AI 中文总结
研究具有恒定涡度的悬垂周期重力水波的存在性，通过共形映射公式和全局分岔框架，辅以计算机辅助证明，首次严格证明其存在，还证实了相关全局解分支及解决了特定猜想。
AI 中文摘要
我们为具有恒定涡度的悬垂周期重力水波提供了一个构造性存在性证明。通过引入共形映射公式，我们对流体域进行参数化以适应多值表面高度，绕过传统域扁平化框架中出现的坐标奇点。我们在固定的宏观物理参数下操作，并采用全局分岔框架，辅以基于牛顿 - 康托罗维奇定理的计算机辅助证明，以构造性地证明精确解分支的存在。至关重要的是，这使我们能够首次在固定的\(O(1)\)物理参数下，沿着从平坦状态出发的有限深度全局分岔分支，严格证明悬垂周期重力水波的存在性。此外，我们的分析证实了存在一个过渡到悬垂剖面的全局解分支，并解决了康斯坦丁、施特劳斯和瓦尔瓦鲁卡关于该分支在波谷线处通过物理自相交的拓扑终止的猜想。
英文摘要
We provide a constructive existence proof for overhanging periodic gravity water waves with constant vorticity. By introducing a conformal mapping formulation, we parameterize the fluid domain to accommodate multi-valued surface heights, bypassing coordinate singularities that arise in traditional domain-flattening frameworks. We operate at fixed, macroscopic physical parameters and employ a global bifurcation framework, supplemented by a computer-assisted proof based on the Newton-Kantorovich theorem, to constructively prove the existence of the branch of exact solutions. Crucially, this allows us to provide the first rigorous existence proof of overhanging periodic gravity water waves obtained along a finite-depth global bifurcation branch from the flat state at fixed $O(1)$ physical parameters. Moreover, our analysis confirms the existence of a global solution branch that transitions to overhanging profiles and resolves the conjecture of Constantin, Strauss, and Varvaruca regarding the topological termination of this branch via physical self-intersection at the trough line.
