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Wednesday, June 20, 2018 - 14:00 in V3-201

Well-posedness for a class of nonlinear SPDEs with strongly continuous perturbation

A talk in the Bielefeld Stochastic Afternoon series by
Aleksandra Zimmermann from Universität Duisburg-Essen

Abstract: We consider the stochastic evolution equation $$du - div(a(x, u, Du) + F (u)) dt = \Phi dW$$ for $T > 0$, on a bounded Lipschitz domain $D$ with homogeneous Dirichlet boundary conditions and initial condition in $L^2 (D)$. The main technical difficulties arise from the nonlinear diffusion-convection operator which is defined by a Carathéodory function $a = a(x, \lambda, \xi)$ satisfying appropriate growth and coercivity assumptions and $F : \mathbb{R} \rightarrow \mathbb{R}^d$ Lipschitz continuous. On the right-hand side, we consider an additive stochastic perturbation with respect to a cylindrical Wiener process with values in $L^2 (D)$. We obtain approximate solutions by a semi-implicit time discretization. Adjusting the method of stochastic compactness to our setting, we are able to pass to the limit in the approximate equation. We show an $L^1$-contraction principle and obtain existence and uniqueness of (stochastically) strong solutions.

Within the CRC this talk is associated to the project(s): B1, B2


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