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June  2021, 41(6): 2829-2871. doi: 10.3934/dcds.2020388

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

Nicolas Dirr 1, , Hubertus Grillmeier 2, and  Günther Grün 2,,

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

Received  October 2019 Revised  July 2020 Published  June 2021 Early access  December 2020

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: Stochastic porous-medium equation, nonnegative solutions, waiting time phenomenon, finite speed of propagation.
Mathematics Subject Classification: 35R60, 35R35, 35K65, 35K10, 37M05, 60H15, 65N30, 76S05.
Citation: Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2829-2871. doi: 10.3934/dcds.2020388
References:
[1]

D. G. Aronson, L. A. Caffarelli and S. Kamin, How an initially stationary interface begins to move in porous medium flow, SIAM J. Math. Anal., 14 (1983), 639-658.  doi: 10.1137/0514049.

[2]

V. Barbu, G. Da Prato and M. Röckner, Stochastic Porous Media Equations, Lecture Notes in Mathematics, vol. 2163, Springer, Cham, 2016. doi: 10.1007/978-3-319-41069-2.

[3]

V. Barbu, G. Da Prato and M. Röckner, Existence and uniqueness of nonnegative solutions to the stochastic porous media equations, Indiana Univ. Math. J., 57 (2008), 187-211. doi: 10.1512/iumj.2008.57.3241.

[4]

V. Barbu and M. Röckner, Localization of solutions to stochastic porous media equations: Finite speed of propagation, Electron. J. Probab., 17 (2012), 1-11.  doi: 10.1214/EJP.v17-1768.

[5]

D. Breit, E. Feireisl and M. Hofmanová, Incompressible limit for compressible fluids with stochastic forcing, Arch. Ration. Mech. Anal., 222 (2016), 895-926.  doi: 10.1007/s00205-016-1014-y.

[6]

Z. Brzeźniak, W. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310.  doi: 10.1016/j.nonrwa.2013.12.005.

[7]

Z. Brzeźniak and M. Ondreját, Strong solutions to stochastic wave equations with values in Riemannian manifolds, J. Funct. Anal., 253 (2007), 449-481.  doi: 10.1016/j.jfa.2007.03.034.

[8]

G. Da Prato, M. Röckner, B. Rozovskii and F. Wang, Strong solutions of stochastic generalized porous media equations: Existence, uniqueness, and ergodicity, Comm. Partial Differential Equations, 31 (2006), 277-291.  doi: 10.1080/03605300500357998.

[9]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., vol. 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[10]

K. Dareiotis, M. Gerencscér and B. Gess, Entropy solutions for stochastic porous media equations, J. Differential Equations, 266 (2019), 3732-3763.  doi: 10.1016/j.jde.2018.09.012.

[11]

K. Dareiotis and B. Gess, Well-posedness of nonlinear diffusion equations with nonlinear conservative noise, preprint, arXiv: 1811.08356v2.

[12]

B. Davidovitch, E. Moro and H. Stone, Spreading of viscous fluid drops on a solid substrate assisted by thermal fluctuations, Phys. Rev. Lett., 95 (2005), 244505. doi: 10.1103/PhysRevLett.95.244505.

[13]

K. C. Djie, On Upper Bounds for Waiting Times for Doubly Nonlinear Parabolic Equations, Ph.D thesis, Rheinisch-Westfälische Technische Hochschule Aachen, 2008.

[14]

B. Fehrman and B. Gess, Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise, preprint, arXiv: 1807.04230.

[15]

B. Fehrman and B. Gess, Well-posedness of nonlinear diffusion equations with nonlinear conservative noise, Arch. Ration. Mech. Anal., 233 (2019), 249-322.  doi: 10.1007/s00205-019-01357-w.

[16]

J. Fischer and G. Grün, Finite speed of propagation and waiting times for the stochastic porous medium equation: A unifying approach, SIAM J. Math. Anal., 47 (2015), 825-854.  doi: 10.1137/140960578.

[17]

J. Fischer and G. Grün, Existence of positive solutions to stochastic thin-film equations, SIAM J. Math. Anal., 50 (2018), 411-455.  doi: 10.1137/16M1098796.

