Nonlinear partial differential equations and applications

非线性偏微分方程及其应用

• 批准号: 0070569
• 负责人: Panagiotis Souganidis
• 金额: $ 9.3万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070569&HistoricalAwards=false
• 关键词: Nonlinear; partial; differential; equations; applications

This is a proposal to develop general methods to study nonlinear, hyperbolic and parabolic/elliptic partial differential equations. More precisely using techniques from analysis, partial differential equations and probability, the PI plans to continue his program working in the following four general areas: A. Fully nonlinear stochastic partial differential equations of first-and second- order . B. Phase Transitions. C. Turbulent reaction-diffusion equations and combustion. D. First- and second-order fully nonlinear, degenerate, elliptic and parabolic equations. Most of the partial differential equations considered in this proposal arise as models in continuum and statistical physics. These models appear in a variety of topics ranging from material sciences and phase transitions (motion of fronts, mesoscopic and macroscopic scales, homogenization), in fluid flows (turbulent reaction-diffusion and combustion), and in stochastic analysis (interacting particle systems, flows in random environments and with random velocities, and stochastic control). The qualitative analysis of these models contributes to the better understanding of the actual physical problems, and provides, in many cases, the foundation for the development of efficient numerical algorithms.

这是一项开发研究非线性、双曲和抛物线/椭圆偏微分方程的通用方法的提案。 更准确地说，利用分析、偏微分方程和概率等技术，PI 计划继续其在以下四个一般领域的项目： A. 一阶和二阶完全非线性随机偏微分方程。 B.相变。 C. 湍流反应扩散方程和燃烧。 D. 一阶和二阶完全非线性、简并、椭圆和抛物线方程。本提案中考虑的大多数偏微分方程都是作为连续介质和统计物理中的模型出现的。 这些模型出现在各种主题中，从材料科学和相变（前沿运动、细观和宏观尺度、均质化）、流体流动（湍流反应扩散和燃烧）到随机分析（相互作用的粒子系统、随机环境中和随机速度的流动以及随机控制）。这些模型的定性分析有助于更好地理解实际物理问题，并在许多情况下为开发有效的数值算法提供基础。