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

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, Brzeźniak, Hausenblas, Tubaro; Stoch. Proc. Appl., 123 (2013)] and [Barbu, Röckner; J. Eur. Math. Soc., 17 (2015)].

23 pages, 54 references, to appear in Journal of Functional Analysis

References in corpus (5)

• Global L_2-solutions of stochastic Navier-Stokes equations
• Stability of solutions to stochastic partial differential equations
• On Itô's formula for elliptic diffusion processes
• Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations
• Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise

Cited by in corpus (2)

• Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions
• Stability and moment estimates for the stochastic singular $\Phi$-Laplace equation