{
  "id": "1807.07551",
  "version": "v1",
  "published": "2018-07-19T17:43:24.000Z",
  "updated": "2018-07-19T17:43:24.000Z",
  "title": "Stability of vacuum for the Landau equation with moderately soft potentials",
  "authors": [
    "Jonathan Luk"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Consider the spatially inhomogeneous Landau equation with moderately soft potentials (i.e. with $\\gamma \\in (-2,0)$) on the whole space $\\mathbb R^3$. We prove that if the initial data $f_{\\mathrm{in}}$ are close to the vacuum solution $f_{\\mathrm{vac}} \\equiv 0$ in an appropriate norm, then the solution $f$ remains regular globally in time. This is the first stability of vacuum result for a binary collisional model featuring a long-range interaction. Moreover, we prove that the solutions in the near-vacuum regime approach solutions to the linear transport equation as $t\\to +\\infty$. Furthermore, in general, solutions do not approach a traveling global Maxwellian as $t \\to +\\infty$. Our proof relies on robust decay estimates captured using weighted energy estimates and the maximum principle for weighted quantities. Importantly, we also make use of a null structure in the nonlinearity of the Landau equation which suppresses the most slowly-decaying interactions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-19T17:43:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "moderately soft potentials",
      "near-vacuum regime approach solutions",
      "linear transport equation",
      "robust decay estimates",
      "vacuum result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}