An existence theorem for the unmodified Vlasov Equation

Abstract

Abstract 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.