Tamtam Proceedings - lamsin

Approximation of the Vlasov equations 37‖µ‖ ∞,ε = 1(2ε) 2d sup {µ(B ∞ ((x, v), ε))} . (13)(x,v)∈R 2dwhere B ∞ ((x, v), ε) is the ball of radius ε centered at (x, v) for the infinite norm. Notethat we may bound ‖µ N (T, ·)‖ ∞,ε by‖µ N (T, ·)‖ ∞,ε ≤ (4 m(T )) 2d . (14)We may also introduce discrete L ∞ norm at other scales by defining in general‖µ‖ ∞,η = 1(2η) 2d sup {µ(B ∞ ((x, v), η))} . (15)(x,v)∈R 2dThe quantities R, K, m will always be assumed to be bounded at the initial time T = 0uniformly in N.3. Main resultsThe main point in the derivation of the Vlasov equation is to obtain a control on theprevious quantities. We first do it for a short time as given byTheorem 3.1 If α < 1, there exists a time T and a constant c depending only on R(0),K(0), m(0) but not on N such that for some α < α ′ < 3R(T ) ≤ 2 (1 + R(0)), K(T ) ≤ 2 (1 + K(0)), m(T ) ≤ 2 m(0),E(T ) ≤ c (m(0)) 2α′ (K(0)) α′ (R(0)) α′ −α ,sup ‖µ N (t, ·)‖ ∞,ε ≤ (8 m(0)) 2d .t≤TRemarkThe constant 2, which appears in the bounds, is of course only a matter of convenience.This means that another theorem could be written with 3 instead of 2 for instance; Thetime T would then be larger. However increasing this value is not really helpful becausethe kind of estimates which we use for this theorem blow up in finite time, no matter howlarge the constant in the bounds is.This theorem can, in fact, be extended on any time intervalTheorem 3.2 For any time T > 0, there exists a function Ñ of R(0), K(0), m(0) and Tand a constant C(R(0), K(0), m(0), T ) such that if N ≥ Ñ thenR(T ), K(T ), m(T ), E(T ) ≤ C(R(0), K(0), m(0), T ).TAMTAM –Tunis– 2005