Smooth Attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities

TL;DR

This paper establishes the existence of smooth global attractors for the 3D Brinkmann-Forchheimer equations with highly nonlinear terms, extending the understanding of long-term behavior in complex fluid models.

Contribution

It proves the existence of regular dissipative solutions and global attractors for equations with polynomial growth nonlinearities, using novel maximal regularity estimates.

Findings

Existence of regular dissipative solutions

Existence of global attractors for the equations

Application to Brinkmann-Forchheimer equations with Navier-Stokes terms

Abstract

We prove the existence of regular dissipative solutions and global attractors for the 3D Brinkmann-Forchheimer equations with the nonlinearity of an arbitrary polynomial growth rate. In order to obtain this result, we prove the maximal regularity estimate for the corresponding semi-linear stationary Stokes problem using some modification of the nonlinear localization technique. The applications of our results to the Brinkmann-Forchheimer equation with the Navier-Stokes inertial term are also considered.