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.
Non-conservative $H^{\frac 12-}$ weak solutions of the incompressible 3D Euler equations
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abstract
For any positive regularity parameter $\beta < \frac 12$, we construct non-conservative weak solutions of the 3D incompressible Euler equations which lie in $H^{\beta}$ uniformly in time. In particular, we construct solutions which have an $L^2$-based regularity index \emph{strictly larger} than $\frac 13$, thus deviating from the $H^{\frac{1}{3}}$-regularity corresponding to the Kolmogorov-Obhukov $\frac 53$ power spectrum in the inertial range.
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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.