Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions

Irena Lasiecka,To Fu Ma,Rodrigo Nunes Monteiro

doi:10.3934/dcdsb.2018141

Abstract

This paper is concerned with long-time dynamics of a full von Karman system subject to nonlinear thermal coupling and free boundary conditions. In contrast with scalar von Karman system, vectorial full von Karman system accounts for both vertical and in plane displacements. The corresponding PDE is of critical interest in flow structure interactions where nonlinear plate/shell dynamics interacts with perturbed flows [vicid or invicid] [8,9,15]. In this paper it is shown that the system admits a global attractor which is also smooth and of finite fractal dimension. The above result is shown to hold for plates without regularizing effects of rotational inertia and without any mechanical dissipation imposed on vertical displacements. This is in contrast with the literature on this topic [15] and references therein. In order to handle highly supercritical nature of the von Karman nonlinearities, new results on 'hidden' trace regularity generated by thermal effects are exploited. These lead to asymptotic compensated compactness of trajectories which then allows to use newly developed tools pertaining to quasi stable dynamical systems [8].

Highlights

This paper is concerned with long-time behavior and theory of global attractors associated with dynamic system of nonlinear elasticity modeled by a full vectorial von Karman system subject to thermal effects
It has been observed that thermal dissipation provides substantial damping mechanism for the oscillations so that there may be no need for mechanical dissipation
While this kind of result is to be expected for the dynamics with an overall smoothing effect, it is much less expected in hyperbolic type of models without strong mechanical dissipation and with highly unbounded nonlinear effects

Summary

Introduction

This paper is concerned with long-time behavior and theory of global attractors associated with dynamic system of nonlinear elasticity modeled by a full vectorial von Karman system subject to thermal effects. This system describes nonlinear oscillations in a plate dynamics which account for both vertical and in plane displacements - denoted respectively by w and u = (u1, u2) –as well as the averaged thermal stresses φ and θ affecting each of these displacements [17, 18, 34, 35]. This is pronounced for plate/shell models without the regularizing effects of

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Conclusion