We prove the existence of a compact, finite dimensional, global attractor for a coupled PDE system comprising a nonlinearly damped semilinear wave equation and a nonlinear system of thermoelastic plate equations, without any mechanical (viscous or structural) dissipation in the plate component. The plate dynamics is modelled following Berger's approach; we investigate both cases when rotational inertia is included into the model and when it is not. A major part in the proof is played by an estimate--known as stabilizability estimate--which shows that the difference of any two trajectories can be exponentially stabilized to zero, modulo a compact perturbation. In particular, this inequality yields bounds for the attractor's fractal dimension which are independent of two key parameters, namely $γ$ and $κ$, the former related to the presence of rotational inertia in the plate model and the latter to the coupling terms. Finally, we show the upper semi-continuity of the attractor with respect to these parameters.
In this paper we study a nonlinear transmission problem for a plate which consists of thermoelastic and isothermal parts. The problem generates a dynamical system in a suitable Hilbert space. Main result is the proof of the asymptotic smoothness of this dynamical system. Also we prove the existence of a compact global attractor in particular cases when the nonlinearity is of Berger type or scalar.
We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in R^n. Well-posedness of solutions reconstructing maximal parabolic regularity in nonlinear thermoelastic plates is established. In addition, exponential decay rates for strong solutions are also shown.
In this paper, we consider generalized thermoelastic plate equations with Fourier's law of heat conduction. By introducing a threshold for decay properties of regularity-loss, we investigate decay estimates of solutions with/without regularity-loss in a framework of weighted $L^1$ spaces. Furthermore, asymptotic profiles of solutions are obtained by using representations of solutions in the Fourier space, which are derived by employing WKB analysis. Next, we study generalized thermoelastic plate equations with additional structural damping, and analysis the influence of structural damping on d…
We study an initial-boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff \& Love-type with parabolic heat conduction due to Fourier, mechanically simply supported and held at the reference temperature on the boundary. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we use barrier techniques to prove the global existence and exponential stability of solutions under a smallness condition on the initial data. It is the first result of this kind established…
We consider the linear thermoelastic plate equations with free boundary conditions in uniform $C^4$-domains, which includes the half-space, bounded and exterior domains. We show that the corresponding operator generates an analytic semigroup in $L^p$-spaces for all $p\in(1,\infty)$ and has maximal $L^q$-$L^p$-regularity on finite time intervals. On bounded $C^4$-domains, we obtain exponential stability.
In this paper we consider a transmission problem for a system of a thermoelastic plate with (or without) rotational inertia term coupled with a membrane with different variants of damping for the plate and/or the membrane. We prove well-posedness of the problem and higher regularity of the solution and study the asymptotic behaviour of the solution, depending on the damping and on the presence of the rotational term.
In this article we study the asymptotic behavior of solutions, in sense of global pullback attractors, of the evolution system $$ \begin{cases} u_{tt} +ηΔ^2 u+a(t)Δθ=f(t,u), & t>τ,\ x\inΩ,\\ θ_t-κΔθ-a(t)Δu_t=0, & t>τ,\ x\inΩ, \end{cases} $$ subject to boundary conditions $$ u=Δu=θ=0,\ t>τ,\ x\in\partialΩ, $$ where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $N\geqslant 2$, which boundary $\partialΩ$ is assumed to be a $\mathcal{C}^4$-hypersurface, $η>0$ and $κ>0$ are constants, $a$ is an Hölder continuous function, $f$ is a dissipative nonlinearity locally Lipschitz in the second variable.
We study dispersive mixed-order systems of pseudodifferential operators in the setting of $L^p$-Sobolev spaces. Under the weak condition of quasi-hyperbolicity, these operators generate a semigroup in the space of tempered distributions. However, if the basic space is a tuple of $L^p$-Sobolev spaces, a strongly continuous semigroup is in many cases only generated if $p=2$ or $n=1$. The results are applied to the linear thermoelastic plate equation inertial term and with Fourier's or Maxwell-Cattaneo's law of heat conduction.
