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1、?41?2Q?CC?Vol.41, No. 2 1998K3?ACTA MATHEMATICA SINICAMarch, 1998D?C?Boltzmann?RoW(?b?D?D?b430072)?FbpZVt?JlsIk?gifPvjKfLXBoltzmannMGHqc?mHqYurU?UOwkeK?QdTkhNn?aOS?fLXBoltzmannMG?n?aOS?Hqc? MR(1991)E?35F25B?O175.29Global Solution Existence of the Cauchy Problem for the Relativistic Boltzmann Equatio
2、n in a Periodic BoxJiang Zhenglu (Department of Mathematics, Wuhan University, Wuhan 430072, China)Abstract In this paper its prove that the Cauchy problem for the relativistic Boltz- mann equation with hard relativistic interactions has a global mild solution of theinitial distribution function sat
3、isfies finite mass, energy and entropy.Keywords Relativistic Boltzmann equation, Global mild solution, Cauchy problem1991 MR Subject Classification 35F25Chinese Library Classification O175.291?G?P?y?ZF?W?l?X?BoltzmannD?(?RBE).ND?f?(?1)f t+?pp0f?x= Q(f,f),(1)SyM?Q?Q(f,f) =M 2?R3R3R3W?p,p 1,p?,p?1?p0p
4、10p?0p?10?f?f?1 ff1?d3p 1d3p?d3p?1.(2)j?x= (t,?x)ep= (p0,?p) ( = 0,1,2,3)T?rLS?Ne?f = f(t,?x,?p), f1= f(t,?x,?p1), f?= f(t,?x,?p?), f?1= f?t,?x,?p?1?; p 0= (M2+ |?p|2)1 2, p10=vQdR?1995-12-31,?OdR?1997-04-20,?dR?1997-06-28?Dr?G376?BB?41?(M2+ |?p1|2)1 2, p?0= (M2+ |?p?|)12, p?10=?M2+ |?p?1|2?12, W(p,
5、p1;p?,p?1)tM?Vj?N?uM?M?Vj?4S?p,p1,p?ep?1; MeM?Wc?H?RBE(1)d?P?y?x?e1yLI?f?N?W?p,p 1;p?,p?1?= S(g,)(4)?p + p1 p? p?1?,(3)SyS = |p1+ p|2, g =1 2|p1 p|,j?k?1000 0100 0010 0001; (g,) thj?C?thj?tYcos = 1 2(p p1)(p p?)(4M2 S)1(4)d0,i?IB?Z(3)?RBE(1)?2yU?RBEtLv?u?RBE(1)W?p?WZ?H?RBEt?U?a?y?J?D?a?d?z?d?ZZ?yT?h
6、?J?X?m?y?ZZ(?3).aBO?G?RBE,?L?d?fG?RBE?r?AiZK. Bichteler (?4)eD. Bancel (?5).?M. Dudy nskieM.L. Ekiel-Jezewskig?B?N?h(?6e2).d6y?khj?C?(g,) 0,a.e.,gS1 2(g,) L1loc(R3 S2),(5) ?BR?S2gS1 2(g,) p0p10dd3p 0(|?p1| +),R (0,+),(6)SyS2tS?ZC, BRt?R?Z?, d = sindd, 0 , 0 2,U?RBEl?d?pD?R?k(5)e(6)?dG?Y?(?7)?kW?FR?B
7、L?Y?B?y?S?W?l?X?Y?2.Y?g?Y?dG?Y?YKb?Pd?k(5)?XDipernaeLions?DC?pDRBE?d?thIn?(Pjd?g?Y?H?(g,)m?oWX?).c?V?RBE?G?xPvGlasseyeStraussd?8y?U?J?IpD?d6y?4.1?pDy?Y(4.5)?S?a?f(x,p,t)?tRBE(2.1)?e?U?d?P?i?RBE?r?m?I?i?I?qi?2y?W?p?RBE(1),?khj?C(g,)?(g,) 0,a.e.,z(z2+ M2)1 2(z,) L1loc(0,+) S2),(7) ?BR?S2gS1 2(g,) p0p2