[18]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.

[19]

B. Gess, Strong solutions for stochastic partial differential equations of gradient type, J. Funct. Anal., 263 (2012), 2355-2383.  doi: 10.1016/j.jfa.2012.07.001.

[20]

B. Gess, Finite speed of propagation for stochastic porous media equations, SIAM J. Math. Anal., 45 (2013), 2734-2766.  doi: 10.1137/120894713.

[21]

L. Giacomelli, R. Dal Passo and A. Shishkov, The thin film equation with nonlinear diffusion, Comm. Partial Differential Equations, 26 (2001), 1509-1557. 

[22]

L. Giacomelli and G. Grün, Lower bounds on waiting times for degenerate parabolic equations and systems, Interfaces Free Bound., 8 (2006), 111-129.  doi: 10.4171/IFB/137.

[23]

H. Grillmeier and G. Grün, Nonnegativity preserving convergent schemes for stochastic porous-medium equations, Math. Comp., 88 (2019), 1021-1059.  doi: 10.1090/mcom/3372.

[24]

G. Grün, K. Mecke and M. Rauscher, Thin-film flow influenced by thermal noise, J. Stat. Phys., 122 (2006), 1261-1291.  doi: 10.1007/s10955-006-9028-8.

[25]

G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Numer. Math., 87 (2000), 113-152.  doi: 10.1007/s002110000197.

[26]

M. Hofmanová, M. Röger and M. von Renesse, Weak solutions for a stochastic mean curvature flow of two-dimensional graphs, Prob. Theory and Related Fields, 168 (2017), 373-408.  doi: 10.1007/s00440-016-0713-5.

[27]

M. Hofmanová and J. Seidler, On weak solutions of stochastic differential equations, Stoch. Anal. Appl., 30 (2012), 100-121.  doi: 10.1080/07362994.2012.628916.

[28]

A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, Theory Probab. Appl., 42 (1997), 167-174.  doi: 10.4213/tvp1769.

[29]

I. Karatzas and S. E. Shreve, Brownian motion, in Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1988, 47–127. doi: 10.1007/978-1-4684-0302-2_2.

[30]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.

[31]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[32]

J. L. Vázquez, The Porous-Medium Equation. Mathematical Theory, The Clarendon Press, Oxford University Press, Oxford, 2007.

show all references

References:
[1]

D. G. Aronson, L. A. Caffarelli and S. Kamin, How an initially stationary interface begins to move in porous medium flow, SIAM J. Math. Anal., 14 (1983), 639-658.  doi: 10.1137/0514049.

[2]

V. Barbu, G. Da Prato and M. Röckner, Stochastic Porous Media Equations, Lecture Notes in Mathematics, vol. 2163, Springer, Cham, 2016. doi: 10.1007/978-3-319-41069-2.

[3]

V. Barbu, G. Da Prato and M. Röckner, Existence and uniqueness of nonnegative solutions to the stochastic porous media equations, Indiana Univ. Math. J., 57 (2008), 187-211. doi: 10.1512/iumj.2008.57.3241.

[4]

V. Barbu and M. Röckner, Localization of solutions to stochastic porous media equations: Finite speed of propagation, Electron. J. Probab., 17 (2012), 1-11.  doi: 10.1214/EJP.v17-1768.

[5]

D. Breit, E. Feireisl and M. Hofmanová, Incompressible limit for compressible fluids with stochastic forcing, Arch. Ration. Mech. Anal., 222 (2016), 895-926.  doi: 10.1007/s00205-016-1014-y.

[6]

Z. Brzeźniak, W. Liu and J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17 (2014), 283-310.  doi: 10.1016/j.nonrwa.2013.12.005.

[7]

Z. Brzeźniak and M. Ondreját, Strong solutions to stochastic wave equations with values in Riemannian manifolds, J. Funct. Anal., 253 (2007), 449-481.  doi: 10.1016/j.jfa.2007.03.034.

[8]

G. Da Prato, M. Röckner, B. Rozovskii and F. Wang, Strong solutions of stochastic generalized porous media equations: Existence, uniqueness, and ergodicity, Comm. Partial Differential Equations, 31 (2006), 277-291.  doi: 10.1080/03605300500357998.

