PDF《SYSTEM WITH L1 DENSITY》

140232 and 156112. 2. Conservation of mass and energy We would like here to recall some basic identities related to the VP system. They hold for smooth solutions, and a priori not for weak solutions, for which only the associated a priori bounds remain valid. Integrating (1.1) with respect to v and noting that the last term is in v-divergence form we obtain the local conservation of mass Btρpt, xq ` divxpJpt, xqq 0, (2.1) where the current J is de?ned by Jpt, xq ? RN vfpt, x, vq dv. (2.2) Integrating again with respect to x, we obtain the global conservation of mass d dt ? RN ?RN fpt, x, vqdxdv d dt ? RN ρpt, xqdx 0. (2.3) LAGRANGIAN SOLUTIONS FOR VLASOV-POISSON WITH L1 DENSITY

3 Multiplying (1.1) by |v|2

2 , integrating in x and v, we get after integration by parts in v d dt ? RN ?RN |v|2

2 fpt, x, vqdxdv ? ? RN ?RN E ¨ vf dxdv 0. (2.4) Using (1.6) and (2.1), one has BtE ` N ? k

1 B Bxk ? ω |SN?1| x |x|N ? Jk B 0, (2.5) or in other words BtE ω?xp?Δxq?1 divxJ, (2.6) which means that BtE is the the gradient component of ?ωJ (Helmholtz projection). We deduce that ? RN E ¨ BtE dx ?ω ? RN E ¨ J dx. (2.7) Using (2.2) in (2.4), we obtain the conservation of energy d dt ? ― C ? RN ?RN |v|2

2 fpt, x, vqdxdv ` ω

2 ? RN |Ept, xq|2 dx ? ? ?

0 . (2.8) The total conserved energy is the sum of the kinetic energy and of the potential energy multiplied by the factor ω ?1. In particular, in the electrostatic case ω `1 we deduce from (2.8) a uniform bound in time on both the kinetic and the potential energy, assuming that they are ?nite initially. In the gravitational case ω ?1 it is not possible to exclude that the individual terms of the kinetic and potential energy become unbounded in ?nite time, while the sum remains constant. Indeed it is known that it does not happen in three dimensions as soon as f0 is su?ciently integrable, but we cannot exclude this a priori for only L1 solutions. Note that the assumption E0 P L2 is satis?ed in

3 dimensions as soon as ρ0 ? ρb P L6{5. However, in one or two dimensions, for E0 to be in L2 it is necessary that ? pρ0 ? ρbqdx 0, as is easily seen in Fourier variable. It is also necessary that ρ0 ? ρb has enough decay at in?nity. Thus in one or two dimensions, in order to have ?nite energy, ρb cannot be zero identically. 3. Regularity of the force field for L1 densities 3.1. Singular integrals. We recall here the basic properties of singular integral operators, that can be found in [36] for example. De?nition 3.1. A function K is a singular kernel of fundamental type in RN if the following properties hold: (1) K|RN zt0u P C1pRN zt0uq. (2) There exists a constant C0 ě

0 such that |Kpxq| ? C0 |x|N , x P RN zt0u. (3.1)

4 A. BOHUN, F. BOUCHUT, AND G. CRIPPA (3) There exists a constant C1 ě

0 such that |?Kpxq| ? C1 |x|N`1 , x P RN zt0u. (3.2) (4) There exists a constant A2 ě

0 such that ˇ ˇ ˇ ˇ ˇ ? R1?|x|?R2 Kpxqdx ˇ ˇ ˇ ˇ ˇ ? A2, (3.3) for every

0 ? R1 ? R2 ? 8. Theorem 3.2 (Calder? on-Zygmund [36]). A singular kernel of fundamental type K has an extension as a distribution on RN (still denoted by K), unique up to a constant times Dirac delta at the origin, such that ? K P L8pRN q. De?ne Su K ? u, for u P L2 pRN q, (3.4) in the sense of multiplication in the Fourier variable. Then we have the estimates for

1 ? p ?

8 ||Su||LppRN q ? CN,ppC0 ` C1 ` || ? K||L8 q||u||LppRN q, u P Lp X L2pRN q. (3.5) If K is a singular kernel of fundamental type, we call the associated oper- ator S a singular integral operator on RN . We de?ne then the Fr? echet space RpRN q XmPN, 1?p?8Wm,ppRN q and its dual R1pRN q ? S 1pRN q, where S 1pRN q is the space of tempered distributions on RN . Since all singular integral operators are bounded on RpRN q, by duality we can de?ne the op- erator S also R1pRN q ? R1pRN q. In particular it enables to de?ne Su for u P L1pR........