1、E39 E4ZZVol.39, No.41996Z7ACTA MATHEMATICA SINICAJuly, 1996WcMaxwell-Schrodinger Cauchy OK (1)i( isd100080)(nqy100088)yetrk H. Pecher njku k x glT: fvnmet Maxwewll-Scohrodpingqer njk CazuchyhMaxwell-SchrodingerCauchykAt Maxwell-Schrodinger njCauchy0ZS1,2, U R HxHrpWAoCLagrangian MKm 1 L =i(t + iA0)

2、2 ( iA) ( iA) V 2(x) K(x) |2 |2 + 1 A + A 1( A)2.(0.1) 2 |2t02a LagrangianLJrp QboMaxwell-Schrodinger Q F = J ,F = A A,+ 1 D D (0.2)iD02 j j= V (x) + K(x) |2 + |2.D = iA ( = 0, 1, , n) r “n+1 H WAFA” A = (A0, A) = (A0, A1, , An)uKX1, 2, 3.nJaLLorentz hPA = 0(0.3)Q (0.2) Cauchy N Coulomb hPdiv A = 0(

3、0.4)1994816 t1995310 ti q y ETe j Lp Pevo Maxwell-Schrodinger Q1986 Y K. Nakumitsu n M. Tsutsumi V p Lp Lp c 4 #WAQO 5 C K(x) 0, 0, V (x) 0K(x) 0, 0 gP o Maxwell-Schrodinger Q Lorentz hP Cauchy vb o Maxwell-Schrodinger Q (0.2) U RD R P$Im %L& gC!H. Pecher l pInV( c QO v l Hardy-Littlewood-Sobolev c

4、zVkzV c BQO w Cbo Maxwell-Schrodinger QLorentz hP Coulomb hP Cauchy)JR *v W se!R ! #Q $+,!%cLv -J(i) .0A (0)=A(x0),= 0(x),A (0) = A1 (x), = 0, 1, , n,(1.2)2= tt DAlembert 3 J0 = |2, Jj = i (Dj Dj), j = 1, 2, , n.lwX-A P = 0A = tA, .(1.1), (1.2) B-vdu + Zu = N (u), dtP0 jAj = 0,u(0) = u0(x) = A0(0x),

5、P 0(0x), A0(x),P 0(x), 0(x) .(1.3)#$ u = (A0, P0, A, P, ), Zu = P0, A0, P, A, 2i , N (u) = (0, J0, 0, J, J), NJ = A j j + 1 P 0 1 A Aj j+ iA 0 iV (x) iK(x) |2 i|2.(1.4)22O(NC)c*/p Maxwell-Schrodinger R+, Cauchy 45 (1)4499L BO & . Lorentz hP6/0uq Q Cauchydu + Zu = N (u), dtu(0) = u0(x).(1.5)1N0Q-7mJ1

6、-,6/P 0(x) jA0(x) = 0,0 0j 002(1.6)jP (x) A (x)+ | (x)| = 0j00 (1.5) !b (1.3) !2X 1.1 ( m 2, PvY m = Hm Hm1 (Hm1)n (Hm2)n Hm2,Xm = Hm+1 Hm (Hm)n (Hm1)n Hmzj! (1.5) ! u(t) C(0,T ); Xm) C1(0,T ); Y m) . ft!(1.7)vbo Maxwell-Schrodinger QU RD%LSamn +2gP 8H (1.5). DbQN m = n + 4 0 Sobolev dwJ h93 Pv X22m

7、 ,Y m zj! (1.5)!.Ghift! k# :%2Q 43( n +2 m n +4 .L 5a j;2222D 1.1 ( m n + 2 , V (x) Hm(Rn, R), K(x) = n|x|, f2n n, (0, 1),(1.8), n (1 = 0) 4b! u(t) J1u C(0,T ); Xm) C1(0,T ); Y m),(1.9)0fsv W6(i) T = , x (ii) T , lim u(t) X n +2 = . -7mJ16/ (1.6)tT20(1.5)! u(t)(1.3)! 15v# !ev-7m6Ve72M t 0, T ,T T ,

8、x0 strong in Xm1 02weak in Xu(t) X n +2 C,uj mu (x), j ,1.2Jj1.1x$6/ (1.8) 3ju.+L2DMm! u (t) x u(t) *sve u (t) weak inXm (t), j .J 1.1 Hsp48 1,(1.8)a7 1.3 n 3 0 6/ (1.8) 5lC6/ (1.8) , N n 4 0 6/ (1.8)1W5l 6/ (1.8) C DbQN 6/ (1.8) uT = n 2 (n 3)bagP #0J1.1Hh93)JRjM Coulomb hP bo Maxwell-Schrodinger Q

