{
  "id": "2309.15690",
  "version": "v1",
  "published": "2023-09-27T14:33:13.000Z",
  "updated": "2023-09-27T14:33:13.000Z",
  "title": "A continuation criterion for the Landau equation with very soft and Coulomb potentials",
  "authors": [
    "Stanley Snelson",
    "Caleb Solomon"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the spatially inhomogeneous Landau equation in the case of very soft and Coulomb potentials, $\\gamma \\in [-3,-2]$. We show that solutions can be continued as long as the following three quantities remain finite, uniformly in $t$ and $x$: (1) the mass density, (2) the velocity moment of order $s$ for any small $s>0$, and (3) the $L^p_v$ norm for any $p>3/(5+\\gamma)$. In particular, we do not require a bound on the energy density. If we specialize our result to the spatially homogeneous case, we recover the best known continuation criterion in that regime.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-27T14:33:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuation criterion",
      "coulomb potentials",
      "quantities remain finite",
      "spatially inhomogeneous landau equation",
      "energy density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}