×

Markov processes and magneto-hydrodynamics equations. (English. Russian original) Zbl 1474.60172

J. Math. Sci., New York 258, No. 6, 739-757 (2021); translation from Zap. Nauchn. Semin. POMI 486, 7-34 (2019).
Summary: 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.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
60J25 Continuous-time Markov processes on general state spaces
76W05 Magnetohydrodynamics and electrohydrodynamics
78A25 Electromagnetic theory (general)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Olesen, P., Integrable version of Burgers equation in magnetohydrodynamics, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 68, 016307 (2003)
[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] Kac, M., Probability and Related Topics in the Physical Sciences (1958), New York: Interscience Publ, New York
[4] McKean, HP, A class of Markov processes associated with non-linear parabolic equations, Proc. Nat. Ac. Sci., 56, 1907-1911 (1966) · Zbl 0149.13501
[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] 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] R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications, Springer (2018). · Zbl 1422.91014
[8] V. Kolokoltsov, Differential Equations on Measures and Functional Spaces, Birkhäuser (2019). · Zbl 1451.34001
[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). · Zbl 1390.60211
[10] Le Cavil, A.; Oudjane, N.; Russo, F., Probabilistic representation of a class of nonconservative nonlinear partial differential equations, ALEA Lat. Am. J. Probab. Math. Stat., 13, 1189-1233 (2016) · Zbl 1355.60089
[11] V. N. Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations, Cambridge Tracts in Mathematics, 182, Cambridge Univ. Press (2010). · Zbl 1222.60003
[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). · Zbl 1448.35385
[13] Ya. Belopolskaya, “Stochastic models for forward systems of nonlinear parabolic equations,” Statist. Papers, 59, 1505-1519 (2018). · Zbl 1408.60056
[14] Ya. Belopolskaya, “Stochastic interpretation of quasilinear parabolic systems with crossdiffusion,” Teor. Veroyatn. Primen., 61, 268-299 (2016).
[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). · Zbl 1290.35289
[16] A. Friedman, Stochastic Differential Equations and Applications, Vol. 1, Probability and Mathematical Statistics, 28, Acad. Press, New York-London (1975). · Zbl 0323.60056
[17] M. A. Shubin, Lectures on Equations of Mathematical Physics [in Russian], Moscow (2003).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.