AI 中文总结
研究证明Beltrami流上存在全局稳定双周期重力-毛细波。核心方法是将稳态水波问题转化为特定分岔问题并应用全局分岔论证。主要贡献是首个真正三维表面波严格存在性结果。
AI 中文摘要
我们证明了Beltrami流上存在一族全局稳定的双周期重力-毛细波。这是关于真正三维表面波（无论有无涡度）在接近简单显式解的微扰区域之外的首个严格存在性结果。证明基于将稳态水波问题重新表述为‘恒等加紧’形式的分岔问题，并在Hölder空间中应用全局分岔论证。主要挑战在于分岔点处线性化的核是二维的，并且两个核元素对于获得真正的三维解都是必要的。由于这阻止了使用经典全局分岔理论，我们使用从层流分岔的局部解族的参数化引入了分岔问题的新颖重新表述。在这个新参数空间中，我们应用解析全局分岔理论的变体。沿着分支，我们随后证明了一个更尖锐的爆破替代方案，即表面梯度在$C^{0,\gamma}$中的爆破。
英文摘要
We prove the existence of a global family of steady, doubly periodic gravity-capillary waves on Beltrami flows. This is the first rigorous existence result for genuinely three-dimensional inviscid surface waves, with or without vorticity, beyond the perturbative regime close to simple explicit solutions. The proof is based on reformulating the steady water wave problem as a bifurcation problem of the form `identity plus compact' and applying a global bifurcation argument in Hölder spaces. The main challenge is that the kernel of the linearisation at the bifurcation point is two-dimensional, and that both kernel elements are necessary to obtain genuinely three-dimensional solutions. Since this prevents the use of classical global bifurcation theory, we introduce a novel reformulation of the bifurcation problem using the parameterisation of a local family of solutions bifurcating from laminar flow. In this new parameter space, we apply a variation of analytic global bifurcation theory. Along the branch, we then prove a sharper blow-up alternative, namely blow-up of the surface gradient in $C^{0,γ}$.
Comments48 pages, 4 figures
