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The Work of Yoshihiro Shibata Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Local Regularity Results for the Instationary Navier-Stokes Equations Based on Besov Space Type Criteria

Author : Reinhard Farwig

Published in: Recent Developments of Mathematical Fluid Mechanics

Publisher: Springer Basel

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Abstract

Consider a weak instationary solution u of the Navier-Stokes equations in a domain \(\Omega \subset \mathbb{R}^{3}\), i.e.,
$$u \in L^{\infty }\big(0,T;L^{2}(\Omega )\big) \cap L^{2}\big(0,T;W_{0}^{1,2}(\Omega )\big)$$
and u solves the Navier-Stokes system in the sense of distributions. It is a famous open problem whether weak solutions are unique and smooth. A main step in the analysis of this problem is to show that the given weak solution is a strong one in the sense of J. Serrin, i.e., \(u \in L^{s}\big(0,T;L^{q}(\Omega )\big)\) where s > 2, q > 3 and \(\frac{2} {s} + \frac{3} {q} = 1\). In this review we report on recent results on this problem, considering first of all optimal initial values u(0) to yield a local in time strong solution, then criteria to prove regularity locally on subintervals of [0, T). Special emphasis is put on results for smooth bounded and also general unbounded domains. Most criteria are based on conditions of Besov space type.

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previous chapter Blow-Up Criterion for 3D Navier-Stokes Equations and Landau-Lifshitz System in a Bounded Domain
next chapter On Global Well/Ill-Posedness of the Euler-Poisson System
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Metadata
Title
Local Regularity Results for the Instationary Navier-Stokes Equations Based on Besov Space Type Criteria
Author
Reinhard Farwig
Copyright Year
2016
Publisher
Springer Basel
Book
Recent Developments of Mathematical Fluid Mechanics

Print ISBN: 978-3-0348-0938-2

Electronic ISBN: 978-3-0348-0939-9

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-0348-0939-9_11

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