Asymptotic Stability of Systems of Coupled Nonlinear Partial Differential Equations arising From Acoustic-Structural and Fluid-Structural Interaction.

摘要

In this thesis, we study the following fundamental qualitative properties of the solutions of systems of differential equations: (i) the existence and uniqueness of solutions given an appropriate initial configuration and (ii) the long time behavior of the solutions, with emphasis on the latter. The analysis of these properties for nonlinear differential equations often requires different techniques from those used for the linear equations.;The particular models under considerations are systems of coupled PDEs arising from the modeling of a structure interacting with acoustic or fluid environments. These models typically couple a hyperbolic PDE with a parabolic PDE. For example, fluid-structural interaction couples a Navier-Stokes equation with the system of elasticity of wave equation (see (1.4.1) below); while acoustic-structural interaction couples the wave equation with nonlinear boundary conditions with a linear elastic plate equation (see (1.3.1) - (1.3.2) below).;One of the main challenges is the analysis of the dynamics on the interface separating the structure and its surrounding environment. The analysis of these questions often involves, in addition to PDE estimates and operator theory, geometric considerations which appear to play a decisive role for the final results.;The models and problems considered are of interest in a multitude of applications ranging from naval and aerospace engineering to cell biology and biomedical engineering. Specific examples include the noise suppression in an acoustic chamber, which has medical or mechanical applications; and the modeling of the deformability and viscoelasticity of cells in human blood strains.