An Onsager type theorem for the Euler-Boussinesq equations in two spatial dimensions

Abstract

In this article, we construct non-trivial weak solutions $(v, θ)$ to the inviscid Euler-Boussinesq system in two spatial dimensions. These solutions exhibit compact temporal support, thereby violating the conservation of the temperature's $L^p$-norm. Furthermore, the pair $(v, θ)$ resides in the Hölder space $C^γ(\mathbb R \times \mathbb T^2) \times C^γ(\mathbb R \times \mathbb T^2)$ for any exponent $γ<1/3$. The methodology integrates a Nash iteration scheme with a linear decoupling technique to achieve these results.