{
  "id": "1511.04850",
  "version": "v1",
  "published": "2015-11-16T07:40:59.000Z",
  "updated": "2015-11-16T07:40:59.000Z",
  "title": "Well-posedness of the Prandtl equation in Sobolev space without monotonicity",
  "authors": [
    "Chao-Jiang Xu",
    "Xu Zhang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the well-posedness theory for the Prandtl boundary layer equation on the half plane with initial data in Sobolev spaces. We consider a class of initial data which admit the non-degenerate critical points, so it is not monotonic. For this kind of initial data, we prove the local-in-time existence, uniqueness and stability of solutions for the nonlinear Prandtl equation in weighted Sobolev space. We use the energy method to prove the existence of solution by a parabolic regularizing approximation. The nonlinear cancellation properties of the Prandtl equations and non-degeneracy of the critical points are the main ingredients to establish a new energy estimate. Our result improves the classical local well-posedness results for the initial data that are monotone, analytic or Gevery class, and it will also help us to understand the ill-posedness and instability results for the Prandtl equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-16T07:40:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sobolev space",
      "initial data",
      "monotonicity",
      "prandtl boundary layer equation",
      "nonlinear prandtl equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151104850X"
    }
  }
}