A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport

TL;DR

This paper introduces a novel structure-preserving spatial discretization for the DLSS equation, leveraging a gradient flow framework that ensures convergence and preserves key properties of the continuous model.

Contribution

It presents a new discretization method based on a gradient flow formulation with respect to a generalized metric, ensuring stability and convergence for the DLSS equation.

Findings

Discrete dynamics inherit gradient flow structure

Convergence proven as mesh size vanishes

Preserves contractivity and monotonicity properties

Abstract

We propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle. Our choice of discretization is motivated by a novel gradient flow formulation with respect to a metric that generalizes martingale transport. The discrete dynamics inherits this gradient flow structure, and in addition further properties, such as an alternative gradient flow formulation in the Wasserstein distance, contractivity in the Hellinger distance, and monotonicity of several Lypunov functionals. Our main result is the convergence in the limit of vanishing mesh size. The proof relies an a discrete version of a nonlinear functional inequality between integral expressions involving second order derivatives.