On Onsager's type conjecture for the inviscid Boussinesq equations

In this paper, we investigate the Cauchy problem for the three dimensional inviscid Boussinesq system in the periodic setting. For $1\le p\le \infty$, we show that the threshold regularity exponent for $L^p$-norm conservation of temperature of this system is $1/3$, consistent with Onsager exponent. More precisely, for $1\le p\le\infty$, every weak solution $(v,\theta)\in C_tC^{\beta}_x$ to the inviscid Boussinesq equations satisfies that $\|\theta(t)\|_{L^p(\mathbb{T}^3)}=\|\theta_0\|_{L^p(\mathbb{T}^3)}$ if $\beta>\frac{1}{3}$, while if $\beta<\frac{1}{3}$, there exist infinitely many weak solutions $(v,\theta)\in C_tC^{\beta}_x$ such that the $L^p$-norm of temperature is not conserved. As a byproduct, we are able to construct many weak solutions in $C_tC^{\beta}_x$ for $\beta<\frac{1}{3}$ displaying wild behavior, such as fast kinetic energy dissipation and high oscillation of velocity. Moreover, we also show that if a weak solution $(v, \theta)$ of this system has at least one interval of regularity, then this weak solution $(v,\theta)$ is not unique in $C_tC^{\beta}_x$ for $\beta<\frac{1}{3}$.