On the Dynamics, Structure and Stability of Certain Nonlinear Systems in Applied Sciences

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This research project investigates problems on the interface between continuum physics and applied partial differential equations, with emphasis on analysis of physically relevant solutions of the nonlinear systems that arise in the description of fluid flow and continuum mechanics. The project has three primary components: (a) The investigation of hydrodynamic models for compressible and incompressible fluids. Issues to be addressed include large-data existence theory for physical systems, issues of stability and asymptotic analysis, and singular limits in hydrodynamics. (b) The analysis of material structures with focus on fluid-solid interaction problems as well as fluid-particle models arising in biology and environmental sciences. (c) The investigation of inviscid models with emphasis on hyperbolic balance laws.The enormous range of spatial and temporal scales that arise in the description of compressible fluids, nonlinear materials, multi-phase flows, astrophysical systems, and biological systems makes analysis of the associated models challenging. Use of explicit integration or even numerical approximation for generic solutions to the governing equations is impractical for the majority of applications. This research explores alternative approaches that will contribute to the further investigation of a wide variety of important physical systems and to the design of high performance computational algorithms. One of the goals of this project is to contribute to the development of the scientific workforce by providing mentoring and advanced training for young researchers and by organizing a framework for interdisciplinary interactions with researchers from related fields.