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

## TL;DR

This paper establishes a robustness of regularity result for the 3D convective Brinkman-Forchheimer equations, showing that strong solutions persist under small perturbations for a range of absorption exponents, extending known results from Navier-Stokes equations.

## Contribution

It proves a robustness of regularity for the 3D convective Brinkman-Forchheimer equations for the first time, extending the concept to a broader class of nonlinear PDEs.

## Key findings

- Strong solutions remain regular under small initial and forcing perturbations.
- The smallness condition parallels those for 3D Navier-Stokes equations.
- Results apply for absorption exponent r in [1, 3].

## Abstract

We prove a robustness of regularity result for the $3$D convective Brinkman-Forchheimer equations $$ \partial_tu -\mu\Delta u + (u \cdot \nabla)u + \nabla p + \alpha u + \beta\abs{u}^{r - 1}u = f, $$ for the range of the absorption exponent $r \in [1, 3]$ (for $r > 3$ there exist global-in-time regular solutions), i.e. we show that strong solutions of these equations remain strong under small enough changes of the 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).