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

In this article, we construct non-trivial weak solutions $(v,\theta )$ 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 $(v,\theta )$ resides in the Hölder space $C^{\gamma }(\mathbb{R}\times {\mathbb{T}}^2)\times C^{\gamma }(\mathbb{R}\times {\mathbb{T}}^2)$ for any exponent $\gamma <1\mathrm{/}3$. The methodology integrates a Nash iteration scheme with a linear decoupling technique to achieve these results.