平面四点涡的碰撞
Collisions of Four Point Vortices in the Plane
摘要：
本文讨论的问题的存在性（不一定自相似）解决方案的4-涡问题，导致全部或部分碰撞。首先，我们表明，能量的考虑意味着，对于一般的$N$ -涡问题，维里为零是一个必要条件的解决方案，都朝着总碰撞和满足一定的规则性条件。对于假定有界的演化，给出了渐近部分碰撞构形的分类。这种分类取决于惯性和涡度的考虑。对于不一定有界的演化，我们讨论了部分和（非正规）总碰撞的维里不为零时，涡度的一般条件成立之间的关系。最后，我们给出了一个正则变换，对于一类满足涡度限制的4涡系统，允许正式应用平均原理，以减少当两个涡度接近二元碰撞时的动态。约化系统是一个一自由度汉密尔顿系统。
英文摘要：
This paper addresses the question of existence of (not necessarily self-similar) solutions to the 4-vortex problem that lead to total or partial collision. We begin by showing that energy considerations alone imply that, for the general $N$-vortex problem, the virial being zero is a necessary condition for a solution to both evolve towards total collision and satisfy certain regularity condition. For evolutions assumed to be bounded, a classification for asymptotic partial collision configurations is offered. This classification depends on inertia and vorticity considerations. For non-necessarily bounded evolutions, we discuss the relationship between partial and (non-regular) total collisions when the virial is not zero and a generic condition on the vorticities holds. Finally, we give a canonical transformation that, for a class of 4-vortex systems satisfying a restriction on the vorticities, allows to formally apply the averaging principle in order reduce the dynamics when two of the vorticities are near a binary collision. The reduced system is a one-degree of freedom hamiltonian system.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
