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arXiv · 2209.01950

On stability estimates for the inviscid Boussinesq equations

Christian Zillinger
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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 $\epsilon^{-4}$.

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Christian Zillinger. 2022-09-05. On stability estimates for the inviscid Boussinesq equations. https://arxiv.org/abs/2209.01950

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