{
  "id": "2102.01248",
  "version": "v1",
  "published": "2021-02-02T01:22:16.000Z",
  "updated": "2021-02-02T01:22:16.000Z",
  "title": "Ill-posedness issues on $(abcd)$-Boussinesq system",
  "authors": [
    "Chulkwang Kwak",
    "Christopher Maulén"
  ],
  "comment": "23 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the Cauchy problem for $(abcd)$-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona, Chen, and Saut, describes a small-amplitude waves on the surface of an inviscid fluid, and derived as a first-order approximation of incompressible, irrotational Euler equations. We mainly establish the ill-posedness of the system under various parameter regimes, which generalize the result of the one-dimensional BBM-BBM case by Chen and Liu. Most of results established here, we obtain the optimal result for two-dimensional BBM-BBM system. The proof follows from an observation of the \\emph{high to low-frequency cascade} present in nonlinearity, motivated by Bejenaru and Tao.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-02T01:22:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boussinesq system",
      "ill-posedness issues",
      "two-dimensional euclidean spaces",
      "irrotational euler equations",
      "one-dimensional bbm-bbm case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}