5 5 9 ( ) JournalofXiamenUniversity(NaturalScience) Vol.5 No.5 Sep.!"#$% ( 365) &':!"#$%&' " %()*./ 3456789:; 犔 < = >?@AB. :C)D E E ; ; ;/ (): O75 *.: A */): 438 479 ()5 87 6 ' FGH I)JK " %()*. / [ ] 狋 div( 狌 )= () ( 狌 ) 狋 div( 狌 狌 ) ( 狆 犕 /)μδ 狌 (λμ ) div 狌 =div( 犕 犕 ) 犕狋 div( 狌 犕 )div( 犕 狌 ) 狏 Δ 犕 = div 犕 = (3) Δ = 狊 (4) LM 狌 犕 N O! / N P Q. 狊 > RST!UV. 狆 = 狆 ( )= 犪 U WXYV 犪 > = >. μ λ Z[ μ >\λμ / LM ).] ^ *_`?ab3c H FG %/ ;H FG/ 犕 ( 犕 =) V`?ab % T RS ` % Navier Stokes Poisson. FG ^()~(4) )"$ # ( 狋 狓 ) 犚 Ω =( ) ΩΩ 犚 U"$ % ^Z[H : ( 狌. 犕 ) 狋 ==( 狇 犕 ) Ω #. 狌 = 犕 = 狀 = 犚 Ω # (5) LM 狀 Ω $% ;. ()~ : 4 7 Email:wei.wei.84@63.com (4)%" :; %. 34 ( 狌 犕 )U ^()~ (4)%34" :; H Z[: ( 狌 犕 ) "_ : 犔 loc( 犚 犔 (Ω)) 犆 烄 ( 犚 犔 weak (Ω)) 狌 犔 loc( 犚 ( 犎 (Ω)) ) 犕 犔 loc( 犚 ( 犎 (Ω)) )div 犕 = 犕 犆 烅 ( 犚 犠 weak(ω)) 犔 loc( 犚 ( 犎 (Ω)) 犆 ( 犚 犠 weak (Ω)) 狌 犆 ( 犚 ( 犔 weak(ω)) ) 狌 犔 loc( 犚 犔 烆 (Ω )) LM 犚 =[ ). 犈 ( 狋 ) H % :; 犈 ( 狋 )= [ ( 狋 ) 狌 ( 狋 ) 犕 ( 狋 ) ( 犪狋 ) ]d 狓 ( 狋 )d 狓. LZ[:; K d 犈 ( 狋 ) d 狋 μ 狌 ( 狋 ) (λμ ) div 狌 ( 狋 ) d 狓 狏 犕 ( 狋 ) d 狓 (6) 犇 ( 犚 )#. (7)? % 狋 犚 ( )Z[ % Poisson 烄 Δ = 狊 Ω # d 狓 = 烅 烆 狀 Ω=. (8) () 犇 ( 犚 犚 )? φ 犇 ( 犚 犚 ) ( 犚 φ 狋 狌 φ ) d 狓 d 狋 =. (9)
88 ( ) ()N (3) ( 犇 ( 犚 Ω))? φ ( 犇 ( 犚 Ω)) N 犚 犚 ( 狌 φ 狋 狌 狌 : φ 狆 divφ ) d 狓 d 狋 = [ μ 狌 : φ(λμ ) div 狌 divφ ] d 狓 d 狋 犕 犕 : φd 狓 d 狋 犚 犚 犚 犚 φ d 狋 犕 divφd 狓 d 狋 () [ 犕 φ 狋 ( 狌 犕 犕 狌 ): φ ] d 狓 d 狋 = 狏 犕 : φd 狓 d 狋 犚 LM φ 犔 loc( 犚 )H φ = φd 狓. () 5 ( 狌 犕 ) ()~(5)%3 4"$:; H( 狌 犕 )Z[#b U K(7) 狋犈 ( 狋 ) [ μ 狌 ( 狋 ) ( μ λ) div 狌 ( 狋 ) d 狓 d 狋 狋狏犕 ( 狋 ) ] d 狓 d 狋 犈 狋 犚 LM 狇 犈 = [ { ( >} 犕 ) d 狓 ] d 狓 < () ( 犕 )Z[ (8) div 犕 = ( 犇 (Ω)) #. (3) XY `{ >}. " % RS. = H 3 /( ) =3Li ons [3] ` DiPerna Lions % [4] " %. * Navier Stokes %. Feire isl [5] Lions% > /.? /&' " ( H ) [6 7].Hu [8] Feire isl [59] % 3 / #. Tan [] Hu % " % /! U ( 狇 犕 )Z[K ()\ (8)N(3) " _ : (i)?