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RD Hence 1+ 1 U ∈ Hloc 2(m+1) (R × RD ) . 3), we see that, in order for the vector field R3 × R3 (vj (ξ), Fj (t, x)) ∈ R3 × R3 to generate a unique characteristic flow, Fj should be locally Lipschitz continuous in x uniformly in t, for each j = 1, . . , N . 8). Unfortunately, the strategy based on the classical velocity averaging lemma as explained above (in the most favorable L2 setting) fails to gain that much regularity— in fact, m = 1 in the Vlasov equation, so that the best one can hope for with this method is a gain of 1 + 14 derivatives in (t, x), which is not enough to allow us to define characteristics for the Vlasov–Maxwell system.

Roy Soc. (London) 157, 49–88 (1866). 1 Introduction This chapter aims to provide a unified presentation of some recent studies (see [CGT99], [CCG00], [BGR05], [CCG05] and [GR05]) on the convergence to equilibrium of the solution of Boltzmann’s equation for Maxwellian pseudomolecules and, in particular, of the solution of Kac’s analog of Boltzmann’s equation. The main feature of these researches, with respect to the other ones, is the pursuit of the optimal rate of exponential convergence both in a weighted χ-metric (see for example [GTW95]) and in the total variation metric for probability measures.

Compare also [BGR05]. 1) with Bn,k (f0 ) = 1 1 − pn,k pn (γ)Cγ (·; f0 ). , If f0 has finite Linnik information L(f0 ), then there is a constant C depending only on k and L(f0 ) for which Bn,k (f0 ) H k/2 (R) C. In view of this fact, in [CCG05] the set Bn,k (Un,k , respectively) is called the set of the “beautiful” (“ugly” respectively) trees. Next, since pn,k becomes negligible as n diverges, in order that the new approach may be successful it would suffice to show that the “discrepancy” between Un,k and the Maxwell distribution is bounded in a sense to be specified.

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10 Lies About The Atkins Diet by Tom Venuto


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