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

  • * Corresponding author: Günther Grün

    * Corresponding author: Günther Grün
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  • 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.

    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 $)

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

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

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

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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