Title: Steady bubbles and drops in inviscid fluids Show Chinese title

Title: 无粘性流体中的稳定气泡和液滴

Abstract: We construct steady non-spherical bubbles and drops, which are traveling wave solutions to the axisymmetric two-phase Euler equations with surface tension, whose inner phase is a bounded connected domain. The solutions have a uniform vorticity distribution in this inner phase and they have a vortex sheet on its surface. Our construction relies on a perturbative approach around an explicit spherical solution, given by Hill's vortex enclosed by a spherical vortex sheet. The construction is sensitive to the Weber numbers describing the flow. At critical Weber numbers, we perform a bifurcation analysis utilizing the Crandall-Rabinowitz theorem in Sobolev spaces on the 2-sphere. Away from these critical numbers, our construction relies on the implicit function theorem. Our results imply that the model containing surface tension is richer than the ordinary one-phase Euler equations, in the sense that for the latter, Hill's spherical vortex is unique (modulo translations) among all axisymmetric simply connected uniform vortices of a given circulation. Show Chinese abstract

Abstract: 我们构造了稳定的非球形气泡和液滴，它们是具有表面张力的轴对称两相欧拉方程的行波解，其内相是一个有界连通区域。 这些解在内相中具有均匀的涡量分布，并且在表面上具有涡量片。 我们的构造依赖于围绕一个显式球形解的摄动方法，该解由被球形涡量片包围的希尔涡流给出。 该构造对描述流动的韦伯数敏感。 在临界韦伯数处，我们利用索伯列夫空间上的Crandall-Rabinowitz定理进行分岔分析。 远离这些临界数时，我们的构造依赖于隐函数定理。 我们的结果表明，包含表面张力的模型比普通的单相欧拉方程更丰富，在这种意义上，对于后者，给定环量下所有轴对称单连通均匀涡量中，希尔球形涡流是唯一的（模平移）。