{
  "id": "2505.08171",
  "version": "v1",
  "published": "2025-05-13T02:17:50.000Z",
  "updated": "2025-05-13T02:17:50.000Z",
  "title": "Asymptotic behavior toward viscous shocks for the outflow problem of barotropic Navier-Stokes equations",
  "authors": [
    "Moon-Jin Kang",
    "HyeonSeop Oh",
    "Yi Wang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the large-time asymptotic stability of viscous shock profile to the outflow problem of barotropic Navier-Stokes equations on a half line. We consider the case when the far-field state as a right-end state of 2-Hugoniot shock curve belongs to the subsonic region or transonic curve. We employ the method of $a$-contraction with shifts, to prove that if the strength of viscous shock wave is small and sufficiently away from the boundary, and if a initial perturbation is small, then the solution asymptotically converges to the viscous shock up to a dynamical shift. We also prove that the speed of time-dependent shift decays to zero as times goes to infinity, the shifted viscous shock still retains its original profile time-asymptotically. Since the outflow problem in the Lagrangian mass coordinate leads to a free boundary value problem due to the absence of a boundary condition for the fluid density, we consider the problem in the Eulerian coordinate instead. Although the $a$-contraction method is technically more complicated in the Eulerian coordinate than in the Lagrangian one, this provides a more favorable framework by avoiding the difficulty arising from a free boundary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-13T02:17:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "viscous shock",
      "barotropic navier-stokes equations",
      "outflow problem",
      "asymptotic behavior",
      "eulerian coordinate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}