9、Cauchy2Aj = n 1t2 AjJj 2n Jj ,2(n 2) 2|x|n2i1i 22t = A div A A iV (x) i| 2 n i2222div A = 0, iK(x) |2 n2 2 (n 2) 1|x|n2 | ,(1.9)A(0) = A0(x),?48 8Sa h61BA(0) = A1(x),(0) = 0(x), 1|x|n2 JjHm C Jj Hm(1.10)J1.10Qk 22 VQ9:ZE!%L&H. Pecher l6b;g pqbJ%plgPYD 2.1 ( 1, 2, , N J1 |j| kjn $kj 0, 1 p, q , j 0,(

10、j = 1, 2, ,N ), 0xN6/b48a = 1 Kj |j | ,j = 1, 2, , N.(2.1)jj qjn(i) aj 0pj=1(ii) aj = 0 (j = 1, , n) fa 1 jj/qj,(2.3).svcaj 0Njpaj 0jNjj=1ujL Cuj W kj,qj .(2.4)pj=11N x(iii)NN0, ( = 1 2 ) f= 1 ,(2.5)aj j., ,Najj=1pj=1j/qjNj=1jj ujLpN Cj=1juj W kj ,qj .(2.6)G6/ (i)01B (2.4)6, 1.H. Pecher lN0QgP0Q (2.

11、6). l pj = (paj )1, . pj J11 pj 0paj 00Q (2.4). k+gP5l(i)C #0l pj = (aj p)1, (aj 0), vNj=1jjLpaj 0j pj j .(2.12)3j ujj ujj ujh6 aj 0 b(iii)gPv r (iii)SobolevdwJ 2(2.4).2YD 2.28 ( | mn $m n + 2 , .M f, g Hm, svc(fg) fg L2 C( f H| g n +2 + f n +2 g H|1 ).(2.13)H 2H 21N M 0 k m scfg Hk C f Hm g H n +1

12、+ f H n +2 g Hm1 + f H n +1 g Hk .(2.14)222Z p (2.14) 2LF 2.3 ( f(x), g(x), h(x) Hm, m 2 n + 2 , M 0 k m sfgh Hk Cf H n +2 g H n +2 h Hm ,(2.15)22f, g, h M f, g, h Gvf,g,h2YD 2.4 ( s1, s2 s 0 f s1 + s2 s n , .scfg Hs C f Hs1 g Hs2 .(2.16)Gpl2.1 :0Qh6fg Hs |+|sC fg L2 ,(2.17)O(NC)c*/p Maxwell-Schrodi

13、nger R+, Cauchy 45 (1)4501l q1 = q2 = 2, k1 = s1, k2 = s2, 1 = 2 = 1, 1 = , 2 = , =;G0l2.16/2 c(2.16). InV( 1%.#l lYD 2.59,10 ( ABanach * X z!(8o T (t):hHSL0 (A),X = D(A) YfMR Au X b4 Banach * (0 1). Fdu + Au = f (u),u(0) = u (x).(2.18)dt0xU Rmf(u) J1f (u) f (v) X C( u X , v X ) u v X ,(2.19)u, v X,

14、 0 0, O 9G Cauchy (2.18) sb! u(t) J13E(?Hu(t) C1(0,T ); X) C(0,T ); D(A).(2.20)L0QJ1.1,vU RD N (u):R N3Z 5Y m zC0- 8oSLk# :O=; p I. Segal J11.#L 0.Z = (I), (I), (I), (I), (I)2 i , N(u) = P0, J0, P, J, J ,mu0(x) Y m+2, 0fu0(x) weakinXu0(x),( 0).(3.2)gZY m z!(8oSLCp D(Z) = Y m+2, Sobolev dwJh93Y m+2 X

15、m. k# rl2.5, 5SMdJ (0, 1), 1 0QN(u) N(v) Y m C( u Y m+2 , v Y m+2 ) u v Y m+2 ,(3.3)u, vY m+2. r# 2/T 0 (3.1)b! uC 0, T ), Y m+2C1 0, T ); Y m . 1N1kzVc;*k :0QJ1.1.YD 3.1J1.16/x u, v Y m+2, .sN(u) N(v) Y m C( u Y m+2 , v Y m+2 ) u v Y m+2 ,(3.4) 0 1, v = (A ,0P , A ,P , ).G h6Nu Nv = P0 P0, J0 J0, P

16、 P ; , J J , J J .(3.5)ptLIqbDwD4pvAj j Ajj Hm2 C( Aj Hm+1 + Hm )( Hm + Aj Aj Hm+1 ). (3.9)k#N5ScK |2 K | |2 Hm2 = I, 2I C|+|=m2(K |2 K | |2) L2 + C|+|=m2(K | |2)( ) L2+ C (K |2 K | |2) L2 + C K | |2 ( ) L2 = I1 + I2 + I3 + I4.B1(x)x M ?iNMDcIj (j = 1, 2, 3, 4).2I3 C K ( ) L2 + C K ( ) L2(3.10)2C 1|

