Stationary self-similar profiles for the two-dimensional inviscid Boussinesq equations

TL;DR

This paper investigates the existence of stationary self-similar solutions to the 2D inviscid Boussinesq equations in a half-plane, establishing conditions for both their existence and non-existence.

Contribution

It provides a comprehensive analysis of ($- ext{alpha}$)-homogeneous solutions, revealing when such solutions can or cannot exist, including regular and singular profiles.

Findings

Non-existence of certain self-similar solutions

Existence of solutions with regular profiles

Existence of solutions with singular profiles

Abstract

We consider ($-\alpha$)-homogeneous solutions (stationary self-similar solutions of degree $-\alpha$) to the two-dimensional inviscid Boussinesq equations in a half-plane. We show their non-existence and existence with both regular and singular profile functions.