . [ABSTRACT FROM AUTHOR]

This paper is devoted to bi-spatial random attractors of the stochastic Fitzhugh-Nagumo equations with additive noise on thin domains when the terminate space is the Sobolev space. We first established the existence of random attractor on regular space and then show that the upper semi-continuity of these attractors under the Sobolev norm when a family of (n+1)-dimensional thin domains degenerates onto an nn-dimensional domain. [ABSTRACT FROM AUTHOR]

We consider an incompressible Bingham flow in a thin domain with rough boundary, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. In mathematical terms, this problem is described by non linear variational inequalities over domains where a small parameter ϵ denotes the thickness of the domain and the roughness periodicity of the boundary. By using an adapted linear unfolding operator we perform a detailed analysis of the asymptotic behavior of the Bingham flow when ϵ tends to zero. We obtain the homogenized limit problem for the velocity and the pressure, which preserves the nonlinear character of the flow, and study the effects of the microstructure in the corresponding effective equations. Finally, we give the interpretation of the limit problem in terms of a non linear Darcy law. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a convection–diffusion–reaction problem in a thin domain endowed with the Robin-type boundary condition describing the reaction catalyzed by the upper wall. Motivated by the microfluidic applications, we allow the oscillating behavior of the upper boundary and analyze the resonant case where the amplitude and period of the oscillation have the same small order as the domain's thickness. Depending on the magnitude of the reaction mechanism, we rigorously derive three different asymptotic models via the unfolding operator method. In particular, we identify the critical case in which the effects of the domain's geometry and all physically relevant processes become balanced. [ABSTRACT FROM AUTHOR]

We study the magnetic Laplacian and the Ginzburg–Landau functional in a thin planar, smooth, tubular domain and with a uniform applied magnetic field. We provide counterexamples to strong diamagnetism, and as a consequence, we prove that the transition from the superconducting to the normal state is non-monotone. In some nonlinear regime, we determine the structure of the order parameter and compute the super-current along the boundary of the sample. Our results are in agreement with what was observed in the Little–Parks experiment, for a thin cylindrical sample. [ABSTRACT FROM AUTHOR]

We study spiral waves in a mathematical model of a nonlinear optical system with a feedback loop. Starting from a delayed scalar diffusion equation in a thin annulus with oblique derivative boundary conditions, we shrink the annulus and derive the limiting equation on a circle. Based on the explicitly constructed normal form of the Hopf bifurcation for the one-dimensional delayed scalar diffusion equation, we make predictions about the existence and stability of two-dimensional spirals that we verify in direct numerical simulations, observing pulsating and rotating spiral waves. [ABSTRACT FROM AUTHOR]

This paper deals with the dynamical behaviour of a stochastic integro-differential equation driven by additive noise defined on thin domains. We prove the existence and uniqueness of random attractors for the equation in an (n + 1) -dimensional narrow domain. Due to the fact that the memory term includes the whole past history of the phenomenon, we are not able to prove compactness of the generated RDS, but its asymptotic compactness can be proved by the splitting method. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic behavior of a class of non-autonomous stochastic FitzHugh–Nagumo systems driven by additive white noise on unbounded thin domains. For this aim, we first show the existence and uniqueness of random attractors for the considered equations and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain. [ABSTRACT FROM AUTHOR]

In this work we consider higher dimensional thin domains with the property that both boundaries, bottom and top, present oscillations of weak type. We consider the Laplace operator with Neumann boundary conditions and analyze the behavior of the solutions as the thin domain shrinks to a fixed domain ω⊂Rn. We obtain the convergence of the resolvent of the elliptic operators in the sense of compact convergence of operators, which in particular implies the convergence of the spectra. This convergence of the resolvent operators will allow us to conclude the global dynamics, in terms of the global attractors of a reaction diffusion equation in the thin domains. In particular, we show the upper semicontinuity of the attractors and stationary states. An important case treated is the case of a quasiperiodic situation, where the bottom and top oscillations are periodic but with period rationally independent. [ABSTRACT FROM AUTHOR]

