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Header Page of 123 B GIO DC V O TO TRNG I HC S PHM H NI * PHM TH TRANG DNG IU TIM CN NGHIM CA MT S H PHNG TRèNH DNG NAVIER-STOKES LUN N TIN S TON HC H NI - 2015 Footer Page of 123 Header Page of 123 B GIO DC V O TO TRNG I HC S PHM H NI * PHM TH TRANG DNG IU TIM CN NGHIM CA MT S H PHNG TRèNH DNG NAVIER-STOKES Chuyờn ngnh: Phng trỡnh vi phõn v tớch phõn Mó s: 62 46 01 03 LUN N TIN S TON HC NGI HNG DN KHOA HC PGS TS Cung Th Anh H NI - 2015 Footer Page of 123 Header Page of 123 LI CAM OAN Tụi xin cam oan õy l cụng trỡnh nghiờn cu ca tụi Cỏc kt qu vit chung vi cỏc tỏc gi khỏc, u ó c s nht trớ ca cỏc ng tỏc gi a vo lun ỏn NCS Phm Th Trang Footer Page of 123 Header Page of 123 LI CM N Lun ỏn ny c thc hin ti B mụn Gii tớch, Khoa Toỏn - Tin, Trng i hc S phm H Ni, di s hng dn nghiờm khc, tn tỡnh, chu ỏo ca PGS.TS Cung Th Anh Tỏc gi xin by t lũng kớnh trng v bit n sõu sc n Thy, ngi ó dn dt tỏc gi vo mt hng nghiờn cu khú khn, vt v nhng thc s thỳ v v cú ý ngha Tỏc gi vụ cựng bit n PGS.TS Trn ỡnh K v cỏc thy cụ B mụn Gii tớch ó c v ng viờn v truyn cho tỏc gi nhiu kinh nghim quý bỏu nghiờn cu khoa hc Tỏc gi trõn trng gi li cm n n Ban Giỏm hiu, Phũng Sau i hc, Ban Ch nhim Khoa Toỏn - Tin, Trng i hc S phm H Ni; Ban Giỏm hiu, Phũng o to, Phũng T chc, Khoa T nhiờn, Trng Cao ng Hi Dng, c bit l cỏc thy cụ giỏo v cỏc anh ch nghiờn cu sinh Seminar ca B mụn Gii tớch, Khoa Toỏn - Tin, Trng i hc S phm H Ni ó luụn giỳp , to iu kin thun li v ng viờn tỏc gi sut quỏ trỡnh hc v nghiờn cu Li cm n sau cựng, xin dnh cho gia ỡnh ca tỏc gi, nhng ngi ó dnh cho tỏc gi tỡnh yờu thng trn vn, tng ngy chia s, ng viờn tỏc gi vt qua mi khú khn hon thnh lun ỏn Tỏc gi thnh kớnh dõng tng mún qu tinh thn ny lờn cỏc bc sinh thnh, nhng ngi tng ngy ún i v hy vng tng bc trng thnh ca tỏc gi Footer Page of 123 Header Page of 123 Mc lc Trang ph bỡa Li cam oan Li cm n Mc lc Mt s kớ hiu dựng lun ỏn M U L DO CHN TI TNG QUAN VN NGHIấN CU MC CH, I TNG V PHM VI NGHIấN CU CA LUN N 14 PHNG PHP NGHIấN CU 15 KT QU CA LUN N 16 CU TRC CA LUN N 17 Chng MT S KIN THC CHUN B 18 1.1 CC KHễNG GIAN HM, TON T V BT NG THC LIấN QUAN N S HNG PHI TUYN 18 1.1.1 Cỏc khụng gian hm 18 1.1.2 Cỏc toỏn t 20 1.1.3 Cỏc bt ng thc liờn quan n s hng phi tuyn 21 TP HT LI 22 1.3 MT S KT QU THNG DNG 26 1.2 Footer Page of 123 Header Page of 123 1.3.1 Mt s bt ng thc thng dựng 26 1.3.2 Mt s b v nh lớ quan trng 29 Chng H PHNG TRèNH NAVIER-STOKES-VOIGT 31 2.1 T BI TON 31 2.2 S TN TI V DUY NHT CA NGHIM YU 33 2.3 S TN TI TP HT LI 40 2.4 NH GI S CHIU FRACTAL CA TP HT LI 47 2.5 MI QUAN H GIA TP HT LI VI TP HT U V TP HT TON CC 56 2.5.1 Mi quan h gia hỳt lựi v hỳt ton cc 56 2.5.2 Mi quan h gia hỳt lựi v hỳt u 57 2.6 TNH TRN CA TP HT LI 59 2.6.1 Tớnh b chn ca hỳt lựi (H ())2 60 2.6.2 Tớnh compact ca hỳt lựi (H ())2 64 2.7 TNH NA LIấN TC TRấN CA TP HT LI SINH BI H PHNG TRèNH NAVIER-STOKES-VOIGT HAI CHIU 68 2.7.1 Tp hỳt lựi ca h phng trỡnh Navier-Stokes hai chiu 69 2.7.2 Tớnh na liờn tc trờn ca hỳt lựi sinh bi h NavierStokes-Voigt hai