Mathematical analysis of bubble rings in ideal fluids

在本项目中，将研究环形气泡涡旋的演化。气泡旋涡是两相流体中出现的一种特殊旋涡。一个典型而引人入胜的例子是由海豚或白鲸在水中制造的气泡环。基本的数学模型由具有表面张力的两相欧拉方程给出。一个主要目标是建立稳定环运动而不改变其形状的彻底的数学结构，以及这些环的扰动。这种行波对于经典的欧拉方程是已知的，但对于依赖于表面张力的模型是未知的。特别令人感兴趣的是表面张力对环形状的作用，这将被研究。这个项目的第二个目标是了解如何利用表面张力的影响来严格证明一个或多个相互作用的气泡环的某些非线性运动规律。对经典欧拉方程的这种运动规律的理解很差，预计表面张力的正则化效应有助于从数学上驯服这个问题。

In this project, the evolution of toroidal bubble vortices is to be investigated. Bubble vortices are special vortices that occur in two-phase fluids. A typical and fascinating example is an air bubble ring in water created by dolphins or beluga whales. The underlying mathematical model is given by the two-phase Euler equations with surface tension. One major goal is a thorough mathematical construction of steady rings that move without changing their shape, and of perturbations of these. Such traveling waves are known for the classical Euler equations, but their existence is unknown for surface tension dependent models. Of particular interest is the role of the surface tension for the shape of the ring, which will be investigated. A second objective of this project is to understand how the effect of surface tension can be exploited to rigorously justify certain nonlinear motion laws of one or more interacting bubble rings. The understanding of such motion laws for the classical Euler equations is poor, and it is expected that the regularising effect of surface tension helps to mathematically tame the problem.