By using dimension reduction and homogenization techniques, we study the steady flow of an incompresible viscoplastic Bingham fluid in a thin porous medium. A main feature of our study is the dependence of the yield stress of the Bingham fluid on the small parameters describing the geometry of the thin porous medium under consideration. Three different problems are obtained in the limit when the small parameter tends to zero, following the ratio between the height of the porous medium and the relative dimension of its periodically distributed pores. We conclude with the interpretation of these limit problems, which all preserve the nonlinear character of the flow. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the effects of a small boundary perturbation on the non-isothermal fluid flow through a thin channel filled with porous medium. Starting from the Darcy–Brinkman–Boussinesq system and employing asymptotic analysis, we derive a higher-order effective model given by the explicit formulae. To observe the effects of the boundary irregularities, we numerically visualize the asymptotic approximation for the temperature, whereas the justification and the order of accuracy of the model is provided by the theoretical error analysis. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of pressure-driven Stokes flow in a thin domain. By letting the thickness of the domain tend to zero we derive a generalized form of the classical Reynolds–Poiseuille law, i.e. the limit velocity field is a linear function of the pressure gradient. By prescribing the external pressure as a normal stress condition, we recover a Dirichlet condition for the limit pressure. In contrast, a Dirichlet condition for the velocity yields a Neumann condition for the limit pressure. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the flow through a thin corrugated domain filled with fluid-saturated porous medium. The porous medium flow is described by the nonlinear Darcy–Lapwood–Brinkman model acknowledging the viscous shear and the inertial effects. The thickness of the domain is assumed to be of the same small order ε as the period of the oscillating boundaries. Depending on the magnitude of the permeability with respect to ε , we rigorously derive different asymptotic models and compare the results with the non-oscillatory case. We employ a homogenization technique based on the adaption of the unfolding method and deduce the influence of the porous structure and boundary oscillations on the effective flow. [ABSTRACT FROM AUTHOR]

This article is concerned with the asymptotic behavior of solutions of a class of non-autonomous stochastic Fitzhugh-Nagumo systems driven by multiplicative white noise on unbounded thin domains. The existence and uniqueness of tempered pullback random attractors are proved for the systems defined on (n + 1) − dimensional unbounded narrow domains. We also establish the upper semicontinuity of these attractors when a family of (n + 1) − dimensional unbounded thin domains collapses onto an n −dimensional unbounded domain. A cut-off method and a decomposition technique are employed to derive the pullback asymptotic compactness of solutions in order to overcome the difficulties caused by the non-compactness of Sobolev embeddings on unbounded domains as well as the lack of regularity of one component of the solutions. [ABSTRACT FROM AUTHOR]

The paper is divided in two parts. In the first one, we consider a linear elliptic partial differential equation, or system, in an unbounded cylinder with a nonpositive term of zero order. We analyze the existence and the dimension of the space of solutions, when we look for solutions which tend exponentially to zero at infinity and for oscillatory solutions which have a potential growth at infinity. This type of problems appears when we study the influence of boundary conditions in homogenization, singular perturbation problems and reduction of dimension problems. As an example, we obtain a corrector result for a wave problem in a thin beam imposing initial and boundary conditions. It was known that the initial conditions produce almost periodic oscillations in time. Here, we show that the boundary conditions also produce almost periodic oscillations in the direction of the axis of the beam. [ABSTRACT FROM AUTHOR]

A system of stochastic retarded reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing on thin domains is considered. Relations between the asymptotic behavior for the stochastic retarded equations defined on thin domains in Rn+1 and an equation on a domain in Rn are investigated. We first show the existence and uniqueness of tempered random attractors for these equations. Then, we analyze convergence properties of the solutions as well as the attractors. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic behavior for non-autonomous stochastic complex Ginzburg-Landau equations with multiplicative noise on thin domains. For this aim, we first show that the existence and uniqueness of random attractors for the considered equations and the limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse onto an interval. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to theoretically investigate 2D wave patterns in a nonlinear optical system with diffractive feedback. We consider a delayed functional differential diffusion equation on a thin annulus with Neumann boundary conditions. To study pattern formation phenomena, we construct the Faria normal form of a Hopf bifurcation, which is necessarily degenerate because the equation exhibits O(2) symmetry. The coefficients of the normal form determine the excitation of rotating or standing waves with the prescribed spatio-temporal characteristics, but these coefficients can be expressed only implicitly. However, for sufficiently thin annuli, the equation corresponds to a limit 1D model on a circle, which was carefully analysed in our previous papers and whose normal form was explicitly calculated. 2D wave stability predictions that we make based on the normal form of the limit 1D model are in good agreement with direct numerical simulations of the 2D model. To accelerate computation, we elaborated an efficient fast finite Hankel transform algorithm for thin annular domains. [ABSTRACT FROM AUTHOR]

