{
  "id": "0811.1193",
  "version": "v1",
  "published": "2008-11-07T20:30:38.000Z",
  "updated": "2008-11-07T20:30:38.000Z",
  "title": "Conditional stability of unstable viscous shocks",
  "authors": [
    "Kevin Zumbrun"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Continuing a line of investigation initiated by Texier and Zumbrun on dynamics of viscous shock and detonation waves, we show that a linearly unstable Lax-type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\\cap H^2$ perturbatoins, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with $p$ unstable eigenvalues, we establish conditional stability on a codimension-$p$ manifold of initial data, with sharp rates of decay in all $L^p$. For $p=0$, we recover the result of unconditional stability obtained by Howard, Mascia, and Zumbrun.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-11-07T20:30:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35"
    ],
    "keywords": [
      "unstable viscous shocks",
      "conditional stability",
      "unstable lax-type viscous shock solution",
      "semilinear strictly parabolic system",
      "conservation laws possesses"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1016/j.jde.2009.02.017",
      "journal": "Journal of Differential Equations",
      "year": 2009,
      "volume": 247,
      "number": 2,
      "pages": 648
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009JDE...247..648Z"
    }
  }
}