On stochastic porous-medium equations with critical-growth conservative multiplicative noise

Nicolas Dirr; Hubertus Grillmeier; Günther Grün

doi:10.3934/dcds.2020388

Article Contents

2021, Volume 41, Issue 6: 2829-2871. Doi: 10.3934/dcds.2020388

This issue Previous Article Properties of multicorrelation sequences and large returns under some ergodicity assumptions Next Article Approximation properties of Lüroth expansions

On stochastic porous-medium equations with critical-growth conservative multiplicative noise

1.
Cardiff University, School of Mathematics, 21-23 Senghennydd Road, Cathays Cardiff, CF24 4AG, UK
2.
Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Mathematics, Cauerstr. 11, 91058 Erlangen, Germany

^* Corresponding author: Günther Grün
^* Corresponding author: Günther Grün

Received: October 2019

Revised: July 2020

Early access: December 2020

Published: June 2021

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Abstract

First, we prove existence, nonnegativity, and pathwise uniqueness of martingale solutions to stochastic porous-medium equations driven by conservative multiplicative power-law noise in the Ito-sense. We rely on an energy approach based on finite-element discretization in space, homogeneity arguments and stochastic compactness. Secondly, we use Monte-Carlo simulations to investigate the impact noise has on waiting times and on free-boundary propagation. We find strong evidence that noise on average significantly accelerates propagation and reduces the size of waiting times – changing in particular scaling laws for the size of waiting times.

Keywords:

Mathematics Subject Classification: 35R60, 35R35, 35K65, 35K10, 37M05, 60H15, 65N30, 76S05.

Citation:

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Figure 1. Log-log plot of the average size of waiting times in terms of $ \bar S $ for different noise amplitudes ($ m = 0.5 $)

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Figure 2. Log-log plot of the average size of waiting times $ T^\ast $ in dependence of the noise amplitude $ \nu $ ($ m = 0.5 $)

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Figure 3. Average value of free-boundary location for $ m = 0.5 $ over the time intervall [0,100]

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Figure 4. Log-log plot of the average free-boundary location in terms of time ($ m = 0.5 $)

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Table 1. Average waiting times $ \cdot 10^3 $

	$ \nu=0 $	$ \nu=0.0125 $	$ \nu=0.025 $	$ \nu= 0.05 $	$ \nu= 0.1 $	$ \nu= 0.2 $
$ \bar S=1 $	63.8	39.0	25.7	15.2	8.36	4.43
$ \bar S=2 $	31.1	20.0	13.6	8.31	4.67	2.54
$ \bar S=4 $	15.0	10.1	7.22	4.51	2.61	1.43
$ \bar S=8 $	7.14	5.05	3.74	2.44	1.44	0.838
$ \bar S=16 $	3.39	2.45	1.86	1.28	0.791	0.491

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Table 2. Estimated variances $ \cdot 10^8 $

	$ \nu=0.0125 $	$ \nu=0.025 $	$ \nu= 0.05 $	$ \nu= 0.1 $	$ \nu= 0.2 $
$ \bar S=1 $	34.6	60.2	45.6	39.2	17.2
$ \bar S=2 $	19.6	20.0	23.8	15.3	9.14
$ \bar S=4 $	8.0	11.2	9.68	6.74	4.62
$ \bar S=8 $	2.73	5.43	5.05	4.68	2.95
$ \bar S=16 $	1.47	1.91	1.65	1.93	1.38

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Table 3. Average scaling of waiting times w.r.t. $ \bar{S}^{-1} $

	$ \nu=0 $	$ \nu=0.0125 $	$ \nu=0.025 $	$ \nu= 0.05 $	$ \nu= 0.1 $	$ \nu= 0.2 $
$ p_\nu $	1.06	0.998	0.947	0.892	0.85	0.793

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