On stable solutions to the Euler equations in convex planar domains

TL;DR

This paper investigates the regularity and geometric structure of stable solutions to the Euler equations in convex planar domains, revealing conditions under which solutions are Holder continuous with convex level curves and describing stagnation points.

Contribution

It provides new insights into the regularity, convexity, and stagnation point structure of stable Euler solutions in convex domains, under specific nondegeneracy conditions.

Findings

Stable solutions are Holder continuous with convex level curves.

Detailed description of stagnation points set.

Solutions with nice level set topology are in the L^-strong closure of the coadjoint orbit.

Abstract

In convex planar domains, given an initial vorticity with one sign, we study the regularity and geometric properties of the dynamically stable solutions to the Euler equations in the coadjoint orbit of the initial vorticity. These flows have elliptic stagnation points. Under some nondegeneracy conditions on the data, we show they are Holder continuous and have convex level curves. We also give a detailed description for the set of stagnation points. If the initial vorticity has nice level set topology, these stable solutions are in the L^\infty-strong closure of the coadjoint orbit. We also demonstrate the sharpness of most assumptions we made.