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2018 | OriginalPaper | Chapter

4. Stokes Problems in Irregular Domains with Various Boundary Conditions

Authors : Sylvie Monniaux, Zhongwei Shen

Published in: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Publisher: Springer International Publishing

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Abstract

Different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in \(\mathbb{R}^{3}\), such as Dirichlet, Neumann, Hodge, or Robin boundary conditions, are presented here. The situation is a little different from the case of smooth domains. The analysis of the problem involves a good comprehension of the behavior near the boundary. The linear Stokes operator associated to the various boundary conditions is first studied. Then a classical fixed-point theorem is used to show how the properties of the operator lead to local solutions or global solutions for small initial data.

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previous chapter The Stokes Equation in the L p -Setting: Well-Posedness and Regularity Properties

next chapter Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow

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Title: Stokes Problems in Irregular Domains with Various Boundary Conditions
Authors: Sylvie Monniaux
Zhongwei Shen
Publisher: Springer International Publishing
Book: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids
Print ISBN: 978-3-319-13343-0

Electronic ISBN: 978-3-319-13344-7

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-13344-7_4

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