Skip to main content
SpringerLink
Log in

Homogenized Dynamics of Stochastic Partial Differential Equations with Dynamical Boundary Conditions

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes.

A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with a static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes’ boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albeverio S., Bernabei S., Rockner M. and Yoshida M.W. (2005). Homogenization with respect to Gibbs measures for periodic drift diffusions on lattices. C. R. Math. Acad. Sci. Paris 341(11): 675–678

    MATH  MathSciNet  Google Scholar 

  2. Allaire G. (1992). Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6): 1482–1518

    Article  MATH  MathSciNet  Google Scholar 

  3. Allaire G., Murat M. and Nandakumar A. (1993). Appendix of “Homogenization of the Neumann problem with nonisolated holes”. Asymptotic Anal. 7(2): 81–95

    MATH  MathSciNet  Google Scholar 

  4. Antontsev S.N., Kazhikhov A.V. and Monakhov V.N. (1990). Boundary value problems in mechanics of nonhomogeneous fluids. North-Holland, Amsterdam New York

    MATH  Google Scholar 

  5. Bensoussan A., Lions J.L. and Papanicolaou G. (1978). Asymptotic Analysis for Periodic Structure. North-Holland, Amsterdam New York

    Google Scholar 

  6. Blanc X., LeBris C. and Loins P.L. (2007). On the energy of some microscopic stochastic lattices. Arch. Rat. Mech. Anal. 184(2): 303–339

    Article  MATH  Google Scholar 

  7. Briane M. and Mazliak L. (1998). Homogenization of two randomly weakly connected materials. Portugaliae Mathematic 55: 187–207

    MATH  MathSciNet  Google Scholar 

  8. Brahim-Otsmane S., Francfort G.A. and Murat F. (1998). Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl. 71: 197–231

    MathSciNet  Google Scholar 

  9. Caffarelli L.A., Souganidis P. and Wang L. (2005). Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm. Pure Appl. Math. LLVIII: 1–43

    MathSciNet  Google Scholar 

  10. Cherkaev A. and Kohn R.V. (1997). Topics in the Mathematical Modeling of Composite Materials. Birkhaeuser, Boston

    Google Scholar 

  11. Chueshov I. and Schmalfuss B. (2004). Parabolic stochastic partial differential equations with dynamical boundary conditions. Diff. and Integ. Eq. 17: 751–780

    MATH  MathSciNet  Google Scholar 

  12. Cioranescu D. and Donato P. (1999). An Introduction to Homogenization. Oxford University Press, New York

    MATH  Google Scholar 

  13. Cioranescu D. and Donato P. (1989). Exact internal controllability in perforated domains. J. Math. Pures Appl. 68: 185–213

    MATH  MathSciNet  Google Scholar 

  14. Cioranescu D. and Donato P. (1996). Homogenization of the Stokes problem with nonhomogeneous slip boundary conditions. Math. Methods in Appl. Sci. 19: 857–881

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Cioranescu D., Donato P., Murat F. and Zuazua E. (1991). Homogenization and correctors results for the wave equation in domains with small holes. Ann. Scuola Norm. Sup. Pisa 18: 251–293

    MATH  MathSciNet  Google Scholar 

  16. Da Prato G. and Zabczyk J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  17. Zabczyk J. and Prato G. (1996). Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  18. Duan J., Gao H. and Schmalfuss B. (2002). Stochastic Dynamics of a Coupled Atmosphere-Ocean Model. Stochastics and Dynamics 2: 357–380

    Article  MATH  MathSciNet  Google Scholar 

  19. Dudley R.M. (2002). Real Analysis and Probability. Cambridge Univ. Press, Cambridge

    MATH  Google Scholar 

  20. Duncan T.E., Maslowski B. and Pasik-Duncan B. (1998). Ergodic boundary/point control of stochastic semilinear systems. SIAM J Control Optim. 36: 1020–1047

