AI 中文总结
本文针对带一般涡量的二维行进重力水波，利用改进的Weiss型单调性公式与Almgren型频率公式，证明了严格双侧线性增长区域外不存在全密度退化驻点，并推广了旋转波的Stokes猜想。
AI 中文摘要
本文重新研究了带涡量的二维行进重力水波在驻点附近自由表面的奇异渐近行为。我们证明了在严格的双侧线性增长区域之外，不存在全密度退化驻点。我们的主要工具是改进的Weiss型单调性公式和改进的Almgren型频率公式，二者结合提供了一种完全避免使用Bessel型微分不等式的新方法，而Bessel型微分不等式是以往文献（Ann. I. H. Poincaré-AN, 29, 861--885, 2012）中证明全密度退化驻点不存在的核心工具。作为结果，我们得到了任意维度下频率归一化爆破的一致界，以及二维情形下的强收敛性。作为应用，我们将旋转波的Stokes猜想推广到更广泛的涡量分布类。
英文摘要
In this paper, we revisit the singular asymptotics of the free surface near stagnation points for two-dimensional traveling gravity water waves with vorticity. We prove the nonexistence of full-density degenerate stagnation points beyond the strict two-sided linear growth regime. Our main tools are a modified Weiss-type monotonicity formula and a modified Almgren-type frequency formula. Together, they provide a new approach that completely avoids the use of a Bessel-type differential inequality, which is an essential tool used in the previous literature to prove the nonexistence of full-density degenerate stagnation points (Ann. I. H. Poincaré-AN, 29, 861--885, 2012). As consequences, we obtain uniform bounds for frequency-normalized blow-ups in arbitrary dimension and strong convergence in dimension two. As an application, we extend the Stokes conjecture for rotational waves to a broader class of vorticity distributions.
Comments26 Pages. Comments and suggestions are welcome
