{
  "id": "2012.13175",
  "version": "v1",
  "published": "2020-12-24T09:26:45.000Z",
  "updated": "2020-12-24T09:26:45.000Z",
  "title": "About lifespan and the continuous dependence for the Navier-Stokes equation in $\\dot{B}^{\\frac{d}{p}-1}_{p,r}$",
  "authors": [
    "Weikui Ye",
    "Zhaoyang Yin",
    "Wei Luo"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2012.03489",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we mainly investigate the Cauchy problem for the Navier-Stokes ($NS$) equation. We first establish the local existence in the Besov space $\\dot{B}^{\\frac{d}{p}-1}_{p,r}$ with $1\\leq r,p<\\infty$. We give a lower bound of the lifespan $T$ which depends on the norm of the Littlewood-Paley decomposition of the initial data $u_0$. Then we prove that if the initial data $u^n_0\\rightarrow u_0$ in $\\dot{B}^{\\frac{d}{p}-1}_{p,r}$, then the corresponding lifespan satisfies $T_n\\rightarrow T$, which implies that the common lower bound of the lifespan. Finally, we prove that the data-to-solutions map is continuous in $\\dot{B}^{\\frac{d}{p}-1}_{p,r}$. So the solutions of Navier-Stokes equation are well-posedness (existence, uniqueness and continuous dependence) in the Hadamard sense. Combining \\cite{wbx,ns7,ns2}, we deduce that $\\dot{B}^{\\frac{d}{p}-1}_{p,\\infty}$ with $1\\leq p<\\infty$ is the critical space which solutions are ill-posedness, while $u\\in\\dot{B}^{\\frac{d}{p}-1}_{p,r}$ with $1\\leq r,p<\\infty$ are well-poseness. Moreover, if we choose the initial data in a subset $\\bar{B}^{\\frac{d}{p}-1}_{p,\\infty}$ of $\\dot{B}^{\\frac{d}{p}-1}_{p,\\infty}$, we can obtain the well-posedness of the solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-24T09:26:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "navier-stokes equation",
      "continuous dependence",
      "initial data",
      "common lower bound",
      "corresponding lifespan satisfies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}