Mathematical Sciences: The Partial Differential Equations ofFluid Dynamics and their Numerical Approximation

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The PI will continue his research program on partial differential equations of fluid dynamics in several directions involving both compressible and incompressible flows. As primary goals, he will focus on the following four topics: 1. Classical weak solutions and measure-valued solutions of the incompressible Euler equations in 2 and 3 dimensions. 2. Breakdown of smooth solutions for the 3-D Euler equations. 3. Multi-dimensional conservation laws and geometric shock dynamics. 4. Nonlinear development of instabilities in supersonic shear layers. The research is in the general area of applied nonlinear analysis which is currently an important and active research area of applied mathematics. The mixture of analytical and numerical approaches to the solutions of research problems enhances the success potential of the research program. One can expect research results from the program with significant impact on both the mathematical science community and the general science and engineering community.