The Stochastic Porous Media Equations in \(\mathbb{R}^{d}\)

  • Viorel Barbu
  • Giuseppe Da Prato
  • Michael Röckner
Part of the Lecture Notes in Mathematics book series (LNM, volume 2163)


Here we shall treat Eq. ( 3.1) in the domain \(\mathcal{O} = \mathbb{R}^{d}\). Though the methods are similar to those used for bounded domains, there are, however, some notable differences and as seen below the dimension d of the space plays a crucial role.


Stochastic Porous Media Equations Finite Extinction Time Maximal Monotone Multivalued Function Main Existence Result Crandall-Liggett Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 6.
    V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces (Springer, New York, 2010)CrossRefzbMATHGoogle Scholar
  2. 15.
    V. Barbu, M. Röckner, On a random scaled porous media equations. J. Differ. Equ. 251, 2494–2514 (2011)CrossRefzbMATHGoogle Scholar
  3. 21.
    V. Barbu, G. Da Prato, M. Röckner, Existence and uniqueness of nonnegative solutions to the stochastic porous media equation. Indiana Univ. Math. J. 57 (1), 187–212 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 24.
    V. Barbu, G. Da Prato, M. Röckner, Stochastic porous media equation and self-organized criticality. Commun. Math. Phys. 285, 901–923 (2009)CrossRefzbMATHGoogle Scholar
  5. 26.
    V. Barbu, M. Röckner, F. Russo, The stochastic porous media equations in \(\mathbb{R}^{d}\). J. Math. Pures Appl. (9), 103 (4), 1024–1052 (2015)Google Scholar
  6. 27.
    V. Barbu, M. Röckner, F. Russo, A stochastic Fokker-Planck equation and double probabilistic representation for the stochastic porous media type equation. arXiv:1404.5120Google Scholar
  7. 36.
    H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer, New York, 2010)CrossRefGoogle Scholar
  8. 51.
    G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, 2nd edn. (Cambridge University Press, Cambridge, 2014)Google Scholar
  9. 67.
    L. Hörmander, The Analysis of Linear Partial Differential Operators (Springer, New York, 1963)zbMATHGoogle Scholar
  10. 71.
    N. Krylov, Itô’s formula for the L p-norm of stochastic W p 1-valued processes. Probab. Theory. Relat. Fields 147, 583–605 (2010)CrossRefzbMATHGoogle Scholar
  11. 82.
    C. Prevot, M. Röckner, A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905 (Springer, Berlin, 2007)Google Scholar
  12. 84.
    J. Ren, M. Röckner, F.Y. Wang, Stochastic generalized porous media and fast diffusion equations. J. Differ. Equ. 238 (1), 118–152 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 85.
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)CrossRefzbMATHGoogle Scholar
  14. 86.
    M. Röckner, F.Y. Wang, Non-monotone stochastic generalized porous media equations. J. Differ. Equ. 245 (12), 3898–3935 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 90.
    J.L. Vazquez, J.R. Esteban, A. Rodriguez, The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane. Adv. Differ. Equ. 1, 21–50 (1996)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Viorel Barbu
    • 1
  • Giuseppe Da Prato
    • 2
  • Michael Röckner
    • 3
  1. 1.Department of MathematicsAl. I. Cuza University & Octav Mayer Institute of Mathematics of the Romanian AcademyIasiRomania
  2. 2.Classe di ScienzeScuola Normale Superiore di PisaPisaItaly
  3. 3.Department of MathematicsUniversity of BielefeldBielefeldGermany

Personalised recommendations