On Stability Estimates for the Inviscid Boussinesq Equations

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}.