For every γ<1/3, there exist compactly supported weak solutions (v,θ) to the 2D Euler-Boussinesq system in C^γ(R×T^2) × C^γ(R×T^2) that violate conservation of the temperature's L^p-norm.
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An Onsager type theorem for the Euler-Boussinesq equations in two spatial dimensions
For every γ<1/3, there exist compactly supported weak solutions (v,θ) to the 2D Euler-Boussinesq system in C^γ(R×T^2) × C^γ(R×T^2) that violate conservation of the temperature's L^p-norm.