Mathematical Sciences: Problems in Harmonic Analysis and PDE

9401782该奖项支持对涉及偏微分方程组问题的数学研究。这项工作的主旨是关于如何在处理线性和非线性系统时使用调和分析。这些方程式有几个来源。一种是涉及对旋转恒星的研究。在建立白矮星自由能泛函平衡解的分岔图方面，将做一些工作。此外，还将努力了解该能量函数临界点的自由边界。其他工作涉及表示球面上二维流体流动的偏微分方程组的旋转对称性，以及对称性可能在哪里被破坏的分析。其他研究方向包括磁流体力学的反问题和实际分析情况下的线性方程的局部可解性问题。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，分析经常开发出近似解的方法和对这些近似的精度的估计。***

9401782 Chanillo This award supports mathematical research on problems involving systems of partial differential equations. The thrust of the work is concerned with how one uses harmonic analysis in the treatment of linear and nonlinear systems. The equations arise from several sources. One is that involving the study of rotating stars. Work will be done in establishing a bifurcation diagram for the equilibrium solutions of the free energy functional of white dwarfs. In addition, efforts will be made to understand the free boundary of the critical points of this energy function. Other work involves the rotational symmetry of partial differential equations which represent 2-dimensional fluid flow on a sphere and the analysis of where symmetry can be expected to break. Additional lines of investigation involve inverse problems from magnetohydrodynamics and local solvability problems for linear equations in the real analytic case. 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 often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***