Global Weak Solutions to the Navier-Stokes-Darcy-Boussinesq System for Thermal Convection in Coupled Free and Porous Media Flows

TL;DR

This paper proves the existence of global weak solutions and a weak-strong uniqueness principle for a coupled Navier-Stokes-Darcy-Boussinesq system modeling thermal convection in free and porous media flows in 2D and 3D.

Contribution

It establishes the first rigorous proof of global weak solutions and weak-strong uniqueness for this complex coupled system under general interface conditions.

Findings

Existence of global weak solutions in 2D and 3D.

Weak-strong uniqueness of solutions.

Application of discretization and compactness methods.

Abstract

We study the Navier-Stokes-Darcy-Boussinesq system that models the thermal convection of a fluid overlying a saturated porous medium in a general decomposed domain. In both two and three spatial dimensions, we first prove the existence of global weak solutions to the initial boundary value problem subject to the Lions and Beavers-Joseph-Saffman-Jones interface conditions. The proof is based on a proper time-implicit discretization scheme combined with the Leray-Schauder principle and compactness arguments. Next, we establish a weak-strong uniqueness result such that a weak solution coincides with a strong solution emanating from the same initial data as long as the latter exists.