1、 sZv2008.9 - 2008.121 = BaBZ=a=ZZ 0LaplaceZ Z 0 .Z Li-Yaus Harnack inequalitywZ 0oZa+ Cauchy5Cauchy-Kowalevski aLZ|H. Lewy 0b5AS I5 Open problemsb I F. John, Partial Difierential Equations, Springer-Verlag, 1982.!w v . u Zv2002 Mb d u sZv2008 Mb1x1: Q sZ PDE1f u(x;y;)PDE F(x;y; ;u;ux;uy; ;uxx;uxy;)
2、= 0 (1)1“ T F 1M x;y; ;f u#u:K Xf b(solution)u (1): Tu(x;y;)# M (1) Tx;y; bW u(1) T1tM b|“dMQ 9 1 px;y; Lu#Z(1)Cu L bW u 1x;y; f b e n H9 8 u 7 59o p ax;y; bWB B a 5b sZF(PDEs):#B+f # sZFBZFb:nf mPDE n m HN HZF 4(under-determined) n :L(linear)dL(nonlinear)80 ); (5)u = u(t;x1; ;xn)b| n = 1 :?o5.locV
3、U.l bn = 2 : 0 obn = 3 : 2o;ob3. MaxwellZ Maxwell equations b O1 f /1 :“Et = curlH;Ht = curlE;divE = divH = 0;(6)“; H sY b H“ b+YT1“ T“Et = curlH; Ht = curlEw Tt = 0 H1“ TdivE = divH = 0 5 T t ( b 4 s Ei;Hk ( c2 =1=“oZ(5)b Y L ZFh H 0 V U .“ b6. ?V(x;y;z) m SchrodingerZ it = 22m()+V; (11)h = 2 Planc
4、k b7. TricomiZuxx = xuyy: (12)6B 1Z uxx = yuyy:5 Z t Es1Tb 0Z :L:b:d:L:Z9 n pdLZ L= 4yN L=LZ V U b/ +dLZ0b8. 3Euclid bW.l. w. . z = u(x;y)YVL7 Kl w / = ELZ(1+u2y)uxx 2uxuyuxy +(1+u2x)uyy = 0: (13)9. 1+nMinkowski bW. w x = x(t; ) 2Rn / = ELZjx j2xtt 2hxt;x ixt +(jxtj2 1)x = 0: (14)10. = . . (x;y) ( s
5、 x; y) / = ELZ(1c22x)xx 2c2xyxy +(1c22y)yy = 0; (15)c qq = p2x +2yXf b Zp = A (16)Z 8 8c2 = 1 12 q2: (17)11.1 V A8T Navier-StokesZ. s uk pWBF sZ8:uit +3Xk=1uixkuk +1pxi = ui (i = 1;2;3);3Xk=1ukxk = 0 (divu = 0);(18) 7 T“ b612. b Z8:t +3Xj=1xj(vj) = 0;t(vi)+3Xj=1xj(vivj +ijp) = 0;t(E)+3Xj=1xj(vjE +pv
6、j) = 0;(19)(t;x)V U 8 v = (v1(t;x);v2(t;x);v3(t;x) p E =E(t;x) = ?b M V M yN pT# = ?E WB 1“ Tp = p(;E) (p = p(;T)b(20) 8 T Vr Tb(20) TY 8: :Z:bi Z(20) T(19) TBB EL sZFb13.f u(t;x) dLZB 0 Korteweg-de VriesZ.ut +cuux +uxxx = 0; (21) o$ n54 V 0 o 0.lb14.wMonge-ApereZS = S2 1S +S: (22) Z ( w q H4bY km I
