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)∂tu−μΔu+(u⋅∇)u+∇p+αu+β|u|r−1u=0 on a 3D 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 4μβ≥1. Additionally, we show that in the critical case every weak solution verifies the energy equality and hence is continuous into the phase space L2. As an application of this result we prove the existence of a strong global attractor, using the theory of evolutionary systems developed by Cheskidov.