{
  "id": "2112.10081",
  "version": "v3",
  "published": "2021-12-19T07:43:51.000Z",
  "updated": "2022-03-06T13:00:49.000Z",
  "title": "Ill-posedness for the Cauchy problem of the Camassa-Holm equation in $B^{1}_{\\infty,1}(\\mathbb{R})$",
  "authors": [
    "Yingying Guo",
    "Weikui Ye",
    "Zhaoyang Yin"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "For the famous Camassa-Holm equation, the well-posedness in $B^{1+\\frac{1}{p}}_{p,1}(\\mathbb{R})$ with $ p\\in [1,\\infty)$ and the ill-posedness in $B^{1+\\frac{1}{p}}_{p,r}(\\mathbb{R})$ with $ p\\in [1,\\infty],\\ r\\in (1,\\infty]$ had been studied in \\cite{d1,d2,glmy,yyg}, that is to say, it only left an open problem in the critical case $B^{1}_{\\infty,1}(\\mathbb{R})$ proposed by Danchin in \\cite{d1,d2}. In this paper, we solve this problem by proving the norm inflation and hence the ill-posedness for the Camassa-Holm equation in $B^{1}_{\\infty,1}(\\mathbb{R})$. Therefore, the well-posedness and ill-posedness for the Camassa-Holm equation in all critial Besov spaces $B^{1+\\frac{1}{p}}_{p,1}(\\mathbb{R})$ with $ p\\in [1,\\infty]$ have been completed. Finally, since the norm inflation occurs by choosing an special initial data $u_0\\in B^{1}_{\\infty,1}(\\mathbb{R})$ but $u^2_{0x}\\notin B^{0}_{\\infty,1}(\\mathbb{R})$ (an example implies $B^{0}_{\\infty,1}(\\mathbb{R})$ is not a Banach algebra), we then prove that this condition is necessary. That is, if $u^2_{0x}\\in B^{0}_{\\infty,1}(\\mathbb{R})$ holds, then the Camassa-Holm equation has a unique solution $u(t,x)\\in \\mathcal{C}_T(B^{1}_{\\infty,1}(\\mathbb{R}))\\cap \\mathcal{C}^{1}_T(B^{0}_{\\infty,1}(\\mathbb{R}))$ and the norm inflation will not occur.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2022-03-06T13:00:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "35G25",
      "35L30"
    ],
    "keywords": [
      "cauchy problem",
      "ill-posedness",
      "norm inflation occurs",
      "critial besov spaces",
      "special initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}