Singular-degenerate multivalued stochastic fast diffusion equations

TL;DR

This paper develops a well-posedness framework for singular-degenerate multivalued stochastic fast diffusion equations, including the stochastic sign fast diffusion equation, using stochastic variational inequalities and establishing existence and regularity of solutions.

Contribution

It generalizes the stochastic variational inequalities approach to stochastic fast diffusion equations and introduces a new proof of well-posedness applicable to general diffusion coefficients.

Findings

Established well-posedness for stochastic sign fast diffusion equations

Proved existence of strong solutions with higher regularity for linear multiplicative noise

Generalized the SVI approach to a broader class of stochastic fast diffusion equations

Abstract

We consider singular-degenerate, multivalued stochastic fast diffusion equations with multiplicative Lipschitz continuous noise. In particular, this includes the stochastic sign fast diffusion equation arising from the Bak-Tang-Wiesenfeld model for self-organized criticality. A well-posedness framework based on stochastic variational inequalities (SVI) is developed, characterizing solutions to the stochastic sign fast diffusion equation, previously obtained in a limiting sense only. Aside from generalizing the SVI approach to stochastic fast diffusion equations we develop a new proof of well-posedness, applicable to general diffusion coefficients. In case of linear multiplicative noise, we prove the existence of (generalized) strong solutions, which entails higher regularity properties of solutions than previously known.