December  2019, 8(4): 867-882. doi: 10.3934/eect.2019042

Existence and extinction in finite time for Stratonovich gradient noise porous media equations

Dipartimento di Informatica, Università degli Studi di Verona, Strada Le Grazie 15, I–73134, Verona, Italy

* Corresponding author: Mattia Turra

Received  November 2018 Published  June 2019

We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $ dX - \bigl( \nu \Delta X + \Delta \psi (X) \bigr) dt = \sum_{i = 1}^N \langle b_i, \nabla X \rangle \circ d\beta_i $ in $ ]0,T[ \times \mathcal{O} $, with $ X(0) = x(\xi) $ in $ \mathcal{O} $ and $ X = 0 $ on $ ]0,T[ \times \partial \mathcal{O} $. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.

Citation: Mattia Turra. Existence and extinction in finite time for Stratonovich gradient noise porous media equations. Evolution Equations & Control Theory, 2019, 8 (4) : 867-882. doi: 10.3934/eect.2019042
References:
[1]

P. BakC. Tang and K. Wiesenfeld, Self-organized criticality: An explanation of the $1/f$ noise, Phys. Rev. Lett., 59 (1987), 381-384.   Google Scholar

[2]

P. BakC. Tang and K. Wiesenfeld, Self-organized criticality, Phys. Rev. A, 38 (1988), 364-374.  doi: 10.1103/PhysRevA.38.364.  Google Scholar

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P. Bantay and M. Janosi, Self-organization and anomalous diffusion, Physica A: Stat. Mech. Appl., 185 (1992), 11-18.  doi: 10.1016/0378-4371(92)90432-P.  Google Scholar

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V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Monographs in Mathematics, Springer-Verlag New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

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V. Barbu, Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion equations, Annu. Rev. Control, 34 (2010), 52-61.  doi: 10.1016/j.arcontrol.2009.12.002.  Google Scholar

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V. BarbuZ. BrzeźniakE. Hausenblas and L. Tubaro, Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise, Stochastic Process. Appl., 123 (2013), 934-951.  doi: 10.1016/j.spa.2012.10.008.  Google Scholar

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V. BarbuZ. Brzeźniak and L. Tubaro, Stochastic nonlinear parabolic equations with Stratonovich gradient noise, Appl. Math. Optim., 78 (2018), 361-377.  doi: 10.1007/s00245-017-9409-1.  Google Scholar

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V. BarbuG. Da Prato and M. Röckner, Finite time extinction for solutions to fast diffusion stochastic porous media equations, C. R. Acad. Sci. Paris, Ser. I, 347 (2009), 81-84.  doi: 10.1016/j.crma.2008.11.018.  Google Scholar

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V. BarbuG. Da Prato and M. Röckner, Stochastic porous media equations and self-organized criticality, Commun. Math. Phys., 285 (2009), 901-923.  doi: 10.1007/s00220-008-0651-x.  Google Scholar

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V. BarbuG. Da Prato and M. Röckner, Finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise, J. Math. Anal. Appl., 389 (2012), 147-164.  doi: 10.1016/j.jmaa.2011.11.045.  Google Scholar

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V. Barbu, G. Da Prato and M. Röckner, Stochastic Porous Media Equations, vol. 2163 of Lecture notes in Mathematics, Springer International Publishing, 2016. doi: 10.1007/978-3-319-41069-2.  Google Scholar

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V. Barbu and M. Röckner, Stochastic porous media equations and self-organized criticality: convergence to the critical state in all dimensions, Commun. Math. Phys., 311 (2012), 539-555.  doi: 10.1007/s00220-012-1429-8.  Google Scholar

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J. Berryman and C. Holland, Nonlinear diffusion problems arising in plasma physics, Phys. Rev. Lett., 40 (1978), 1720-1722.  doi: 10.1103/PhysRevLett.40.1720.  Google Scholar

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J. Berryman and C. Holland, Asymptotic behavior of the nonlinear diffusion equation $n_t = (n^{-1}n_x)_x$, J. Math. Phys., 23 (1982), 983-987.  doi: 10.1063/1.525466.  Google Scholar

