加权空间中一阶格点系统的统计解及其Kolmogorov熵.pdf

2023,43A(5):15591574http:/?m?:X?O)9 Kolmogorov?qU/(n?325035):T?m?:X?O)9 Kolmogorov?.ky?:X?K3?mk?N5,)N?)YL3xCBorel V,?XyTxCv Liouville n,T:X?O),?O)?Kolmogorov?O.c:X;.f;?m;O);Kolmogorov?.MR(2010)Ka:35B41;34D35;76F20a:O175.8zI:A?:1003-3998(2023)05-1559-161O)?Vg5?uOn,R,(1.1)ui()=ui,i Z,(1.2)Z L?8,ui=ui(t),?,fi(t,ui)gi(t)?,R m,ui,.?m2=?u=(ui)iZ:XiZiu2if17,?O18,”?n19,20,zAn21?.:X?C1?2,Xz 22,23 y?:X?Nf3?;z 2426?g:X3;?f?,9?f?Y5?O,nAuN?:n;z 27?:Xf?35;z 2830?:X?f;z 31,32?:Xff?35K.,z 3335?m:X?.f!f?35Y59g.d:X?.c1;C,z 36?m:X(1.1)?f,?E,?5 fi(t,uim,ui+m)A?.?,8cvkz?m:X?O)9 Kolmogorov?K.?3?m:X(1.1)?O)?359 Kolmogorov?O.d,kyK(1.1)(1.2)3?m 2?N5,?Xy)N?3 2)?YL3.8.f,umk-Y5(n 4.1)3x Borel V,yTVv Liouville n,:X(1.1)?O).?,O)?Kolmogorov?9.?O.p?,z 37?:X(1.1)?CK,fi(t,ui)=fi(ui)m t,m2=?u=(ui)iZ:XiZu2i+,ui R?.(1.5)z 37,?3?m 2?:X?O)9 Kolmogorov?,?3y.f?35,93?y)?Lum?-Y5?)?(J.w,m 2 2Au i 1?A/.d,C?35?,?z 37(J?2.,?,g?m:X?O)?,3y.f?xC Borel VO)?,?O)?Kolmogorov?,y.?O.2?N53?!,k0?b?f,?yK(1.1)(1.2)?)3?m 2k?N5.k,?2(1.3)?m,:Z (0,+),i (i)=i?,u|i|4,u i Z,ve0 (i)=i c0,R,(2.8)u()=(ui,)iZ=u 2.(2.9)?yK(2.8)(2.9)?N5(?y.f?35),b?,9(2.2),(2.3)?c1,c2 f(t,u)=(fi(t,ui)iZ,g(t)=(gi(t)iZv(H1)?c1,c2v c1 1,c26 2 (1.1)?v:=c1c2 0.(2.10)1562n?Vol.43 A(H2)z i Z,k fi(t,s)C(R2),fi(t,s)s C(R2),fi(t,s)s 0,s R,3Y :R+R+?supiZsuptRmaxsr,r|fi(t,s)s|6(r),r R+.(2.11)(H3)b?g(t)=(gi(t)iZ C(R;2),Zteskg(s)k2ds 0 1/2,uT?,?c0=1 K(2.1).w,(y1)(y)C(R)limy(y1)(y)=lim|i|(i1)(i)=1,?supyR(y1)(y)1,?c1=supiZ(i1)(i),K c1 1,(2.2).,?,dn3?|(i 1)(i)|=|0(i )|6(i )6 max(i+1),(i 1)6 c1(i),i Z.?c2=c1,4 1/2?,/?,K c26 2,(2.3).,?,i Z,P fi(t,s)=s3|sint|,?(s)=4s2,K(fi(t,s)iZv(H2).uv(H3)?g(t)=(gi(t)iZ?35z 38.eyK(2.8)(2.9)?N)?35.n 2.1?(H1)(H3).(1)uz?u 2 R,K(2.8)(2.9)3?)u C(,T),2)C1(?,T?,2),(2.13)T ,XJ T .(2.14)ydu A:27 2k.5f,g(t)C(R;2),Iy f(t,u):R272u u 2 Lipschitz N?=.?B 2?k.4,?u=(ui)iZ,v=(vi)iZ B,Kuz i Z?t R,d(H2)n3i(0,1)?fi(t,ui)fi(t,vi)=(ui vi)f0i(t,i),(2.15)i=iui+(1 i)vi B.l?d(2.11)kkf(t,u)f(t,v)k2=XiZi?fi?t,ui?fi?t,vi?2=XiZi?ui vi?2?f0i?t,i?2No.5qU?:?m?:X?O)9 Kolmogorov?15636 supiZsuptRmaxB|f0i(t,)|2ku vk26(rB)ku vk2,(2.16)rBk.8 B?.?;n,n?