Asymptotic log-Harnack inequality for the stochastic convective Brinkman−Forchheimer equations with degenerate noise

In this work, we study the two- and three-dimensional stochastic convective Brinkman−Forchheimer (SCBF) equations and investigate the asymptotic behavior of their strong solutions. By means of the asymptotic coupling method, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with the SCBF equations driven by additive and multiplicative degenerate noise. As consequences of this inequality, we obtain gradient estimates, asymptotic irreducibility, the asymptotically strong Feller property, asymptotic heat kernel estimates, and ergodicity. Moreover, for fast-growing nonlinearities corresponding to absorption exponents r∈(3,∞), the asymptotic log-Harnack inequality is proved without any restrictions on the Brinkman coefficient (effective viscosity) μ>0, the Darcy coefficient α>0, or the Forchheimer coefficient β>0.