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


This is an introductory chapter mainly devoted to the formulation of problems, models and some preliminaries on convex and infinite dimensional analysis, indispensable for understanding the sequel.


Maximal Monotone Maximal Monotone Operator Porous Media Equation Convex Lower Semicontinuous Function Yosida Approximation 
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. 4.
    P. Bak, K. Chen, Self-organized criticality. Sci. Am. 264, 40 (1991)CrossRefGoogle Scholar
  2. 5.
    P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381–394 (1987); Phys. Rev. A 38, 364–375 (1988)Google Scholar
  3. 6.
    V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces (Springer, New York, 2010)CrossRefzbMATHGoogle Scholar
  4. 7.
    V. Barbu, Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion problems. Annu. Rev. Control. JARAP 340, 52–61 (2010)CrossRefGoogle Scholar
  5. 14.
    V. Barbu, Th. Precupanu, Convexity and Optimization in Banach Spaces (Springer, New York, 2011)zbMATHGoogle Scholar
  6. 28.
    J.G. Berryman, C.J. Holland, Nonlinear diffusion problems arising in plasma physics. Phys. Rev. Lett. 40, 1720–1722 (1978)MathSciNetCrossRefGoogle Scholar
  7. 29.
    J.G. Berryman, C.J. Holland, Asymptotic behavior of the nonlinear diffusion equations. J. Math. Phys. 54, 425–426 (1983)MathSciNetGoogle Scholar
  8. 35.
    H. Brezis, Operatéurs Maximaux Monotones (North-Holland, Amsterdam, 1973)zbMATHGoogle Scholar
  9. 38.
    Z. Brzezniak, J.van Neerven, M. Veraar, L. Weis, L. Itô’s formula in UMD Banach spaces and regularity of solutions of the Zakai equation. J. Differ. Equ. 245 (1), 30–58 (2008)Google Scholar
  10. 42.
    J.M. Carlson, E.R. Changes, E.R. Grannan, G.H. Swindle, Self-organized criticality and singular diffusions. Phys. Rev. Lett. 65, 2547–2550 (1990)CrossRefGoogle Scholar
  11. 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
  12. 55.
    C.M. Elliot, Free Boundary Problems (Pitman, London, 1982)Google Scholar
  13. 66.
    D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary. Commun. Partial Differ. Equ 27 (7–8), 1283–1299 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 67.
    L. Hörmander, The Analysis of Linear Partial Differential Operators (Springer, New York, 1963)zbMATHGoogle Scholar
  15. 69.
    K. Kim, C. Mueller, R. Sowers, A stochastic moving boundary value problem. Ill. J. Math. 54 (3), 927–962 (2010)MathSciNetzbMATHGoogle Scholar
  16. 70.
    K. Kim, Z. Zheng, R. Sowers, A stochastic Stefan problem. J. Theor. Probab. 25 (4), 1040–1080 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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
  18. 73.
    P. Li, S.T. You, On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88, 309–318 (1983)CrossRefGoogle Scholar
  19. 75.
    R.S. Liptser, A.N. Shiryayev, Theory of Martingales. (Translated from the Russian by K. Dzjaparidze [Kacha Dzhaparidze]). Mathematics and Its Applications (Soviet Series), vol. 49 (Kluwer, Dordrecht, 1989)Google Scholar
  20. 82.
    C. Prevot, M. Röckner, A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905 (Springer, Berlin, 2007)Google Scholar
  21. 88.
    Ph. Rosneau, Fast and super fast diffusion processes. Phys. Rev. Lett. 74, 1057–1059 (1995)Google Scholar
  22. 89.
    J.L.Vazquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type. Oxford Lecture Series in Mathematics and Its Applications, vol. 33 (Oxford University Press, Oxford, 2006)Google Scholar
  23. 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
  24. 91.
    H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). (German) Math. Ann. 71 (4), 441–479 (1912)Google 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

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