Asymptotic patterns and singular limits in nonlinear evolution problems

几个世纪以来，偏微分方程（PDE）在科学和工程中发挥了重要作用，通过构造解和分析具有足够精度的特征来解释所考虑的现象。在许多情况下，该理论可以胜任这项任务，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，相关偏微分方程的正则解可能开始集中在低维区域，最终爆破。找到有趣的渐近模式或奇点的解决方案，这个建议的主题，往往是一个困难的问题。近年来，我们已经开发出胶合技术，以实现这一点在经典问题的椭圆和抛物方程。在不可压缩流体中，许多基本现象还没有被数学证明，我们相信胶合方法可以揭示惊人的特征。我们将重点关注集中奇点形成挑战中的四个主题。我们建议阐明的基本法律的动态的涡laments的欧拉方程，建立真正的解决方案，与他们一致。特别地，我们要建立1904年Da Rios的“涡丝猜想”和1858年Helmholtz的涡环蛙跳定律。在经典的2D水波问题的恒定涡量，我们建议建立悬垂行波通过一种机制类似于去奇异CMC表面。我们还提出了长期的涡旋和尖锐的前的相互作用的演变和相关的爆破方案，包括在凯勒-西格尔趋化系统的II型爆破解决方案的分析。

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.