chiu 80 Chng H PHNG TRèNH KELVIN-VOIGT- BRINKMAN-FORCHHEIMER 90 3.1 T BI TON 90 3.2 S TN TI V DUY NHT CA NGHIM YU 92 3.3 S TN TI TP D -HT LI 102 3.4 S TN TI V TNH N NH CA NGHIM DNG 113 KT LUN 118 CC KT QU T C 118 KIN NGH MT S VN NGHIấN CU TIP THEO 118 Footer Page of 123 Header Page of 123 DANH MC CC CễNG TRèNH CễNG B C S DNG TRONG LUN N 119 TI LIU THAM KHO 120 Footer Page of 123 Header Page of 123 MT S K HIU THNG DNG TRONG LUN N H, V cỏc khụng gian hm dựng nghiờn cu h Navier-Stokes, Navier-Stokes-Voigt, Kelvin-Voigt-Brinkman-Forchheimer (xin xem chi tit tr 19) V khụng gian i ngu ca khụng gian V (ã, ã), | ã | tớch vụ hng v chun khụng gian H ((ã, ã)), ã ã ã, ã | ã |p tớch vụ hng v chun khụng gian V chun khụng gian V i ngu gia V v V chun khụng gian Lp (), vi p Id ỏnh x ng nht A, As , B cỏc toỏn t dựng nghiờn cu h Navier-Stokes, Navier-Stokes-Voigt, Kelvin-Voigt-Brinkman-Forchheimer (xin xem chi tit tr 20, 21) D(As ) xỏc nh ca toỏn t As hi t yu Y X P(X) bao úng ca Y X h cỏc b chn ca X dF (K) s chiu fractal ca compact K dist(A, B) na khong cỏch Hausdorff gia hai A, B Footer Page of 123 Header Page of 123 M U L DO CHN TI Phng trỡnh o hm riờng bt u c nghiờn cu vo gia th k XVIII v phỏt trin mnh m t gia th k XIX cho n Nú c coi nh chic cu ni gia toỏn hc v ng dng Rt nhiu phng trỡnh o hm riờng l mụ hỡnh toỏn ca cỏc bi toỏn thc t, c bit l cỏc phng trỡnh v h phng trỡnh c hc cht lng Lp phng trỡnh ny xut hin mụ t chuyn ng ca cỏc cht lng v khớ nh nc, khụng khớ, du m, di nhng iu kin tng i tng quỏt Chỳng cng xut hin nghiờn cu nhiu hin tng quan trng khoa hc k thut nh khoa hc hng khụng, khớ tng hc, cụng nghip du m, vt lớ plasma, Mt nhng lp h phng trỡnh c bn, quan trng c hc cht lng l h Navier-Stokes, mụ t dũng chy ca cht lng thun nht, nht, khụng nộn c, c xõy dng t cỏc nh lut bo ton lng, ng lng v cú dng u u + (u ã )u + p = g(x, t), t ã u = 0, ú u = u(x, t), p = p(x, t) tng ng l hm vect tc v hm ỏp sut cn tỡm, = const > l h s nht v g l hm ngoi lc H phng trỡnh Navier-Stokes a ln u tiờn nm 1822, v c bt u nghiờn cu mnh t na u th k XX vi cỏc cụng trỡnh nn múng ca Leray (1934) v Hopf (1951) Sau gn mt th k phỏt trin, lớ thuyt h phng trỡnh Navier-Stokes ó t c nhiu kt qu sõu sc (xem, chng Footer Page of 123 Header Page 10 of 123 hn, cỏc cun chuyờn kho [14, 47, 48] v cỏc bi tng quan [4, 50]) Tuy nhiờn, cũn rt nhiu cõu hi m cha c gii quyt, ú ni bt l tớnh nht ca nghim yu v s tn ti ton cc ca nghim mnh ca h Navier-Stokes ba chiu Nhng n lc gii quyt bi toỏn ny ó lm phỏt sinh nhiu hng nghiờn cu mi thỳ v Mt s ú l nghiờn cu cỏc bin dng ca h phng trỡnh Navier-Stokes Nhng h nh vy xut hin mụ t chuyn ng ca cỏc cht lu cỏc iu kin vt lớ nht nh, chng hn h Navier-Stokes-Voigt (trong mt s ti liu vit l Voight) xut hin nghiờn cu chuyn ng ca cht lng nht n hi [38], h Navier-Stokes vi s hng tt dn [6], h Brinkman-Forchheimer i lu xut hin nghiờn cu cỏc dũng cht lu cỏc tng xp bóo hũa [33], h g-Navier-Stokes hai chiu xut hin nghiờn cu h Navier-Stokes mng [43], cỏc -mụ hỡnh c hc cht lng [16, 23, 25], h cht lu loi hai xut hin nghiờn cu cht lng khụng