Long-time behaviour of a fully discrete Lagrangian scheme for a family of fourth order

A fully discrete Lagrangian scheme for solving a family of fourth order equations numerically is presented. The discretization is based on the equation's underlying gradient flow structure w.r.t. the $L^2$-Wasserstein distance, and adapts numerous of its most important structural properties by construction, as conservation of mass and entropy-dissipation. In this paper, the long-time behaviour of our discretization is analyzed: We show that discrete solutions decay exponentially to equilibrium at the same rate as smooth solutions of the origin problem. Moreover, we give a proof of convergence of discrete entropy minimizers towards Barenblatt-profiles or Gaussians, respectively, using $Γ$-convergence.