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Stochastic non-resistive magnetohydrodynamic system with Lévy noise. (English) Zbl 1372.76030
Summary: In this work we prove the existence and uniqueness of path-wise solutions up to a maximal time for the viscous, non-resistive stochastic magnetohydrodynamic system perturbed by Lévy noise in two and three dimensions. The local-in-time existence and uniqueness result for the ideal and non-viscous MHD equations with Lévy noise is also addressed.

MSC:
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q35 PDEs in connection with fluid mechanics
76M35 Stochastic analysis applied to problems in fluid mechanics
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[1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Stud. Adv. Math. 93, Cambridge University Press, Cambridge, 2004.
[2] V. Barbu and G. Da Prato, Existence and ergodicity for the two-dimensional stochastic magneto-hydrodynamic equations, Appl. Math. Optim. 56 (2007), 145-168. · Zbl 1187.76727
[3] H. Bessaih, E. Hausenblas and P. Razafimandimby, Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type, NoDEA Nonlinear Differential Equations Appl. 22 (2015), no. 6, 1661-1697. · Zbl 1326.60089
[4] Z. Brzeźniak, E. Hausenblas and P. Razafimandimby, Stochastic nonparabolic dissipative systems modeling the flow of liquid crystals: Strong solution, Math. Anal. Incompressible Flow 1875 (2014), 41-72.
[5] H. Cabannes, Theoretical Magnetofluid-Dynamics, Academic Press, New York, 1970.
[6] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Internat. Ser. Monogr. Phys., Clarendon Press, Oxford, 1961.
[7] J.-Y. Chemin, D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math. 286 (2016), 1-31. · Zbl 1333.35183
[8] T. G. Cowling, Magnetohydrodynamics, Interscience Tracts Phys. Astron. 4, Interscience Publishers, New York, 1957.
[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge University Press, Cambridge, 1992.
[10] G. Duvant and J-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Math. Anal. 46 (1972), 241-279. · Zbl 0264.73027
[11] H. L. Elliott and M. Loss, Analysis, 2nd ed., Grad. Texts in Math. 14, American Mathematical Society, Providence, 2001.
[12] C. L. Fefferman, D. S. McCormick, J. C. Robinson and J. L. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal. 267 (2014), no. 4, 1035-1056. · Zbl 1296.35142
[13] B. P. W. Fernando and S. S. Sritharan, Nonlinear filtering of stochastic Navier-Stokes equation with Itô-Lévy noise, Stoch. Anal. Appl. 31 (2013), no. 3, 381-426. · Zbl 1311.60068
[14] L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations, Probab. Appl. (N. Y.), Springer, Heidelberg, 2011. · Zbl 1228.60002
[15] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891-907. · Zbl 0671.35066
[16] J. U. Kim, Existence of a local smooth solution in probability to the stochastic Euler equations in \mathbfR^3, J. Funct. Anal. 256 (2009), no. 11, 3660-3687. · Zbl 1166.60036
[17] J. U. Kim, On the stochastic quasi-linear symmetric hyperbolic system, J. Differential Equations 250 (2011), no. 3, 1650-1684. · Zbl 1211.35174
[18] H. Kozono, Weak and classical solutions of the two-dimensional magnetohydrodynamic equations, Tohoku Math. J. (2) 41 (1989), no. 3, 471-488. · Zbl 0683.76103
[19] O. A. Ladyženskaya and V. A. Solonnikov, On the solvability of unsteady motion problems in magnetohydrodynamics, Dokl. Akad. Nauk SSSR 124 (1959), 26-28. · Zbl 0091.19804
[20] O. A. Ladyzhenskaya and V. A. Solonnikov, Solution of some non-stationary problems of magnetohydrodynamics for a viscous incompressible fluid, Tr. Mat. Inst. Steklova 59 (1960), 115-173.
[21] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Course Theoret. Phys. 8, Pergamon Press, Oxford, 1960.
[22] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Springer, New York, 1972. · Zbl 0223.35039
[23] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts Appl. Math. 27, Cambridge University Press, Cambridge, 2002.
