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An existence theorem for the unmodified Vlasov Equation
Author(s) -
Illner R.,
Neunzert H.
Publication year - 1979
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670010410
Subject(s) - mathematics , vlasov equation , initial value problem , kinetic energy , mathematical analysis , conservation law , energy (signal processing) , energy method , weak solution , plasma , physics , classical mechanics , statistics , quantum mechanics
The initial value problem for the modified Vlasov equation with a mollification parameter δ > 0, as introduced by Batt, has a unique global solution in the weak sense whenever f 0 ε L 1 and f 0 ≧ 0 λ‐a.e. Assuming boundedness of f 0 and boundedness of the kinetic energy, it is shown that, as δ → 0, there are subsequences δ n → 0 such that the corresponding solutions converge weakly in the measure‐theoretical sense. The limits are shown to be global weak solutions of the initial value problem for Vlasov's equation, and these solutions are seen to be weakly continuous with respect to t. For the plasma physical case, boundedness of the kinetic energy is a consequence of energy conservation.