Boussinesq equations can be used to describe the coupling nature of thermodynamics and fluid dynamics. We show the existence of Hölder continuous periodic weak solution of the Boussinesq equations, which approximates the Onsager’s critical spatial regularity \(\sigma =\frac{1}{3}\) and satisfies the prescribed kinetic energy. This work extends the result of Buckmaster et al. (Commun Pure Appl Math 72(2):229–274, 2019) to the Boussinesq equations. We overcome the difficulty of interactions between velocity and temperature by constructing a modified building block and push forward \(C^{1/5-\epsilon }\) -solution of Tao and Zhang (Acta Math Sci Ser B 38(5):1591–1616, 2018) to \(C^{1/3-\epsilon }\) . We also present the sharp energy regularity for a \(\sigma \) -Hölder continous weak solution \((v,\theta )\) with kinetic energy \(e_v\in C^{2\sigma /(1-\sigma )}\) in the case \(\sigma <\frac{1}{3}\) , in which we modify some mistakes of choosing the parameters in the work of De Rosa and Tione (Anal PDE 15(2):405–428, 2022).