(1)?y.yn?(2),u(t)=(ui(t)iZ 2(2.8)3 2S,?12ddtku(t)k2+ku(t)k2+(Au(t),u(t)+(f(t,u(t),u(t)=(g(t),u(t),t .(2.17)P wi(t)=iui(t),w(t)=(wi(t)iZ,?Ok(Au,u)=XiZ(Bu)i(Bw)i,(2.18)|(Bw)i i(Bu)i|=|ui+1(i+1 i)|6 c2i|ui+1|.(2.19)5?c26 2,k(Au,u)=XiZ(Bu)i(Bw)i=XiZni(Bu)i)2+(Bu)i(Bw)i i(Bu)ioXiZni(Bu)i)2 c2i|ui+1|(Bu)i|oXiZn(i12c2i)(Bu)i)212c2i|ui+1|2o c22XiZiu2i+1 c1c22XiZi+1u2i+1=12c1c2kuk2.(2.20),db?(H1)Cauchy?(f(t,u(t),u(t)=XiZifi(t,ui(t)ui(t)6 0,(2.21)(g(t),u(t)6ku(t)k22+kg(t)k22.(2.22)n(2.17)(2.20)(2.22)?ddtku(t)k2+ku(t)k26kg(t)k2,t ,(2.23)d?0 d(2.10).d(2.23)Gronwall?,=?(2.14).y.dn 2.1,z?u 2,K(2.8)(2.9)3?N).d,)N?U(t,):23 u7 u(t)=U(t,)u 2,t R,3m 2)L U(t,)t.ey U(t,)t?Y5.n 2.2?(H1)(H3),K U(t,)t 2?YL,=u?t,U(t,):27 2YN?.y?B 2?k.8.u?R,?u(k)B(k=1,2)?,Pu(k)(t)=U(t,)u(k)OK(2.8)(2.9)Au u(k)?).P u()=u(1)()u(2)(),K u vd u(t)dt+A u(t)+u(t)+f(t,u(1)(t)f(t,u(2)(t)=0,t ,(2.24)1564n?Vol.43 A u()=u(1)u(2).(2.25)u (2.24)3 2S,?12ddtk u(t)k2+k u(t)k2+(A u(t),u(t)+(f(t,u(1)(t)f(t,u(2)(t),u(t)=0.(2.26)d(2.14),(2.16)(H2),(H3)3 L(t,B)0,?|(f(t,u(1)(t)f(t,u(2)(t),u(t)|6 L(t,B)k u(t)k2.(2.27)(2.7),(2.26)(2.27)?ddtk u(t)k2+2?2p2(1+c1)L(t,B)?k u(t)k26 0.(2.28)(2.28)A Gronwall?,?k u(t)k26 k u()k2e222(1+c1)L(t,B)(t).y?.3.f?353!,yL U(t,)t3 23k.8.,?yU(t,)tk.C5.D-f.k.8.f?z 38,39.?PD=?bD=D(s)|s R|limsessupuD(s)kuk2=0?.(3.1)w,2?k.8 B u8a D.e?nL U(t,)t3 23k.8.n 3.1?(H1)(H3).K3cB0=B0(s)|s R,?t R,bD=D(s)|s R D,0=0(t,bD)6 t?U(t,)D()B0(t),t 0,B0(s)=B0(0;R(s)L 2?:%,R(s)?4.yP R(t)0?R2(t)=1+etZteskg(s)k2ds,t R.(3.2)Kd(2.14)(3.2)cB0=B0(0;R(t)|t R U(t,)t3 2?k.8.y?.?e5yL U(t,)t3 2k.D-C?5.n 3.2?(H1)(H3).Ku?t R,0 bD=D(s)|s R D,I0=I0(t,bD)N 0=0(t,bD)6 t,?supuD()X|i|I0i|(U(t,)u)i|26 2,6 0.(3.3)No.5qU?:?m?:X?O)9 Kolmogorov?1565y Urysohn n31w()C1(R+,R+)?(x)=0,0 6 x 6 1,0 6(x)6 1,1 6 x 6 2,(x)=1,x 2,|0(x)|6 0,x 0,(3.4)0 0.?bD=D(s)|s R D,?t,R,t ,Pu(t)=u(t;,u)=U(t,)uK(2.8)-(2.9)?u D()?).?g,I 0,Pwi=iui,vi=(|i|I)ui,zi=ivi,i Z,w=(wi)iZ,v=(vi)iZ,z=(zi)iZ.v=(vi)iZ(2.8)3 2S,?