Mathematical Sciences: "Nonlinear Problems in Fluid Mechanics and Related Partial Differential Equations"

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9623203 Chen This proposal is directed toward research on nonlinear problems in fluid mechanics and related partial differential equations, along with the analysis and development of efficient nonlinear methods. The objective of this proposed research program is threefold: investigate important multidimensional nonlinear problems such as geometric flows, transonic flows, and the Riemann problem to find the correct function spaces in which to pose the multidimensional hyperbolic conservation laws and in which to develop the numerical algorithms that converge stably and rapidly; analyze and develop existing shock capturing methods and related potential techniques to formulate new, more efficient nonlinear methods and to solve global solutions of the nonlinear systems; probe in the regularizing mechanism for nonlinear partial differential equations to gain new physical insights and to guide the formulation of the efficient nonlinear methods. The nonlinear problems arise in all areas of science and engineering. In particular, they arise in material sciences, biophysics, and environmental and global change processes, in which fluid motions are essential features. Such features are predominant in such areas as elasticity, plasticity, semiconductor, biological transport of ions, phase transition, multi-phase process, and combustion. The fluid motions are generally governed by the principles of fluid mechanics and, in turn, by certain nonlinear systems of partial differential equations. Understanding solutions of such nonlinear systems enables ones to understand the corresponding fluid motions, to gain new physical insights, to discover the intrinsic laws in these areas. The proposed nonlinear problems include the well-posedness of solutions of these nonlinear systems and the qualitative behavior of new nonlinear phenomena as well as the analysis and development of numerical computational methods and related potential nonlinear methods.