Asymptotic patterns and singular limits in nonlinear evolution problems

非线性演化问题中的渐近模式和奇异极限

• 批准号: EP/Z000394/1
• 负责人: Manuel Del Pino
• 金额: $ 197.71万
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FZ000394%2F1
• 关键词: Asymptotic; patterns; singular; limits; nonlinear

For centuries, partial differential equations (PDE) have played an important role in science and engineering by constructing solutions and analysing features with sufficient accuracy to explain the phenomena under consideration. In many cases, the theory is up to that task, but more recently it has been challenged to account for increasingly subtle nonlinear natural phenomena. When parameters of the model, or time, approach critical values, regular solutions of the associated PDE may begin to concentrate at lower dimensional regions, eventually blowing up. Finding solutions with interesting asymptotic patterns or singularities, the topic of this proposal, is often a difficult problem. In recent years, we have developed gluing techniques to achieve this in classical problems in elliptic and parabolic equations. In incompressible fluids, many fundamental phenomena have not been mathematically justied, and we believe that gluing methods can lead to the unveiling of striking features. We will focus on four topics in the concentration-singularity formation challenge. We propose to elucidate fundamental laws on the dynamics of vortex laments of the Euler equations, building true solutions in agreement with them. In particular, we want to establish the 1904 Da Rios "vortex filament conjecture" and 1858 Helmholtz leapfrogging law for vortex rings. In the classical 2d water wave problem with constant vorticity, we propose to build overhanging travelling waves through a mechanism similar to desingularization in CMC surfaces. We also propose the analysis of long-term vortex and sharp-fronts interaction-evolution and associated blow-up scenarios, including type II blow-up solutions in the Keller-Segel chemotaxis system.

几个世纪以来，偏微分方程 (PDE) 通过以足够的精度构建解决方案和分析特征来解释所考虑的现象，在科学和工程领域发挥着重要作用。在许多情况下，该理论能够胜任这项任务，但最近它受到了解释日益微妙的非线性自然现象的挑战。当模型的参数或时间接近临界值时，相关偏微分方程的常规解可能开始集中在较低维区域，最终爆炸。寻找具有有趣的渐近模式或奇点的解决方案（本提案的主题）通常是一个难题。近年来，我们开发了粘合技术来解决椭圆方程和抛物线方程的经典问题。在不可压缩流体中，许多基本现象尚未在数学上得到证明，我们相信粘合方法可以揭示显着的特征。我们将重点关注集中奇点形成挑战中的四个主题。我们建议阐明欧拉方程涡流动力学的基本定律，并建立与其一致的真实解。特别是，我们想建立1904年达里奥斯的“涡丝猜想”和1858年涡环的亥姆霍兹蛙跳定律。在具有恒定涡度的经典二维水波问题中，我们建议通过类似于 CMC 表面去奇异化的机制来构建悬垂行波。我们还提出了对长期涡流和锐锋相互作用演化以及相关爆炸场景的分析，包括 Keller-Segel 趋化系统中的 II 型爆炸解决方案。