Hydrodynamic limits and hypocoercivity for kinetic equations with heavy tails

This expository article, written for the proceedings of the Journées EDP 2023, presents mainly joint works with Dolbeault and Laflèche [1] and Mouhot [3]. We will review some results about long time behaviour of linear kinetic equations for which the microscopic equilibrium (that is, the kernel of the reorientation operator) is typically a density with polynomial decay. There will be no space confinement and the reorientation operator could be of scattering, Fokker–Planck or Levy–Fokker–Planck types. We will first present a spectral approach a la Ellis and Pinsky that yields to a unified treatment of the macroscopic limits for this kind of equations and then focus on re-shaping the Dolbeault–Mouhot–Schmeiser L 2 -hypocoercivity method to get explicit rates of decay to zero in suitable weighted norms.