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

双曲、动力学和对流-反应-扩散系统的调制、动力学和图案形成前沿

• 批准号: 2154387
• 负责人: Kevin Zumbrun
• 金额: $ 23.57万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154387&HistoricalAwards=false
• 关键词: Frontiers; Modulation; Dynamics; Pattern; Formation

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.

滚动波是在快速移动的倾斜流中形成的大规模周期性脉冲，例如在运河或大坝溢洪道上。作为潜在的破坏性现象，从水力工程的角度来看，了解其发生的条件和特征很重要。这转化为有关此类波浪的稳定性或持久性的问题。直到最近，此类问题仍未达到现有的数学工具。 PI和合作者为这项研究开发了一个实质性的工具包，该工具包将在实际应用上携带。第二个同样具有挑战性的话题涉及通过Boltzmann方程对许多粒子系统的研究，并再次使用最近开发的技术工具。最后，现代生物形态模型的研究有望为血管生成和相关生物学过程的开始和新兴动态阶段提供新的见解。该项目还将为研究生提供研究培训机会。 PI将研究有关放松，动力学方程和生物力学模式形成的关键开放问题。特别令人感兴趣的是关于不连续的无关周期性波和多维氢化性冲击的非线性时间 - 对称稳定性的开放问题，用于稳定的Boltzmann方程的不变歧管以及在具有保护法的生物形态学模型中的Turing模式的分叉和稳定性。该项目的目的是开发新的理论方法，以解决浅水流，气体动态和形态发生的实际兴趣问题。方法包括有限和无限维动态系统工具的混合物与来自冲击和双曲保存法的专业技术的混合物：特别是Kreiss对称和伪差异技术。该奖项反映了NSF的立法任务，并通过使用基础的智力综述和广泛的评估来评估支持NSF的立法任务，并被认为是诚实的。