June  2020, 9(2): 375-398. doi: 10.3934/eect.2020010

Stochastic porous media equations with divergence Itô noise

Normandie University, INSA de Rouen Normandie, LMI, 7600 Rouen, France

* Corresponding author: ioana.ciotir@insa-rouen.fr

Received  January 2019 Published  June 2020 Early access  August 2019

Fund Project: The author is partially supported by the European Union with the European regional development fund (ERDF, HN0002137) and by the Normandie Regional Council (via the M2NUM and M2SiNum projects) and by the ANR Project QUTE-HPC Quantum Turbulence Exploration by High-Performance Computing (ANR-18-CE46-0013)

We study the existence and uniqueness of solution to stochastic porous media equations with divergence Itô noise in infinite dimensions. The first result prove existence of a stochastic strong solution and it is essentially based on the non-local character of the noise. The second result proves existence of at least one martingale solution for the critical case corresponding to the Dirac distribution.

Citation: Ioana Ciotir. Stochastic porous media equations with divergence Itô noise. Evolution Equations and Control Theory, 2020, 9 (2) : 375-398. doi: 10.3934/eect.2020010
References:
[1]

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

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, 190. Academic Press, Inc., Boston, MA, 1993.

[3]

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

[4]

V. Barbu, The fast logarithmic equation with multiplicative Gaussian noise, Annals of the University of Bucharest (Mathematical Series), 3 (2012), 145-153. 

[5]

V. Barbu, Stochastic porous media equations, Stochastic Analysis: A Series of Lectures, of the Series Progress in Probability, 68 (2015), 101-133.  doi: 10.1007/978-3-0348-0909-2_4.

[6]

V. BarbuG. Da Prato and M. Röckner, Existence of strong solution for stochastic porous media equations under general motonocity conditions, Annals of Probability, 37 (2009), 428-452.  doi: 10.1214/08-AOP408.

[7]

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

[8]

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

[9]

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

[10]

V. BarbuG. Da Prato and M. Röckner, Finite time extinction for solutions to fast diffusion stochastic porous media equations, Comptes Rendus Mathematiques, 347 (2009), 81-84.  doi: 10.1016/j.crma.2008.11.018.

[11]

V. Barbu, S. Bonaccorsi and L. Tubaro, Stochastic differential equations with variable structure driven by multiplicative Gaussian noise and sliding mode dynamic, Mathematics of Control Signals and Systems, 28 (2016), Art. 26, 28 pp. doi: 10.1007/s00498-016-0178-1.

[12]

I. Ciotir, Existence and Uniqueness of Solutions to the Stochastic Porous Media Equations of Saturated Flows, Appl. Math. Optim., 61 (2010), 129-143.  doi: 10.1007/s00245-009-9078-9.

[13]

I. Ciotir, Convergence of the solutions for the stochastic porous media equations and homogenization, Journal of Evolution Equation, 11 (2011), 339-370.  doi: 10.1007/s00028-010-0094-7.

[14]

I. Ciotir, Existence and uniqueness of the solution for stochastic super-fast diffusion equations with multiplicative noise, J. Math. Anal. Appl., 452 (2017), 595-610.  doi: 10.1016/j.jmaa.2017.03.018.

[15]

I. Ciotir and J. M. Tölle, Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise, Journal of Functional Analysis, 271 (2016), 1764-1792.  doi: 10.1016/j.jfa.2016.05.013.

[16]

I. Ciotir and J. M. Tölle, Convergence of invariant measures for singular stochastic diffusion equations, Stochastic Processes and their Applications, 122 (2012), 1998-2017.  doi: 10.1016/j.spa.2011.11.011.

[17]

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

[18]

G. Da PratoM. RöcknerB. L. Rozovskii and F.-Y. Wang, Strong Solution of Stochastic Generalized Porous Media Equations: Existence, Uniqueness and Ergoticity, Comm. PDEs, 31 (2006), 277-291.  doi: 10.1080/03605300500357998.

[19]

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

[20]

B. Gess, Finite Time Extinction for Stochastic Sign Fast Diffusion and Self-Organized Criticality, Communications in Mathematical Physics, 335 (2015), 309-344.  doi: 10.1007/s00220-014-2225-4.

[21]

B. Gess and M. Röckner, Singular-degenerate multivalued stochastic fast diffusion equations, SIAM Journal on Mathematical Analysis, 47 (2015), 4058-4090.  doi: 10.1137/151003726.

