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

Abstract

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

17 pages

Cited by in corpus (11)

• Stochastic convective Brinkman-Forchheimer equations on general Unbounded Domains
• Weak pullback mean random attractors for the stochastic convective Brinkman-Forchheimer equations and locally monotone stochastic partial differential equations
• Wentzell-Freidlin Large Deviation Principle for the stochastic convective Brinkman-Forchheimer equations
• Asymptotic analysis of the 2D convective Brinkman-Forchheimer equations in unbounded domains: Global attractors and upper semicontinuity
• Random attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations in some unbounded domains
• Optimal Control of the 3D Damped Navier-Stokes-Voigt Equations with Control Constraints
• Long term behavior of 2D and 3D non-autonomous random convective Brinkman-Forchheimer equations driven by colored noise
• Existence and upper semicontinuity of random pullback attractors for 2D and 3D non-autonomous stochastic convective Brinkman-Forchheimer equations on whole domain
• Averaging principle for the stochastic convective Brinkman-Forchheimer equations
• Slightly Compressible Forchheimer Flows in Rotating Porous Media
• Martingale solution, invariant measure and ergodicity for stochastic convective Brinkman-Forchheimer equations on general domains in ${\mathbb{R}}^d$