In this paper we study Cauchy problem for thermoelastic plate equations with friction or structural damping in $\mathbb{R}^n$, $n\geq1$, where the heat conduction is modeled by Fourier's law. We explain some qualitative properties of solutions influenced by different damping mechanisms. We show which damping in the model has a dominant influence on smoothing effect, energy estimates, $L^p-L^q$ estimates not necessary on the conjugate line, and on diffusion phenomena. Moreover, we derive asymptotic profiles of solutions in a framework of weighted $L^1$ data. In particular, sharp decay estimates…
In this paper we study the interaction problem between a nonlinear thermoelastic plate and a compressible viscous fluid with the adiabatic constant $γ>12/7$. The existence of a weak solution for this problem is obtained by constructing a time-continuous operator splitting scheme that decouples the fluid and the structure. The fluid sub-problem is given on a fixed reference domain in the arbitrary Lagrangian-Eulerian (ALE) formulation, and the continuity equation is damped on this domain as well. This allows the majority of the analysis to be performed on the fixed reference domain, while the c…
We consider the non-linear thermoelastic plate equation in rectangular domains $Ω$. More precisely, $Ω$ is considered to be given as the Cartesian product of whole or half spaces and a cube. First the linearized equation is treated as an abstract Cauchy problem in $L^p$-spaces. We take advantage of the structure of $Ω$ and apply operator-valued Fourier multiplier results to infer an $\mathcal R$-bounded $\mathcal H^\infty$-calculus. With the help of maximal $L^p$-regularity existence and uniqueness of local real-analytic strong solutions together with analytic dependency on the data is shown.
In the present article, we consider a thermoelastic plate of Reissner-Mindlin-Timoshenko type with the hyperbolic heat conduction arising from Cattaneo's law. In the absense of any additional mechanical dissipations, the system is often not even strongly stable unless restricted to the rotationally symmetric case, etc. We present a well-posedness result for the linear problem under general mixed boundary conditions for the elastic and thermal parts. For the case of a clamped, thermally isolated plate, we show an exponential energy decay rate under a full damping for all elastic variables. Rest…
We consider an initial-boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law giving rise to a 'second sound' effect. We study the local well-posedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev $H^{k}$-spaces. Exploiting the sole source of energy dissipation entering the system throu…
The present work intends to complement the study of the regularity of the solutions of the thermoelastic plate with rotacional forces. The rotational forces involve the spectral fractional Laplacian, with power parameter $τ\in [0,1]$ ( $γ(-Δ)^τu_{tt}$). Previous research regarding regularity showed that, as for the analyticity of the semigroup $S(t)=e^{\mathbb{B}t}$ for the Euler-Bernoulli Plate($τ=0$) model, the first result was established by Liu and Renardy, \cite{LiuR95} in the case of hinged and clamped boundary conditions, for the case $τ=1$ (Plate Kirchoff-Love) Lasiecka and Triggiani s…
Identification of thermal expansion coefficient in a thermoelastic plate from final time-measured displacement
Hanbat National University（汉南国立大学）；Austrian Academy of Sciences（奥地利科学院）；Indian Institute of Space Science and Technology（印度空间科学与技术学院）；Kocaeli university（科贾埃利大学）
This article is concerned with bending plate theory for thermoelastic diffusion materials under Green-Naghdi theory. First, we present the basic equations which characterize the bending of thin thermoelastic diffusion plates for type II and III models. The theory allows for the effect of transverse shear deformation without any shear correction factor, and permits the propagation of waves at a finite speed without energy dissipation for type II model and with energy dissipation for type III model. By the semigroup theory of linear operators, we prove the well-posedness of the both models and t…
The purpose of this paper is to demonstrate how different types of boundary conditions do not impact the asymptotic behaviour of the solutions of thermoelastic wave model. For an initial-boundary value problem associated with this system, we prove a global well-posedness result in a certain topology under appropriate regularity conditions on the data. Further, we show that under particular classes of boundary conditions, the energy associated to the system decays polynomially to zero and not exponentially.
We study the large time behavior of solutions to a linear transmission problem in one space dimension. The problem at hand models a thermoelastic material with second sound confined by a purely elastic one. We shall characterize all equilibrium states of the considered system and prove that every solution approaches one designated equilibrium state with an exponential rate as time goes to infinity. Hereto, we apply methods from the theory of strongly continuous semigroups. In particular, we obtain uniform resolvent bounds for the underlying generator. This removes the largeness assumption of e…
A mathematical formulation of an estimation problem of a cavity inside a three-dimensional thermoelastic body using time domain data is considered. The governing equation of the problem is given by a system of equations in the linear theory of thermoelasticity which is a coupled system of the elastic wave and heat equations. A new version of the enclosure method in the time domain which is originally developed for the classical wave equation is established. For a comparison, the results in the decoupled case are also given.
In this study we consider a coupled system of partial differential equations (PDE's) which describes a certain structural acoustics interaction. One component of this PDE system is a wave equation, which serves to model the interior acoustic wave medium within a given three dimensional chamber $% Ω$. This acoustic wave equation is coupled on a boundary interface ($% Γ_{0}$) to a two dimensional system of thermoelasticity: this thermoelastic PDE comprises a structural beam or plate equation, which governs the vibrations of flexible wall portion $Γ_{0}$ of the chamber $Ω$; the elastic dynamics i…
We consider the transmission problem for a coupled system of undamped and structurally damped plate equations in two sufficiently smooth and bounded subdomains. It is shown that, independently of the size of the damped part, the damping is strong enough to produce uniform exponential decay of the energy of the coupled system.
