Frontiers in Modulation, Dynamics, and Pattern Formation for Hyperbolic, Kinetic, and Convection-Reaction-Diffusion Systems

翻滚波是在快速移动的倾斜水流中形成的大规模周期性脉冲，例如在运河或水坝的溢洪道上。作为一种潜在的破坏性现象，从水利工程的角度了解其发生的条件和特征是很重要的。这就转化为这种波在小扰动下的稳定性或持久性的问题。直到最近，现有的数学工具还无法解决这些问题。PI和合作者已经为这项研究开发了一个实质性的工具包，将用于实际应用。第二个同样具有挑战性的课题，涉及通过玻尔兹曼方程研究多粒子系统，同样使用最近开发的技术工具。最后，现代生物形态学模型的研究有望为血管发生和相关生物过程的启动和涌现动力学阶段提供新的见解。该项目还将为研究生提供研究培训机会。PI将调查一些关于松弛、动力学方程和生物力学模式形成的关键开放性问题。特别感兴趣的是关于不连续无粘周期波和多维液压冲击的非线性时间渐近稳定性的开放问题，稳定玻尔兹曼方程的不变流形，以及具有守恒定律的生物形态学模型中图灵模式的分岔和稳定性。该项目的目标是发展新的理论方法，以解决在浅水流动，气体动力学和形态发生方面尚未解决的实际问题。方法包括有限维和无限维动力系统工具与来自冲击和双曲守恒定律的专门技术的混合：特别是Kreiss对称器和伪微分技术。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Roll waves are large-scale periodic pulses that form in a rapidly moving inclined flow, such as on a canal or dam spillway. As potentially destructive phenomena, it is important from a hydroengineering standpoint to understand the conditions and the characteristics for their occurrence. This translates into questions on the stability, or persistence under small disturbances, of such waves. Until recently such questions were out of reach of existing mathematical tools. The PI and collaborators have developed a substantial toolkit for this study, which will be brought to bear on practical applications. A second, equally challenging topic, concerns the study of many-particle systems via Boltzmann's equation, again using recently developed technical tools. Finally, the study of modern biomorphology models promises to give new insights into initiation and emergent dynamics phases of vasculogenesis and related biological processes. The project will also provide research training opportunities for graduate students. The PI will investigate a selection of key open questions on relaxation, kinetic equations, and biomechanical pattern formation. Of particularly interest are open questions on nonlinear time-asymptotic stability of discontinuous inviscid periodic waves and multi-dimensional hydraulic shocks, invariant manifolds for steady Boltzmann’s equation, and bifurcation and stability of Turing patterns in biomorphology models possessing conservation laws. The objective of the project is the development of new theoretical approaches to unresolved questions of practical interest in shallow-water flow, gas dynamics, and morphogenesis. Methods include a blend of finite- and infinite-dimensional dynamical systems tools with specialized techniques coming from shocks and hyperbolic conservation laws: in particular, Kreiss symmetrizer and pseudodifferential techniques.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.