Ill-posedness in $B^s_{p,\infty}$ of the Euler equations: Non-continuous dependence
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arXiv
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| Format: | Preprint |
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2025
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| author | Li, Jinlu Yu, Yanghai |
| author_facet | Li, Jinlu Yu, Yanghai |
| contents | In this paper, we solve an open problem left in the monographs \cite[Bahouri-Chemin-Danchin, (2011)]{BCD}. Precisely speaking, it was obtained in \cite[Theorem 7.1 on pp293, (2011)]{BCD} the existence and uniqueness of $B^s_{p,\infty}$ solution for the Euler equations. We furthermore prove that the solution map of the Euler equation is not continuous in the Besov spaces from $B^s_{p,\infty}$ to $L_T^\infty B^s_{p,\infty}$ for $s>1+d/p$ with $1\leq p\leq \infty$ and in the Hölder spaces from $C^{k,α}$ to $L_T^\infty C^{k,α}$ with $k\in \mathbb{N}^+$ and $α\in(0,1)$, which later covers particularly the ill-posedness of $C^{1,α}$ solution in \cite[Trans. Amer. Math. Soc., (2018)]{MYtams}. Beyond purely technical aspects on the choice of initial data, a remarkable novelty of the proof is the construction of an approximate solution to the Burgers equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12619 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ill-posedness in $B^s_{p,\infty}$ of the Euler equations: Non-continuous dependence Li, Jinlu Yu, Yanghai Analysis of PDEs In this paper, we solve an open problem left in the monographs \cite[Bahouri-Chemin-Danchin, (2011)]{BCD}. Precisely speaking, it was obtained in \cite[Theorem 7.1 on pp293, (2011)]{BCD} the existence and uniqueness of $B^s_{p,\infty}$ solution for the Euler equations. We furthermore prove that the solution map of the Euler equation is not continuous in the Besov spaces from $B^s_{p,\infty}$ to $L_T^\infty B^s_{p,\infty}$ for $s>1+d/p$ with $1\leq p\leq \infty$ and in the Hölder spaces from $C^{k,α}$ to $L_T^\infty C^{k,α}$ with $k\in \mathbb{N}^+$ and $α\in(0,1)$, which later covers particularly the ill-posedness of $C^{1,α}$ solution in \cite[Trans. Amer. Math. Soc., (2018)]{MYtams}. Beyond purely technical aspects on the choice of initial data, a remarkable novelty of the proof is the construction of an approximate solution to the Burgers equation. |
| title | Ill-posedness in $B^s_{p,\infty}$ of the Euler equations: Non-continuous dependence |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.12619 |