Mathematical Sciences: Partial Differential Equations: Free Boundary Problems

小行星9401251 这个项目是关于偏微分方程组的自由边界问题。 需要研究三类问题：（i）非线性抛物型反应扩散方程组的Stefan型问题，（ii）Navier-Stokes方程的自由边界问题（具有与固定边界分离的自由边界），等离子体问题（让人想起涡流补丁问题）时产生的带电质量分布的运动耦合到带电质量产生的电场。 在这些问题的每一个领域都取得了一些初步进展，但需要发展新的想法和方法以取得进一步进展。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 ***

9401251 Friedman This project is concerned with free boundary problems for systems of partial differential equations. Three classes of problems are to be investigated: (i) Stefan type problems for a reaction-diffusion system of nonlinear parabolic equations, (ii) free boundary problems for Navier-Stokes equation (with a free boundary which separates from the fixed boundary), and (iii) plasma-type problems (reminiscent of the vortex patch problem) arising when the motion of charged mass distribution is coupled to the electric field generated by charged mass. Some initial progress has been made in each of these problems areas, but new ideas and methods need to be developed for further progress. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis oftendevelops methods for approximation of solutions and estimates on the accuracy of these approximations. ***