(u(t),v(t)+(u(t),v(t)+(Au(t),v(t)+(f(t,u(t),v(t)=(g(t),v(t).(3.5)e(3.5)?1O?O.k,(u(t),v(t)=12ddtXiZi(|i|I)u2i(t).(3.6)u(Au(t),v(t),k(Au(t),v(t)=XiZ(Bu)i(Bz)i=XiZ(|i|I)(Bu)i(Bw)i+XiZ(Bu)i?(Bz)i(|i|I)(Bw)i?.(3.7)d(2.2)(2.20),XiZ(|i|I)(Bu)i(Bw)i c22XiZi(|i|I)u2i+1 c1c22XiZi+1(|i|I)u2i+1=c1c22XiZi+1(|i|I)(|i+1|I)+(|i+1|I)u2i+1.(3.8)An(3.4),?XiZi+1(|i|I)(|i+1|I)u2i+160IXiZi+1u2i+1=0Ikuk2.(3.9)(3.9)“(3.8),kXiZ(|i|I)(Bu)i(Bw)i 12c1c2?0Ikuk2+XiZi(|i|I)u2i?.(3.10)?O?XiZ(Bu)i(Bz)i(|i|I)(Bw)i=XiZ(Bu)i(|i+1|I)(|i|I)i+1ui+11566n?Vol.43 A 0IXiZ|(Bu)i|i+1|ui+1|02IXiZi+1|ui+1|2+(Bu)i)2=02I(XiZi+1|ui+1|2+XiZi+1(Bu)i)2).(3.11)d(2.2)(2.6),XiZi+1(Bu)i)26 c1XiZi(Bu)i)2=c1kBuk26 2c1(c1+1)kuk2.(3.12)(3.12)“(3.11),KXiZ(Bu)i(Bz)i(|i|I)(Bw)i 02I(kuk2+2c1(c1+1)kuk2).(3.13)2(3.10)(3.13)“(3.7),?(Au(t),v(t)12c1c2?0Ikuk2+XiZi(|i|I)u2i?02I(kuk2+2c1(c1+1)kuk2)=12c1c2XiZi(|i|I)u2i(c1c2+1+2c1(c1+1)02Ikuk2.(3.14),A(H2)9 Cauchy?(f(t,u(t),v(t)=XiZi(|i|I)fi(t,ui)ui 0,(3.15)(g(t),v(t)=XiZi(|i|I)gi(t)ui(t)612XiZi(|i|I)g2i(t)+2XiZi(|i|I)u2i(t).(3.16)(3.5),(3.6),(3.14)(3.16)9.5,kddtXiZi(|i|I)u2i(t)+XiZi(|i|I)u2i(t)61XiZi(|i|I)g2i(t)+(c1c2+1+2c1(c1+1)0R2(t)I,6 1,(3.17)1=1(t,bD)n 3.1?.m.y?0,d(H3)9(3.2)I1=I1(t,)N,?(c1c2+1+2c1(c1+1)0R2(t)I623,I I1.(3.18)(3.18)“(3.17),kddtXiZi(|i|I)u2i(t)+XiZi(|i|I)u2i(t)61XiZi(|i|I)g2i(t)+23,6 1,I I1.(3.19)No.5qU?:?m?:X?O)9 Kolmogorov?1567(3.19)A Gronwall?,?XiZi(|i|I)u2i(t)6XiZi(|i|I)u2i()e(t)+etZtesX|i|Iig2i(s)ds+23.(3.20),d(H3),0,I2=I2(t,)N,?ZtesX|i|Iig2i(s)ds 6X|i|IZtesig2i(s)ds 623,I I2.(3.21)2d(3.1),0,2=2(t,bD),?XiZi(|i|I)u2i()e(t)6 e(t)ku()k26 e(t)supuD()ku()k23 23.D-f(P)AD=AD(t):t R,v(1);5t R,AD(t)2?;f8;(2)C5U(t,)AD()=AD(t),t ;(3).5bD=D(t)|t R D,klimdist2(U(t,)D(),AD(t)=0,t R,dist2(,)L 2?Hausdorffl.4CO)?353!,kyL U(t,)t3 2k-Y5,?3xC Borel V ttR,?yTx(2.8)?O)v Liouville n.n 4.1?(H1)(H3).KL U(t,)t3 2k-Y5,=u?t R u 2,2-N?7 U(t,)u3(,t Yk.y?t R u 2,d(2.14)?kU(t,s)uk26 e(ts)kuk2+etZtsekg()k2d6 kuk2+etZtekg()k2d,t s.