Mathematical Sciences: Topics on Free Boundary Problems and Singular Parabolic Equations

本项目将集中研究非线性退化或奇异发展方程的局部和渐近行为。将尝试哈纳克型估计的局部正则性、奇点的出现以及固有的局部和全局版本。预计将不得不分析关于具有极限索波列夫指数的抛物型方程的新型弱解和有趣方面的问题。所要研究的方程是作为非线性现象的模型出现的，如等离子体中的快速扩散、相变和不混溶流体在多孔介质中的流动。将考虑金属表面涂覆胶体流体和传导-对流的问题。

This project will focus on local and asymptotic behavior of nonlinear degenerate or singular evolution equations. Local regularity, appearance of singularities, and intrinsic local and global versions of Harnack type of estimates will be attempted. It is anticipated that questions concerning new type of weak solutions and intriguing aspects of parabolic equations with limiting Sobolev exponent will have to be analyzed. The equations to be studied arise as models of nonlinear phenomena such as fast diffusion in plasma, transition of phase, and flow of immiscible fluids in a porus medium. Problems concerning coating of metal surfaces with colloidal fluids and conduction-convection will be considered.