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Archive for Rational Mechanics and Analysis

, Volume 234, Issue 2, pp 911–923 | Cite as

Well-Posedness and Ill-Posedness Problems of the Stationary Navier–Stokes Equations in Scaling Invariant Besov Spaces

  • Hiroyuki TsurumiEmail author
Article
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Abstract

We consider the stationary Navier–Stokes equations in \(\mathbb {R}^n\) for \(n\geqq 3\) in the scaling invariant Besov spaces. It is proved that if \(n<p\leqq \infty \) and \(1\leqq q\leqq \infty \), or \(p=n\) and \(2<q\leqq \infty \), then some sequence of external forces converging to zero in \(\dot{B}^{-3+\frac{n}{p}}_{p,q}\) can admit a sequence of solutions which never converges to zero in \(\dot{B}^{-1}_{\infty ,\infty }\), especially in \(\dot{B}^{-1+\frac{n}{p}}_{p,q}\). Our result may be regarded as showing the borderline case between ill-posedness and well-posedness, the latter of which Kaneko–Kozono–Shimizu proved when \(1\leqq p<n\) and \(1\leqq q\leqq \infty \).

Notes

Acknowledgements

The author was partly supported by Grant-in-Aid for JSPS Research Fellow (Grant Number: JP19J11499), Top Global University Project of Waseda University, and by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and EngineeringWaseda UniversityTokyoJapan

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