7、 n sZ. .b Z T Esb sZil: a:5 s 8 # il Ub7BcBZBZ B K sZ a #/ 0 bBa p+LZE(t;x)- l:+:L:Bdxdt = c: (1.2)“+LB iBHLxct = const: , ; (1.3)Z(1.1)u dudt =ddtu(t;ct+) = ut +cux = 0: (1.4)yN“BHLu usLBL 1b Z(1.1):Y: / Tu(t;x) = u(0;) , f() = f(xct); (1.5)f() B if V Uu. S.b TV Yu Su(0;x) = f(x) (1.6)B bQ Tf C1(R)
8、 5 (1.5)f A (1.1) Sfb iu i(t;x) Sf fM1 = x ct H17 V(x;t)+L S wLx-USbu(t;x)1 SG. . u.Fb) S.Y.+L(1.3) u(t;x)bnm1.1b-6t(t;x)x0 xct = m1.1:+LT B% HWt(x;u)- mV Uf u ?Ct = T Hm t = 0 Hm x-Z_ McT u(x;0) = u(x+cT;T) = f(x): (1.7)m B c_.l7Mo.nm1.2b-6ucxx x+cTu(0;x) u(T;x)m1.2:o M|oZ LoLZ dLodLZb=a pKsZE2/ A
9、U 0.K.s.Z.E. p sZ t QbxZ_htZ_k (t;x)- b I nx h t k (t;x)b p sZv(t+k;x)v(t;x)k +cv(t;x+h)v(t;x)h = 0 (1.8)9 sZ(1.1) 1 bh;k ! 0 HZ T Mvt +cvx = 0: 1 1h;kl H (1.8) Sv(0;x) = f(x) (1.9)v5(1.1)(1.6)M $ s5 = k=h:“(1.8) T Vw Tv(t+k;x) = (1+c)v(t;x)cv(t;x+h): (1.10) TV Vt H YvV Ut+k H Yvb .M. .0. EEf(x) = f
10、(x+h): (1.11) (1.10) TMv(t+k;x) = (1+c)cE)v(t;x); (1.12)t = nkbNYV1(1.8)5v(t;x) = v(nk;x) = (1+c)cE)nv(0;x)=nXm=0Cmn 1+cm(cE)nmf(x)=nXm=0Cmn 1+cm(c)nmf(x+(nm)h):(1.13)3A v(t;x) = v(nk;x)G u x- x; x+h; x+2h; ; x+nh = x+ t; (1.14)F t (xx + nhWb7 sZG u = xct = xcnhF uWx;x + nhbh;k ! 0 H A ?v l sZ y/v(t
11、;x) H u(t;x)11f()Q7 u(t;x)x1“f uWx;x + t1bs T Courant-Friedrichs-Lewy.5.:s:Z:G: : u:K:A:c: :s:Z:G: : u:bs T(1.8).9 ab5 f 7 O 9 B 9. V ? 79lbV(1.13) TA V A Tf ap|iOf V“ * T| V ?v(t;x) = v(nk;x)vl“nXm=0Cmn (1+c)m(c)nm = (1+2c)n“ (1.15)byN1% Hv V ? 3|tZ_ n f 9bB as T _ v(t+k;x)v(t;x)k +cv(t;x)v(t;xh)h
12、= 0; (1.16):|v(t+k;x) = (1c)+cE1)v(t;x): (1.17)(1.17)5Mv(t;x) = v(nk;x)=nXm=0Cmn 1cm(c)nmf(x(nm)h):(1.18) Tv(t;x)1fG ux; xh; x2h; ; xnh = x t (1.19)4Fb t (xt1xWb 7h;k ! 071 M5“(1.19)K x uWx t;xb uWc = xct1 c 1 (1.20)H TCourant-Friedrichs-Lewy5bHq(1.20)/ T9 VV/ Y L A(1.18) Sf f V“|Pv(t;x) = v(nk;x) 3
13、Kv V ? “nXm=0Cmn (1c)m(c)nm = “(1c)+c)n = “: (1.21)V 1Hq(1.20) i OfB= 5h;k !07 Ok=h = M H(1.18) v L l u(t;x) = f(xct)b Y L iu(t;x) ju(t+k;x)(1c)u(t;x)cu(t;xh)j= jf(xctck)(1c)f(xct)cf(xcth)j Kh2;(1.22)K = 12(c22 +c)supjf00j: (1.23) T9 f fxct)TaylorZ 7 Tb 7w =uv jw(t+k;x)(1c)w(t;x)cw(t;xh)j Kh2: (1.24
