毛细重力波携带涡旋的存在性与稳定性
Existence and stability of capillary-gravity wave-borne vortices
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中文总结 AI 辅助
本文证明有限深二维水体中毛细重力波可携带点涡旋（亚临界弗劳德数、超临界邦德数下存在且条件轨道稳定）及空腔涡旋，波速均为O(1)。
中文摘要 AI 辅助
本文研究了在有限深度的二维水体中传播且在其内部携带涡旋的波的存在性与稳定性。这些波受重力作用，位于恒定气压的空气区域下方，气-水界面为自由边界，并沿该边界纳入毛细效应。我们考虑两类涡旋：点涡旋，其中涡度形式上是狄拉克δ函数；以及空腔涡旋，其中涡核是流体域外恒定压力的区域，且绕其存在非零环量。首先，我们证明对于任何亚临界弗劳德数和超临界邦德数，稳态小振幅波携带点涡旋均存在，并进一步证明它们具有条件轨道稳定性。其次，通过涡旋去奇异化论证，我们构造了毛细重力波携带的空腔涡旋。值得注意的是，这两类涡旋的波速均为O(1)。
英文摘要
In this paper, we consider the existence and stability of waves progressing through a finite-depth two-dimensional body of water that carry a vortex in their bulk. The waves are acted upon by gravity, and sit below a region of air at constant pressure, with the air--water interface being a free boundary along which capillary effects are incorporated. We consider two classes of vortices: point vortices, where formally the vorticity is a Dirac $δ$, and hollow vortices, where the vortex core is a region of constant pressure outside the fluid domain and about which there is a nonzero circulation. First, we prove that steady small-amplitude wave-borne point vortices exist for any subcritical Froude number and supercritical Bond number, and then show they are conditionally orbitally stable. Second, through a vortex desingularization argument, we construct capillary-gravity wave-borne hollow vortices. Notably, the wave speed is $O(1)$ for both these families.
发表机构
- University of Washington Tacoma（华盛顿大学塔科马分校）
- University of Missouri（密苏里大学）
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