8、10dd3p 0(|?p1| +),R (0,+).(8)f?(g,) = gsin,j? 2, 0 0,?dk0 0,o?|?p1| k0m?BRS2gS1 2(g,) p0p10d3pd k0msup t0,Tsup n?Lnk(fn) Ln(fn)? L1(BR) sup t0,Tsup n?R3R3?BRS2gS1 2(g,) p0p10d3pd? 1|?p1|kfn 1? d3xd3p1 sup t0,Tsup n?R3R3p10fn1d3xd3p1.?aX(29),eQ?P?eQ?L n(fn)?+n=1dL1(0,T) BR)yg?(R (0,+).?A2Q+n(fn,fn)1+
9、fn(n = 1,2,3,)?L1(0,T) BR)yFLg?u(R,T (0,+).Z?q?Qn(fn,fn) Qn(fn,fn) +en lnk, (k 1)Syen?Sf?en=M 2? 1 +1 n?R3fd3p?1?R3R3R3Wn?p,p1;p?,p?1? p0p10p?0p?10K(fn)d3p1d3p?d3p?1,j?K(fn) =?fn?fn1? fnfn1?ln?fn?fn 1?fnfn1?,?Y?x1?q?x2?A3?n +m?t?d?p1: |?p1| kiLv?BRS2gS1 2 p0p10(g,) n(g,)|p0+p10n(p0+ p10)d3pd 0(R,k).
10、JpD?x3,u?p?n +m?t?d?p1: |?p1| kiLv? ?BRS2gS1 2(g,) n(g,)d3pd 0(R,k).Jpi?t?w?LS?Sd0,1 0,1i?a?f(x,y) =(x2+ 1#y2+ 1 xy 1)1 2x#y2+ 1 yx2+ 1,?x ?= ym;22,?x = ym.gaXLS?n? = xy, =1 2(x2+ y2),?eQp?f(x,y)d0,1 0,1it?384?BB?41?j?sup|?p1|k?BRS2gS1 2(g,) n(g,)d3pd(a) 2 sup|?p1|k?11d?R0?S2gS1 2(g,) n(g,)d? r2dr?(
11、b) 2 sup|?p1|k?11d?h00?S2gS1 2(g,) n(g,)d? ?4gp0r2 rp10 r1p0? rRdg?(c) C ?h00?S2z(M2+ z2)1 2(z,) n(z,)ddz 0(n +).Syr = |?p|, r1= |?p1|, =?p?p1 rr1, 2g2= p0p10rr1M2, p0=r2+ M2, p10=#r21+ M2, CtLSuM?Rek?n?Pi?qHpf?(a)tq?3?y?Z?n? (b)tY?jSsn?w?bRU?r1e,?ger?mZ4gdg = (rp10/p0r1)dr;(c)tU?Sd0,10,1i?f(x,y)t?i
12、?t?tY(7)e?wu?P?iP?ye?J?yZ?jS?U?i?hZT?X?td?O?lt?2Q?o?Q?z?BoltzmannE?s?m?e?385?1 De Groot S R, Van Leeuwen W A, Van Weert Ch G. Relativistic Kinetic Theory. New York: North-Holland Publishing Company, 1980 2 Marek Dudy nski, Maria L Ekiel-Je zewska. On the linearized relativistic Boltzmann equation. C
13、ommun Math Phys, 1988, 115: 6076293 LiboffR L. Kinetic Theory: Classical Quantum, and Relativistic Descriptions. 1990 4 Bichteler K. On the Cauchy problem of the relativistic Boltzmann equation. Commun Math Phys, 1967, 4: 352364 5 Bancel D. Probl eme de Cauchy pour l equation de Botzmann en relativi
14、t e g en eral. Ann Inst Henri Poincar e, 1973, 18: 263284 6 Marek Dudy nski, Maria L Ekiel-Je zewska.Global existence proof for relativistic Boltzmann equation. Journal of Stat Phys, 1992, 66(3/4): 9911001 7 Diperna R J, Lions P L. On the Cauchy problem for Boltzmann equations: Global existence and weakly stability. Ann Math, 1989, 130: 321366 8 Gl