This paper is concerned with the regular random dynamics for the reaction-diffusion equation defined on a thin domain and perturbed by rough noise, where the usual Winner process is replaced by a general stochastic process satisfied the basic convergence. A bi-spatial attractor is obtained when the non-initial space is p-times Lebesgue space or Sobolev space. The measurability of the solution operator is proved, which leads to the measurability of the attractor in both state spaces. Finally, the upper semi-continuity of attractors under the p-norm is established when the narrow domain degenerates onto a lower dimensional set. Both methods of symbolical truncation and spectral decomposition provide all required uniform estimates in both Lebesgue and Sobolev spaces. [ABSTRACT FROM AUTHOR]

We study the 3-D compressible barotropic radiation fluid dynamics system describing the motion of the compressible rotating viscous fluid with gravitation and radiation confined to a straight layer Ωϵ=ω×(0,ϵ), where ω is a 2-D domain. We show that weak solutions in the 3-D domain converge to the strong solution of—the rotating 2-D Navier-Stokes-Poisson system with radiation in ω as ϵ→0 for all times less than the maximal life time of the strong solution of the 2-D system when the Froude number is small (Fr=O(ϵ)),—the rotating pure 2-D Navier-Stokes system with radiation in ω as ϵ→0 when Fr=O(1). [ABSTRACT FROM AUTHOR]

Motivated by the applications from chemical engineering, in this paper we present a rigorous derivation of the effective model describing the convection-diffusion-reaction process in a thin domain. The problem is described by a nonlinear elliptic problem with nonlinearity appearing both in the governing equation as well in the boundary condition. Using rigorous analysis in appropriate functional setting, we show that the starting singular problem posed in a two-dimensional region can be approximated with one which is regular, one-dimensional and captures the effects of all physical processes which are relevant for the original problem. [ABSTRACT FROM AUTHOR]

In this work we consider a dissipative reaction–diffusion equation in a d -dimensional thin domain shrinking to a one dimensional segment and obtain good rates for the convergence of the attractors. To accomplish this, we use estimates on the convergence of inertial manifolds as developed previously in [6] and Shadowing theory. [ABSTRACT FROM AUTHOR]

Many physical systems are well described on domains which are relatively large in some directions but relatively thin in other directions. In this scenario, we typically expect the system to have emergent structures that vary slowly over the large dimensions. For practical mathematical modelling of such systems we require efficient and accurate methodologies for reducing the dimension of the original system and extracting the emergent dynamics. Common mathematical approximations for determining the emergent dynamics often rely on self-consistency arguments or limits as the aspect ratio of the 'large' and 'thin' dimensions becomes unphysically infinite. Here we build on a new approach, previously establish for systems which are large in only one dimension, which analyses the dynamics at each crosssection of the domain with a rigorous multivariate Taylor series. Then centre manifold theory supports the local modelling of the system's emergent dynamics with coupling to neighbouring cross-sections treated as a non-autonomous forcing. The union over all cross-sections then provides powerful support for the existence and emergence of a centre manifold model global in the large finite domain. Quantitative error estimates are determined from the interactions between the cross-section coupling and both fast and slow dynamics. Two examples provide practical details of our methodology. The approach developed here may be used to quantify the accuracy of known approximations, to extend such approximations to mixed order modelling and to open previously intractable modelling issues to new tools and insights. [ABSTRACT FROM AUTHOR]

This paper deals with the limiting behavior of stochastic reaction–diffusion equations driven by multiplicative noise and deterministic non-autonomous terms defined on thin domains. We first prove the existence, uniqueness and periodicity of pullback tempered random attractors for the equations in an ( n + 1 ) -dimensional narrow domain, and then establish the upper semicontinuity of these attractors when a family of ( n + 1 ) -dimensional thin domains collapses onto an n -dimensional domain. [ABSTRACT FROM AUTHOR]