Symmetric doubly periodic gravity-capillary waves with small vorticity (Q6540262)

This manuscript addresses a significant problem in fluid dynamics concerning the existence and behaviour of small-amplitude doubly periodic water waves influenced by both gravity and surface tension in a three-dimensional setting.\N\NThe primary problem tackled in this paper involves the construction and analysis of steady-state gravity-capillary waves in a three-dimensional fluid domain with a free surface. These waves are influenced by gravity and surface tension, and the fluid itself exhibits small, nonzero vorticity. This represents a significant departure from the classical studies of irrotational water waves, as the inclusion of vorticity introduces additional complexity to the wave dynamics and the mathematical formulation.\N\NTo address this problem, the authors employ a mathematical framework that leverages bifurcation theory and advanced techniques in the analysis of partial differential equations. They construct solutions by assuming a small perturbation from a uniform flow, and demonstrate that these solutions can exhibit a doubly periodic structure in the horizontal plane. The methods used include a global representation of the vorticity as the cross product of two gradients, inspired by techniques from magnetohydrodynamics, specifically the work of \textit{D. Lortz} [Z. Angew. Math. Phys. 21, 196--211 (1970; Zbl 0198.30505)] on magnetohydrostatic equilibria. The authors successfully adapt these ideas to the fluid dynamics context, overcoming significant mathematical challenges posed by the non-elliptic nature of the free boundary problem and the loss of regularity under Fréchet differentiation.\N\NThe main findings of this paper include the establishment of the existence of small-amplitude, symmetric, doubly periodic gravity-capillary waves with nonzero vorticity. The authors show that these waves bifurcate from uniform flows and maintain periodicity in both horizontal directions. They also provide a detailed analysis of the wave structure, demonstrating that the presence of surface tension is crucial for the existence of these solutions. The mathematical results are corroborated by rigorous proofs that include a detailed discussion of the local bifurcation theory and the specific conditions under which the solutions exist.\N\NThe significance of this research lies in its contribution to the understanding of complex wave phenomena in fluid dynamics. The construction of three-dimensional gravity-capillary waves with vorticity expands the theoretical framework for studying water waves, providing new insights into how vorticity affects wave formation and stability. This work not only advances the mathematical theory of water waves but also has potential for practical applications in oceanography and engineering, where understanding wave behaviour is critical for modelling and predicting sea conditions.