Long time behavior of solutions to flow-structure interactions arising in modeling of subsonic and supersonic flows of gas
A talk with Irena Lasiecka, University of Memphis, part of the CAMS Colloquium Series.
Description:Irena Lasiecka, University of Memphis
Abstract: We consider flow - structure interaction comprising of a modified wave equation coupled to a nonlinear plate equation. The model has no damping imposed neither on the structure nor the flow. The regime of the parameters considered includes supersonic flows -the latter known for depleting ellipticity from the corresponding static model. Thus, both well-posedness of finite energy solutions and long time behavior of the model have been open questions in the literature. The results presented include:
• Existence, uniqueness and Hadamard wellposedness of finite energy solutions.
• Existence of global and finite dimensional attracting sets for the structure in the absence of any mechanical dissipation.
The key "hidden regularity type" inequalities, responsible for proving existence of nonlinear semigroup, are derived by using microlocal analysis.
The existence of an attracting set is proved without imposing any form of dissipation on the model. This is achieved by exploiting "compensated compactness" related to the dispersive character of the flow equation. To our knowledge, this is the first complete exhibition and rigorous justification of this fact -previously known experimentally only. In order to resolve the difficulty, we follow the decoupling method of  which reduces the problem to a study of nonlinear plates with the delay terms. This presentation is based on a joint work with Igor Chueshov, Kharkov University and Justin Webster, Oregon State University.