Some Problems in Applied Nonlinear Partial Differential Equations

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The purpose of the project is to probe the dissipationmechanism in viscous approximation of multidimensional compressiblepotential flow by studying the regularity property, positivity of density,and dynamics of convective singularity surfaces, to justify the relationbetween variety of plate theories and three dimensional elasticity theory byderiving the asymptotic limit of three dimensional nonlinear elasticity,to deepen the understanding of optimal transport mechanism innon-turbulent semi-geostrophic equations, to investigate evolution ofinterface in multi-phase crystal growth in material science, stochasticsolutions of Hamilton-Jacobi equations which model many interestingphysical processes such as front propagation in combustion. Rigorous mathematics in the study of the nonlinear partialdifferential equtions addressed in the project make it possible both toconduct stable numerical computation in, and to understand the qualitativefeatures of the phenomena addressed above. One of our objectives is tostudy viscous oompressible potential flow and semi-geostrophic flow. Theadvance in the knowledge of global large solutions of mathematicalequations has important impact on our fundamental understanding of airflow. The second objective is to analyze the front propagation of crystalgrowth and thin-film blistering in the semiconducting materials from whichcomputing devices are manufactured. The third objective of this project isthe training of graduate students and postdoctoral fellows in themathematical analysis of applied problems.