Dynamics and Stable Structures in Some Nonlinear PDEs

该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）反应流的Boussinesq模型中的行波：Boussinesq系统是展示重力分层介质中预混火焰行为的最简单方程系统。(ii)在薄的三维区域中Navier-Stokes方程的吸引子：在该领域以前研究的工作条件是，随着区域厚度的消失，极限几何形状是平坦的。 本项目将研究技术上更复杂的非平坦极限几何形状的情况。 这项调查的部分动机是调整模型的应用，例如在海洋学。(iii)自相似的，奇异的解决方案，复杂的金斯堡-朗道方程：这个方程描述了各种现象，从非线性波到二阶相变。 兴趣也源于与三维Navier-Stokes方程和三维超临界非线性薛定谔方程的类比。(iv)稀疏波解决方案严格双曲型系统的守恒律与大数据（以下的结果NSF资助DMS-0306201）。稳定的模式所产生的解决方案的数学物理方程，特别是有关流体或气体动力学，是中央感兴趣的科学家和工程师。 稳定的模式是那些预期在实验中观察到的。 它们可以是连续波、跳跃（例如在所研究的量的密度中）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，为观察到的模式的时间演变提供了重要的见解。 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，到相变现象的研究，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.