[9]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., vol. 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[10]

K. Dareiotis, M. Gerencscér and B. Gess, Entropy solutions for stochastic porous media equations, J. Differential Equations, 266 (2019), 3732-3763.  doi: 10.1016/j.jde.2018.09.012.

[11]

K. Dareiotis and B. Gess, Well-posedness of nonlinear diffusion equations with nonlinear conservative noise, preprint, arXiv: 1811.08356v2.

[12]

B. Davidovitch, E. Moro and H. Stone, Spreading of viscous fluid drops on a solid substrate assisted by thermal fluctuations, Phys. Rev. Lett., 95 (2005), 244505. doi: 10.1103/PhysRevLett.95.244505.

[13]

K. C. Djie, On Upper Bounds for Waiting Times for Doubly Nonlinear Parabolic Equations, Ph.D thesis, Rheinisch-Westfälische Technische Hochschule Aachen, 2008.

[14]

B. Fehrman and B. Gess, Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise, preprint, arXiv: 1807.04230.

[15]

B. Fehrman and B. Gess, Well-posedness of nonlinear diffusion equations with nonlinear conservative noise, Arch. Ration. Mech. Anal., 233 (2019), 249-322.  doi: 10.1007/s00205-019-01357-w.

[16]

J. Fischer and G. Grün, Finite speed of propagation and waiting times for the stochastic porous medium equation: A unifying approach, SIAM J. Math. Anal., 47 (2015), 825-854.  doi: 10.1137/140960578.

[17]

J. Fischer and G. Grün, Existence of positive solutions to stochastic thin-film equations, SIAM J. Math. Anal., 50 (2018), 411-455.  doi: 10.1137/16M1098796.

[18]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.

[19]

B. Gess, Strong solutions for stochastic partial differential equations of gradient type, J. Funct. Anal., 263 (2012), 2355-2383.  doi: 10.1016/j.jfa.2012.07.001.

[20]

B. Gess, Finite speed of propagation for stochastic porous media equations, SIAM J. Math. Anal., 45 (2013), 2734-2766.  doi: 10.1137/120894713.

[21]

L. Giacomelli, R. Dal Passo and A. Shishkov, The thin film equation with nonlinear diffusion, Comm. Partial Differential Equations, 26 (2001), 1509-1557. 

[22]

L. Giacomelli and G. Grün, Lower bounds on waiting times for degenerate parabolic equations and systems, Interfaces Free Bound., 8 (2006), 111-129.  doi: 10.4171/IFB/137.

[23]

H. Grillmeier and G. Grün, Nonnegativity preserving convergent schemes for stochastic porous-medium equations, Math. Comp., 88 (2019), 1021-1059.  doi: 10.1090/mcom/3372.

[24]

G. Grün, K. Mecke and M. Rauscher, Thin-film flow influenced by thermal noise, J. Stat. Phys., 122 (2006), 1261-1291.  doi: 10.1007/s10955-006-9028-8.

[25]

G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Numer. Math., 87 (2000), 113-152.  doi: 10.1007/s002110000197.

[26]

M. Hofmanová, M. Röger and M. von Renesse, Weak solutions for a stochastic mean curvature flow of two-dimensional graphs, Prob. Theory and Related Fields, 168 (2017), 373-408.  doi: 10.1007/s00440-016-0713-5.

[27]

M. Hofmanová and J. Seidler, On weak solutions of stochastic differential equations, Stoch. Anal. Appl., 30 (2012), 100-121.  doi: 10.1080/07362994.2012.628916.

[28]

A. Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces, Theory Probab. Appl., 42 (1997), 167-174.  doi: 10.4213/tvp1769.

[29]

I. Karatzas and S. E. Shreve, Brownian motion, in Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1988, 47–127. doi: 10.1007/978-1-4684-0302-2_2.

[30]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.

[31]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[32]

J. L. Vázquez, The Porous-Medium Equation. Mathematical Theory, The Clarendon Press, Oxford University Press, Oxford, 2007.

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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$ \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
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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$ \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
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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$ \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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