Time periodic solutions close to localized radial monotone profiles for the 2D Euler equations

TL;DR

This paper proves the existence of time periodic solutions near radial vortex profiles for the 2D Euler equations, distinguishing between focusing and defocusing cases, and introduces a new spectral analysis approach without requiring explicit radial profiles.

Contribution

It develops a novel, flexible method for finding time periodic solutions near radial vortices without explicit profiles, using spectral analysis of Sturm-Liouville problems.

Findings

Focusing case yields a countable family of symmetric bifurcating solutions.

Defocusing case shows scarcity of bifurcating solutions with lower symmetry.

Deep spectral structure related to positivity of integral operators is uncovered.

Abstract

In this paper, we address for the 2D Euler equations the existence of rigid time periodic solutions close to stationary radial vortices of type $f_0(|x|){\bf 1}_{\mathbb{D}}(x)$, with $\mathbb{D}$ the unit disc and $f_0$ being a strictly monotonic profile with constant sign. We distinguish two scenarios according to the sign of the profile: defocusing and focusing. In the first regime, we have scarcity of the bifurcating curves associated with lower symmetry. However in the focusing case we get a countable family of bifurcating solutions associated with large symmetry. The approach developed in this work is new and flexible, and the explicit expression of the radial profile is no longer required as in [41] with the quadratic shape. The alternative for that is a refined study of the associated spectral problem based on Sturm-Liouville differential equation with a variable potential that changes the sign depending on the shape of the profile and the location of the time period. Deep hidden structure on positive definiteness of some intermediate integral operators are also discovered and used in a crucial way. Notice that a special study will be performed for the linear problem associated with the first mode founded on Pr\"{u}fer transformation and Kneser's Theorem on the non-oscillation phenomenon.