14、)yNic 1, supxjw(t+k;x)j (1c)supxjw(t;x)j+csupxjw(t;xh)j+Kh2= supxjw(x;t)j+Kh2:(1.25)yw(x;0) = 0 9 T(1.25)Lwt = nk Hju(t;x)v(t;x)j supxjw(nk;x)j supxjw(0;x)j+nKh2 = Kth : (1.26)5 h ! 0 Hw(t;x) ! 0b s T(1.16)v l sZub51.f f L !/ % c1h ! 0 H(1.16) S fv l u.4 U fMV“ HuvM (V“B Y Lb2. I n V ? 9(1.17)7L !v
15、jv(t+k;x)(1c)v(t;x)cv(t;xh)j :dXdt = U(2.6)dUdt = 0b(2.7)ZF(2.6)-(2.7) V HqV U(X;U) = (X(0)+tU(0);U(0), (2.8)1s TX(t) = X(0)+tU(0)U(t) = U(0)b(2.9)+Y:X(0) = fib U(0) = u(0;X(0) = (fi)b(2.10)“(2.9) T VX(t) = fi +t(fi), U(t) = (fi)b(2.11)L ! B(t;x) T VVx = fi +t(fi) (2.12)Qfi:fi = fi(t;x)(2.13)5|(2.13
16、) T U(t) = (fi) VCauchy5(2.1)-(2.2)u(t;x) = (fi(t;x)b(2.14) (2.11) T Tl Cauchy5(2.1)-(2.2)b |(x) = sinxb(2.15) i%t 2 0;1)Vx = fi + tsinfi 9 V pfi :fi = fi(t;x)b N HCauchy5(2.1)-(2.2)u(t;x) = sinfi(t;x)8(t;x) 2 0;1)Rb(2.16) |(x) = tanhxb i%t 2R+Vx = fi +ttanhfi 9 V pfi = fi(t;x)b VCauchy5(2.1)-(2.2)u
17、(t;x) = tanh fi(t;x)8(t;x) 2R+ Rb2 /ZE:+:L:Z:E wLx = X(t):+:Lb+LZEKv+ :|: :s:Z:5: p:s:Z: F:M:5b O0(x) 0; 8x 2R (2.17)H i%t 2 R+ VV(2.12) TQfi = fi(t;x)b“Hq(2.17)/, V +LZE /Cauchy5(2.1)-(2.2)b Hq(2.17)/Cauchy5(2.1)-(2.2) i8b +Y (2.17) T H +LZE? b yt 20;k0(x)k1C0 xfi = 1+0(fi)t 1k0(x)kC0t 0b(2.18) i%t
18、 20;k0(x)k1C0 9 VV(2.12) TQfi =fi(t;x)b9 Cauchy5(2.1)-(2.2) u0; k0(x)k1C0 R ib6BZ (2.17) H5Cauchy5(2.1)-(2.2)Bi8b QE b T(2.17) T ifi1#fi2 (fi1 (fi2)b(2.19) Ti“V(0;fi1)(0;fi2)+LX1(t) = fi1 +t(fi1); X2(t) = fi2 +t(fi2) (2.20)AK H Y =Mbiu H+L sY |(fi1)(fi2)yN )B b“i8L ! bHq(2.17) TB 0 ) V(x) = tanhx./
19、dB/(2.17) T+ il(2.17) TV +L(1.52)| q(fi)1fi 2R9f Vx- +LB ? 9 +LM nm2.1(a)b T(2.17) T 5B+L| nm2.1(b)b7 L H YKl9 Y3 o S)? 3W(2.21)ujt=0 = (x) ; (2.22)a(u) 1uC1f 7(x)x 2RC1f i O C1 b 2.1 Cauchy5(2.21)-(2.22) R+ R iB8C1 sA1Hq da(x)dx 0; 8x 2Rb (2.23) Z(2.21)+LZdxdt = a(u)b(2.24)“+LCauchy5(2.21)-(2.22)C1
