We study decay properties of solutions to the Cauchy problem for the collision-less Vlasov–Poisson system which appears Vlasov plasma physics and stems from Liouville’s equation coupled with Poisson’s equation for the determining the self-consistent electrostatics or gravitational forces

Ono, Kosuke

Tokushima University Institutional Repository

On Decay Properties of Solutions forthe Vlasov{Poisson SystemByKosuke Ono Department of Mathematical SciencesThe University of TokushimaTokushima 770-8502, JAPANe-mail : ono@ias.tokushima-u.ac.jp(Received September 30, 2009)AbstractWe study decay properties of solutions to the Cauchy problemfor the collision-less Vlasov{Poisson system which appears Vlasovplasma physics and stems from Liouville's equation coupled withPoisson's equation for the determining the self-consistent electro-statics or gravitational forces.2000 Mathematics Subject Classication. 82D10, 82C40, 35B401 IntroductionWe consider the Cauchy problem for the following kinetic system@tf + v  rxf + E  rvf = 0 in RN  RN  (0;1) (1.1)E(x; t) =  rxU(x; t) in RN  (0;1) (1.2)f(x; v; 0) = (x; v)  0 ; (1.3)where U = U(x; t) is a potential which generates the force eld E = E(x; t).Then, the system (1.1){(1.3) describes the evolution of a microscopic densityf = f(x; v; t)  0 of particles subject to the action of the force eld E. Wewill be mainly interested in the Vlasov{Poisson system where the force eld isself-consistent and given by xU(x; t) = (x; t) ; U(x; t)! 0 as jxj ! 1 ; (1.4)(x; t) =Zf(x; v; t) dv :This work was in part supported by Grant-in-Aid for Scientic Research (C) of JSPS(Japan Society for the Promotion of Science).1where rx = (@x1 ;    ; @xN ), rv = (@v1 ;    ; @vN ), x is the Laplacian in thex variable, and  is a constant. The sign  = +1 represents to electrostatic(repulsive) interaction between the particles of the same species, while  =  1represents gravitational (attractive) interaction (see Risken [11], Glassy [5] forphysical interpretations).From (1.2) and (1.4), we haveE(x; t) =SN 1xjxjN  (x; t) ; (1.5)where SN 1 is (N   1)-dimensional volume of the N -dimensional unit sphere,and the symbol  is the convolution in the x variable.The existence of local solutions of the system is known for every N 2 N(e.g. [3], [4], [6], [8]). The Global existence problem has been studied by severalauthors under suitable restrictions (see [1], [2], [6], [7], [12], [14]).In this paper we study decay properties of solutions to the Cauchy problemfor the Vlasov{Poisson system.Let f = f(x; v; t)  0 be a strong solution of the Vlasov{Poisson systemwith non-negative initial datum (x; v) 2 C10 (RN RN ), where C10 (RN RN )denotes the space of compactly supported, continuously dierentiable functions(see [9], [10]).Our main result is as follows.Theorem 1.1 Let N  4 and  > 0. Then the solution f = f(x; v; t)  0 ofthe Vlasov{Poisson system satises thatkjx=t  vj2fkL1x;v  C1t 2 ; t > 0 ; (1.6)and for 1  q  1 + 2=N ,k(t)kLqx  C1t N(1 1=q) ; t > 0 ; (1.7)and for N=(N   1) < p  N(N + 2)=(N2  N   2),kE(t)kLpx  C1t N(1 1=N 1=p) ; t > 0 ; (1.8)where C1 = C1(k(1 + jxj2)kL1x;v ; kkL1x;v ) is a constant depending on k(1 +jxj2)kL1x;v and kkL1x;v .Finally we x some notation. The function spaces Lpx;v and Lpx meanLp(RN RN ) and Lp(RN ) with usual norms k kLpx;v and k kLpx for 1  p  1,respectively. Positive constants will be denoted by C and will change from lineto line.22 ProofWe rst state the well-known convolution inequality (see for instance [13]).Lemma 2.1 (Hardy{Littlewood{Sobolev inequality)Let 0 <  < N and 1 < q < p <1. Thenkjxj   f(x)kLpx  CkfkLqx for f 2 Lqxwith 1 + 1=p = =N + 1=q.The following proposition plays an important role in the proof of Theorem1.1.Proposition 2.2(1)ddtkE(t)k2L2x =  2ZE  j dx ; j =Zvf dv(2)N   22kE(t)k2L2x =Zx  Edx ;  =Zf dvProof. (1) Using (1.2) and integrating by parts, we observe thatddtkE(t)k2L2x =ddtZjrxU j2dx =  2ZUUt dx=  2ZU@t dx = 2ZUrx  j dx= 2ZrxU  j dx =  2ZE  j dx ;where we used the fact @t+rx  j = 0, indeed, @t =R@tf dv =  R(v rxf+E  rvf) dv =  r Rvf dv =  rx  j.(2) Using (1.2) and (1.4) and integrating by parts, we observe thatZx  Edv = 1Zx  rxUxU dx = 1Xk;jZxkUxkUxjxj dx=   1Xk;jZ@xj (xkUxk)Uxj dx=   10@Z jrxU j2 dx+ 12Xk;jZxk@xk(U2xj ) dx1A=   10@kE(t)k2L2x   12Xk;jZU2xj dx1A=   1kE(t)k2L2x  N2ZjrxU j2 dx=N   22kE(t)k2L2x :3Proof of Theorem 1.1 Using the Vlasov{Poisson system and integrating byparts, we observe thatddtkjx  tvj2fkL1x;v=  2ZZ(x  tv)  vf dvdx ZZjx  tvj2(v  rxf + E  rvf) dvdx=  ZZjx  tvj2E  rvf dvdx =  2tZZ(x  tv)  Ef dvdx=  2tZx  Edx+ 2t2ZE  j dx ;where  =Rf dv and j =Rvf dv.From Proposition 2.2, we haveddtkjx  tvj2fkL1x;v =  N   2tkE(t)k2L2x  1t2ddtkE(t)k2L2xorddtkjx  tvj2fkL1x;v +1t2ddtkE(t)k2L2x=  N   4tkE(t)k2L2x :When  > 0 and N  4, we seekjx  tvj2fkL1x;v +1t2ddtkE(t)k2L2x  kjxj2kL1x;vorkjx=t  vj2fkL1x;v +1ddtkE(t)k2L2x  kjxj2kL1x;v t 2 ; t > 0 ; (2.1)which gives the estimate (1.6).For a  1 and R > 0, we observeZf dvZjx=t vjRf dv +Zjx=t vjR(R 2jx=t  vj2f)1=af1 1=a dv CRNkfkL1x;v +R 2=aZjx=t  vj2f dv1=aZf dv1 1=a:Optimizing the above estimate in R, that is, takingRN+2=a =CkfkL1x;v 1Zjx=t  vj2f dv1=aZf dv1 1=a;4we have thatZf dv C kfk 1L1x;vZjx=t  vj2f dv1=aZf dv1 1=a!aN=(aN+2)kfkL1x;v ;and from the Holder inequality,kZf dvkL(aN+2)=(aN)x Ckfk2=(aN+2)L1x;v Z Zjx=t  vj2f dv1=aZf dv1 1=adx!aN=(aN+2) Ckfk2=(aN)L1x;v kjx=t  vj2fk1=aL1x;vkfk1 1=aL1x;vaN=(aN+2):Putting q = (aN + 2)=(aN) (i.e. a = 2=(N(q   1))), we obtain that for1  q  1 + 2=N ,kZf dvkLqx  Ckfkq 1L1x;vkjx=t  vj2fkN2 (q 1)L1x;v kfk1 N2 (q 1)L1x;v1=q:Here, we note that kfkL1x;v = kkL1x;v , indeed, ddtkfkL1x;v =RR@tf dvdx =  RR (v  rxf + E  rvf) dvdx = 0. And, f is a constant along characteristics,indeed, since f is an integral of the system of ordinary dierential equations_X = V ; _V = E(X; t) ; t  0 ;f satises thatf(X(t); V (t); t) = f(X(0); V (0); 0) = (X(0); V (0)) ; t  0 ;and hence, kfkL1x;v  kkL1x;v (see [9], [10]).Thus, we have that for 1  q  1 + 2=N ,kZf dvkLqx  C1kjx=t  vj2fkN2 (1 1=q)L1x;v;and from (2.1),k(t)kLqx = kZf dvkLqx  C1t N(1 1=q) ; t > 0 ;which implies the estimate (1.7), where C1 = C1(k(1 + jxj2)kL1x;v ; kkL1x;v ) isa constant depending on k(1 + jxj2)kL1x;v and kkL1x;v .5Moreover, using Lemma 2.1 with  = N   1, we obtainkE(t)kLpx  CkxjxjN  (t)kLpx  Ck(t)kLqx C1t N(1 1=N 1=p) ; t > 0with 1=p = 1=q   1=N , 1 < q  (N + 2)=N , i.e. N=(N   1) < p  N(N +2)=(N2  N   2), which implies the estimate (1.8). References[1] C. Bardos and P. Degond, Global existence for the Vlasov{Poisson equation in 3 spacevariables with small initial data, Ann. Inst. H. Poincare Anal. Non Lineaire 2 (1985)101{118.[2] J. 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On Decay Properties of Solutions for the Vlasov-Poisson System

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On Decay Properties of Solutions for the Vlasov-Poisson System

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