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.

这是开发一般方法来研究非线性，双曲线和抛物线/椭圆形偏微分方程的建议。 PI更精确地使用分析，部分微分方程方程和概率的技术，计划继续在以下四个一般领域中工作的程序：A。完全非线性随机的局部偏微分方程第一和第二阶。 B.相变。 C.湍流反应扩散方程和燃烧。 D.一阶和二阶非线性，退化，椭圆形和抛物线方程。该提案中考虑的大多数部分微分方程都是连续和统计物理学中的模型。 这些模型出现在各种主题中，包括物质科学和相变（前面的运动，介质和宏观尺度的运动，均质化），流体流动（湍流反应 - 扩散和燃烧）以及随机环境中的相互作用粒子系统，以及随机环境以及随机速度和速度对照，以及与随机的速度和Stochocative Controction和Stochchostic Controctions相互作用）。这些模型的定性分析有助于更好地理解实际的物理问题，并在许多情况下为开发有效的数值算法的基础提供了基础。