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Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces


Authors: Gerard Misiołek and Tsuyoshi Yoneda
Journal: Trans. Amer. Math. Soc. 370 (2018), 4709-4730
MSC (2010): Primary 35Q35; Secondary 35B30
DOI: https://doi.org/10.1090/tran/7101
Published electronically: November 14, 2017
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Abstract: We construct an example showing that the solution map of the Euler equations is not continuous in the Hölder space from $ C^{1,\alpha }$ to $ L^\infty _tC^{1,\alpha }_x$ for any $ 0<\alpha <1$. On the other hand we show that it is continuous when restricted to the little Hölder subspace $ c^{1,\alpha }$. We apply the latter to prove an ill-posedness result for solutions of the vorticity equations in Besov spaces near the critical space $ B^1_{2,1}$. As a consequence we show that a sequence of best constants of the Sobolev embedding theorem near the critical function space is not continuous.


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Additional Information

Gerard Misiołek
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: gmisiole@nd.edu

Tsuyoshi Yoneda
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1 Meguro, Tokyo 153-8914, Japan
Email: yoneda@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/tran/7101
Keywords: Euler equations, H\"older spaces, solution map
Received by editor(s): January 29, 2016
Received by editor(s) in revised form: September 23, 2016
Published electronically: November 14, 2017
Additional Notes: The second author was partially supported by JSPS KAKENHI Grant Number 25870004.
Article copyright: © Copyright 2017 American Mathematical Society

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