Large-amplitude periodic solutions to the steady Euler equations with piecewise constant vorticity

TL;DR

This paper constructs and analyzes large-amplitude steady solutions to the 2D Euler equations with piecewise constant vorticity, revealing complex interface behaviors and stagnation points through bifurcation theory and novel formulations.

Contribution

It introduces a new local formulation for multi-layer Euler flows that accommodates overhanging profiles and stagnation points, advancing understanding of complex steady solutions.

Findings

Solutions terminate with stagnation or conformal breakdown.

Interface can form corners or develop thin regions.

Numerical evidence supports diverse interface behaviors.

Abstract

We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation theory, we rigorously construct curves of solutions that terminate either with stagnation on the interface or when the conformal equivalence between one of the layers and a strip breaks down in a $C^1$ sense. We give numerical evidence that, depending on parameters, these occur either as a corner forming on the interface or as one of the layers developing regions of arbitrarily thin width. Our proof relies on a novel formulation of the problem as an elliptic system for the velocity components in each layer, conformal mappings for each layer, and a horizontal distortion which makes these mappings agree on the interface. This appears to be the first local formulation for a multi-layer problem which allows for both overhanging wave profiles and stagnation points.