On stability estimates for the inviscid Boussinesq equations

TL;DR

This paper investigates the stability of the inviscid 2D Boussinesq equations near specific shear flow and stratification, revealing explicit solutions that grow rapidly and providing bounds on norm inflation for the linearized problem.

Contribution

It constructs explicit solutions demonstrating instability and derives improved bounds on norm inflation for the linearized inviscid Boussinesq equations.

Findings

Explicit solutions grow from size ε to 1 in time ε^{-2}.

Linearized problem shows improved bounds on norm inflation.

Instability occurs near shear flow with stratification α > 1/4.

Abstract

We consider the (in)stability problem of the inviscid 2D Boussinesq equations near a combination of a shear flow $v=(y,0)$ and a stratified temperature $\theta=\alpha y$ with $\alpha>\frac{1}{4}$. We show that for any $\epsilon>0$ there exist non-trivial explicit solutions, which are initially perturbations of size $\epsilon$, and grow to size $1$ on a time scale $\epsilon^{-2}$. Moreover, the (simplified) linearized problem around these non-trivial states exhibits improved upper bounds on the possible size of norm inflation for frequencies larger and smaller than $\epsilon^{-4}$.