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

TL;DR

This work proves an Onsager-type flexibility result for the two-dimensional Euler-Boussinesq equations by constructing compactly supported, Hölder continuous weak solutions with exponent $\gamma<1/3$. The authors adapt a two-stage Nash-N interpolation, combining Newton linearization and Mikado-type perturbations, to decouple the velocity-thermal coupling and control Reynolds-like stresses. The main achievement is the existence of nontrivial weak solutions in $C^{\gamma}(\mathbb{R}\times \mathbb{T}^2)\times C^{\gamma}(\mathbb{R}\times \mathbb{T}^2)$ that dissipate the temperature norm, while maintaining rigorous estimates on the iterative perturbations, Reynolds stresses, and pressure. This extends Onsager-type nonrigidity to the 2D Euler-Boussinesq system and provides a robust framework for future explorations of energy/temperature dissipation in geophysical fluid models. The results rely on a refined convex integration scheme with Newton decoupling and time-oscillation techniques that prevent interactions across directions in 2D, yielding sharp Hölder thresholds and rigorous well-posedness components for the linearized EB system embedded in the construction.

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.

Theorems & Definitions (75)

• Theorem 1.1
• Theorem 1.2: Main Theorem
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Lemma 2.4
• Lemma 2.5
• Proposition 2.6
• Proposition 2.7
• Proposition 2.8