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

Abstract

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

44 pages