20、 9 du(t;x(t)dt = ut +uxdxdt = ut +a(u)ux = 0b(2.25) TV “+Lu = u(t;x) bi+LZ(2.24)B+L| q HWt17+Lx-4bWUS19 +LLbiHq(2.22) TV i(0;fi)+LALx = fi +a(fi)t; (2.26)7+L u |u = (fi)b(2.27)A1:L !Cauchy5(2.21)-(2.22)R+ R i8C15 1 (2.23) TB bQE bL !(2.23) T 5Ai fi1fi2 fi1 a(fi2): (2.28) Ti“V(0;fi1)+Lx = fi1 + a(fi1
21、)tV(0;fi2)+Lx = fi2 +a(fi2)tAK HW =Mbiu H+L sY |(fi1)(fi2)yN) ?B bi8C1L ! bA1b s:i(2.23) T,+LZE V i%t 2R+ VV(2.26) TQfi VbY L i(2.23) T(2.26) T Vxfi = 1+ da(fi)dfi t 1 0; 8t 0b(2.29) i%t 2R+ TV x 1fi9f b 6BZ ia(fi) Va (x)C0 fi !1 H x !1b(2.30)(2.29)(2.30)V i%t 2R+, (2.26)l BVRRC1sb“ L VV(2.26) TQfi:
22、fi = fi(t;x)| (2.27) TL Cauchy5(2.21)-(2.22) BC1u = a(fi(t;x)b sb8b5(2.23) T+ il (2.23) T V(0;fi)+L| qa(fi) 1fi 2 R9f bVx- +LB ? b +LMb:2.1: +Y 0: : : H:W breaking timeb/ ) HWb i VCauchy5(2.21)-(2.22)i C1biS = B(t;x),YV V_/SBH+L:x = (;t;x)biaC1;C1 V+L| qb +Ax-M:(0;fi)b u(t;x) = (fi) (2.31)#(;t;x) =
23、fi +a(fi)b(2.32)+Y = t H x = fi +a(fi)tb(2.33)|Z(2.31) T 1x p Vux(t;x) = 0(fi)fixb(2.34)6BZ Z(2.33) H1fi pxfi = 1+da(fi)dfi tb(2.35)|(2.35) T (2.34) Tux = 0(fi)1+ da(fi)dfi tb(2.36)“) ux HM kv5) (2.36) Ts H ,5bTda(fi)dfi 0; 8fi 2R; (2.37)5t 2R+ H(2.36) Ts v1bN H lM? 3b7T tfi 2Rda(fi)dfi | * lM|K H Y
24、 =? 3b y“fitV09Fhda(fi)dfii1H(2.36) Ts Vv t0b AK* HW fi0 Pda(fi)dfi |Kla(fi0) = minfi2Rda(fi)dfib(2.38)“ HW V/ Ttb = 1da(fi)dfiflflflfi=fi0; (2.39)fi0 (2.38) TbA B f /“fi0BBb2.1 p/ Cauchy5 HW8:ut +uux = 0;t = 0 : u = expfx2gb(2.40)N Ha(u) = u; (x) = expfx2gb V(0;fi)+L| q a(fi) = a(expffi2g) = expffi
25、2gb p HW (2.39) Tsf(fi) , ddfia(fi) = ddfi expffi2g = 2fiexpffi2gKlbif 0(fi) = (2+4fi2)expffi2g;f f(fi) “ 1p27fi0 = 1p2)f(fi) |KlbCauchy5(2.40) HWtb = 12fi0 expffi20g= 1p2expf12g=re2 t1:16b6i+LVr T(2.33) LK* bWUSxb = fi0 +a(fi0)tb = 1p2 +re2 exp12= p2b“ p Cauchy5(2.40) HW#K* (pe2;p2)b) xb):2.2YV ) ?C:2.1 dLZ+LBK HW =MC b dLZLZ)b51. p/ Cauchy5 HW#K* 8:ut +uux = 0;ujt=0 = sinx;8:ut +uux = 0;ujt=0 = tanhx8:ut +u2ux = 0;ujt=0 = (1+x2)1:9x3
