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Attractors and Long Time Behavior of von Karman Thermoelastic Plates

Attractors and Long Time Behavior of von Karman Thermoelastic Plates This paper undertakes a study of asymptotic behavior of solutions corresponding to von Karman thermoelastic plates. A distinct feature of the work is that the model considered has no added dissipation —particularly mechanical dissipation typically added to plate equation when long time-behavior is considered. Thus, the model consists of undamped oscillatory plate equation strongly coupled with heat equation. Nevertheless we are able to show that the ultimate (asymptotic) behavior of the von Karman evolution is described by finite dimensional global attractor. In addition, the obtained estimate for the dimension and the size of the attractor are independent of the rotational inertia parameter γ and heat/thermal capacity κ , where the former is known to change the character of dynamics from hyperbolic ( γ >0) to parabolic like ( γ =0). Other properties of attractors such as additional smoothness and upper-semicontinuity with respect to parameters γ and κ are also established. The main ingredients of the proofs are (i) sharp regularity of Airy’s stress function, and (ii) newly developed (Chueshov and Lasiecka in Memoirs of AMS, in press) “compensated” compactness methods applicable to non-compact dynamics. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Attractors and Long Time Behavior of von Karman Thermoelastic Plates

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Publisher
Springer Journals
Copyright
Copyright © 2008 by Springer Science+Business Media, LLC
Subject
Mathematics; Numerical and Computational Methods ; Mathematical Methods in Physics; Mathematical and Computational Physics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s00245-007-9031-8
Publisher site
See Article on Publisher Site

Abstract

This paper undertakes a study of asymptotic behavior of solutions corresponding to von Karman thermoelastic plates. A distinct feature of the work is that the model considered has no added dissipation —particularly mechanical dissipation typically added to plate equation when long time-behavior is considered. Thus, the model consists of undamped oscillatory plate equation strongly coupled with heat equation. Nevertheless we are able to show that the ultimate (asymptotic) behavior of the von Karman evolution is described by finite dimensional global attractor. In addition, the obtained estimate for the dimension and the size of the attractor are independent of the rotational inertia parameter γ and heat/thermal capacity κ , where the former is known to change the character of dynamics from hyperbolic ( γ >0) to parabolic like ( γ =0). Other properties of attractors such as additional smoothness and upper-semicontinuity with respect to parameters γ and κ are also established. The main ingredients of the proofs are (i) sharp regularity of Airy’s stress function, and (ii) newly developed (Chueshov and Lasiecka in Memoirs of AMS, in press) “compensated” compactness methods applicable to non-compact dynamics.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Oct 1, 2008

References