Regularity theory for nonlinear systems of SPDEs

2013/11/30 by Dominic Breit
Engineering · Mathematics · #Algebraic geometry #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Nonlinear system #Number theory #Stability and Controllability of Differential Equations #Stochastic partial differential equation #Stochastic process #Type (biology) #math.AP #msc:35B65 #msc:35D30 #msc:35K55 #msc:35R60 #msc:60H15

paper · pdf · doi:10.1007/s00229-014-0704-8

openalex publication_date 2014/09/20 · arxiv created 2016/12/31 · openalex created_date 2019/06/27 · arxiv updated 2020/05/15 · openalex updated_date 2026/08/05

Abstract

We consider systems of stochastic evolutionary equations of the type du=div S(∇ u) dt+Φ(u)dW_t where S is a non-linear operator, for instance the p-Laplacian S(ξ)=(1+|ξ|)^p-2ξ, ξ∈\mathbb R^{d× D}, with p∈(1,∞) and Φ grows linearly. We extend known results about the deterministic problem to the stochastic situation. First we verify the natural regularity: \mathbb E[sup_t∈(0,T)∫_G'|∇ u(t)|² dx+∫₀^T∫_G'|∇ F(∇ u)|² dx dt]<∞, where F(ξ)=(1+|ξ|)^(p-2)/(2)ξ. If we have Uhlenbeck-structure then \mathbb E[‖∇ u‖_q^q] is finite for all q<∞.

Citations

• Elliptic Partial Differential Equations of Second Order
• Regularity of minima: An invitation to the dark side of the calculus of variations
• SPDE in Hilbert space with locally monotone coefficients
• Parabolic Systems with Polynomial Growth and Regularity
• Direct Methods in the Calculus of Variations