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

9404379迪贝内代托该奖项支持对由某些退化和奇异的发展方程建模的问题进行应用数学研究。特别地，我们将得到时间和空间变量中解的内禀Harnack型估计、Holder连续性和渐近性。对于方程组，我们将分析解的边界行为。这些研究产生的新技术允许对一类应用和工业问题进行数学处理。这些课程包括了解金属涂层中的薄膜破裂(溶液的存在、破裂和变量值较大时的行为)、三维体中裂纹的识别(椭圆型不适定问题的稳定性估计)、热敏电阻问题以及当简并度接近饱和点时多孔材料中不相容流体的流动问题。进一步的工作将集中在遵循Navier-Stokes系统或达西定律的流体的导热/对流相变问题上。这涉及到对温度可能奇点的Hausdorff维度的估计。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，分析常常发展出近似解的方法，并估计这些近似的精度。***

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. ***