Nonlinear Problems in Conservation Laws and Fluid Dynamics and Related Partial Differential Equations

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DMS Award AbstractAward #: 0204225PI: Chen, Gui-QiangInstitution: Northwestern University Program: Applied MathematicsProgram Manager: Catherine MavriplisTitle: Nonlinear Problems in Conservation Laws and Fluid Dynamics and Related Partial Differential EquationsThe investigator continues studies of nonlinear problems in conservation laws and fluid dynamics and related nonlinear partial differential equations and their applications, along with the analysis and development of efficient nonlinear methods. The objective of this research program is twofold: (1) to investigate important nonlinear problems such as multidimensional transonic shocks and free boundary problems, vacuum problems, asymptotic stability problems, compressible fluids with various constitutive relations, singular limit problems, multiphase problems, and the Riemann problem to gain new physical insights, to guide the formulation of efficient nonlinear methods, and to find the correct function spaces in which to pose the nonlinear conservation laws and develop the numerical methods that converge stably and rapidly; and (2) to analyze and develop nonlinear methods including free boundary methods, kinetic methods, geometric measure methods, weak convergence methods, shock capturing techniques, energy methods, and related potential techniques to formulate new, more efficient nonlinear methods and to solve various more important nonlinear problems in conservation laws and fluid dynamics.The nonlinear problems and related partial differential equations in this research program arise in such areas as gas dynamics, hydraulics, combustion, magnetohydrodynamics, semiconductor, elasticity, multiphase flow, phase transitions, kinetic theory, biophysics, and material science. The award will support research on the solvability of these nonlinear problems and related partial differential equations, the qualitative behavior of their solutions and related applications, as well as the analysis and development of nonlinear methods in applied analysis and numerical analysis. This research will lead to a deeper understanding of nonlinear phenomena and will provide more efficient nonlinear methods and theories for applications.Date: May 1, 2002