The non-existence of horizontally flat singularity for steady axisymmetric free surface flows near stagnation points

TL;DR

This paper proves that horizontally flat singularities cannot occur at stagnation points in steady axisymmetric gravity flows, narrowing the possible asymptotic behaviors to Stokes corners and horizontal cusps.

Contribution

It demonstrates the non-existence of horizontally flat singularities at stagnation points in such flows, refining the understanding of flow asymptotics near stagnation points.

Findings

Horizontally flat singularities are impossible at stagnation points.

Stokes corner and horizontal cusp are the only possible asymptotics.

Analysis suggests similar singular profiles in rotational gravity flows and 2D waves with vorticity.

Abstract

In a recent research on degenerate points of steady axisymmetric gravity flows with general vorticity, it has been shown that the possible asymptotics near any stagnation point must be the "Stokes corner", the "horizontal cusp", or the "horizontal flatness" (Theorem 1.1, Du, Huang, Pu, Commun. Math. Phys., 400, 2137-2179, 2023). In this paper, we focus on the horizontally flat singularity and show that it is not possible, and therefore the "Stokes corner" and the "cusp" are the only possible asymptotics at the stagnation points. The basic idea of our proof relies on a perturbation of the frequency formula for the two-dimensional problem (Varvaruca, Weiss, Acta Math., 206, 363-403, 2011). Our analysis also suggests that, for steady axisymmetric rotational gravity flows, the singular asymptotic profiles at stagnation points are similar to the scenario observed in two-dimensional waves with vorticity (Varvaruca, Weiss, Ann. I. H. Poincare-AN, 29, 861-885, 2012).