Finsler structure for variable exponent Wasserstein space and gradient flows

TL;DR

This paper develops a variational framework using Finsler geometry for variable exponent Wasserstein spaces to analyze gradient flows of parabolic equations involving the $q(x)$-Laplacian, extending known results for constant $q$.

Contribution

It introduces a novel Finsler metric-based approach on Wasserstein space for variable exponent PDEs, enabling existence and uniqueness results.

Findings

Established a new Finsler structure on probability space.

Derived gradient flow formulations for variable $q(x)$-Laplacian equations.

Extended known results to variable exponent cases.

Abstract

In this paper, we propose a variational approach based on optimal transportation to study the existence and unicity of solution for a class of parabolic equations involving $q(x)$-Laplacian operator \begin{equation*}\label{equation variable q(x)} \frac{\partial \rho(t,x)}{\partial t}=div_x\left(\rho(t,x)|\nabla_x G^{'}(\rho(t,x))|^{q(x)-2}\nabla_x G^{'}(\rho(t,x)) \right) .\end{equation*} The variational approach requires the setting of new tools such as appropiate distance on the probability space and an introduction of a Finsler metric in this space. The class of parabolic equations is derived as the flow of a gradient with respect the Finsler structure. For $q(x)\equiv q$ constant, we recover some known results existing in the literature for the $q$-Laplacian operator.