> /( ) ()~(4) 34 " :; Z[ (5). (i)? <<4/3#) ()~(4) 3 4" :; Z[ (5)LMK()M% $ φ %&' φ : = divφd 狓 狓犻 φd 狓 狊 φd 狓 (4) 犻 =? φ ( 犇 ( 犚 Ω)) 犔 loc ( 犐 ( 犠 (Ω)) ). (i)h ( ~ ~ )U %34 : 烄 狆 ( ~ )= ~ ~ Ω # 烅 Δ ~ = ~ 狊 Ω # 烆 ~ 狀 = Ω # Z[ ~ d 狓 = ( ~ ~ )=( 狊 ). (5) (iv)z[ ()~(5)% () ( 狊 ): lim ( 狋 ) 狊 犔 (Ω)= 狋 lim ( 狋 ) 犠 (Ω)= 狋 lim esssup ( 狋 τ> 狋 (τ) 狌 ( 狋 ) 犕 ( 狋 ) )d 狓 = LM>. ] Jiang [] 3 9 槡 33 # b4 *M%K(4) ()? 狋 犚. "#$ " %()*./ 3456789:;../3c ( ~6) ()~(4)% 犔 < _= JK ()@AB. % *_ U []M.?* Navier Stokes % %. : " %()*./ #3/ \ L 45%L ; 6%. 7`* Navier Stokes % 89:; 6 犕 / div( 犕 犕 ). ] 3 \ PQ ] 3c %%. % 3 []M. Fan [] % * _ % ` " % ( )*./ #. 36 Ω U 犚 "$% Lipschi
5 <<: " */ % 89 tz ( 狌 犕 )U ()~(4)%34" :;.] H =4V 犈 N >Z[ 珋 limsup 犈 ( 狋 ) 犈 (6) 狋 N a.e.( 狋 狓 ) 犚 Ω. (7)? 狋 犚 [ ] ( 狌 犕 ( 狋 ) )( 狊 ) d 狓 犮 ( 犈 Ω)exp 犮 ( Ω) 狋 LM=4 V 犮 ( 犈 Ω)N 犮 ( Ω) ( 狌 犕 ) I. 5 FG/ 犕 = >! U? H! " # $ % % Navier Stokes Poisson% [3] % 犔 < =?@AB. c @ ()~(4)!"#$%. 53 AB [4]M% C7. [5]M %4.\ 犕 犆 ( 犚 犔 weak (Ω)) :; 犈 ( 狋 )? 犚 U DE %. 54!( 犕 )Z[K()\ (8)N(3) ()~(5) U" :;!"$:;. [] % &' * F % LM"$:; % * G F [4]% 7.7H I [6]M%..J ] K 5 3 *_ ()~(4)%" :; Z [ (5)N(6) lim sup 犈 ( 狋 ) 犈 L 狋 M 犈 K()L. 76 4LM ~5% * [7]MM. 76 Ω U 犚 M " $ % Lipschitz 犔 ( 犚 Ω) 狌 犔 loc( 犚 ( 犎 (Ω)) )Z[ 狋 div( 狌 )= 犇 ( 犚 Ω)# (8) 狋 div( 狌 )= 犇 ( 犚 犚 )# (9) LM( 狌 )N Ω O. 76 H K (9) 犆 ( 犚 犔 weak (Ω))( 狌 ) ( 犔 loc( 犚 犔 (Ω))) 狌 ( 犔 loc( 犚 犔 (Ω))). C;? 狋 犚 ( 狋 )d 狓 = 狊 Ω. () 763 狆 < 狇 犳 犔狆 ( 犚 犔狇 ( 犚 )).H { 犛犈 [ 犳 ]} 犈 > U 犳 % 3 3 ) 犛犈 [ 犳 ] 犆 ( 犚 犔狇 ( 犚 )) H 犳 犔狆 ( 犚 犔狇 ( 犚 )) 犛犈 [ 犳 ] 犔狆 ( 犚 犔狇 ( 犚 ))W()? 犳. 764 Ω 犚 U"$% Lipschitz V 狆 狉 ( ). 34"$; β (V Bogovski ) { } 狆 β=( β β ): 犺 犔 (Ω) 犺 d 狓 = 狆 ( 犠 (Ω)) Z [ β ( 犺 狆 ) 犠 ( Ω) 犮 ( 狆 Ω) 犺 犔狆 (Ω) 狏 =β ( 犺 )Udiv 狏 = 犺 a.e. Ω # 狏 Ω= %. H 犺 犔狆 (Ω)*P H JK: 犺 =div 犵 狉?4 犵 ( 犔 (Ω)) 犵 狀 Ω= β ( 犺 ) 犔 犮 ( 狆 狉狉 Ω) 犵 犔 (Ω). 765 狉 > 狉 > U V 狉犳 ( 犺 狉 狉 )= 狉 d 犺 狉 [ 狉 ]. 狉 犺 QRS? 狉 N 狉 %V 犽 犽 T 狉 (Ω) 犽 ( 狉 狉 ) 犳 ( 狉 ) 犽 ( 狉 狉 ) 狉 [ 狉 ]. 766 < 狆 < Ω U 3 4 " $.H Ω 犆 狆 犫 ( 犔狆 (Ω)) 犠 (Ω) Z[ ηd 狓 = 犫 η η 犆 ( 犚 ) 犔狆 (Ω) 犮 ( 狆 Ω) 犫 犔狆 (Ω). 8M% JH 犱 犱 犚. 89 UV [4] 4.7. 3 36!89 W L %. X :; K()P 3Y%JK. ] 3 4 % ( 狋 ) 犇 ( 犚 )Z[ ( 狋 ) :; K ()? [ 犚 μ 狌 ( μ λ) div 狌 狏 犕 ]d 狓 d 狋 狋犚 犪 ] 犕 φ ( 狋 ) ) [ ( 狌 d 狓 d 狋. () 狊 U34 V AB ( 狋 )d 狓 U 狋 I %" 犪 φ 狋 s 犚 } s d 狓 d 狋 =. ()
