Mathematical Sciences: Partial Differential Equations: Free Boundary Problems

9401251弗里德曼这个项目是关于偏微分方程组的自由边界问题。我们将研究三类问题：(I)非线性抛物方程反应扩散系统的Stefan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)，以及(Iii)当带电质量分布的运动与带电质量产生的电场耦合时产生的等离子体类型问题(使人想起涡片问题)。在每个问题领域都取得了一些初步进展，但需要发展新的思路和方法才能取得进一步进展。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常发展出近似解的方法和对这些近似的精度的估计。***

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. ***