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Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case
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 |
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.
References:
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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. Google Scholar |
[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. Google Scholar |
[14] |
B. Fehrman and B. Gess, Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise, preprint, arXiv: 1807.04230. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[14] |
B. Fehrman and B. Gess, Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise, preprint, arXiv: 1807.04230. Google Scholar |
[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. |




63.8 | 39.0 | 25.7 | 15.2 | 8.36 | 4.43 | |
31.1 | 20.0 | 13.6 | 8.31 | 4.67 | 2.54 | |
15.0 | 10.1 | 7.22 | 4.51 | 2.61 | 1.43 | |
7.14 | 5.05 | 3.74 | 2.44 | 1.44 | 0.838 | |
3.39 | 2.45 | 1.86 | 1.28 | 0.791 | 0.491 |
63.8 | 39.0 | 25.7 | 15.2 | 8.36 | 4.43 | |
31.1 | 20.0 | 13.6 | 8.31 | 4.67 | 2.54 | |
15.0 | 10.1 | 7.22 | 4.51 | 2.61 | 1.43 | |
7.14 | 5.05 | 3.74 | 2.44 | 1.44 | 0.838 | |
3.39 | 2.45 | 1.86 | 1.28 | 0.791 | 0.491 |
34.6 | 60.2 | 45.6 | 39.2 | 17.2 | |
19.6 | 20.0 | 23.8 | 15.3 | 9.14 | |
8.0 | 11.2 | 9.68 | 6.74 | 4.62 | |
2.73 | 5.43 | 5.05 | 4.68 | 2.95 | |
1.47 | 1.91 | 1.65 | 1.93 | 1.38 |
34.6 | 60.2 | 45.6 | 39.2 | 17.2 | |
19.6 | 20.0 | 23.8 | 15.3 | 9.14 | |
8.0 | 11.2 | 9.68 | 6.74 | 4.62 | |
2.73 | 5.43 | 5.05 | 4.68 | 2.95 | |
1.47 | 1.91 | 1.65 | 1.93 | 1.38 |
1.06 | 0.998 | 0.947 | 0.892 | 0.85 | 0.793 |
1.06 | 0.998 | 0.947 | 0.892 | 0.85 | 0.793 |
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