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On the Cauchy problem associated with the Brinkman flow in $\mathbb {R}_{+}^{3}$

Part of: Incompressible viscous fluids Partial differential equations

Published online by Cambridge University Press:  07 September 2021

Michel Molina Del Sol ,
Eduardo Arbieto Alarcon  and
Rafael José Iorio Junior
Michel Molina Del Sol
Affiliation:
Facultad de Ciencias, Departamento de Matemática, Universidad Católica del Norte (UCN), Avenida Angamos 0610, Antofagasta, Chile (mmolina01@ucn.cl)
Eduardo Arbieto Alarcon
Affiliation:
Instituto de Matemática y Estatística (IME), Universidade Federal de Goiás, Goiania, Brazil (alarcon@mat.ufg.br)
Rafael José Iorio Junior
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil (rafael@impa.br)
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Abstract

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space

\[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \]
under the assumption that the plane $z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.

Keywords

Cauchy problemintegral-differential nonlinear evolution equationslocal and global well-posednesspartial differential equations

MSC classification

Primary: 35A01: Existence problems
Secondary: 35B51: Comparison principles 76D07: Stokes and related (Oseen, etc.) flows
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 5 , October 2022 , pp. 1089 - 1108
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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