{
  "id": "0812.1627",
  "version": "v1",
  "published": "2008-12-09T06:25:53.000Z",
  "updated": "2008-12-09T06:25:53.000Z",
  "title": "Long time behaviour of viscous scalar conservation laws",
  "authors": [
    "Anne-Laure Dalibard"
  ],
  "comment": "36 pages",
  "journal": "Indiana University Mathematics Journal 59, 1 (2010) 257-300",
  "doi": "10.1512/iumj.2010.59.3874",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is concerned with the stability of stationary solutions of the conservation law $\\partial_t u + \\mathrm{div}_y A(y,u) -\\Delta_y u=0$, where the flux $A$ is periodic with respect to its first variable. Essentially two kinds of asymptotic behaviours are studied here: the case when the equation is set on $\\R$, and the case when it is endowed with periodic boundary conditions. In the whole space case, we first prove the existence of viscous stationary shocks - also called standing shocks - which connect two different periodic stationary solutions to one another. We prove that standing shocks are stable in $L^1$, provided the initial disturbance satisfies some appropriate boundedness conditions. We also extend this result to arbitrary initial data, but with some restrictions on the flux $A$. In the periodic case, we prove that periodic stationary solutions are always stable. The proof of this result relies on the derivation of uniform $L^\\infty$ bounds on the solution of the conservation law, and on sub- and super-solution techniques.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-12-09T06:25:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35B40",
      "76L05"
    ],
    "keywords": [
      "viscous scalar conservation laws",
      "long time behaviour",
      "periodic stationary solutions",
      "standing shocks",
      "periodic boundary conditions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.1627D"
    }
  }
}