8 ( ) # =4K " φ [ 犚 μ 狌 ( μ λ) div 狌 狏 犕 ]d 狓 d 狋 狋犚 犕 φ ( 狋 ) )d ] 狓 d 狋 [ ( 狌 犺 狋犪 狊 犚 d 犺 d 狓 d 狋. (3) 狊犺 ` %3) 犛 ε (9)M Z ; 狋.? φ ( 狋 ) 犇 ( 犚 ) *[\34W ] 犐 Z[supp ( 狋 ) 犐 犚.? φ 犇 ( 犐 犚 )\ ε (inf 犐 )" 狋犛 ε [ ] div 犛 ε [ 狌 ]= 犇 ( 犐 犚 )#.(4) (6)" 犛 ε [ 狌 ] 犆 ( 犐 ( 犔狇 ( 犚 )) ) 犛 ε [ ] 狋犛 ε [ ] 犆 (I L ( 犚 ))div 犛 ε [ 狌 ] 犆 ( 犐 犔 ( 犚 ))\ 犛 ε [ 狌 ] 犆 ( 犐 犈 (Ω)) LM 3 狇 = ; = 狇 >. 犈 (Ω)={ 犵 ( 犔 (Ω)) div 犵 犔 (Ω) 犵 狀 Ω = % Z[. φ ( 狋 狓 )= ( 狋 ) β ε ( 狋 狓 ) βε=β 犛 [ ε ] 犛 [ ε ] d狓 Ω LM 犇 ( 犚 ) β U Bogoviki =( )../ %^ _ φ ( 狋 狓 )*_ K()%`. K(4) ab * 狋 φ= βε ( 狋 狓 )βε ( 狋 狓 )= ( βε 狋 狓 )β 狋犛 [ ε ] 狋犛 [ ε ] d狓 = Ω βε ( 狋 狓 )β ( div 犛 ε [ 狌 ]) (5) 犚 狋 狌 βεd 狓 d 狋 犚 狌 β(div 犛 [ ε 狌 ]d 狓 d 狋 犚 狌 狌 βεd 狓 d 狋 μ 狌 βεd 狓 d 狋 犚 ( 狆 ( 犚 ) 狆 ( 狊 )) 犛 ε[ ] 犛 [ ε ] d狓 d 狓 d 狋 Ω (λμ ) 犚 div 狌 d 狓 d 狋 犕 犚 犛 [ ε ] 犛 [ ε ] d狓 Ω ( 犛 [ ε ] 犛 [ ε ] d狓 ) d 狓 d 狋 Ω 犕 犕 βεd 狓 d 狋 犚 犚 βε βεd 狓 d 狋 = (6) K(6)M ε c 3~4\( 狌 犕 )% " 犚 狋 狌 β( 狊 )d 狓 d 狋 犚 狌 β(div( 狌 ))d 狓 d 狋 犚 狌 狌 β ( 狊 )d 狓 d 狋 μ 犚 狌 β ( 狊 )d 狓 d 狋 ( 狆 ( 犚 ) 狆 ( 狊 ))( 狊 )d 狓 d 狋 (λμ ) 犚 ( 狊 )div 狌 d 狓 d 狋 犕 ( 犚 狊 )d 狓 d 狋 犚 β ( 狊 )d 狓 d 狋 犕 犕犚 β ( 狊 )d 狓 d 狋 =. (7) < ` _K(7) F :; K(3)c * 狋犚 犪 狊 [ ( 狌 犕 ( 狋 ) ) 犺 狊 d 犺犺 狌 β( ) 狊 ] d 狓 d 狋 ( 犚 μ 狌 (λμ ) div 狌 ν 犕 )d 狓 d 狋 犚 狌 β(div( 狌 ))d 狓 d 狋 犚 狌 狌 β ( 狊 )d 狓 d 狋 μ 犚 狌 β ( 狊 )d 狓 d 狋 犚 ( 狆 ( ) 狆 ( 狊 ))( 狊 )d 狓 d 狋 (λμ ) 犚 ( 狊 )div 狌 d 狓 d 狋 犕 ( 犚 狊 )d 狓 d 狋 犚 β ( 狊 )d 狓 d 狋 犕 犕犚 β ( 狊 )d 狓 d 狋. (8)
5 <<: " */ % 8 犞 = 犪 狊 [ ( 狌 犕 ( 狋 ) ) 犺 狊 d 犺犺 狌 β( ) 狊 ] d 狓 (9) 犠 δ= ( μ 狌 (λμ ) div 狌 ν 犕 )d 狓 狌 狌 : β ( 狊 )d 狓 μ 狌 : β ( 狊 )d 狓 ( 狆 ( ) 狆 ( 狊 ))( 狊 )d 狓 (λμ ) ( 狊 )div 狌 d 狓 犕 ( 犚 狊 )d 狓 d 狋 狌 β(div(( 狌 ))d 狓 d 狋 犕 犕犚 β ( 狊 )d 狓 d 狋 犚 β ( 狊 )d 狓 d 狋. (3) ` β % C\ K* 狌 β( 狊 )d 狓 d 狋 狌 d 狓 犮 (Ω) ( 狊 ) d 狓.