PDF《SYSTEM WITH L1 DENSITY》

LAGRANGIAN SOLUTIONS TO THE VLASOV-POISSON SYSTEM WITH L1 DENSITY ANNA BOHUN, FRANC ?OIS BOUCHUT, AND GIANLUCA CRIPPA Abstract.

The recently developed theory of Lagrangian ?ows for trans- port equations with low regularity coe?cients enables to consider non BV vector ?elds. We apply this theory to prove existence and stability of global Lagrangian solutions to the repulsive Vlasov-Poisson system with only integrable initial distribution function with ?nite energy. These so- lutions have a well-de?ned Lagrangian ?ow. An a priori estimate on the smallness of the superlevels of the ?ow in three dimensions is established in order to control the characteristics. 1. Introduction We consider the Cauchy problem for the classical Vlasov-Poisson system Btf ` v ¨ ?xf ` E ¨ ?vf 0, (1.1) fp0, x, vq f0 px, vq , (1.2) where fpt, x, vq ě

0 is the distribution function, t ě 0, x, v P RN , and Ept, xq ??xUpt, xq (1.3) is the force ?eld. The potential U satis?es the Poisson equation ?ΔxU ωpρpt, xq ? ρbpxqq, (1.4) with ω `1 for the electrostatic (repulsive) case, ω ?1 for the grav- itational (attractive) case, and where the density ρ of particles is de?ned through ρpt, xq ? RN fpt, x, vqdv, (1.5) and ρb ě 0, ρb P L1pRN q is an autonomous background density. Since we are in the whole space, the relation (1.3) together with the Poisson equation (1.4) yield the equivalent relation Ept, xq ω |SN?1| x |x|N ? pρpt, xq ? ρbpxqq, (1.6) where the convolution is in the space variable. The Vlasov-Poisson system has been studied for long. Existence of local in time smooth solutions in dimension N

3 has been obtained in [21] after the results of [8]. Global smooth solutions have been proved to exist in [9, 32, 24], with improvements on the growth in time in [33, 37, 19, 34, 28, 30]. Related results are [35, 27, 15]. These solutions need a su?ciently smooth Key words and phrases. Vlasov-Poisson system, Lagrangian ?ows, non BV vector ?elds, superlevels, weakly convergent initial data.

1 2 A. BOHUN, F. BOUCHUT, AND G. CRIPPA initial datum f0. In particular, the following theorem is a classical result due to Pfa?elmoser [32]. Theorem 1.1. If N 3, let f0 be a non-negative C1 function of compact support de?ned on R6. Then there are a non-negative f P C1pR7q and U P C2pR4q, which tends to zero at in?nity for every ?xed t, satisfying equations (1.1)-(1.5) with ρb 0. For each ?xed t, the function fpt, x, vq has compact support. The solution is determined uniquely by the initial datum f0. In a di?erent spirit, global weak solutions were proved to exist in [6, 16, 18], with only f0 P L1pR6q, f0 log` f0 P L1, |v|2f0 P L1, E0 P L2 (and ρb 0, ω `1). Related results with weak initial data have been obtained in [31, 22, 38, 29]. In this paper we would like mainly to extend the existence result of [16] to initial data in L1 with ?nite energy (in the repulsive case ω `1), avoiding the L log` L assumption. Our existence result is Theorem 8.4. It involves a well-de?ned ?ow. Even weaker solutions were considered in [39, 25, 26], where the distribution function is a measure. However, these solutions do not have well-de?ned characteristics. Our approach uses the theory of Lagrangian ?ows for transport equations with vector ?elds having weak regularity, developed in [17, 1, 4, 14, 5], and recently in [13, 2, 10]. It enables to consider force ?elds that are not in W1,1 loc , nor in BVloc. In this context we prove stability results with strongly or weakly convergent initial distribution function. The ?ow is proved to converge strongly anyway. Our main results were announced in [12]. Related results can be found in [3]. Acknowledgment. This research has been partially supported by the SNSF grants