ON THE DYNAMICS OF CERTAIN NONLINEAR SYSTEMS IN APPLIED SCIENCES: TRANSPORT, MOTION AND MIXING

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This project focuses on the fundamental challenges of modeling, analysis and computation for nonlinear models in applied sciences involving hyperbolic balance laws, kinetic models, and mixtures of simple and complex fluids. The project has two primary components: (a) The investigation of complex flows with emphasis on fluid-particle interaction models and polymeric fluids. Issues to be addressed include large-data existence theory for physical systems, issues of stability and asymptotic analysis, hydrodynamic and diffusive limits as well as aspects of regularity. (b) The investigation of multidimensional systems with focus on hyperbolic systems of conservation laws and models of compressible fluids with dispersive effects. Issues to be addressed include large-data well-posedness theory for physical systems, the investigation of the structure of solutions as well as the analysis of singular limits.Advances in this interdisciplinary research will rely substantially on the development of new analytical techniques in nonlinear partial differential equations. Analytical and numerical methods in this field have developed together; analytical understanding contributes to the construction of accurate and efficient numerical schemes, while numerical experiments often lead the theoretical analysis. Progress in this research program will contribute to the successful investigation of a wide variety of important physical systems arising in biology and materials science as well as to the design of high performance computational algorithms. Furthermore, this project will 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.