Asymptotic patterns and singular limits in nonlinear evolution problems

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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.