Asymptotic patterns and singular limits in nonlinear evolution problems

几个世纪以来，偏微分方程(PDE)在科学和工程中扮演着重要的角色，它以足够的精度构造解和分析特征，以解释所考虑的现象。在许多情况下，这一理论都能胜任这项任务，但最近，它在解释日益微妙的非线性自然现象方面受到了挑战。当模型的参数或时间接近临界值时，相关PDE的规则解可能开始集中在低维区域，最终爆炸。寻找具有有趣的渐近模式或奇点的解决方案，这是这个提议的主题，通常是一个困难的问题。近年来，我们在椭圆和抛物型方程的经典问题中发展了粘合技术来实现这一点。在不可压缩流体中，许多基本现象还没有得到数学上的证明，我们相信粘合方法可以导致显著特征的揭示。我们将集中讨论集中-奇点形成挑战中的四个主题。我们建议阐明欧拉方程涡旋动力学的基本规律，并建立与之相一致的真实解。特别是，我们想要建立1904年Da Rios“涡丝猜想”和1858年Helmholtz涡环跳跃定律。在经典的二维常涡度水波问题中，我们提出了通过一种类似于CMC表面去奇异的机制来构造悬垂行波。我们还建议分析长期涡旋和锋面的相互作用-演化和相关的爆破情景，包括Keller-Segel趋化系统中的第二类爆破解决方案。

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.