Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise

TL;DR

This paper establishes existence and uniqueness of solutions for complex stochastic nonlinear diffusion equations with singular diffusivity and gradient noise, addressing an open problem in the mathematical analysis of such equations.

Contribution

It introduces a novel variational framework for these equations, deriving a commutator relation under geometric conditions, and connects boundary conditions with noise coefficients.

Findings

Proved existence and uniqueness of solutions.

Derived a commutator relation for noise coefficients.

Connected boundary conditions with noise structure.

Abstract

We study existence and uniqueness of a variational solution in terms of stochastic variational inequalities (SVI) to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative Stratonovich gradient-type noise. We derive a commutator relation for the unbounded noise coefficients in terms of a geometric Killing vector condition. The drift term is given by the total variation flow, respectively, by a singular $p$-Laplace-type operator. We impose nonlinear zero Neumann boundary conditions and precisely investigate their connection with the coefficient fields of the noise. This solves an open problem posed in [Barbu, Brze\'{z}niak, Hausenblas, Tubaro; Stoch. Proc. Appl., 123 (2013)] and [Barbu, R\"ockner; J. Eur. Math. Soc., 17 (2015)].