Skip to main content
SpringerLink
Log in

Unique solvability of the stationary Fokker-Planck equation in a class of positive functions

  • Partial Differential Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We study the stationary Focker-Planck equation Δu − div(u f) = 0 with a given vector field f of the class C ∞0 (R n) on the basis of a fixed point principle that generalizes the contraction mapping method. Next, we introduce a parameter in the equation and prove the unique solvability of the equation Δu − div(uγ f) = 0 with the parameter in the class of positive slowly increasing functions. We reveal the analytic dependence of the positive solution u on the parameter γ. Pointwise estimates for positive solutions are proved.

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. Noarov, A.I., On the Solvability of the Stationary Fokker-Planck Equations That Are Close to the Laplace Equation, Differ. Uravn., 2006, vol. 42, no. 4, pp. 521–530.

    MathSciNet  Google Scholar 

  2. Mikhailov, V.P., Differentsial’nye uravneniya v chastnykh proizvodnykh (Partial Differential Equations), Moscow: Nauka, 1983.

    Google Scholar 

  3. Mizohata, S., The Theory of Partial Differential Equations, New York: Cambridge Univ., 1973. Translated under the title Teoriya uravnenii v chastnykh proizvodnykh, Moscow: Mir, 1977.

    MATH  Google Scholar 

  4. Yosida, K., Functional Analysis, Berlin: Springer-Verlag, 1965. Translated under the title Funktsional’nyi analiz, Moscow: Mir, 1967.

    MATH  Google Scholar 

  5. Krylov, N.V., Lektsii po ellipticheskim i parabolicheskim uravneniyam v prostranstvakh Gel’dera (Lectures on Elliptic and Parabolic Equations in Hölder Spaces), Novosibirsk: Nauchnaya Kniga, 1998.

    Google Scholar 

  6. Noarov, A.I., Generalized Solvability of the Stationary Fokker-Planck Equation, Differ. Uravn., 2007, vol. 43, no. 6, pp. 813–819.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia

    A. I. Noarov

Authors
  1. A. I. Noarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © A.I. Noarov, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45, No. 2, pp. 191–202.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Noarov, A.I. Unique solvability of the stationary Fokker-Planck equation in a class of positive functions. Diff Equat 45, 197–208 (2009). https://doi.org/10.1134/S0012266109020062

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266109020062

Keywords

Search

Navigation