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

TL;DR

This paper introduces a fully discrete Lagrangian scheme for fourth order equations that preserves key structural properties and demonstrates exponential decay to equilibrium and convergence to known profiles.

Contribution

The paper develops a novel fully discrete Lagrangian scheme that maintains the gradient flow structure and proves its long-time decay and convergence properties.

Findings

Discrete solutions decay exponentially at the same rate as smooth solutions.

Discrete entropy minimizers converge to Barenblatt-profiles or Gaussians.

The scheme preserves conservation of mass and entropy-dissipation.

Abstract

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 $\Gamma$-convergence.