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I. Ciotir and J. Tölle, Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise, J. Funct. Anal., 271 (2016), 1764-1792.  doi: 10.1016/j.jfa.2016.05.013.  Google Scholar

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K. Dareiotis and B. Gess, Nonlinear diffusion equations with nonlinear gradient noise, preprint. arXiv: 1811.08356 Google Scholar

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B. Fehrman and B. Gess, Well-posedness of stochastic porous media equations with nonlinear, conservative noise, preprint. arXiv: 1712.05775 Google Scholar

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B. Gess, Finite time extinction for stochastic sign fast diffusion and self-organized criticality, Commun. Math. Phys., 335 (2015), 309-344.  doi: 10.1007/s00220-014-2225-4.  Google Scholar

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N. Krylov and B. Rozovskii, Stochastic evolution equations, J. Math. Sci., 16 (1981), 1233-1277.  doi: 10.1007/BF01084893.  Google Scholar

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L. Leibenzon, The motion of a gas in a porous medium, in Complete Works, vol. 2, Acad. Sciences URSS, 1930. Google Scholar

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

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I. Munteanu and M. Röckner, Total variation flow perturbed by gradient linear multiplicative noise, Infin. Dimens. Anal. Quantum Probab. Rel. Top., 21 (2018), 1850003, 28pp. doi: 10.1142/S0219025718500030.  Google Scholar

[23]

M. Muskat, The flow of homogeneous fluids through porous media, Soil Science, 46 (1938), 169. doi: 10.1097/00010694-193808000-00008.  Google Scholar

[24]

B. Sixou, L. Wang and F. Peyrin, Stochastic diffusion equation with singular diffusivity and gradient-dependent noise in binary tomography, J. Phys.: Conf. Ser., 542 (2014), 012001. doi: 10.1088/1742-6596/542/1/012001.  Google Scholar

[25]

J. Tölle, Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions, preprint. arXiv: 1803.07005v3 Google Scholar

[26] J. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type, Oxford University Press, 2006.  doi: 10.1093/acprof:oso/9780199202973.001.0001.  Google Scholar
[27] J. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, 2007.  doi: 10.1093/acprof:oso/9780198569039.001.0001.  Google Scholar
[28]

L. Wang, B. Sixou and F. Peyrin, Filtered stochastic optimization for binary tomography, in 2015 IEEE 12th ISBI, 2015, 1604–1607. doi: 10.1109/ISBI.2015.7164187.  Google Scholar

show all references

References:
[1]

P. BakC. Tang and K. Wiesenfeld, Self-organized criticality: An explanation of the $1/f$ noise, Phys. Rev. Lett., 59 (1987), 381-384.   Google Scholar

[2]

P. BakC. Tang and K. Wiesenfeld, Self-organized criticality, Phys. Rev. A, 38 (1988), 364-374.  doi: 10.1103/PhysRevA.38.364.  Google Scholar

[3]

P. Bantay and M. Janosi, Self-organization and anomalous diffusion, Physica A: Stat. Mech. Appl., 185 (1992), 11-18.  doi: 10.1016/0378-4371(92)90432-P.  Google Scholar

[4]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Monographs in Mathematics, Springer-Verlag New York, 2010. doi: 10.1007/978-1-4419-5542-5.  Google Scholar

[5]

V. Barbu, Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion equations, Annu. Rev. Control, 34 (2010), 52-61.  doi: 10.1016/j.arcontrol.2009.12.002.  Google Scholar

[6]

V. BarbuZ. BrzeźniakE. Hausenblas and L. Tubaro, Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise, Stochastic Process. Appl., 123 (2013), 934-951.  doi: 10.1016/j.spa.2012.10.008.  Google Scholar

[7]

V. BarbuZ. Brzeźniak and L. Tubaro, Stochastic nonlinear parabolic equations with Stratonovich gradient noise, Appl. Math. Optim., 78 (2018), 361-377.  doi: 10.1007/s00245-017-9409-1.  Google Scholar

[8]

V. BarbuG. Da Prato and M. Röckner, Finite time extinction for solutions to fast diffusion stochastic porous media equations, C. R. Acad. Sci. Paris, Ser. I, 347 (2009), 81-84.  doi: 10.1016/j.crma.2008.11.018.  Google Scholar