    Article  MATH  MathSciNet  Google Scholar 

  21. E, W., Li, X., Vanden-Eijnden, E.: Some recent progress in multiscale modeling. In: Multiscale modeling and simulation, Lect. Notes Comput. Sci. Eng. 39, Berlin: Springer, 2004, pp. 3–21

  22. Escher J. (1993). Quasilinear parabolic systems with dynamical boundary. Comm. Part. Diff. Eq. 18: 1309–1364

    Article  MATH  MathSciNet  Google Scholar 

  23. Escher J. (1995). On the qualitative behavior of some semilinear parabolic problem. Diff. and Integ. Eq. 8(2): 247–267

    MATH  MathSciNet  Google Scholar 

  24. Fusco N. and Moscariello G. (1987). On the homogenization of quasilinear divergence structure operators. Ann. Math. Pura Appl. 164(4): 1–13

    MathSciNet  Google Scholar 

  25. Hintermann T. (1989). Evolution equations with dynamic boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 113: 43–60

    MATH  MathSciNet  Google Scholar 

  26. Huang Z. and Yan J. (1997). Introduction to Infinite Dimensional Stochastic Analysis. Science Press/Kluwer Academic Pub., Beijing/New York

    Google Scholar 

  27. Imkeller, P., Monahan, A. (eds.): Stochastic Climate Dynamics, a Special Issue in the journal Stochastics and Dynamics, Vol. 2, No. 3 (2002)

  28. Jikov V.V., Kozlov S.M. and Oleinik O.A. (1994). Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin

    Google Scholar 

  29. Kleptsyna M.L. and Piatnitski A.L. (2002). Homogenization of a random non-stationary convection-diffusion problem. Russ. Math. Surve. 57: 729–751

    Article  MATH  Google Scholar 

  30. Kushner H.J. and Huang H. (1985). Limits for parabolic partial differential equations with wide band stochastic coefficients and an application to filtering theory. Stochastics 14(2): 115–148

    MATH  MathSciNet  Google Scholar 

  31. Langer R.E. (1932). A problem in diffusion or in the flow of heat for a solid in contact with a fluid. Tohoku Math. J. 35: 260–275

    MATH  Google Scholar 

  32. Lapidus L., Amundson N.(eds) (1977). Chemical Reactor Theory, Englewood Cliffs, NJ, Prentice-Hall, Englewood Cliffs NJ

    Google Scholar 

  33. Lions J.L. (1969). Quelques méthodes de résolution des problèmes non linéaires. Dunod, Paris

    MATH  Google Scholar 

  34. Lions P.L. and Masmoudi N. (2005). Homogenization of the Euler system in a 2D porous medium. J. Math. Pures Appl. 84: 1–20

    MATH  MathSciNet  Google Scholar 

  35. Marchenko V.A. and Khruslov Ya E. (2006). Homogenization of partial differential equations. Birkhauser, Boston

    MATH  Google Scholar 

  36. Maslowski B. (1995). Stability of semilinear equations with boundary and pointwise noise. Annali Scuola Normale Superiore di Pisa Scienze Fisiche e Matematiche 22: 55–93

    MATH  MathSciNet  Google Scholar 

  37. Maso G.D. and Modica L. (1986). Nonlinear stochastic homogenization and ergodic theory. J. Rei. Ang. Math. B. 368: 27–42

    Google Scholar 

  38. Mikelić A. and Paloi L. (1999). Homogenization of the invisicid incompressible fluid flow through a 2D porous medium. Proc. Amer. Math. Soc. 127: 2019–2028

    Article  MATH  MathSciNet  Google Scholar 

  39. Nandakumaran A.K. and Rajesh M. (2002). Homogenization of a parabolic equation in a perforated domain with Neumann boundary condition. Proc. Indian Acad. Sci. (Math. Sci.) 112: 195–207

    MATH  MathSciNet  Google Scholar 

  40. Nandakumaran A.K. and Rajesh M. (2002). Homogenization of a parabolic equation in a perforated domain with Dirichlet boundary condition. Proc. Indian Acad. Sci. (Math. Sci.) 112: 425–439

