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

Niklas Sapountzoglou; Aleksandra Zimmermann

doi:10.3934/dcds.2020367

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2021, Volume 41, Issue 5: 2341-2376. Doi: 10.3934/dcds.2020367

This issue Previous Article On the asymptotic properties for stationary solutions to the Navier-Stokes equations Next Article On the vanishing discount problem from the negative direction

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

Niklas Sapountzoglou and
Aleksandra Zimmermann^,

University of Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany

^* Corresponding author
^* Corresponding author

Received: November 30, 2019

Revised: July 31, 2020

Early access: November 2020

Published: May 2021

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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 Itô. 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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Mathematics Subject Classification: 35K92, 60H15.

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