Mathematical Sciences: Partial Differential Equations: Free Boundary Problems

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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. ***