Stochastic Interpretation of the MHD-Burgers System

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We construct a stochastic interpretation of a generalized solution of the Cauchy problem for the simplest magneto-hydrodynamics system, namely, a system including the Burgers equation with a pressure due to a magnetic field. The probabilistic representation constructed in the paper can be used for numerical computations.

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  1. 1.

    Ya. Belopolskaya and Yu. Dalecky, “Investigation of the Cauchy problem for quasilinear systems with finite and infinite number of arguments by means of Markov random processes,” Izv. VUZ Matematika, 12, 5–17 (1978).

  2. 2.

    Ya. Belopolskaya and Yu. Dalecky, Stochastic Equations and Differential Geometry, Kluwer Academic Publishers (1990).

  3. 3.

    E. Pardoux, F. Pradeilles, and R. Rao, “Probabilistic interpretation of a system of semilinear parabolic partial differential equations,” Ann. Inst. Henri Poincaré, 33, 467–490 (1997).

  4. 4.

    S. Peng, “Probabilistic interpretation for systems of quasilinear parabolic partial differential equations,” Stoch. Stoch. Rep., 37, 61–74 (1991).

  5. 5.

    Ya. Belopolskaya, “Probabilistic models of the conservation and balance laws in switching regimes,” J. Mat. Sci., 229, 601–625 (2018).

  6. 6.

    R. Griego and R. Hersh, “Random evolutions, Markov chains and systems of partial differential equations,” Proc. Nat. Acad. Sci. USA, 62, 305–308 (1969).

  7. 7.

    A. Jüngel, “Diffusive and nondiffusive population models,” in: Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, 397–425 (2010).

  8. 8.

    L. Desvillettes, T. Lepoutre, and A. Moussa, “Entropy, duality and cross diffusion,” SIAM J. Math. Anal., 46, 820–853 ( 2014).

  9. 9.

    H. Jin, Z. Wang, and L. Xiong, “Cauchy problem of the magnetohydrodynamic Burgers system,” Commun. Math. Sci., 13, 127–151 (2015).

  10. 10.

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

  11. 11.

    Ya. Belopolskaya, “Stochastic interpretation of quasilinear parabolic systems with crossdiffusion,” Theor. Prob. Appl., 61, 208–234 (2017).

  12. 12.

    Ya. Belopolskaya, “Probabilistic models of the dynamics of the growth of cells under contact inhibition,” Math. Notes, 101, 406–416 (2017).

  13. 13.

    H. Kunita, “Stochastic flows acting on Schwartz distributions,” J. Theor. Probab., 7, 247–278 (1994).

  14. 14.

    H. Kunita, “Generalized solutions of stochastic partial differential equations,” J. Theor. Probab., 7, 279–308 (1994).

  15. 15.

    Ya. Belopolskaya and W. Woyczynski, “Generalized solutions of nonlinear parabolic equations and diffusion processes,” Acta Appl. Math., 96, 55–69 (2007).

  16. 16.

    Ya. Belopolskaya and W. Woyczynski, “Generalized solution of the Cauchy problem for systems of nonlinear parabolic equations and diffusion processes,” Stoch. Dyn., 11, 1–31 (2012).

  17. 17.

    P. Protter, Stochastic Integration and Differential Equations, Springer (2010).

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Correspondence to Ya. I. Belopolskaya.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 466, 2017, pp. 7–29.

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Belopolskaya, Y.I., Stepanova, A.O. Stochastic Interpretation of the MHD-Burgers System. J Math Sci 244, 703–717 (2020) doi:10.1007/s10958-020-04643-1

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