Taiwanese Journal of Mathematics

A Poisson Problem of Transmission-type for the Stokes and Generalized Brinkman Systems in Complementary Lipschitz Domains in $\mathbb{R}^3$

Andrei-Florin Albişoru

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Abstract

The purpose of this paper is to give a well-posedness result for a boundary value problem of transmission type for the Stokes and generalized Brinkman systems in two complementary Lipschitz domains in $\mathbb{R}^3$. In the first part of the paper, we have introduced the classical and weighted $L^2$-based Sobolev spaces on Lipschitz domains in $\mathbb{R}^3$. Afterwards, the trace and conormal derivative operators are defined in the case of both Stokes and generalized Brinkman systems. Also, a summary of the main properties of the layer potential operators for the Stokes system, is provided. In the second part of the work, we exploit the well-posedness of another transmission problem concerning the Stokes system on two complementary Lipschitz domains in $\mathbb{R}^3$ which is based on the Potential Theory for the Stokes system. Then, certain properties of Fredholm operators will allow us to show our main well-posedness result in $L^2$-based Sobolev spaces.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 24 pages.

Dates
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1557799218

Digital Object Identifier
doi:10.11650/tjm/190408

Subjects
Primary: 35J25: Boundary value problems for second-order elliptic equations 35Q35: PDEs in connection with fluid mechanics 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Keywords
Stokes system Brinkman system transmission problems layer potentials Fredholm operator Sobolev spaces

Citation

Albişoru, Andrei-Florin. A Poisson Problem of Transmission-type for the Stokes and Generalized Brinkman Systems in Complementary Lipschitz Domains in $\mathbb{R}^3$. Taiwanese J. Math., advance publication, 14 May 2019. doi:10.11650/tjm/190408. https://projecteuclid.org/euclid.twjm/1557799218


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