Singularities of Steady Axisymmetric Free Surface Flows with Gravity

Abstract

We consider a steady axisymmetric solution of the Euler equations describing the irrotational flow without swirl of an incompressible inviscid fluid acted on by gravity and with a free surface. We analyze stagnation points as well as points on the axis of symmetry. At points on the axis of symmetry that are not stagnation points, constant velocity motion is the only blowup profile consistent with the invariant scaling of the equation. This suggests the presence of downward‐pointing cusps at those points. At stagnation points on the axis of symmetry, the unique blowup profile consistent with the invariant scaling of the equation is the Garabedian pointed bubble solution with water above air. Thus at stagnation points on the axis of symmetry with no water above the stagnation point, the invariant scaling of the equation cannot be the right scaling. A finer blowup analysis of the velocity field yields that in the case when the surface is described by an injective curve, the velocity field scales almost like and is asymptotically given by urn:x-wiley:10429832:media:cpa21514:cpa21514-math-0002 with a nonzero constant c0. The last result relies on a frequency formula in combination with a concentration compactness result for the axially symmetric Euler equations by Delort. While the concentration compactness result alone does not lead to strong convergence in general, we prove the convergence to be strong in our application. © 2014 Wiley Periodicals, Inc.