17、 |dy L B1(x) |x y|+ C 1| | |dy L B1(x) |x y|+ C 1| |dy L RnB1(x) |x y|+ C 1| | |dy L RnB1(x) |x y|HH22C 2 m + 2 m Hm .(3.11)?I4 C 2 m + 2 m Hm .(3.12)HHI1 =h6|1|+|2|+|=m2K 1 ( )2 L2 + K 1 2 ( ) L2 . (3.13)h6 (3.13)?D=?L5ScbQC (vDbD). V;gP:1QO(NC)c*/p Maxwell-Schrodinger R+, Cauchy 45 (1)4503IR I | 2

18、. l= 11n + 2 2 1 = n + 2 2 (3.14)p2 2n,p2n.vr Holder 1BHardy-Littlewood-Sobolev 1B 7K 1 ( )2 L2 K 1 ( )2 Lp Lp C 1 ( )2 Lq Hm C Hm 2 Hm .(3.15)p6 1 1 m| , 1 = 1 n , 0fO(3.15) IqbA1B48 FV6/p2nqpn1 1 m |1| + 1 m|2| ,q2n2n(3.16)1 1 12 .nn + n + 2 2 n(3.17)2n#b r6/ (1.8) 1.22IR II 3 | m 2.h6 n + 2 m n +

19、 4 , k#l 2 21 = 1n + 2 | 1 = n + 2 | 2(3.18)p2 n,pn,| 1 n | , 0fO (3.19) Iqb48 FV6/B-v2qpnp2nn + 2 | + n 12m 2 |1| |2| 2n,(3.20)n2n + 2 |n + n #b r6/ (1.8) 7N (3.19) 48 0 (3.1)b! u(x, t) C 0, T); Y m+2 C1 0, T); Y m .LB1k;*k :0QJ1.1.# 8v l2YD 3.2 ( m n + 2 , u, u1, u2 Xm, .X 2XRe(N(u), u) C( u n +2

20、) u 2 m ,(3.24) Re(N(u1) N(u2), u1 u2) C( u1 Xm , u2 Xm ) u1 u2 Xm1 ,(3.25)(,) Xm VzG c(3.24) rll2.2HH 2H 2H 2HmH 2H 2H 2Re |2, P0 C 2 n +2 P0 Hm + C P0 n +2 Hm n +2 ,(3.26) Re(J, P )Hm1 C P Hm1 A n +2 n +2 Hm1 + A Hm1 2 n +2 )+ C Hm1 ( H n +2 P Hm1 + Hm1 P H n +2 ),(3.27)22H 2H 2H 2H 2H 2H 2Re(P0,

21、)Hm C 2 n +2 P0 Hm + C P0 n +2 n +2 Hm ,(3.28) Re(iA0, )Hm C A0 Hm 2 n +2 + C A0 n +2 n +2 Hm ,(3.29)H 2Re(iAjAj, )Hm C Hm Aj 2 n +2 Hm+ Aj Hm AjH n +22H n +2 ,(3.30)2Re(iV (x), )Hm C V (x) Hm 2 m ,(3.31)Re(i|2, )Hm C 2 m H2 n +2 ,(3.32)HH 21Re(Ajj, )Hm = Re|m (Ajj) Ajj, L2 2 RejA , |mHH C( A Hm n +

22、2 + 2 A n +2 Hm ) Hm ,(3.33)22fRe(K |2 , )Hm CK |2 L2 Hm + C K |2 2 L2|+|=mH 2 I + 3 n +2 .(3.34)2:cI, V?;gP:c(i) | 3. #0 l 1 = 1 n +22 , 1 = n +22 , (0 2q2n1+2 n 2,(3.36)q 3. h6I =2| n +2|1|+|2|+|=mK 1 2 L L2 +23| n +2|1|+|2|+|=mK 1 2 L2= I1 + I2.(3.37)p (3.11) c QO12K L |1 | |2 |dy +RnB1(x)|x y|B1

23、(x)|1|2 | dy|x y| C 1 L2 1 L2 + C 1 L 2 L .(3.38)k # r Sobolev dwJ&JH 2I1 C 2 n +2 Hm .(3.39)DbQN l1 =1n + 2 |1 2,n + 2 |= 2,3 | 0 Ou 2 n +2 y(t),t 0, T) 0, T0.(3.48)X 2:T( T T0, k# r (3.46)#$ C l y(t) 0, T0 z zIu(t) 2 m u0 2 m eCt,(3.49)XXDbQN = u1 u2 , 7N J1d + 1(I ) + Z = (1 2)(I )u + N(u ) N (u

24、),1 2.(3.50)dt212p x (3.50)Xm1 Vz l=?v (3.46)s1 d222 dt Xm1 |1 2| u1 Xm Xm + C( u1 Xm , u2 Xm ) Xm1 .(3.51)h6 (0) = 0 (3.49) sX 2 m1l 1 0, r Gronwall 1B C1 + CtX 2 m1 d.(3.52)0strong in Xm1uu C(0,T0;Xac (3.49)yPsm), 0.(3.53)u weak in Xm u L 0,T ; Xm , 0,(3.54)0fsco 0Xmu(t) 2 m u0|2 X eCt,t 0, T0.(3.

