Mathematical Sciences: Problems in Harmonic Analysis and PDE

小行星9401782 该奖项支持涉及偏微分方程系统问题的数学研究。 工作的重点是如何使用谐波分析在线性和非线性系统的治疗。 这些方程有几个来源。 一个是涉及旋转恒星的研究。 我们的工作是建立白色矮星自由能泛函平衡解的分岔图。 此外，将努力了解这个能量函数的临界点的自由边界。 其他工作涉及旋转对称的偏微分方程代表二维流体流动的一个领域和分析的对称性可以预期打破。 另外的调查线涉及磁流体动力学的逆问题和线性方程在真实的解析情况下的局部可解性问题。 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。 ***

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. ***