Mathematical Sciences: Topics in Applied Mathematics and the Degenerate Evolution Equations

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9404379 DiBenedetto This award supports applied mathematics research on problems modeled by certain degenerate and singular evolution equations. In particular, intrinsic Harnack-type estimates, Holder continuity and asymptotic behavior of solutions in the time and space variables will be developed. For systems of equations, work will be done analyzing the boundary behavior of solutions. The new techniques generated by these studies permit the mathematical treatment of a class of applied and industrial problems. These include understanding film rupture in the coating of metals (existence of solutions, rupture and behavior for large values of the variables), identification of cracks in three-dimensional bodies (stability estimates for elliptic ill-posed problems) the thermistor problem and problems in the flow of immiscible fluids in a porous material as the degeneracies approach the saturation point. Further work will concentrate on problems of conduction/convection with change of phase for fluids obeying either the Navier-Stokes system or Darcy's law. These involve an estimation of the Hausdorff dimension of the possible singularities of the temperature. 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 onthe accuracy of these approximations. ***