Nonlinear Partial Differential Equations in Continuum Mechanics

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This award will support research on nonlinear evolution partial differentialequations arising from continuum mechanics. Examples include the Euler and Navier-Stokes equations for 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 of rarefied gases, and equations for combustion, multiphase flows, and fluids with chemical reactions.The main objective is to gain better qualitative and quantitative understanding of the behavior of solutions, using mathematically rigorous analysis and numerical simulations. The focus will be on understanding the effects of small scale dissipations andrelaxations on the large scale fluid motions; development of singularitiesand their stuctures from smooth flows; nonlinear stability or instability and interactions for basic linear and nonlinear waves; design and analysis ofhigh resolution numerical methods for large scale calculations of discontinous flows;fluid-dynamic limit problems for various kinetic models; and problems in multi-spacedimensions and in thermo-nonequilibrium. One of the main ideas is to overcome the difficulties that are encountered at the macroscopic level bydesigning and analyzing approximate models which are physically more fundamental and mathematically tractable. Many interesting phenomena in the mechanics of continua can be described by mathematical models that involve nonlinear partial differential equations. A better understanding of these models has important applications in many sciences such as mechanics, turbulence theory, geophysics, meteorology, aeronautics, chemical engineering, etc. This award will support work on these fundamental models.The goal is to gain better qualitative understanding and to improve the foundationsfor effective and accurate numerical simulations.