25、55)k# u(t) 0, T0 6 Xm zy6V Xl Y m+2, F(Zu + N(u), )Y m = (u, (I )Y m + (u,Z)Y m + (N(u), )Y m,(3.56)h6 Xm Y m1 l 3.2(Zu + N (u), )Y m (Zu + N (u)Y m , 0, t 0, T0.(3.57)vr (3.56)bR5Spl3.2Gronwall 1B2rv u(t)Xmy6V 0flim u(t)Xm u(0) Xm ,(3.61)k# Sobolev dwebRh93 u(t)0, T0) zt6V NQ (3.1) evA tXu(t)(0, T

26、z6V 07Nu(t)0C 0, T ; Xm .ev! u(t) 6Ve7-7m0Q=?v u u ( 0)0QkL M2rh u0 X n +2 L, T0 5e7v L, k#p_QO/ I k* 0, T ) O0fsv W6u(t) C(0,T ); Xm) C1(0,T ); Y m).(3.62)(i) T = , x (ii) T , lim u(t) X n +2 = .tT2Iq:0Q-7mJ1-,6/ (1.6) 0(1.5)!(1.3)!f = P0 jAj,g = jPj A0 + |2.(3.63)h6Q !_=; 3f = g,g =f.(3.64)ttp f,

27、g V x (3.64) ?AQ L2 Vz 1 df 2 + g 2 + f= fgdx f 2+ g 2.(3.65)Rn2 dtL2L2L 2h6 f(0) = g(0) 0, 7Nr Gronwall 1BL2L2f g 0,t 0, T ).0O(NC)c*/p Maxwell-Schrodinger R+, Cauchy 45 (1)45072D 1.2 G #0 K(x) = n|x|1,L5SI =K 1 2 |1|+|2 |+|=m L2 c w Vm?J 1.1. rvJ 1.1 5lC n = 2 0J 1.2 gP k#L:T( n 3. l2n2nk+ = n 1 ,

28、k = n 1+ (0 1).(3.66)Cp H1 Lk+ , H1 Lk ,h6I |=m+K L2 + K L2K 1 2 L2 = I1 + I2.(3.67)|1|+|2|+|=m|1|m,|2|mh6 I1?D!_zbk#K 2 | |dy L2dy 2 +LRnB1(x)|x y|B1(x) |x y|1 Hm 22 + n +2 L2 Hm dy2LH 2B1(x)|x y|2H 2 C Hm 2 n +2 .(3.68)DbQN M |1|, |2| 0, sK 1 2 L2 =|x|1|x|a |1 2 | L2 + |x|1|x|a |1 2 | L2 C a 1 k+2

29、 k+ + a 1 k 2 k L2 ,LLE E z % m XlL L (3.69) 1 1a = 1 Lk 2 Lk21 Lk+ 2Lk+2 ,(3.70)h6 |1|, |2|, |s?A m 2, 7Nr (3.69), (3.70)K 1 2 1L C 1 2 k12 2 k12 2 k11 2 k 2 L2k# r (3.67), (3.68), (3.71)7NJ 1.2 0L +L +L L H 2 C Hm 2 n +2 .(3.71)I C Hm 2 n +2 .(3.72)H 2L M. Tsutsumi |j sj8HCMP1 Bogoliubov N N, Skir

30、kov D V. Quantum Fields. The Benjamin: Cumming Publishing Company Inc, 1982.2 Schi L I. Quantum Mechanics. New York: McGraw-Hill, 1968. 3 P_O B I s qS|A I Fd1994,5.4 Tsutsumi M, Nakamitsu K. Global existence of solutions to the Cauchy problem for coupled M-S equation in two space dimensions. In: Phy

31、s Math and Nonlinear PDE. New York: Marcel Dekker, 1988.5 Nakamitsu K, Tsutsumi M. The Cauchy problem for the couple M-S equations. J Math Phys, 1986, 27: 211 216.6 Pecher H, Wahl W. Time dependent nonlinear Schrodinger equations. Manuscripta Math, 1979, 21: 133.7 Stein E M. Singular Integral and Dierential Property of Function. Princeton Universi