In this paper, we consider a weakly coupled system consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain. By assuming a more general type of relaxation functions, we establish explicit and general decay rate results, using the multiplier method and some properties of the convex functions.
In this paper we consider a mathematical model for photoacoustic imaging which takes into account attenuation due to thermodynamic dissipation. The propagation of acoustic (compressional) waves is governed by a scalar wave equation coupled to the heat equation for the excess temperature. We seek to recover the initial acoustic profile from knowledge of acoustic measurements at the boundary. We recognize that this inverse problem is a special case of boundary observability for a thermoelastic system. This leads to the use of control/observability tools to prove the unique and stable recovery of…
We consider a coupled linear system describing a thermoviscoelastic plate with hereditary effects. The system consists of a hyperbolic integrodifferential equation, governing the temperature, which is linearly coupled with the partial differential equation ruling the evolution of the vertical deflection, presenting a convolution term accounting for memory effects. It is also assumed that the thermal power contains a memory term characterized by a relaxation kernel. We prove that the system is exponentially stable and we obtain a closeness estimate between the system with memory effects and the…
This paper is the second part of a work devoted to the modelling of thin elastic plates with small, periodically distributed piezoelectric inclusions. We consider the equations of linear elasticity coupled with the electrostatic equation, with various kinds of electric boundary conditions. We derive the corresponding effective models when the thickness $a$ of the plate and the characteristic dimension $\varepsilon $ of the inclusions tend\ together to zero, in the two following situations: first when $a \simeq \varepsilon$, second when $a/\varepsilon$ tends to zero.
We consider a transmission problem where a structurally damped plate equation is coupled with a damped or undamped wave equation by transmission conditions. We show that exponential stability holds in the damped-damped situation and polynomial stability (but no exponential stability) holds in the damped-undamped case. Additionally, we show that the solutions first defined by the weak formulation, in fact have higher Sobolev space regularity.
We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. The main peculiarity of the model is the assumption that the transversal displacements of the plate are negligible relative to in-plane displacements. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. We also show that the corresponding linearized sys…
A no-contact result for a plate-fluid interaction system in dimension three
University of Zagreb Faculty of Electrical Engineering and Computing（萨格勒布大学电气工程和计算学院）；Department of Mathematics, University of Southern California（南加州大学数学系）；Department of Mathematics, University of California Los Angeles（加州大学洛杉矶分校数学系）；University of Zagreb, Faculty of Science, Department of Mathematics（萨格勒布大学理学院数学系）
The regularity of the coupled system between an electrical network with fractional dissipation and a plate equation with fractional inertial rotational
We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and a classical (nonlinear) elastic plate equation for transversal displacement on a flexible flat part of the boundary. We show that this problem generates a semiflow on appropriate phase space. Our main result states the existence of a compact finite-dimensional global attractor for this semiflow. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stab…
We consider the dynamics of an electrostatically actuated thin elastic plate being clamped at its boundary above a rigid plate. The model includes the harmonic electrostatic potential in the three-dimensional time-varying region between the plates along with a fourth-order semilinear parabolic equation for the elastic plate deflection which is coupled to the square of the gradient trace of the electrostatic potential on this plate. The strength of the coupling is tuned by a parameter $λ$ proportional to the square of the applied voltage. We prove that this free boundary problem is locally well…
Under the action of a time-periodic external forces we prove the existence of at least one time-periodic weak solution for the interaction between a three-dimensional incompressible fluid, governed by the Navier- Stokes equation and a two dimensional elastic plate. The challenge is that the Eulerian domain for the fluid changes in time and is a part of the solution. We introduce a two fixed-point methodology: First we construct a time-periodic solutions for a given variable time-periodic geometry. Then in a second step a (set-valued) fixed point is performed w.r.t.\ the geometry of the domain.…
In this paper we investigate the limit behavior of the solution to quasi-static Biot's equations in thin poroelastic plates as the thickness tends to zero. We choose Terzaghi's time corresponding to the plate thickness and obtain the strong convergence of the three-dimensional solid displacement, fluid pressure and total poroelastic stress to the solution of the new class of plate equations. In the new equations the in-plane stretching is described by the 2D Navier's linear elasticity equations, with elastic moduli depending on Gassmann's and Biot's coefficients. The bending equation is couple…