[22]

N. V. Krylov and B. L. Rozovski, Stochastic evolution equations, J. Soviet Mat., 14 (1979), 71-147. 

[23]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, 2007.

[24]

J. G. RenM. Röckner and F.-Y. Wang, Stochastic generalized porous media and fast diffusions equations, J. Differential Equations, 238 (2007), 118-152.  doi: 10.1016/j.jde.2007.03.027.

[25]

J. M. Tölle, Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions, preprint, 2018.

show all references

References:
[1]

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

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Mathematics in Science and Engineering, 190. Academic Press, Inc., Boston, MA, 1993.

[3]

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

[4]

V. Barbu, The fast logarithmic equation with multiplicative Gaussian noise, Annals of the University of Bucharest (Mathematical Series), 3 (2012), 145-153. 

[5]

V. Barbu, Stochastic porous media equations, Stochastic Analysis: A Series of Lectures, of the Series Progress in Probability, 68 (2015), 101-133.  doi: 10.1007/978-3-0348-0909-2_4.

[6]

V. BarbuG. Da Prato and M. Röckner, Existence of strong solution for stochastic porous media equations under general motonocity conditions, Annals of Probability, 37 (2009), 428-452.  doi: 10.1214/08-AOP408.

[7]

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

[8]

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

[9]

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

[10]

V. BarbuG. Da Prato and M. Röckner, Finite time extinction for solutions to fast diffusion stochastic porous media equations, Comptes Rendus Mathematiques, 347 (2009), 81-84.  doi: 10.1016/j.crma.2008.11.018.

[11]

V. Barbu, S. Bonaccorsi and L. Tubaro, Stochastic differential equations with variable structure driven by multiplicative Gaussian noise and sliding mode dynamic, Mathematics of Control Signals and Systems, 28 (2016), Art. 26, 28 pp. doi: 10.1007/s00498-016-0178-1.

[12]

I. Ciotir, Existence and Uniqueness of Solutions to the Stochastic Porous Media Equations of Saturated Flows, Appl. Math. Optim., 61 (2010), 129-143.  doi: 10.1007/s00245-009-9078-9.

[13]

I. Ciotir, Convergence of the solutions for the stochastic porous media equations and homogenization, Journal of Evolution Equation, 11 (2011), 339-370.  doi: 10.1007/s00028-010-0094-7.

[14]

I. Ciotir, Existence and uniqueness of the solution for stochastic super-fast diffusion equations with multiplicative noise, J. Math. Anal. Appl., 452 (2017), 595-610.  doi: 10.1016/j.jmaa.2017.03.018.

[15]

I. Ciotir and J. M. Tölle, Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise, Journal of Functional Analysis, 271 (2016), 1764-1792.  doi: 10.1016/j.jfa.2016.05.013.

[16]

I. Ciotir and J. M. Tölle, Convergence of invariant measures for singular stochastic diffusion equations, Stochastic Processes and their Applications, 122 (2012), 1998-2017.  doi: 10.1016/j.spa.2011.11.011.

[17]

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

[18]

G. Da PratoM. RöcknerB. L. Rozovskii and F.-Y. Wang, Strong Solution of Stochastic Generalized Porous Media Equations: Existence, Uniqueness and Ergoticity, Comm. PDEs, 31 (2006), 277-291.  doi: 10.1080/03605300500357998.

[19]

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

[20]

B. Gess, Finite Time Extinction for Stochastic Sign Fast Diffusion and Self-Organized Criticality, Communications in Mathematical Physics, 335 (2015), 309-344.  doi: 10.1007/s00220-014-2225-4.

[21]

B. Gess and M. Röckner, Singular-degenerate multivalued stochastic fast diffusion equations, SIAM Journal on Mathematical Analysis, 47 (2015), 4058-4090.  doi: 10.1137/151003726.

[22]

N. V. Krylov and B. L. Rozovski, Stochastic evolution equations, J. Soviet Mat., 14 (1979), 71-147. 

[23]

C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Springer, 2007.

[24]

J. G. RenM. Röckner and F.-Y. Wang, Stochastic generalized porous media and fast diffusions equations, J. Differential Equations, 238 (2007), 118-152.  doi: 10.1016/j.jde.2007.03.027.

[25]

J. M. Tölle, Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions, preprint, 2018.

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