    MATH  MathSciNet  Google Scholar 

  41. Pardoux E. and Piatnitski A.L. (2003). Homogenization of a nonlinear random parabolic partial differential equation. Stochastic Process Appl. 104: 1–27

    Article  MATH  MathSciNet  Google Scholar 

  42. Peixoto J.P. and Oort A.H. (1992). Physics of Climate. Springer, New York

    Google Scholar 

  43. Rockner, M.: Introduction to Stochastic Partial Differential Equations. Preprint 2006, to appear as text notes in Math. 1905, Springer, 2007

  44. Rozovskii B.L. (1990). Stochastic Evolution Equations. Kluwer Academic Publishers, Boston

    Google Scholar 

  45. Sanchez-Palencia, E.: Non Homogeneous Media and Vibration Theory. Lecture Notes in Physics 127, Berlin: Springer-Verlag, 1980

  46. Souza J. and Kist A. (2002). Homogenization and correctors results for a nonlinear reaction-diffusion equation in domains with small holes. The 7th Workshop on Partial Differential Equations II Mat. Contemp. 23: 161–183

    MATH  MathSciNet  Google Scholar 

  47. Timofte C. (2004). Homogenization results for parabolic problems with dynamical boundary conditions. Romanian Rep. Phys. 56: 131–140

    Google Scholar 

  48. Taghite M.B., Taous K. and Maurice G. (2002). Heat equations in a perforated composite plate: Influence of a coating. Int J. Eng. Sci. 40: 1611–1645

    Article  MathSciNet  Google Scholar 

  49. Triebel H. (1978). Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam

    Google Scholar 

  50. Watanabe H. (1988). Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients. Prob. Theory & Related Fields 77: 359–378

    Article  MATH  Google Scholar 

  51. Waymire, E., Duan, J.(eds.): Probability and Partial Differential Equations in Modern Applied Mathematics. IMA Volume 140, New York: Springer-Verlag, 2005

  52. Wang W., Cao D. and Duan J. (2007). Effective macroscopic dynamics of stochastic partial differential equations in perforated domains. SIAM J. Math. Anal. 38: 1508–1527

    Article  MATH  MathSciNet  Google Scholar 

  53. Wright S. (2000). Time-dependent Stokes flow through a randomly perforated porous medium. Asymptot. Anal. 23(3-4): 257–272

    MATH  MathSciNet  Google Scholar 

  54. Yosida K. (1978). Functional Analysis. Springer-Verlag, Berlin

    MATH  Google Scholar 

  55. Vanninathan M. (1981). Homogenization of eigenvalues problems in perforated domains. Proc. Indian Acad. Sci. 90: 239–271

    Article  MATH  MathSciNet  Google Scholar 

  56. Vold R. and Vold M. (1983). Colloid and Interface Chemistry. Addison-Wesley, Reading MA

    Google Scholar 

  57. Yang D. and Duan J. (2007). An impact of stochastic dynamic boundary conditions on the evolution of the Cahn-Hilliard system. Stoch. Anal. and Appl. 25(3): 613–639

    Article  MATH  MathSciNet  Google Scholar 

  58. Zhikov V.V. (1993). On homogenization in random perforated domains of general type. Matem. Zametki 53: 41–58

    MATH  MathSciNet  Google Scholar 

  59. Zhikov V.V. (1994). On homogenization of nonlinear variational problems in perforated domains. Russ. J Math. Phys. 2: 393–408

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, 100080, China

    Wei Wang

  2. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA

    Jinqiao Duan

Authors
  1. Wei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinqiao Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinqiao Duan.

Additional information

Communicated by P. Constantin

Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, W., Duan, J. Homogenized Dynamics of Stochastic Partial Differential Equations with Dynamical Boundary Conditions. Commun. Math. Phys. 275, 163–186 (2007). https://doi.org/10.1007/s00220-007-0301-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0301-8

Keywords

Search

Navigation