{
  "id": "2304.11698",
  "version": "v1",
  "published": "2023-04-23T16:53:53.000Z",
  "updated": "2023-04-23T16:53:53.000Z",
  "title": "Hydrodynamic limits for conservative kinetic equations: a spectral and unified approach in the presence of a spectral gap",
  "authors": [
    "Pierre Gervais",
    "Bertrand Lods"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Triggered by the fact that, in the hydrodynamical limit, several different kinetic equations of physical interest all lead to the same Navier-Stokes-Fourier system, we develop in the paper an abstract framework which allows to explain this phenomenon. The method we develop can be seen as a significative improvement of known approaches for which we fully exploit some structural assumptions on the linear and nonlinear collision operators as well as a good knowledge of the Cauchy theory for the limiting equation. We adopt a perturbative framework in a Hilbert space setting and first develop a general and fine spectral analysis of the linearized operator and its associated semigroup. Then, we introduce a splitting adapted to the various regimes (kinetic, acoustic, hydrodynamic) present in the kinetic equation which allows, by a fixed point argument, to construct a solution to the kinetic equation and prove the convergence towards suitable solutions to the Navier-Stokes-Fourier system. Our approach is robust enough to treat, in the same formalism, the case of the Boltzmann equation with hard and moderately soft potentials, with and without cut-off assumptions, as well as the Landau equation for hard and moderately soft potentials in presence of a spectral gap. We also show that our method applies, at least at the linear level, to quantum kinetic equations for Fermi-Dirac or Bose-Einstein particles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-23T16:53:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "conservative kinetic equations",
      "spectral gap",
      "hydrodynamic limits",
      "unified approach",
      "moderately soft potentials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}