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تسجيل مستخدم جديدEnergy equality for the 3D critical convective Brinkman-Forchheimer equations
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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) $$ partial_tu -muDelta u + (u cdot abla)u + abla p + alpha u + beta|u|^{r - 1}u = 0 $$ 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 $4mubeta geq 1$. 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.
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