International Research Fellowship Program: Hadamard Wellposedness and Asymptotic Stability of Finite Energy Solutions for a Structural Acoustic Interaction Modeled by Nonlinear

Bociu Lorena, Charlottesville VA

Investigators

Abstract

0802187 Bociu The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad. This award will support a twenty-four month research fellowship by Dr. Lorena Bociu to work with Dr. Jean-Paul Zolesio at Centre National de la Recherche Scientifique, Institut Non-Lineaire de Nice, Sophia Antipolis, and with Dr. John Cagnol, at Pole Universitaire Leonard de Vinci, in Paris, France. The proposed research is focused on wellposedness and stability of finite energy solutions to nonlinear structural acoustic models with curved walls. Structural acoustic interactions are described by a system of coupled equations: the wave equation, which models the acoustic medium in a 3-D chamber, and the dynamic shell equation, describing the flexible wall of the chamber. In turn, the motion of a dynamic shell is described by a set of coupled nonlinear partial differential equations, both of hyperbolic type: an elastic wave for the in-plane displacement, and a nonlinear Kirchhoff equation for the scalar normal displacement. Structural acoustic models, due to their large spectrum of engineering applications, have received a lot of attention in engineering and mathematical literature. However, most of the analysis has been performed on linear models with flat walls. The main novelty of the proposed research is that it will account for nonlinear displacements of the curved wall (i.e. fully nonlinear shell) in a coupling with a nonlinear acoustic medium (nonlinear wave equation). Thus, both nonlinear (topological) and geometric aspects will be at the focus of the proposed research, with Euclidean flat geometry being replaced by Riemannian geometry. More specifically, the following issues, recognized as open problems in the literature, will be addressed: First, for a nonlinear shell with nonlinear boundary sources: local and global existence (or blow-up in finite time), uniqueness and regularity of finite energy solutions. Second, for a 3-D structural acoustic nonlinear model with viscous damping and involving shells on the interface between the media: (i) Hadamard wellposedness of finite energy solutions driven by critical and supercritical sources, along with stability of solutions in the presence of boundary (geometrically restricted) damping, and (ii) quantification of the level of nonlinearity of the damping that is sufficient to ensure that finite energy solutions be global. Thus, nonlinearity of the damping is at the heart of the problem. The solution to this problem will not only provide a novel and important contribution to PDEs and their control, but will also have far-reaching potential for transferable research into engineering-based design. The project will use the dynamic shell model based on intrinsic geometry developed by M.Delfour and J.P.Zolesio, which offers great advantages for an analytic formulation of the problem. Host J.P.Zolesio is also an expert in analysis and control of interactive structures - a dominant theme in the proposal. The project will also benefit from strong interaction with J. Cagnol, who is well experienced in shell analysis, including computations with intrinsic geometry-based codes. The proposed research will provide a mathematical solution to a physical problem that is fundamental in application (noise suppression in an acoustic environment). It should also stimulate new approaches in engineering design, eventually impacting society. The proposed methods could be applied to other PDE models sharing common properties: propagation of singularities, finite speed of propagation and supercriticality. Moreover, good wellposedness theory is fundamental for control theory methods to be applied.