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On the continuity of the solutions to the Navier–Stokes equations with initial data in critical Besov spaces

  • Reinhard FarwigEmail author
  • Yoshikazu Giga
  • Pen-Yuan Hsu
Article
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Abstract

It is well known that there exists a unique local-in-time strong solution u of the initial boundary value problem for the Navier–Stokes system in a three-dimensional smooth bounded domain when the initial velocity \(u_0\) belongs to critical Besov spaces. A typical space is \(B=B^{-1+3/q}_{q,s}\) with \(3<q<\infty \), \(2<s<\infty \) satisfying \(2/s+3/q \le 1\) or \(B=\mathring{B}^{-1+3/q}_{q,\infty }\). In this paper, we show that the solution u is continuous in time up to initial time with values in B. Moreover, the solution map \(u_0\mapsto u\) is locally Lipschitz from B to \(C\left( [0,T];B\right) \). This implies that in the range \(3<q<\infty \), \(2<s\le \infty \) with \(3/q +2/s \le 1\) the problem is well posed which is in strong contrast to norm inflation phenomena in the space \(B^{-1}_{\infty ,s}\), \(1\le s <\infty \) proved in the last years for the whole space case.

Keywords

Nonstationary Navier–Stokes system Initial values Weighted Serrin condition Limiting type of Besov space Continuity of solutions Stability of solutions 

Mathematics Subject Classification

35Q30 76D05 

Notes

Acknowledgements

This work is partly supported by the Japan Society for the Promotion of Science (JSPS) and the German Research Foundation through Japanese–German Graduate Externship IRTG 1529. The first and third authors are supported in part by the 7th European Framework Programme IRSES “FLUX”, Grant Agreement No. PIRSES-GA-2012-319012. The second author is partly supported by JSPS through the Grant Kiban S (26220702), Kiban A (17H01091), Kiban B (16H03948), Challenging Research (Pioneering) (18H05323). Moreover, the third author gratefully acknowledges the support by the Iwanami Fujukai Foundation. We also would like to thank the anonymous reviewer for helpful comments to improve the style of this article.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoMeguro-kuJapan

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