Spectral Stability and Oscillations of Dynamical Systems, Boltzmann-Like Models

这个项目的目的是开发数学工具，帮助研究影响偏微分方程解的稳定性和不稳定机制。特别是，这些工具可以用来评估非线性偏微分方程不同状态的稳定性，这是理解由该方程建模的物理系统的行为的关键一步。在这里，稳定性指的是动力学对来自特定状态的初始条件的摄动的鲁棒性。区别状态可以是在光学、流体、神经科学、生态学、化学反应或浅水动力学等应用中产生的非线性波、图案或相干结构。给定状态的稳定性表明它的物理可实现性，而任何不稳定意味着更复杂的动力学。了解这种不稳定性的本质可以作为理解远离失稳状态的非线性动力学的组织的起点。该项目包括将培养本科生的研究活动。研究人员的目标是推广与偏微分方程模型有关的特征值问题的振荡型结果。这些问题从双曲型平衡律系统激波剖面谱稳定性中的非线性特征值问题到多维空间域中的问题。特别是，他利用了一套新的思想，将一般的多维问题投射到动力系统框架中，从而提供了解的存在性和稳定性这一根本问题的新表征。该项目的重点方向之一是引入空间进化系统，这可以被视为空间动力学框架向一般多维领域的深远扩展。本科生从事该项目的研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project aims to develop the mathematical tools that can help investigate stability and instability mechanisms affecting the solutions of partial differential equations. In particular, these tools can be used to assess stability of distinguished states of a nonlinear partial differential equation, which is a key step in understanding the behavior of the physical system modeled by the equation. Stability here means the robustness of the dynamics to perturbations in initial conditions from a particular state. The distinguished state may be a nonlinear wave, pattern, or coherent structure arising in applications such as optics, fluids, neuroscience, ecology, chemical reactions, or shallow water dynamics. The stability of the given state indicates its physical realizability, while any instability suggests more complex dynamics. Understanding the nature of such instabilities can be used as a starting point for understanding the organization of the nonlinear dynamics away from the unstable state. The project includes research activities that will train undergraduate studentThe investigator's goal is to generalize the oscillation-type results for eigenvalue problems that are associated with partial differential equation models. These range from nonlinear eigenvalue problems arising in spectral stability of shock profiles of hyperbolic systems of balance laws to problems in multi-dimensional spatial domains. In particular, he exploits a new set of ideas that cast general multi-dimensional problems in a dynamical systems framework and thus offer a new characterization of the underlying issues of existence and stability of solutions. One of the key directions of the project is the introduction of the Spatial Evolutionary System, which can be viewed as a far-reaching extension of the spatial dynamics framework to general multi-dimensional domains. Undergraduate students are engaged in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.