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

该基金支持的工作涵盖了在双曲型和抛物型非线性偏微分方程领域中出现的广泛问题。将强调三个主要主题。首先，我们要研究守恒定律的运动论。目标是使用这些系统的动力学型公式/近似，获得守恒定律系统的新的规则性和紧性结果。其次，将研究一阶和二阶完全非线性退化椭圆型和抛物型方程。重点讨论了不连续控制问题的分析，收敛性的证明，数值格式的误差估计，最后研究了弱解的正则性和唯一性问题。最后，将对广义前传播和相变进行研究。大部分的焦点将集中在研究移动锋面超过其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，粒子系统的宏观（流体动力）极限和湍流燃烧的大尺度锋面动力学。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

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. ***