(3) ( )UK(8)% `* Δ =divβ ( 狊 ) ] c 4N6 * 犔 ( Ω) 犮 (Ω) 狊 犔 ( Ω). (3) K(3)(3)\5* 犮 ( Ω) [ 狌 犕 ( 狊 ) ( 狋 ) ] d 狓 犞 犮 3( Ω) [ 狌 犕 ( 狊 ) ]d 狓 (33) LMX O. 3 H lder KPoincaré K Cauchy K\4" 狌 βdiv( 狌 ))d 狓 犮 4(Ω) 狌 犔 ( Ω) (34) 狌 狌 β ( 狊 )d 狓 犮 5(Ω) 狌 犔 ( Ω) (35) μ 狌 β ( 狊 )d 狓 μ 狌 d 狓 4 μ 犮 6(Ω) 狊 d 狓 (36) (λμ ) ( 狊 )div 狌 d 狓 [ ] (λμ ). div 狌 d 狓 狊 4 d狓 (37) ` H lder K Poincaré K \ 4* _ 犕 %6 H : 犕 ( 狊 )d 狓 犕 犕 β ( 狊 )d 狓 β ( 狊 ) 犔 ( 犕 ) 犕 d 狓 (Ω) β ( 狊 ) 犠 狇 (Ω) 犕 犔 ( Ω) 犕 犔 ( Ω) 犮狇 7(Ω) 狊 犔 (Ω) 犕 犔 ( Ω) 犮 8(Ω) 犕 犔 ( Ω) 狇 >3 (38) ] ( 狊 )( 狊 )d 狓 ( 狊 ) d 狓. (39) _#64 ` Poincaré K\6" 犠 犮 9( Ω) [ 狌 犕 ( 狊 ) ]d 狓 犮 ( Ω) [ 狌 犕 ( 狊 ) ]d 狓 (4) LMU O %. J] K(33)N(4) O V 犮 ( Ω)Z[ 犮 ( Ω) 犞 犠 ( 4) c K(8) (4)* 犚 狋 ( 狋 ) 犞 ( 狋 )d 狋 犮 ( Ω) 犚 狋 ( 狋 ) 犞 ( 狋 )d 狋 (4) LM 犇 ( )U % Z[. [α β ]U ( )% K (4)M ( 狋 )=ηε ( 狋 τ)xy η ε " 狋犛 ε [ 犞 ] 犮 ( Ω) 犛 ε [ 犞 ] (43)? τ [α β ] LMε O. AB K(43)? < 狊 < 狋 < 犛 ε [ 犞 ]( 狋 ) 犛 ε [ 犞 ]( 狊 )exp{ 犮 ( Ω)( 狋 狊 )}. 犞 ( 狋 ) 犔 loc( 犚 ) K(44)M ε " 犞 ( 狋 ) 犞 ( 狊 )exp{ 犮 ( Ω)( 狋 狊 )} a.e.< 狊 < 狋 < ]Kc K(6) (44) 犞 ( 狋 ) 犈 exp{ 犮 ( Ω) 狋 }a.e.< 狋 < (45)
8 ( ) K (45)(33)\ε( 狋 )% D E (V 3) K(8). _. :;*: [] LiTQin T.Physicsandpartialdiferentialequations [M].nded.Beijing:HigherEducationPress5. [] MoreauR.Magnetohydrodynamics[M].Berlin:Springer 99. [3] LionsP.Mathematicaltopicsinfluidmechanics:compres siblemodels[m].usa:oxforduniversitypress998. [4] DiPernaRLionsPL.Ordinarydiferentialequationstrans porttheoryandsobolevspaces [J].InventMath98998: 5 547. [5] FeireislENovotn ya.petzeltov ah.ontheexistenceof globalydefined weaksolutionstothe Navier Stokese quations[j].journalof MathematicalFluid Mechanics 3:358 39. [6] JiangSZhangP.Onsphericalysymmetricsolutionsof thecompressibleisentropic Navier Stokesequations[J]. Commun MathPhys5:559 58. [7] JiangSZhangP.Axisymmetricsolutionsofthe3 D Navier Stokes equationsforcompressibleisentropicfluids[j].jmath PuresAppl38:949 