(4.1)d(H3)?m s?k.d,kU(t,)uk3(,t k.ey 2-N?7 U(t,)u3(,t?Y5.d,?s(,t,Iy U(t,)u3 =s?Y,=y 0,=(,s,t,u)0,?|r s|=kU(t,r)u U(t,s)uk.(4.2)”?r s s,(4.3)u(s)=U(s,r)u u,(4.4)d(4.1)3 L1=L1(t,s,u)0?ku(1)(t)k2+ku(2)(t)k26 L1.(4.5)u (4.3)3 2S?12ddtk u(t)k2+k u(t)k2+(A u(t),u(t)+(f(t,u(1)f(t,u(2),u(t)=0.(4.6)d(H2),(2.16)(4.5)3 L2=L2(L1)0?|(f(t,u(1)f(t,u(2),u(t)|6 L2k u(t)k2.(4.7)d,(2.7),(4.6)(4.7)?ddtk u(t)k26 2(L2+2p2(1+c1)k u(t)k2.(4.8)A Gronwall?,?k u(t)k26 k u(s)k2expnZts2(L2+2p2(1+c1)do=kU(s,r)u uk2expnZts2(L2+2p2(1+c1)do.(4.9)d(2.13)?|r s|v?,(4.9)m?kU(s,r)u uk2U/?.y.e?z 5 2 Banach 4?.4.1?F 3 R?k.?N.e3 F?5(LIMtL)v(1)?(,+)?K h()k LIMth(t)0;(2)elimth(t)3,K LIMth(t)=limth(t).K LIMt2 Banach 4.(z 5,n 3.1!n 4.1 y?n 3.1 n 4.1,?e(.n 4.1?(H1)(H3).Ku?22 Banach 4 LIMs,3 23?x Borel V ttR,?t?|83 AD(t),kLIM1t Zt?U(t,s)(s)?ds=ZAD(t)(u)dt(u),(4.10)2?Yk.,ttR3eekC5ZAD(t)(u)dt(u)=ZAD()?U(t,)u?d(u),t .(4.11)No.5qU?:?m?:X?O)9 Kolmogorov?1569e(2.8)?O)?y35.d,r(2.8)?dudt=F(u,t):=g(t)Au u f(t,u),t R.(4.12)ln 2.1?y F(u,t):2 R 7 2YN?.e0?a?,?a?vddt(u(t)=(0(u),F(u,t),(4.13)u(t)(4.12)?).4.2?T ve?N:3 2?,3 2?k.f8k.,v(1)?u 2,(u)?Frech et?(P 0(u)3,=z u 2,3 2?0(u),?limkvk0|(u+v)(u)(0(u),v)|kvk=0,v 2;(2)N?u 7 0(u)l 2?2?Yk.N?;(3)u(4.12)?z)u(t)k(4.13).T (4.12)?a,T?(4.12)?.u(4.12)?35z 37.4.3m 2?x Borel V ttR(4.12)?O),ev(1)z3 2?Yk.,N?t 7R2(u)dt(u)Y?;(2)z t R,N?u 7(F(u,t),)z 2 t,z 2N?t 7Z2(F(u,t),)dt(u)L1loc(R);(3)?T,kZ2(u)dt(u)Z2(u)d(u)=ZtZ2?0(u),F(u,s)?ds(u)ds,t,R.u(4.12)?O)?35,ke(J.n 4.2?(H1)(H3).Kn 4.1?x Borel V ttR(4.12)?O).yyn 4.1?x Borel V ttRv 4.3.P Cb(2)3 2?Yk.?N.k,z?t R,ylimttZ2(u)dt(u)=Z2(u)dt(u),Cb(2).(4.14),d(4.10)(4.11)Z2(u)dt(u)Z2(u)dt(u)=ZAD(t)(U(t,t)u)(u)dt(u),t t.(4.15)1570n?Vol.43 A5?t t+k kU(t,t)u uk 0,Cb(2)AD(t)2?;8,?(4.15)Llimtt+Z2(u)dt(u)=Z2(u)dt(u),Cb(2).aqylimttZ2(u)dt(u)=Z2(u)dt(u),Cb(2).(4.14)?y.g,z t R,?t?|8u AD(t)2.uz 2,():27 R(u)=(F(u,t),).(4.16)ey()3 2?Y,=y()C(2).d,?u 2,u 2v ku uk6 1.?Ok|(u)(u)|=|(F(u,t)F(u,t),)|6|(A(u u),)|+|(u u,)|+|(f(t,u)f(t,u),)|6(p8(1+c1)+)ku ukkk+kf(t,u)f(t,u)kkk.