Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc

TL;DR

This paper proves the existence of time periodic solutions for the generalized surface quasi-geostrophic equation in a disc, using bifurcation theory and novel analytical techniques to handle boundary effects and complex frequency calculations.

Contribution

It introduces a new approach to establish time periodic solutions for the generalized SQG equation in a bounded domain, overcoming challenges with Green functions and frequency analysis.

Findings

Existence of countably many bifurcating curves from radial patches.

Development of a splitting method for Green function analysis.

Use of Sneddon's formula for frequency integral representation.

Abstract

In this paper we address the existence of time periodic solutions for the generalized inviscid SQG equation in the unit disc with homogeneous Dirichlet boundary condition when $\alpha\in (0,1)$. We show the existence of a countable family of bifurcating curves from the radial patches. In contrast with the preceding studies in active scalar equations, the Green function is no longer explicit and we circumvent this issue by a suitable splitting into a singular explicit part (which coincides with the planar one) and a smooth implicit one induced by the boundary of the domain. Another problem is connected to the analysis of the linear frequencies which admit a complicated form through a discrete sum involving Bessel functions and their zeros. We overcome this difficulty by using Sneddon's formula leading to a suitable integral representation of the frequencies.