[9]

V. BarbuG. Da Prato and M. Röckner, Stochastic porous media equations and self-organized criticality, Commun. Math. Phys., 285 (2009), 901-923.  doi: 10.1007/s00220-008-0651-x.  Google Scholar

[10]

V. BarbuG. Da Prato and M. Röckner, Finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise, J. Math. Anal. Appl., 389 (2012), 147-164.  doi: 10.1016/j.jmaa.2011.11.045.  Google Scholar

[11]

V. Barbu, G. Da Prato and M. Röckner, Stochastic Porous Media Equations, vol. 2163 of Lecture notes in Mathematics, Springer International Publishing, 2016. doi: 10.1007/978-3-319-41069-2.  Google Scholar

[12]

V. Barbu and M. Röckner, Stochastic porous media equations and self-organized criticality: convergence to the critical state in all dimensions, Commun. Math. Phys., 311 (2012), 539-555.  doi: 10.1007/s00220-012-1429-8.  Google Scholar

[13]

J. Berryman and C. Holland, Nonlinear diffusion problems arising in plasma physics, Phys. Rev. Lett., 40 (1978), 1720-1722.  doi: 10.1103/PhysRevLett.40.1720.  Google Scholar

[14]

J. Berryman and C. Holland, Asymptotic behavior of the nonlinear diffusion equation $n_t = (n^{-1}n_x)_x$, J. Math. Phys., 23 (1982), 983-987.  doi: 10.1063/1.525466.  Google Scholar

[15]

I. Ciotir and J. Tölle, Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise, J. Funct. Anal., 271 (2016), 1764-1792.  doi: 10.1016/j.jfa.2016.05.013.  Google Scholar

[16]

K. Dareiotis and B. Gess, Nonlinear diffusion equations with nonlinear gradient noise, preprint. arXiv: 1811.08356 Google Scholar

[17]

B. Fehrman and B. Gess, Well-posedness of stochastic porous media equations with nonlinear, conservative noise, preprint. arXiv: 1712.05775 Google Scholar

[18]

B. Gess, Finite time extinction for stochastic sign fast diffusion and self-organized criticality, Commun. Math. Phys., 335 (2015), 309-344.  doi: 10.1007/s00220-014-2225-4.  Google Scholar

[19]

N. Krylov and B. Rozovskii, Stochastic evolution equations, J. Math. Sci., 16 (1981), 1233-1277.  doi: 10.1007/BF01084893.  Google Scholar

[20]

L. Leibenzon, The motion of a gas in a porous medium, in Complete Works, vol. 2, Acad. Sciences URSS, 1930. Google Scholar

[21]

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

[22]

I. Munteanu and M. Röckner, Total variation flow perturbed by gradient linear multiplicative noise, Infin. Dimens. Anal. Quantum Probab. Rel. Top., 21 (2018), 1850003, 28pp. doi: 10.1142/S0219025718500030.  Google Scholar

[23]

M. Muskat, The flow of homogeneous fluids through porous media, Soil Science, 46 (1938), 169. doi: 10.1097/00010694-193808000-00008.  Google Scholar

[24]

B. Sixou, L. Wang and F. Peyrin, Stochastic diffusion equation with singular diffusivity and gradient-dependent noise in binary tomography, J. Phys.: Conf. Ser., 542 (2014), 012001. doi: 10.1088/1742-6596/542/1/012001.  Google Scholar

[25]

J. Tölle, Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions, preprint. arXiv: 1803.07005v3 Google Scholar

[26] J. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type, Oxford University Press, 2006.  doi: 10.1093/acprof:oso/9780199202973.001.0001.  Google Scholar
[27] J. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, 2007.  doi: 10.1093/acprof:oso/9780198569039.001.0001.  Google Scholar
[28]

L. Wang, B. Sixou and F. Peyrin, Filtered stochastic optimization for binary tomography, in 2015 IEEE 12th ISBI, 2015, 1604–1607. doi: 10.1109/ISBI.2015.7164187.  Google Scholar

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