{
  "id": "2207.14797",
  "version": "v1",
  "published": "2022-07-29T17:24:16.000Z",
  "updated": "2022-07-29T17:24:16.000Z",
  "title": "On the norm equivalence of Lyapunov exponents for regularizing linear evolution equations",
  "authors": [
    "Alex Blumenthal",
    "Sam Punshon-Smith"
  ],
  "comment": "45 pages",
  "categories": [
    "math.DS",
    "math.AP"
  ],
  "abstract": "We consider the top Lyapunov exponent associated to a dissipative linear evolution equation posed on a separable Hilbert or Banach space. In many applications in partial differential equations, such equations are often posed on a scale of nonequivalent spaces mitigating, e.g., integrability ($L^p$) or differentiability ($W^{s, p}$). In contrast to finite dimensions, the Lyapunov exponent could apriori depend on the choice of norm used. In this paper we show that under quite general conditions, the Lyapunov exponent of a cocycle of compact linear operators is independent of the norm used. We apply this result to two important problems from fluid mechanics: the enhanced dissipation rate for the advection diffusion equation with ergodic velocity field; and the Lyapunov exponent for the 2d Navier-Stokes equations with stochastic or periodic forcing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-29T17:24:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37L30",
      "37H15",
      "37D25",
      "35G05",
      "35Q35",
      "35R60"
    ],
    "keywords": [
      "lyapunov exponent",
      "regularizing linear evolution equations",
      "norm equivalence",
      "2d navier-stokes equations",
      "partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}