Skip to main content

Markov Processes and Magneto-Hydrodynamics Equations

We derive a probabilistic interpretation of a generalized solution of the Cauchy problem for a three-dimensional system of magneto-hydrodynamics equations called the MHD-Burgers system. First we regularize the system under consideration and prove that there exists a unique measurevalued solution of the Cauchy problem for the regularized system. Next we justify a limiting procedure with respect to the regularization parameter and, as a consequence, prove the existence and uniqueness of a solution to the Cauchy problem of the original MHD-Burgers system. Finally, we derive a probabilistic representation of the Cauchy problem solution for the MHD-Burgers system.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    P. Olesen, “Integrable version of Burgers equation in magnetohydrodynamics,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 68, 016307 (2003).

    MathSciNet  Article  Google Scholar 

  2. 2.

    M. Kac, “Foundations of kinetic theory,” In: Proceedings of the Third Berkeley Symposiumon Mathematical Statistics and Probability, 1954–1955, 3, Univ. of California Press, Berkeley and Los Angeles (1956), pp. 171–197.

  3. 3.

    M. Kac, Probability and Related Topics in the Physical Sciences, Interscience Publ., New York (1958).

    Google Scholar 

  4. 4.

    H. P. McKean, “A class of Markov processes associated with non-linear parabolic equations,” Proc. Nat. Ac. Sci., 56, 1907–1911 (1966).

    Article  Google Scholar 

  5. 5.

    H. P. McKean, Jr., “Propagation of chaos for a class of nonlinear parabolic equations,” Lect. Series in Diff. Eq., Catholic Univ., 7, 41–57 (1967).

  6. 6.

    V. I. Bogachev, N. V. Krylov, M. R¨ockner, and S. V. Shaposhnikov, Fokker–Planck–Kolmogorov Equations, Amer. Math. Soc., Providence, R.I. (2015).

  7. 7.

    R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications, Springer (2018).

  8. 8.

    V. Kolokoltsov, Differential Equations on Measures and Functional Spaces, Birkhäuser (2019).

  9. 9.

    A. Le Cavil, N. Oudjane, and F. Russo, “Forward Feynman–Kac type representation for semilinear nonconservative partial differential equations,” Preprint hal-01353757, version 3 (2017).

  10. 10.

    A. Le Cavil, N. Oudjane, and F. Russo, “Probabilistic representation of a class of nonconservative nonlinear partial differential equations,” ALEA Lat. Am. J. Probab. Math. Stat., 13, 1189–1233 (2016).

    MathSciNet  Article  Google Scholar 

  11. 11.

    V. N. Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations, Cambridge Tracts in Mathematics, 182, Cambridge Univ. Press (2010).

  12. 12.

    Ya. I. Belopolskaya and A. O. Stepanova, “Stochastic interpretation of the Burgers-MHD system,” Zap. Nauchn. Semin. POMI, 466, 7–29 (2017); English transl. J. Math. Sci., 244, No. 5, 703–717 (2020).

  13. 13.

    Ya. Belopolskaya, “Stochastic models for forward systems of nonlinear parabolic equations,” Statist. Papers, 59, 1505–1519 (2018).

  14. 14.

    Ya. Belopolskaya, “Stochastic interpretation of quasilinear parabolic systems with crossdiffusion,” Teor. Veroyatn. Primen., 61, 268–299 (2016).

  15. 15.

    V. I. Bogachev, M. R¨ockner, and S. V. Shaposhnikov, “On uniqueness problems related to elliptic equations for measures,” J. Math. Sci., 176, 759–773 (2011).

  16. 16.

    A. Friedman, Stochastic Differential Equations and Applications, Vol. 1, Probability and Mathematical Statistics, 28, Acad. Press, New York–London (1975).

  17. 17.

    M. A. Shubin, Lectures on Equations of Mathematical Physics [in Russian], Moscow (2003).

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ya. I. Belopolskaya.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 486, 2019, pp. 7–34.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Belopolskaya, Y.I. Markov Processes and Magneto-Hydrodynamics Equations. J Math Sci 258, 739–757 (2021). https://doi.org/10.1007/s10958-021-05577-y

Download citation