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arXiv · 1507.02576

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

Ioana Ciotir·Jonas M. Tölle
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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)].

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Ioana Ciotir, Jonas M. Tölle. 2015-07-09. Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise. https://doi.org/10.1016/j.jfa.2016.05.013

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