Dynamics and Stable Structures in Some Nonlinear PDEs

一些非线性偏微分方程中的动力学和稳定结构

• 批准号: 1142369
• 负责人: Marta Lewicka
• 金额: $ 0.77万
• 依托单位: University of Pittsburgh
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1142369&HistoricalAwards=false
• 关键词: Dynamics; Stable; Structures; Some; Nonlinear

The goal of this project is to study four stable structures arising in various physical phenomena, modeled by nonlinear partial differential equations:(i) Traveling waves in the Boussinesq model of reactive flows: The Boussinesq system is the simplest system of equations exhibiting behavior of premixed flames in a gravitationally stratified medium.(ii) Attractors to Navier-Stokes equations in thin three-dimensional domains: The working condition of previous research in the area has been that the limiting geometry, as the thickness of the domain vanishes, is flat. This project will investigate the technically more involved case of non-flat limit geometries. This investigation is partially motivated by adapting the model to applications, e.g. in oceanography.(iii) Self-similar, singular solutions to the complex Ginzburg-Landau equation: This equation describes a variety of phenomena, from nonlinear waves to second-order phase transitions. Interest also stems from analogies with the three-dimensional Navier-Stokes equation and the three-dimensional supercritical nonlinear Schrodinger equation.(iv) Rarefaction wave solutions to strictly hyperbolic systems of conservation laws with large data (following on results of NSF grant DMS-0306201).The stability of patterns arising as solutions to equations of mathematical physics, notably related to fluid or gas dynamics, is of central interest to scientists and engineers. The stable patterns are those expected to be observed in experiments. They may be continuous waves, jumps (for example in the density of the studied quantities), or other singularities. Analysis of unstable patterns, solutions of the equations that are non-observable physically, gives important insight into the time evolution of the observed ones. This project analyzes patterns in solutions of several important systems of equations. The applications range from meteorology, blood circulation, lubrication, and combustion in gases, to studies of phase transition phenomena such as super-conductivity, super-fluidity, and liquid crystals.

该项目的目标是研究在各种物理现象中出现的四种稳定结构，通过非线性偏微分方程建模：（i）反应流 Boussinesq 模型中的行波：Boussinesq 系统是表现预混合行为的最简单的方程组重力分层介质中的火焰。（ii）薄三维域中纳维-斯托克斯方程的吸引子：该领域先前研究的工作条件是极限几何，当域的厚度消失时，是平坦的。 该项目将研究技术上更复杂的非平坦极限几何形状的情况。 这项研究的部分动机是使模型适应应用程序，例如(iii) 复杂的 Ginzburg-Landau 方程的自相似奇异解：该方程描述了从非线性波到二阶相变的各种现象。 人们的兴趣还源于与三维纳维-斯托克斯方程和三维超临界非线性薛定谔方程的类比。(iv) 大数据守恒定律严格双曲系统的稀疏波解（根据 NSF 拨款 DMS-0306201 的结果） ）。作为数学物理方程的解而产生的模式的稳定性，特别是与流体或气体动力学相关的方程，是科学家和工程师的主要兴趣。 稳定的模式是预期在实验中观察到的模式。 它们可能是连续波、跳跃（例如所研究数量的密度）或其他奇点。 对不稳定模式的分析，即物理上不可观察的方程的解，可以为观察到的方程的时间演化提供重要的见解。 该项目分析了几个重要方程组的解的模式。 应用范围从气象学、血液循环、润滑和气体燃烧，到超导性、超流动性和液晶等相变现象的研究。