Energy equality for the 3D critical convective Brinkman-Forchheimer equations

In this paper we give a simple proof of the existence of global-in-time smooth solutions for the convective Brinkman-Forchheimer equations (also called in the literature the tamed Navier-Stokes equations) $${\mathrm{\partial }}_{t}u-\mu \mathrm{\Delta }u+(u\cdot \mathrm{\nabla })u+\mathrm{\nabla }p+\alpha u+\beta \mathrm{\mid }u{\mathrm{\mid }}^{r-1}u=0$$ on a $3$D periodic domain, for values of the absorption exponent $r$ larger than $3$. Furthermore, we prove that global, regular solutions exist also for the critical value of exponent $r=3$, provided that the coefficients satisfy the relation $4\mu \beta \ge 1$. Additionally, we show that in the critical case every weak solution verifies the energy equality and hence is continuous into the phase space $L^2$. As an application of this result we prove the existence of a strong global attractor, using the theory of evolutionary systems developed by Cheskidov.