Long Time Behavior of Multidimensional Systems

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Long Time Behavior of Multidimensional SystemsAbstract of Proposed ResearchThomas C. Sideris The research project focuses on the analysis of multidimensional systems of nonlinear partial differential equations that arise in the motion of compressible and incompressible materials. The work will concentrate on studying the existence and long time behavior of solutions to initial value problems. Our analysis will exploit the detailed structure of the models, and in particular, the effect of nonlinear interactions. We shall study the motion of nonlinear visco-elastic materials, the relationship between compressible and incompressible materials, the formation of shock waves in compressible materials, and the confinement of vortical motion in planar incompressible ideal fluids. Classical ideas of deterministic behavior usually are regarded as saying that the evolutionary laws of a physical system together with the initial configuration determine the state of the system at all future times. Mathematically speaking, the laws of physics are encoded in a system of partial differential equations and their solutions should exist and depend uniquely, and continuously, on the initial data. Creating the mathematical framework for a given system and verifying its well-posedness (in the sense above) is thus a problem of fundamental importance. Mathematical analysis offers important guidance in the specification of strain energy relationships beyond purely phenomenological assumptions. Much of the research in this proposal is devoted to answering these basic questions for models describing the dynamics of broad classes of elastic materials.