Mathematical Sciences: Nonlinear Partial Differential Equations in Continuum Mechanics

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9600137 Xin The PI proposes to study a class of nonlinear partial differential equations arising from Continuum Mechanics and other branches of natural sciences, which includes the Euler and Navier-Stokes equations for both compressible and incompressible, inviscid and viscous fluids, the equations of electro-magneto field dynamics for electrically conducting compressible viscous fluids, the equations of elasticity, nonlinear Boltzmann type equations in kinetic theory, and equations for combustion, multi-phase flows, etc. The main goal is to gain understanding of the dynamics and interactions of basic nonlinear waves such as shock waves, expansion waves, solitons, diffusive waves, vortex sheets, and boundary layers, which play important roles in the study of the general flows. In particular, it is proposed to analyze the effects of the small scale dissipation or relaxation processes on large scale physical quantities and their motions. Other questions such as global existence of both strong and weak solutions, nonlinear stability of basic wave patterns, uniqueness of weak entropy solutions, layer behavior of solutions, numerical methods for the computation of discontinuous flows, and fluid-dynamic limit for kinetic models are also to be explored. It is planned to continue the current effort to expand the existing techniques and to develop new tools for studying some of these problems. One of the main idea is to refine the conventional methods designed to study smooth flows or unrelaxed systems by taking into account more internal structures of the underlying discontinuous flows, and study these problems both theoretically and numerically. %%% Nonlinear evolution of partial differential equations arising from physical balance laws govern many of the most widely encountered phenomena in nature. Solutions to these nonlinear systems remain a central topic in the study of wave propagations and are fundamental to the understanding of basic physical phenomena ranging from compressible and incompressible fluids, acoustic waves and electromagnetic scattering to wave propagations in nonlinear composite materials, in reactive gas mixtures, in fiber optics, as well as in magnetohydrodynamics and general relativity. In this project, It is proposed to study some of the basic questions for these nonlinear systems, which are motivated directly by important applications to many fields such as image processing, aeronautics, composite material, sonar, radar, medical imaging, geophysical exploration, weather prediction, oil reservoir simulations etc.. For example, the proposed study of design of high resolution multi-dimensional numerical methods for calculating sharp fronts for mixed type partial differential equations is extremely important for the oil reservoir simulations, design of composite materials, imaging processing, and propagation of pollutants such as oil in water and air; the proposed study of boundary layer theory is one of the key ingredients in the understanding of turbulence, and the design of supersonic aircrafts, etc.. Thus the accomplishment of this project should be of considerable importance in the improvement and new development of technologies in the above mentioned fields. ***