Attractors and Long Time Behavior of von Karman Thermoelastic Plates

Journal Article · Wed Oct 15 04:00:00 EDT 2008 · Applied Mathematics and Optimization

DOI:https://doi.org/10.1007/S00245-007-9031-8· OSTI ID:21241970

Chueshov, Igor ^[1]; Lasiecka, Irena ^[2]

---

1. Kharkov University, Department of Mechanics and Mathematics (Ukraine)
2. University of Virginia, Department of Mathematics (United States)

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 {gamma} and heat/thermal capacity {kappa}, where the former is known to change the character of dynamics from hyperbolic ({gamma}>0) to parabolic like ({gamma}=0). Other properties of attractors such as additional smoothness and upper-semicontinuity with respect to parameters {gamma} and {kappa} 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.

• OSTI ID: 21241970
• Journal Information: Applied Mathematics and Optimization, Journal Name: Applied Mathematics and Optimization Journal Issue: 2 Vol. 58; ISSN 0095-4616
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
99 GENERAL AND MISCELLANEOUS
ASYMPTOTIC SOLUTIONS
ATTRACTORS
EQUATIONS
FUNCTIONS
HEAT
MATHEMATICAL EVOLUTION
MOMENT OF INERTIA
PLATES
ROUGHNESS