On energy conservation of weak solutions to the α$\alpha$‐type Euler models.

Very recently, Boutros and Titi considered Onsager's conjecture of α$\alpha$‐type Euler models in Besov spaces B3,∞β$B^{\beta }_{3,\infty }$ and B3,∞γ$B^{\gamma }_{3,\infty }$ with β>0,γ>1$\beta >0, \gamma >1$. In this paper, we study the energy conservation of weak solutions to these systems in Lebesgue space L3(T3)$L^{3}(\mathbb {T}^{3})$ and homogenous Sobolev space Ẇ1,3(T3)$\dot{W}^{1,3}(\mathbb {T}^{3})$. Moreover, the corresponding results for the energy and the cross‐helicity of the inviscid α$\alpha$‐magnetohydrodynamic equations are also obtained. [ABSTRACT FROM AUTHOR]