Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces
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- by Gerard Misiołek and Tsuyoshi Yoneda PDF
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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.References
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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
- 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.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 4709-4730
- MSC (2010): Primary 35Q35; Secondary 35B30
- DOI: https://doi.org/10.1090/tran/7101
- MathSciNet review: 3812093