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

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• 作者：Ioana Ciotir^a ; ¹ ; ^{ioana.ciotir@insa-rouen.fr" class="auth_mail" title="E-mail the corresponding author} ; Jonas M. Tö ; lle^b ; ² ; ^{jonas.tolle@aalto.fi" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35K55 ; 35K92 ; 60H15 ; secondary ; 49J40 ; 49J45 ; 60H25
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 October 2016
• 年：2016
• 卷：271
• 期：7
• 页码：1764-1792
• 全文大小：499 K

文摘

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 et al. (2013) [7] and Barbu and Röckner (2015) [10].