Mathematical Sciences: Nonlinear Partial Differential Equations with Applications to Phase Transitions, Front Propagation and Mechanics

这笔赠款支持的Souganidis 9403412工作涵盖了在双曲型和抛物型非线性偏微分方程式领域中出现的广泛问题。将强调三个主要主题。第一，研究守恒定律的动力学理论。其目的是利用守恒律系统的动态型公式/近似，获得这些系统的新的正则性和紧致性结果。第二，研究一阶和二阶完全非线性退化椭圆型和抛物型方程。重点是不连续控制问题的分析，证明收敛，获得数值格式的误差估计，最后研究关于弱解的正则性和唯一性的若干问题。最后，对广义波前传播和相变进行了研究。大部分的重点将是研究移动的前锋越过它们的奇点的规律性，以及相变理论。后者包括反应扩散方程和系统的奇异极限、颗粒系统的宏观(流体力学)极限和湍流燃烧的大尺度前沿动力学。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，分析经常开发出近似解的方法和对这些近似的精度的估计。***

Souganidis 9403412 Work supported by this grant covers a broad range of problems arising in the areas of hyperbolic and parabolic nonlinear partial differential equations. Three primary themes will be emphasized. First, work will be done on the kinetic theory of conservation laws. The goal is to obtain new regularity and compactness results for systems of conservationlaws, using a kinetic type formulation/approximation for these systems. Second, studies will be made on first- and second-order fully nonlinear degenerate elliptic and parabolic equations. The emphasis is on the analysis of discontinuous control problems, on proving convergence, on obtaining error estimates for numerical schemes and finally, on studying several issues regarding the regularity and uniqueness of weak solutions. Lastly, work will be done on generalized front propagation and phase transitions. Much of the focus will be on studying the regularity of moving fronts past their singularities as well as the theory of phase transitions. The latter includes singular limits of reaction-diffusion equations and systems, macroscopic (hydrodynamic) limits of particle systems and large scale front dynamics for turbulent combustion. 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. ***