{
  "id": "0910.1562",
  "version": "v2",
  "published": "2009-10-08T18:11:18.000Z",
  "updated": "2009-11-12T06:44:09.000Z",
  "title": "Approximate Solutions to Second Order Parabolic Equations I: analytic estimates",
  "authors": [
    "Radu Constantinescu",
    "Nick Costanzino",
    "Anna L Mazzucato",
    "Victor Nistor"
  ],
  "comment": "42 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We establish a new type of local asymptotic formula for the Green's function ${\\mathcal G}_t(x,y)$ of a uniformly parabolic linear operator $\\partial_t - L$ with non-constant coefficients using dilations and Taylor expansions at a point $z=z(x,y)$, for a function $z$ with bounded derivatives such that $z(x,x)=x \\in {\\mathbb R}^N$. For $z(x,y) =x$, we recover the known, classical expansion obtained via pseudo-differential calculus. Our method is based on dilation at $z$, Dyson and Taylor series expansions, and the Baker-Campbell-Hausdorff commutator formula. Our procedure leads to an elementary, algorithmic construction of approximate solutions to parabolic equations which are accurate to arbitrary prescribed order in the short-time limit. We establish mapping properties and precise error estimates in the exponentially weighted, $L^{p}$-type Sobolev spaces $W^{s,p}_a({\\mathbb R}^N)$ that appear in practice.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-11-12T06:44:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K10",
      "35K08",
      "35Q84",
      "35S05",
      "02.30.Mv"
    ],
    "keywords": [
      "second order parabolic equations",
      "approximate solutions",
      "analytic estimates",
      "type sobolev spaces",
      "precise error estimates"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1063/1.3486357",
      "journal": "Journal of Mathematical Physics",
      "year": 2010,
      "month": "Oct",
      "volume": 51,
      "number": 10,
      "pages": 3502
    },
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010JMP....51j3502C"
    }
  }
}