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Approximation of the Vlasov equations 35is a solution of the Vlasov equation in the sense of distributions. And the question iswhether a weak limit f of µ N solves (3) or not. If F is C 1 with compact support, then itis indeed the case (it is proved in the book by Spohn [14] for example). The purpose ofthis paper is to justify this limit if|F (x)| ≤ CC, |∇F (x)| ≤|x|α|x| 1+α |∇ 2 F (x)| ≤ C , ∀x ≠ 0, (4)|x|2+αfor α < 1, which is the first rigorous proof of the limit in a case where F is not necessarilybounded.Before being more precise concerning our result, let us explain what is the meaningof (1) in view of the singularity in F . Here we assume either that we restrict ourselvesto the initial configurations for which there are no collisions between particles over atime interval [0, T ] with a fixed T , independent of N. Or we assume that F is regularor regularized but that the norm ‖F ‖ W 1,∞ may depend on N; This procedure is wellpresented in [1] and it is the usual one in numerical simulations (see [15] and [16]). Inboth cases, we have classical solutions to (1) but the only bound we may use is (4).Other possible approaches would consist in justifying that the set of initial configurationsX 1 (0), . . . , X N (0), V 1 (0), . . . , V N (0) for which there is at least one collision, isnegligible or that it is possible to define a solution (unique or not) to the dynamics evenwith collisions.Finally notice that the condition α < 1 is not unphysical. Indeed if F derives froma potential, α = 1 is the critical exponent for which repulsive and attractive forces seemvery different. In other words, this is the point where the behavior of the force when twoparticles are very close takes all its importance.2. Important quantitiesThe derivation of the limit requires a control on many quantities. Although some ofthem are important only at the discrete level, many were already used to get the existenceof strong solutions to the Vlasov-Poisson equation (we refer to [7], [8] and [12], [13] asbeing the closest from our method).The first two quantities are quite natural and are bounds on the size of the support ofthe initial data in space and velocity,R(T ) =sup |X i (t)|, K(T ) = sup |V i (t)|. (5)t∈[0,T ], i=1,...Nt∈[0,T ], i=1,...NOf course R is trivially controlled by K sinceR(T ) ≤ R(0) + T K(T ). (6)Now a very important and new parameter is the discrete scale of the problem denotedε. This quantity represents roughly the minimal distance between two particles or theTAMTAM –Tunis– 2005