Some Problems in Applied Nonlinear Partial Differential Equations

本项目的目的是通过研究对流奇异面的正则性、密度正性和动力学特性，探索多维可压缩势流粘性近似中的耗散机制，通过推导三维非线性弹性力学的渐近极限来证明各种平板理论和三维弹性理论之间的关系，加深对非湍流半地转方程中最优输运机制的理解，研究材料科学中多相晶体生长中界面的演化，以及模拟燃烧中前沿传播等许多有趣物理过程的Hamilton-Jacobi方程的随机解。在研究项目中涉及的非线性偏微分方程组的严密数学使得在中进行稳定的数值计算和理解上述现象的定性特征成为可能。我们的目标之一是研究粘性可压势流和半地转流。数学方程整体大解知识的进步对我们对气流的基本理解有着重要的影响。第二个目标是分析用于制造计算设备的半导体材料中晶体生长和薄膜起泡的前沿传播。该项目的第三个目标是培训研究生和博士后研究员对应用问题进行数学分析。

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.