Nonlinear Partial Differential Equations with Applications

9706231苏加尼迪斯和平研究所建议研究范围广泛的问题，其中一些问题的动机是相变、湍流燃烧等方面的应用，这些问题出现在双曲型和抛物型/椭圆型非线性偏微分方程组和概率领域。更确切地说，利用分析、偏微分方程式和概率论的技术，PI建议继续他的计划，在以下五个一般领域工作：A.动力学守恒定律理论。完全非线性椭圆型和抛物型偏微分方程的粘性解。完全非线性随机偏微分方程解。D.相变。即湍流反应扩散方程。尽管湍流反应扩散方程的大部分工作涉及建模和计算，但这个项目中要研究的大多数问题都是理论性质的。与最后一个主题有关的问题以及与相变和完全非线性随机偏微分方程有关的问题在材料和环境/海洋科学中是重要的。事实上，这些问题和预期的答案应该为研究这些科学中的基本问题提供适当的数学模型和见解。

9706231 Souganidis The PI proposes to study a broad range of problems, some motivated by applications in phase transitions, turbulent combustion, etc., which arise in the areas of hyperbolic and parabolic/elliptic nonlinear partial differential equations and probability. More precisely, using techniques from analysis, partial differential equations, and probability, the PI proposes to continue his program working in the following five general areas: A. Kinetic theory of conservation laws. B. Viscosity solutions of fully nonlinear elliptic and parabolic partial differential equations. C. Fully nonlinear stochastic partial differential equations. D. Phase transitions. E. Turbulent reaction-diffusion equations. Most of the questions to be studied in this project are of a theoretical nature, although much of the work in turbulent reaction-diffusion equations deals with modelling and computations. The questions related to this last topic as well as those related to phase transitions and fully nonlinear stochastic pde are important in material and environmental/oceanic sciences. Indeed these questions and the anticipated answers should provide appropriate mathematical models and insight for the study of basic questions in these sciences.