[24] V. Mandrekar and B. Rüdiger, Existence and uniqueness of path wise solutions for stochastic integral equations driven by Lévy noise on separable Banach spaces, Stochastics 78 (2006), no. 4, 189-212. · Zbl 1119.60040
[25] V. Mandrekar and B. Rüdiger, Lévy noises and stochastic integrals on Banach spaces, Stochastic Partial Differential Equations and Applications—VII, Lect. Notes Pure Appl. Math. 245, Chapman & Hall/CRC, Boca Raton (2006), 193-213. · Zbl 1096.60024
[26] V. Mandrekar and B. Rüdiger, Stochastic Integration in Banach Spaces, Probab. Theory Stoch. Model. 73, Springer, Cham, 2015.
[27] U. Manna and M. T. Mohan, Two-dimensional magneto-hydrodynamic system with jump processes: Well posedness and invariant measures, Commun. Stoch. Anal. 7 (2013), no. 1, 153-178.
[28] C. Marinelli and M. Röckner, On the maximal inequalities of Burkholder, Davis and Gundy, Expo. Math. 34 (2016), no. 1, 1-26. · Zbl 1335.60064
[29] J.-L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation, Appl. Math. Optim. 46 (2002), no. 1, 31-53. · Zbl 1016.35072
[30] M. Métivier, Semimartingales. A Course on Stochastic Processes, De Gruyter Stud. Math. 2, Walter de Gruyter & Co., Berlin, 1982.
[31] M. Métivier, Stochastic Partial Differential Equations in Infinite-Dimensional Spaces, Scuola Normale Superiore, Pisa, 1988. · Zbl 0664.60062
[32] M. T. Mohan and S. S. Sritharan, New methods for local solvability of quasilinear symmetric hyperbolic systems, Evol. Equ. Control Theory 5 (2016), no. 2, 273-302. · Zbl 1346.35126
[33] M. T. Mohan and S. S. Sritharan, Stochastic Euler equations of fluid dynamics with Lévy noise, Asymptot. Anal. 99 (2016), no. 1-2, 67-103. · Zbl 1348.35177
[34] M. T. Mohan and S. S. Sritharan, Frequency truncation method for quasilinear symmetrizable hyperbolic systems, J. Anal. (2017), 10.1007/s41478-017-0049-2.
[35] M. T. Mohan and S. S. Sritharan, Stochastic quasilinear symmetric hyperbolic system perturbed by Lévy noise, Pure Appl. Funct. Anal., to appear.
[36] N. S. Papageorgiou and S. T. Kyritsi-Yiallourou, Handbook of Applied Analysis, Adv. Mech. Math. 19, Springer, New York, 2009. · Zbl 1189.49003
[37] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. An Evolution Equation Approach, Encyclopedia Math. Appl. 113, Cambridge University Press, Cambridge, 2007. · Zbl 1205.60122
[38] P. E. Protter, Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability, 2nd ed., Appl. Math. (New York) 21, Springer, Berlin, 2004.
[39] J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2001. · Zbl 0980.35001
[40] B. Rüdiger and G. Ziglio, Itô formula for stochastic integrals w.r.t. compensated poisson random measures on separable banach spaces, Stochastics 78 (2006), 377-410. · Zbl 1117.60056
[41] S. Sahaev and V. A. Solonnikov, Estimations of the solutions of a certain boundary value problem of magnetohydrodynamics, Tr. Mat. Inst. Steklova 127 (1975), 76-92. · Zbl 0366.76091
[42] K. Sakthivel and S. S. Sritharan, Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise, Evol. Equ. Control Theory 1 (2012), no. 2, 355-392. · Zbl 1260.35128
[43] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math. 36 (1983), 635-664. · Zbl 0524.76099
[44] S. S. Sritharan and M. T. Mohan, \mathbbL^p-solutions of the stochastic Navier-Stokes equations subject to Lévy noise with \mathbbL^m(\mathbbR^m) initial data, Evol. Equ. Control Theory 6 (2017), no. 3, 409-425. · Zbl 1366.76022
[45] S. S. Sritharan and P. Sundar, The stochastic magneto-hydrodynamic system, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999), no. 2, 241-265. · Zbl 0998.76099
[46] P. Sundar, Stochastic magneto-hydrodynamic system perturbed by general noise, Commun. Stoch. Anal. 4 (2010), no. 2, 253-269. · Zbl 1331.76136
[47] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer, New York, 1988. · Zbl 0662.35001
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