973. [8] HuXWangD.Globalexistenceandlarge timebehavior ofsolutionstothethree dimensionalequationsofcm pressiblemagnetohydrodynamicflows[j].arch Rational MechAnal97:3 38. [9] FeireislEPetzeltov a H.Large timedehaviourofsolu tionstothenavier Stokesequationsofcompressibleflow [J].ArchiveforRationalMechanicsand Analysis999 5:77 96. [] TanZWangY.Globalexistenceandlarge timebehavior ofweaksolutionstothecompressiblemagnetohydrody namicequationswithcoulombforce[j].nonlinearanal ysis97:5866 5884. [] JiangFTanZWang H.A noteonglobalexistenceof weaksolutionstothecompressiblemagnetohydrodynam icequationswithcoulombforce[j].j MathAnalAppl 379:36 34. [] FangDZiRZhangT.Decayestimatesforisentropiccom pressiblenavier Stokesequationsinboundeddomain [J]. Journalof MathematicalAnalysisand Applications 386:939 947. [3] KobayashiTSuzukiT.Weaksolutionstothe Navier Stokes Poissonequation[J].Advancesin Mathematical SciencesandApplications88:4 68. [4] NovotnyAStraskrabaI.Introductiontothemathemati caltheoryofcompressibleflow [M].USA:OxfordUni versitypress4. [5] JiangFTanZ.Completeboundedtrajectoriesandatractors forcompressiblebarotropicself gravitatingfluid[j].jmath AnalAppl935:48 47. [6] JiangFTanZ.Onradialysymmetricsolutionsofthe compressibleisentropicself gravitatingfluid[j].nonlin earanalysis:tma7:3463 3483. [7] AbdalahMJiangFTanT.Decayestimatesforisentro pio compressible magnetohy drodynamic equatins in boundeddomain[j].acta mathematicascientic 3B(6):48 4. 犇犲犮犪狔犈狊狋犻犿犪狋犲狊犳狅狉犆狅犿狆狉犲狊狊狊犻犫犾犲犕犪犵狀犲狋狅犺狔犱狉狅犱狔狀犪犿犻犮犈狇狌犪狋犻狅狀狊狑犻狋犺犆狅狌犾狅犿犫犉狅狉犮犲 WANG Wei wei (SchoolofMathenaticalSciencesXiamenUniversityXiamen365China) 犃犫狊狋狉犪犮狋 : Underthehypothesisthat isupperboundedweconstructalyapunovfunctionalforthe multidimensionalisentropic compressiblemagnetohydrodynamicequationswithcoulombforceandshowthattheweaksolutionsdecayexponentialytotheequi libriumstatein 犔 norm. 犓犲狔狑狅狉犱狊 : Navier Stokesequations ;weaksolution;decayestimates;magnetohydrodynamic