The singularity at degenerate points in steady axisymmetric compressible free surface flows with gravity

TL;DR

This paper analyzes the singular behavior of free boundaries in three-dimensional axisymmetric compressible flows with gravity, proving asymptotic behaviors at degenerate points and establishing new monotonicity formulas for quasilinear Bernoulli free boundary problems.

Contribution

It introduces the first monotonicity formulas for quasilinear Bernoulli free boundary problems and resolves the Stokes conjecture in a generalized compressible setting.

Findings

Asymptotic behavior at degenerate points is characterized by Stokes corner flow.

Free boundary geometries include cusps and pointed bubbles.

New nonlinear frequency formula at flat points is established.

Abstract

In this paper, we analyze the singular shape of the free boundary at degenerate points in a three dimensional axisymmetric compressible gravity flow. For all possible degenerate points on the free surface, we prove that the only nontrivial asymptotic behavior of the free surface at the stagnation points away from the axis of symmetry is the Stokes corner flow. The possible geometries for free boundaries at the non-stagnation axis points are downward pointing or upward pointing cusps. At the origin, there are only two nontrivial asymptotics possible: the Garabedian's pointed bubble or a horizontal flat surface. The problem is associated with the analysis of the degenerate points of a quasilinear free boundary problem of the Bernoulli type, and the main obstacles are the absence of a Weiss-type monotonicity formula. To achieve our goal, we establish for the first time monotonicity formulas for quasilinear Bernoulli type free boundary problems. Our formula works both when the equation becomes singular and when the free boundary condition is degenerate. Moreover, we establish a new nonlinear frequency formula at the horizontal flat points at the origin and integrate it with the compensated compactness theory for Euler equations, ensuring the strong convergence of variational solutions. Our results resolve the Stokes conjecture [Mathematical and Physical Papers, Vol. I., 1880] in a generalized compressible, three dimensional axisymmetric framework. In addition, it can also be realized as a compressible counterpart to V\v{a}rv\v{a}ruc\v{a} and Weiss [Comm. Pure Appl. Math., (67), 2014]. Our approach is completely new and gives a systematic approach for studying singularities of a singular Bernoulli type quasilinear free boundary problem.