Robustness of Regularity for the $3$D Convective Brinkman--Forchheimer Equations.

We prove a robustness of regularity result for the $3$D convective Brinkman--Forchheimer equations $$ \partial_tu -\mu\Delta u + (u \cdot \nabla)u + \nabla p + \alpha u + \beta|u|^{r - 1}u = f\mbox{,} $$ for the range of the absorption exponent $r \in [1, 3]$ [for $r > 3$ there exist global-in-time regular solutions as shown for example in Hajduk & Robinson (2017)], i.e. we show that strong solutions of these equations remain strong under small enough changes of initial condition and forcing function. We provide a smallness condition which is similar to the robustness conditions given for the $3$D incompressible Navier--Stokes equations by Chernyshenko et al. (2007) and Dashti & Robinson 2008).