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 <= p <= infinity, we show that the threshold regularity exponent for Lp-norm conservation of temperature of this system is 1/3, consistent with Onsager exponent. More precisely, for 1 <= p <= infinity, every weak solution (v, theta) is an element of CtCx beta to the inviscid Boussinesq equations satisfies that II theta(t)IILp ((T3)) = II theta 0IILp(T3) if beta > 1\3, while if beta < 1\3, there exist infinitely many weak solutions (v, theta) is an element of CtCx beta such that the L-p-norm of temperature is not conserved. As a byproduct, we are able to construct many weak solutions in CtCx beta for beta < 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 CtCx beta for beta < 13. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar