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**Example text**

3. 1. Let 0 < α ≤ 2, 2 ≤ p < ∞ and θ be a smooth function on Rn . We have: 2 | α (|θ | p/2 )|2 d x ≤ p where sgn(θ ) denotes the sign of θ . sgn(θ )|θ | p−1 2α θ d x, Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or H˙ −1/2 53 Proof. The integrals in the desired inequality can be rewritten in the following form: 2 | α (|θ | p/2 )|2 d x = 2 2α (|θ | p/2 )|θ | p/2 d x = − d −t e dt 2α |θ | p/2 22 |t=0+ and sgn(θ )|θ | p−1 p 2α θdx = − d −t e dt 2α θ p p |t=0+ . Let F(t) = e−t θ p and G(t) = e−t |θ | p/2 22 .

217, Providence, RI: Amer. Math. , 1998, pp. : Linear boltzmann equation as the weak coupling limit of a random schrödinger equation. Commun. Pure Appl. Math. 53(6), 667–735 (2000) 44 [ESY] [H] [HL] [IP] [L] [KP] [LP] [MRS] [RS] [RV] [Sp] [U] [UU] [W] D. Benedetto, F. Castella, R. Esposito, M. : On the quantum boltzmann equation. J. Stat. Phys. : Derivation of the boltzmann equation for a fermi gas. J. Stat. Phys. : Fermi gas in a lattice in the van hove limit. J. Stat. Phys. : Global Validity of the Boltzmann equation for a two-dimensional rare gas in the vacuum.

In the particular case when m − 1 = n/2, this reduces to T (t, ξ1 , η1 ) ≤ O(1). An additional argument is needed to prove that right non-ordered graphs, or wrong graphs with m − 1 > n/2, are actually vanishing as ε → 0. We start by considering right non-ordered graphs, recalling that they involve as many n = m − 1). e. e. between the father and the son). However the time ordering of the recollision is not the right one: at least one recollision does not happen immediately after the associated creation.