{
  "id": "1711.05067",
  "version": "v1",
  "published": "2017-11-14T12:00:40.000Z",
  "updated": "2017-11-14T12:00:40.000Z",
  "title": "Existence and uniqueness of $W^{1,r}_{loc}$-solutions for stochastic transport equations",
  "authors": [
    "Jinlong Wei",
    "Jinqiao Duan",
    "Hongjun Gao",
    "Guangying Lv"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We investigate a stochastic transport equation driven by a multiplicative noise. For $L^q(0,T;W^{1,p}({\\mathbb R}^d;{\\mathbb R}^d))$ drift coefficient and $W^{1,r}({\\mathbb R}^d)$ initial data, we obtain the existence and uniqueness of stochastic strong solutions (in $W^{1,r}_{loc}({\\mathbb R}^d))$.In particular, when $r=\\infty$, we establish a Lipschitz estimate for solutions and this question is opened by Fedrizzi and Flandoli in case of $L^q(0,T;L^p({\\mathbb R}^d;{\\mathbb R}^d))$ drift coefficient. Moreover, opposite to the deterministic case where $L^q(0,T;W^{1,p}({\\mathbb R}^d;{\\mathbb R}^d))$ drift coefficient and $W^{1,p}({\\mathbb R}^d)$ initial data may induce non-existence for strong solutions (in $W^{1,p}_{loc}({\\mathbb R}^d)$), we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. It is an interesting example of a deterministic PDE that becomes well-posed under the influence of a multiplicative Brownian type noise. We extend the existing results \\cite{FF2,FGP1} partially.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-14T12:00:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "35A01",
      "35L02"
    ],
    "keywords": [
      "drift coefficient",
      "uniqueness",
      "stochastic transport equation driven",
      "initial data",
      "stochastic strong solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}