Dynamics and Stable Structures in Some Nonlinear PDEs

本项目的目标是研究由非线性偏微分方程模拟的各种物理现象中产生的四种稳定结构：(i)反应流动的Boussinesq模型中的行波：Boussinesq系统是最简单的方程组，显示了重力分层介质中预混火焰的行为。（ii）三维薄域内Navier-Stokes方程的吸引子：该领域以往研究的工作条件是，随着区域厚度的消失，极限几何是平坦的。这个项目将研究技术上更复杂的非平面极限几何的情况。这项调查的部分动机是使模型适应应用，例如在海洋学中。（iii）复金兹堡-朗道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。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也引起了人们的兴趣。（iv）具有大数据的严格双曲守恒律系统的稀疏波解（根据NSF拨款DMS-0306201的结果）。数学物理方程（特别是与流体或气体动力学有关的方程）的解所产生的模式的稳定性是科学家和工程师最感兴趣的问题。稳定的模式是那些预期在实验中观察到的。它们可能是连续波、跳跃（例如所研究量的密度）或其他奇点。对不稳定模式的分析，即物理上不可观测的方程的解，对观测到的方程的时间演化提供了重要的见解。这个项目分析了几个重要方程组的解的模式。其应用范围从气象学、血液循环、润滑和气体燃烧，到研究相变现象，如超导性、超流动性和液晶。

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.