Transmission problems for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems in Lipschitz domains on compact Riemannian manifolds

TL;DR

This paper investigates boundary transmission problems for Navier-Stokes and Darcy-Forchheimer-Brinkman systems on Lipschitz domains within compact Riemannian manifolds, establishing existence and uniqueness under small data conditions.

Contribution

It introduces a layer potential approach combined with fixed point techniques to solve transmission boundary value problems for these fluid systems on Riemannian manifolds.

Findings

Proves existence and uniqueness of solutions for small data

Develops a layer potential method for these systems

Extends analysis to Lipschitz domains on Riemannian manifolds

Abstract

The purpose of this paper is to study boundary value problems of transmission type for the Navier-Stokes and Darcy-Forchheimer-Brinkman systems in two complementary Lipschitz domains on a compact Riemannian manifold of dimension 2 or 3. We exploit a layer potential method combined with a fixed point theorem in order to show existence and uniqueness results when the given data are suitably small in $L^2$-based Sobolev spaces.