Well-posedness of SVI solutions to singular-degenerate stochastic porous media equations arising in self-organised criticality

TL;DR

This paper proves the existence and uniqueness of SVI solutions for a class of stochastic porous media equations with applications to models of self-organised criticality, analyzing the energy functional in detail.

Contribution

It establishes well-posedness of SVI solutions for singular-degenerate stochastic porous media equations with Lipschitz noise, a novel result in this context.

Findings

Unique SVI solutions depend continuously on initial data.

Energy functional analysis enables proof of uniqueness.

Results apply to models exhibiting self-organised criticality.

Abstract

We consider a class of generalised stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organised criticality. We show that these SPDEs have unique SVI solutions which depend continuously on the initial value. In order to formulate this notion of solution and to prove uniqueness in the case of a slowly growing nonlinearity, the arising energy functional is analysed in detail.