Nonlinear Thermoelastic Plate Research Articles

On the existence theory for the nonlinear thermoelastic plate equation

In this paper, we analyze the nonlinear thermoelastic plates, with Fourier heat conduction, and consider a polynomial-type nonlinearity. We first develop a theoretical analysis of the corresponding linear system to derive time decay estimates in L ∞ ( R n ) and H s ( R n ) . Then, using that set of decay estimates and controlling the nonlinearity, we prove the existence and uniqueness of local solutions with initial data ( u ( 0 ) , u t ( 0 ) , θ ( 0 ) ) = ( u 0 , Δ u 1 , Δ θ 1 ) , with u 0 ∈ H s , and u 1 , θ 1 ∈ H s + 1 , for s > n 2 + 1 .

Stability of a nonlinear thermoelastic plate

The problem of loss of stability of a circular plate under lateral compression in an inhomogeneous temperature field is considered. The theory of superposition of a small deformation on a finite one is used. A similar approach to the study of the equilibrium bifurcation of nonlinear thermoelastic bodies was used in the following works.

Mathematical modeling of the effect of an insulating stiffener on a nonlinear thermo-elastic plate

This paper deals with the mathematical modeling of the behavior of a reinforced rectangular thermo-elastic plate with a thin insulating stiffener. We use a variational asymptotic analysis, with respect to the thickness of the inserted body, in order to identify limit models that reflect its effect on the plate. We carry out a mathematical modeling for a stiffener of high rigidity and a stiffener of moderate rigidity.

Weak solution to the incompressible viscous fluid and a thermoelastic plate interaction problem in 3D

In this paper we deal with a nonlinear interaction problem between an incompressible viscous fluid and a nonlinear thermoelastic plate. The nonlinearity in the plate equation corresponds to nonlinear elastic force in various physically relevant semilinear and quasilinear plate models. We prove the existence of a weak solution for this problem by constructing a hybrid approximation scheme that, via operator splitting, decouples the system into two sub-problems, one piece-wise stationary for the fluid and one time-continuous and in a finite basis for the structure. To prove the convergence of the approximate quasilinear elastic force, we develop a compensated compactness method that relies on the maximal monotonicity property of this nonlinear function.

On some nonlinear problem for the thermoplate equations

In this paper, we prove the local and global well-posedness of some nonlinear thermoelastic plate equations with Dirichlet boundary conditions. The main tool for proving the local well-posedness is the maximal \begin{document}$ L_p $\end{document} - \begin{document}$ L_q $\end{document} regularity theorem for the linearized equations, and the main tool for proving the global well-posedness is the exponential stability of \begin{document}$ C_0 $\end{document} analytic semigroup associated with linear thermoelastic plate equations with Dirichlet boundary conditions.

Nonlinear thermoelastic plate equations – Global existence and decay rates for the Cauchy problem

We consider the Cauchy problem in R n for quasilinear thermoelastic Kirchhoff-type plate equations where the heat conduction is modeled by either the Cattaneo law or by the Fourier law. Additionally, we take into account possible inertial effects. Considering nonlinearities which are of fourth-order in the space variable, we deal with a quasilinear system which triggers difficulties typical for nonlinear Schrödinger equations. The different models considered are systems of mixed type comparable to Schrödinger–parabolic or Schrödinger–hyperbolic systems. The main task consists in proving sophisticated a priori estimates leading to obtaining the global existence of solutions for small data, neither known nor expected for the Cauchy problem in pure plate theory nor available before for the coupled system under investigation, where only special cases (bounded domains within with analytic semigroup setting, or the Cauchy problem with semilinear nonlinearities) had been treated before.

Local strong solutions for the non-linear thermoelastic plate equation on rectangular domains in L p -spaces

We consider the non-linear thermoelastic plate equation in rectangular domains $\Omega$. More precisely, $\Omega$ 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 $\Omega$ 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.

Initial-boundary value problems for a class of nonlinear thermoelastic plate equations

This paper studies initial-boundary value problems for a class of nonlinear thermoelastic plate equations. Under some certain initial data and boundary conditions, it obtains an existence and uniqueness theorem of global weak solutions of the nonlinear thermoelstic plate equations, by means of the Galerkin method. Moreover, it also proves the existence of strong and classical solutions.

Existence and Exponential Decay of Solutions to a Quasilinear Thermoelastic Plate System

We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in Rn, n ≤ 3. Existence of finite energy solutions describing the dynamics of a nonlinear thermoelastic plate is established. In addition asymptotic long time behavior of weak solutions is discussed. It is shown that finite energy solutions decay exponentially to zero with the rate depending only on the (finite energy) size of initial conditions. The proofs are based on methods of weak compactness along with nonlocal partial differential operator multipliers which supply the sought after “recovery” inequalities. Regularity of solutions is also discussed by exploiting the underlying analyticity of the linearized semigroup along with a related maximal parabolic regularity [1, 16, 44].

Concerning the exact null controllability of a nonlinear thermoelastic plate

Abstract In this paper, we consider the null controllability problem for a partial differential equation (PDE), which models thin von Karman plate dynamics in the absence of rotational inertia. Using the fact that thelinearization of this thermoelastic PDE is null controllable, we proceed to show that the full von Karman dynamics are locally null controllable. In obtaining the relevant observability inequalities, key use is made of the abstract multiplier which is apparently intrinsic to control theoretic studies of thermoelastic plates.

On the Energy Decay of a Linear Thermoelastic Bar and Plate

It is shown that the energy of a thermoelastic bar and plate decays exponentially fast. The energy method, combined with a multiplier technique and compactness property, is used.