Mathematical Sciences: Problems in Harmonic Analysis and PDE

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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. ***