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) $$\partial_tu -\mu\Delta u + (u \cdot \nabla)u + \nabla p + \alpha u + \beta|u|^{r - 1}u = 0$$ 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 $4\mu\beta \geq 1$. 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.