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

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2018, Volume 23, Issue 3: 1037-1072. Doi: 10.3934/dcdsb.2018141

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

1.
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA and IBS-Polish Academy of Sciences, Warsaw, Poland
2.
Institute of Mathematical and Computer Sciences, University of São Paulo, 13566-590 São Carlos, SP, Brazil

* Corresponding author: Irena Lasiecka
* Corresponding author: Irena Lasiecka

Received: December 31, 2016

Revised: May 31, 2017

Early access: February 2018

Published: June 2018

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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].

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Mathematics Subject Classification: Primary: 35B41; Secondary: 74K20.

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