A mathematical analogy between certain hollow vortex and bubble flows - existence, uniqueness, and identical boundary shapes

Abstract

We study two physically distinct steady free-boundary problems in a two-dimensional inviscid fluid: a hollow vortex in a constant-vorticity flow subject to simple shear, third order and linear strain at infinity, without surface tension, and an irrotational flow past a constant-pressure bubble subject to surface tension, linear strain, and circulation at infinity. Building on the mathematical vortex-bubble correspondence recently developed by Crowdy and Tanveer, we show that both problems are linked by a common mathematical formulation. For the vortex problem, we extend the sequence-space fixed-point approach of Crowdy and Tanveer to obtain new local existence, uniqueness, and analyticity results. For the bubble problem, we develop a separate existence theory, reducing it to a nonlinear singular integral operator fixed-point problem and deriving a scalar solvability condition that defines a realanalytic surface in the physical parameter space. On this surface we obtain locally unique bubble solutions with real-analytic boundaries and holomorphically extendible velocity fields. Finally, we prove that the scalar solvability condition is equivalent to a functional conformal mapping identity. The scalar condition between the parameters is enough for the vortex and bubble problems to have the same free-boundary shape, despite their different exterior flows.