(4.17)yd(2.16)3 L3=L3(u)0?kf(t,u)f(t,u)k6 L3.d(4.17)L|(u)(u)|6(p8(1+c1)+L3)ku ukkk.?()C(2).d(4.14)(4.16)?2N?u 7(F(u,t),)=(u)t?.,d?yt 7Z2(F(u,t),)dt(u)=Z2(u)dt(u)R?Y,w,u L1loc(R).?,?T,d(4.13)?(u(t)(u()=Zt?0(u(),F(u(),)?d.(4.18)y?s s.d(4.18)?(U(t,s)u)(U(,s)u)=Zt(0(U(,s)u),F(U(,s)u,)d.(4.19)A(4.13),(4.19)9 Fubini n?Z2(u)dt(u)Z2(u)d(u)=ZAD(t)(u)dt(u)ZAD()(u)d(u)=LIMM1 MZMZ2?(U(t,s)u)(U(,s)u)?ds(u)dsNo.5qU?:?m?:X?O)9 Kolmogorov?1571=LIMM1 MZMZ2Zt(0(U(,s)u),F(U(,s)u,)dds(u)ds=LIMM1 MZMZtZ2(0(U(,s)u),F(U(,s)u,)ds(u)dds.(4.20)d(4.11)9L?C5 U(,s)=U(,)U(,s)?Z2(0(U(,s)u),F(U(,s)u,)ds(u)=Z2(0(U(,)U(,s)u),F(U(,)U(,s)u,)ds(u)=Z2(0(U(,)u),F(U(,)u,)d(u).5?m s,dkZAD(t)(u)dt(u)ZAD()(u)d(u)=ZtZ2(0(U(,)u),F(U(,)u,)d(u)d=ZtZ2(0(u),F(u(s),s)ds(u)ds.(4.21)y?.5O)?Kolmogorov?!O)ttR?Kolmogorov-?,?y.?O.?supptLV t?|8.5.1?ttRn 4.2?O).?0,P N(t,2)=N(t)2?CX suppt?I?.K(t)=K(t,2)=lnN(t)O)ttR3 2?Kolmogorov-?.?,z t R k suppt AD(t).d,2?CX t AD(t)?7CX suppt.duz t R,AD(t)2;8,?z 0,25CX AD(t)?7k.?k0 6 K(t)6 K(AD(t)I()i|ui|2?1262?.ydn 3.2 z t R?0,3g,I=I(,t,R(t),?sup(ui)iZ=uB0(t)X|i|I()i|ui|2624.(5.3)?u=(ui)iZ AD(t)B0(t),Pu=v+w=(vi)iZ+(wi)iZ,(5.4)vi=ui,|i|6 I(),0,|i|I(),wi=0,|i|6 I(),ui,|i|I().(5.5)Kkkwk=?X|i|I()i|ui|2?1262,(5.6)kvk2=XiZi|vi|2=X|i|6I()i|vi|2=X|i|6I()i|ui|26 kuk26 R2(t).(5.7)(5.7)L|vi|6R(t)i6R(t)I(),|i|6 I().n 5.1 L?5N=?=(i)|i|6I():i R,|i|6R(t)I()?R2I()+13m R2I()+1?e?R2I()+1N()=?hR(t)I()2p2I()+1i+1?2I()+12?CX,P?%vk=(vki)|i|6I()R2I()+1,k=1,2,N().,P vk=(vki)iZ=vki,|i|6 I(),0,|i|I(),k=1,2,N(),v=(vi)|i|6I(),Kk vk 2,k=1,2,N(),v R2I()+1.yu(5.4),(5.5)?v=(vi)iZ,d?3,vk(1 6 k 6 N(),?kv vkk=k v vkkR2I()+162.No.5qU?:?m?:X?O)9 Kolmogorov?1573d,u?u=(ui)iZ AD(t)B0(t),kku vkk=kv+w vkk6 kv vkk+kwk62+2=.L AD(t)?2 N()%3 vk?CX.d 5.1 (5.1),n 5.1?y.z1 Foias C,Prodi G.Sur les solutions statistiques des equations de Naiver-Stokes.Ann Mat Pur Appl,1976,111:3073302 Foias C,Manley O,Rosa R,Temam R.Navier-Stokes Equations and Turbulence.Cambridge:CambridgeUniversity Press,20013 Vishik M,Fursikov 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