某些中空涡与气泡流之间的数学类比——存在性、唯一性及相同边界形状
A mathematical analogy between certain hollow vortex and bubble flows - existence, uniqueness, and identical boundary shapes
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- Ohio State University（俄亥俄州立大学）
- Imperial College London（帝国理工学院）
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中文总结 AI 辅助
本文通过数学对应关系证明二维无粘流中的中空涡与恒压气泡问题共享同一数学表述，并分别建立存在性与唯一性理论，最终揭示两者在特定参数条件下具有相同的自由边界形状。
中文摘要 AI 辅助
我们研究二维无粘流体中两个物理上不同的稳态自由边界问题：一个是在无穷远处受简单剪切、三阶应变和线性应变作用且无表面张力的常涡量流中的中空涡，另一个是在无穷远处受表面张力、线性应变和环量作用的无旋流经过恒压气泡。基于Crowdy和Tanveer最近发展的数学涡-气泡对应关系，我们证明这两个问题通过一个共同的数学表述相联系。对于涡问题，我们扩展了Crowdy和Tanveer的序列空间不动点方法，获得了新的局部存在性、唯一性和解析性结果。对于气泡问题，我们发展了一个独立的存在性理论，将其归结为非线性奇异积分算子不动点问题，并推导出一个标量可解性条件，该条件在物理参数空间中定义了一个实解析曲面。在此曲面上，我们获得具有实解析边界和可全纯延拓速度场的局部唯一气泡解。最后，我们证明该标量可解性条件等价于一个函数共形映射恒等式。参数之间的标量条件足以使涡问题和气泡问题具有相同的自由边界形状，尽管它们的外部流动不同。
英文摘要
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.
