标题： 三维完全局域化重力毛细水波的存在性和条件能量稳定性 显示英文标题

标题： Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves

摘要： 在本文中，我们证明了具有强表面张力效应的三维水波的流体力学问题存在一个完全局部化的孤立波，该波在每个水平方向上都衰减到未扰动的水状态。 证明基于经典的变分原理，即这种类型的孤立波是能量的一个临界点，同时动量保持固定。 我们证明了在动量保持固定且较小的约束下，能量存在一个极小值。 因此，小振幅孤立波的存在性得到了保证，而且由于能量和动量都是守恒量，可以使用标准论证来确立所有极小值的整体稳定性。 然而，由于缺乏三维水波的全局适定性理论，"稳定性"的理解是有条件的。 显示英文摘要

摘要： In this paper we show that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction. The proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy subject to the constraint that the momentum is fixed. We prove the existence of a minimiser of the energy subject to the constraint that the momentum is fixed and small. The existence of a small-amplitude solitary wave is thus assured, and since the energy and momentum are both conserved quantities a standard argument may be used to establish the stability of the set of minimisers as a whole. `Stability' is however understood in a qualified sense due to the lack of a global well-posedness theory for three-dimensional water waves.