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

TL;DR

This paper constructs non-trivial weak solutions to the 2D Euler-Boussinesq equations that violate temperature conservation and are highly regular, using advanced iterative and decoupling methods.

Contribution

It introduces a novel construction of weak solutions with compact support and specific regularity for the Euler-Boussinesq system, extending Onsager-type results.

Findings

Solutions have compact temporal support.

Temperature's $L^p$-norm is not conserved.

Solutions are in $C^eta$ for any $eta<1/3$.

Abstract

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 H\"older space $C^\gamma(\mathbb R \times \mathbb T^2) \times C^\gamma (\mathbb R \times \mathbb T^2)$ for any exponent $\gamma<1/3$. The methodology integrates a Nash iteration scheme with a linear decoupling technique to achieve these results.