发表机构
Institut für Analysis und Numerik, Universität Münster; ETH Zürich, Department of Mathematics（明斯特大学分析与数值研究所; 苏黎世联邦理工学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明无重力三维两相欧拉方程在表面张力下存在螺旋上升气泡解，构造了对称破缺的螺旋相对平衡分支，并给出了无表面张力时此类运动的不存在性定理。
AI 中文摘要
浮力竖直向上，然而在没有外力的情况下，上升的气泡在水中可能沿螺旋轨迹运动。这一现象已被列奥纳多·达·芬奇观察到。我们证明，无重力、三维、两相、不可压缩且具有表面张力的欧拉方程已经允许此类运动。对于每个非负的内外密度比和每个足够小的正轴向韦伯数，我们构造了具有紧致界面和分相无旋流动的光滑近球形气泡和液滴。它们在沿固定轴平移并绕该轴旋转的参考系中是静止的，而它们的质心偏离该轴，因此在实验室坐标中描绘出真正的圆形螺旋线。这些螺旋相对平衡构成一个对称破缺分支，该分支源于轴对称的直线平移解族，并由球谐函数的二阶方位角数一模产生。数学构造利用了超定自由边界公式、其变分结构以及SO(2)-等变Lyapunov-Schmidt约化。在离散的密度比集合上会出现共振形式的特殊困难，包括经常研究的内部密度为零的真空情形。与我们的存在性结果互补，在没有表面张力且内部密度不大于外部密度的情况下，我们证明该模型既不允许非平凡的平移相对平衡，也不允许具有螺旋质心轨迹的相对平衡。作为副产品，我们证明对于齐次三维欧拉方程，球形拓扑的每个正则有限能量平稳涡面（两侧均为无旋流动）都是平凡的。
英文摘要
Buoyancy points straight upward, yet in the absence of external forces, a rising air bubble may trace a spiral through the water. This phenomenon was already observed by Leonardo da Vinci. We show that the gravity-free, three-dimensional two-phase incompressible Euler equations with surface tension already admit such a motion. For every nonnegative inner-to-outer density ratio and every sufficiently small positive axial Weber number, we construct smooth near-spherical bubbles and drops with compact interfaces and phasewise irrotational flow. They are stationary in a frame translating along and rotating about a fixed axis, while their centroids lie off that axis and therefore trace genuine circular helices in laboratory coordinates. These helical relative equilibria form a symmetry-breaking branch that issues from the axisymmetric family of rectilinearly translating solutions and is generated by the spherical harmonic mode of degree two and azimuthal number one. The mathematical construction exploits an overdetermined free-boundary formulation, its variational structure, and an $\mathrm{SO}(2)$-equivariant Lyapunov-Schmidt reduction. Particular difficulties in the form of resonances arise at a discrete set of density ratios, including the frequently studied vacuum case with vanishing inner density. Complementary to our existence result, in the absence of surface tension and if the inner density is not larger than the outer density we prove that the model admits neither nontrivial translating relative equilibria nor relative equilibria with helical centroid trajectories. As a byproduct, we show that every regular finite-energy stationary vortex sheet of spherical topology for the homogeneous three-dimensional Euler equations, with irrotational flow on both sides, is trivial.
