{
  "id": "1609.02772",
  "version": "v1",
  "published": "2016-09-09T12:49:40.000Z",
  "updated": "2016-09-09T12:49:40.000Z",
  "title": "On Rank Two Toda System with Arbitrary Singularities: Local Mass and New Estimates",
  "authors": [
    "Changshou Lin",
    "Juncheng Wei",
    "Wen Yang",
    "Lei Zhang"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "For all rank two Toda systems with an arbitrary singular source, we use a unified approach to prove: (i) The pair of local masses $(\\sigma_1,\\sigma_2)$ at each blowup point has the expression $$\\sigma_i=2(N_{i1}\\mu_1+N_{i2}\\mu_2+N_{i3}),$$ where $N_{ij}\\in\\mathbb{Z},~i=1,2,~j=1,2,3.$ (ii) Suppose at each vortex point $p_t$, $(\\alpha_1^t,\\alpha_2^t)$ are integers and $\\rho_i\\notin 4\\pi\\mathbb{N}$, then all the solutions of Toda systems are uniformly bounded. (iii) If the blow up point $q$ is not a vortex point, then $$u^k(x)+2\\log|x-x^k|\\leq C,$$ where $x^k$ is the local maximum point of $u^k$ near $q$. (iv) If the blow up point $q$ is a vortex point $p_t$ and $\\alpha_t^1,\\alpha_t^2$ and $1$ are linearly independent over $Q$, then $$u^k(x)+2\\log|x-p_t|\\leq C.$$ The Harnack type inequalities of (iii) or (iv) is important for studying the bubbling behaves near each blow up point.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-09T12:49:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35J55"
    ],
    "keywords": [
      "toda system",
      "local mass",
      "arbitrary singularities",
      "vortex point",
      "harnack type inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}