We prove the global-in-time existence of large-data finite-energy weaksolutions to an incompressible hybrid Vlasov-magnetohydrodynamic model in threespace dimensions. The model couples three essential ingredients of magnetizedplasmas: a transport equation for the probability density function, whichmodels energetic rarefied particles of one species; the incompressibleNavier--Stokes system for the bulk fluid; and a parabolic evolution equation,involving magnetic diffusivity, for the magnetic field. The physical derivationof our model is given. It is also shown that the weak solution, whose existenceis established, has nonincreasing total energy, and that it satisfies a numberof physically relevant properties, including conservation of the totalmomentum, conservation of the total mass, and nonnegativity of the probabilitydensity function for the energetic particles. The proof is based on a one-levelapproximation scheme, which is carefully devised to avoid increase of the totalenergy for the sequence of approximating solutions, in conjunction with a weakcompactness argument for the sequence of approximating solutions. The keytechnical challenges in the analysis of the mathematical model are thenondissipative nature of the Vlasov-type particle equation and passage to theweak limits in the multilinear coupling terms.
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