Well-posedness of renormalized solutions for a stochastic $p$-Laplace equation with $L^1$-initial data

Niklas Sapountzoglou; Aleksandra Zimmermann

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arXiv · 1908.11186

Well-posedness of renormalized solutions for a stochastic $p$-Laplace equation with $L^1$-initial data

Niklas Sapountzoglou·Aleksandra Zimmermann

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Abstract

We consider a $p$-Laplace evolution problem with stochastic forcing on a bounded domain $D\subset\mathbb{R}^d$ with homogeneous Dirichlet boundary conditions for $1<p<\infty$. The additive noise term is given by a stochastic integral in the sense of It\^{o}. The technical difficulties arise from the merely integrable random initial data $u_0$ under consideration. Due to the poor regularity of the initial data, estimates in $W^{1,p}_0(D)$ are available with respect to truncations of the solution only and therefore well-posedness results have to be formulated in the sense of generalized solutions. We extend the notion of renormalized solution for this type of SPDEs, show well-posedness in this setting and study the Markov properties of solutions.

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BibTeX RIS

Niklas Sapountzoglou, Aleksandra Zimmermann. 2019-08-29. Well-posedness of renormalized solutions for a stochastic $p$-Laplace equation with $L^1$-initial data. https://arxiv.org/abs/1908.11186

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