Analytic Structure of Stationary Flows of an Ideal Fluid with a Stagnation Point

TL;DR

This paper analyzes the structure of 2D stationary ideal fluid flows with a stagnation point, introducing a framework to describe flow lines analytically despite degeneracies at the stagnation point.

Contribution

It extends previous work by adapting the analytic flow line description to flows with a stagnation point, including the position of the stagnation point as an unknown in the solution.

Findings

Established a function space framework for degenerate elliptic equations.

Proved local existence and uniqueness of solutions near circular flows.

Demonstrated solutions depend analytically on parameters.

Abstract

The flow of an ideal fluid possesses a remarkable property: despite limited regularity of the velocity field, its particle trajectories are analytic curves. In our previous work, this fact was used to introduce the structure of an analytic Banach manifold in the set of 2D stationary flows having no stagnation points. The main feature of our description was to regard the stationary flow as a collection of its analytic flow lines, parameterized non-analytically by values of the stream function $\psi$. In this work, we adapt this description to the case of 2D stationary flows which have a single elliptic stagnation point. Namely, we consider flows in a domain bounded by the graph of analytic function $\rho = b(\varphi)$, where $(\rho,\varphi)$ are polar coordinates centred at the origin. The position $p$ of the stagnation point is an unknown and must be included in the solution. In polar coordinates $(r,\theta)$ centred at $p$, the flow lines are described by graphs of $r=a(\psi,\theta)$, where $a$ is a `partially-analytic' function (analytic in $\theta$, of finite regularity in $\psi$). The equation of stationary flow $\Delta \psi = F(\psi)$ is transformed to the quasilinear elliptic equation $\Xi(a) = F(\psi)$ for the flow lines. The analysis is complicated by the fact that the ellipticity of $\Xi$ degenerates at the stagnation point. We introduce function spaces for the partially-analytic family of flow lines, modelled on the weighted Kondratev spaces, appropriate for the degenerate setting. The equation of stationary flow is thus regarded as an analytic operator equation in complex Banach spaces, with local solution given by the implicit function theorem. In particular, we show that near the circular flow of constant vorticity, the equation has unique solution $p, a(\psi,\theta)$ depending analytically on parameters $b(\varphi)$ and $F(\psi)$.