{
  "id": "2312.13606",
  "version": "v1",
  "published": "2023-12-21T06:49:49.000Z",
  "updated": "2023-12-21T06:49:49.000Z",
  "title": "Scattering for 2d semi-relativistic Hartree equations with short range potential",
  "authors": [
    "Changhun Yang"
  ],
  "comment": "16pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the long time behavior of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is convolved with the singular potential $|x|^{-\\gamma}$ for $1<\\gamma<2$, which is referred to as short-range interaction potential in the sense of scattering phenomenon. We establish the scattering results for small solutions in a weighted space, in other words, we prove that the nonlinear solutions exist globally and behave asymptotically like a linear solution whenever the initial data is sufficiently small. To achieve this, we should obtain time decay estimates for the nonlinear term which is integrable. A key observation is that the loss in time in the course of weighted energy estimates can be recovered by the space resonance method with the help of null structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-21T06:49:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P25",
      "35R11",
      "35Q40",
      "35B40"
    ],
    "keywords": [
      "2d semi-relativistic hartree equations",
      "short range potential",
      "scattering",
      "small solutions",
      "nonlinear term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}