Newton [39], h mụ t chuyn ng ca cht lu vi ỏp sut ph thuc nht [5], õy l mt hng nghiờn cu mi v rt thi s, thu hỳt c s quan tõm ca nhiu nh toỏn hc trờn th gii nhng nm gn õy, ý ngha v tm quan trng ca chỳng, cng nh nhng khú khn thỏch thc v mt toỏn hc t nghiờn cu Tuy nhiờn theo hiu bit ca chỳng tụi, nhng kt qu t c v s tn ti v dỏng iu tim cn nghim ca cỏc h trờn ch yu mi dng li trng hp ngoi lc khụng ph thuc thi gian (trng hp ụtụnụm) v xột phng trỡnh l b chn (xin xem thờm phn Tng quan nghiờn cu di õy) Vic phỏt trin nhng kt qu ny cho trng hp khụng ụtụnụm v khụng b chn l nhng lớ thỳ, cú nhiu ý ngha thc tin, nhng khú vỡ ũi hi nhng cỏch tip cn v cụng c k thut mi Chỳng tụi s chn nghiờn cu ny i vi mt s h phng trỡnh dng Navier-Stokes, xut hin c hc cht lng, lm ti nghiờn cu cho lun ỏn tin s ca mỡnh Footer Page 10 of 123 111 Header Page 113 of 123 Kt hp vi gi thit e(st) g(s) L2 (t k, t; V ) ta thu c t lim sup n tk e(st) g(s), U (s, t k)U (t k, n )u0n ds t = tk t Li cú, e(st) g(s), U (s, t k)wk ds e(st) [v(s)]2 ds cng xỏc nh mt chun L2 (tk, t; V ), tk tng ng vi chun thng dựng, suy t tk e(st) [U (s, t k)wk ]21 ds t lim inf n tk e(st) [U (s, t k)U (t k, n )u0n ]21 ds Tip theo, cng B 3.1, ta cú U (s, t k)wk yu Lr+1 (t k, t; Lr+1 ()), U (s, t k)U (t k, n )u0n v U (s, tk)U (tk, n )u0n U (s, tk)wk mnh L2 (tk, t; (H01 )loc ()), nờn t tk e(st) f (U (s, t k)wk ) ã U (s, t k)wk dxds t lim inf n tk e(st) f (x, U (s, t k)U (t k, n )u0n ) ã U (s, t k)U (t k, n )u0n dxds T cỏc ỏnh giỏ trờn, ta d dng cú c lim sup[U (t, n )u0n ]22 n ek + R2 (t k) t e(st) +2 tk g(s), U (s, t k)wk [U (s, t k)wk ]21 Footer Page 113 of 123 f (x, U (s, t k)wk ) ã U (s, t k)wk dx ds 112 Header Page 114 of 123 Mt khỏc, ta cú [w0 ]22 = [U (t, t k)wk ]22 k =e [wk ]22 t 2 tk t tk t +2 tk e(st) g(s), U (s, t k)wk ds e(st) [U (s, t k)wk ]21 ds e(st) f (x, U (s, t k)wk ) ã U (s, t k)wk dxds Vỡ vy, lim sup[U (t, t n )u0n ]22 ek n ek + R2 (t k) + [w0 ]22 ek [wk ]22 1 + R2 (t k) + [w0 ]22 M ta li cú k e R2 (t 2et k) = k +, suy tk es g(s) ds lim sup[U (t, n )u0n ]22 [w0 ]22 n nh lớ ó c chng minh Nhn xột 3.1 Nh mt h qu trc tip ca nh lớ trờn, ta s ch c s tn ti hỳt ton cc ca quỏ trỡnh sinh bi bi toỏn ngoi lc g V khụng ph thuc vo bin thi gian t Khi ú, ta nh ngha mt na nhúm liờn tc S(t) : V V cho bi S(t)u0 = u(t), ú u(t) l nghim yu nht ca bi toỏn (3.1) vi iu kin ban u u0 D thy rng S(t)u0 = U (t, 0)u0 = U (t + , )u0 , vi R Vỡ vy, t ỏnh giỏ (3.17) ta cú c hỡnh cu B0 = u V : u Footer Page 114 of 123 2 L1 + g 2 113 Header Page 115 of 123 l mt hp th b chn ca S(t), ngha l vi b chn B bt kỡ, tn ti T (B) cho S(t)B B0 vi mi t T (B) Mt khỏc, vi tn + v un B bt kỡ, dóy S(tn )un = U (tn , 0)un = U (0, tn )un l compact tng i V (do phn ii) chng minh nh lớ 3.2) Vỡ vy, S(t) l compact tim cn V Do ú, theo nh lớ ó bit v s tn ti nht hỳt ton cc (xem, chng hn, [49, nh lớ 1.1]), na nhúm S(t) cú mt hỳt ton cc compact liờn thụng A V 3.4 S TN TI V TNH N NH CA NGHIM DNG Gi s ngoi lc g V khụng ph thuc vo bin thi gian t Ta cú nh ngha nghim dng tng ng ca bi toỏn (3.1) nh sau: nh ngha 3.2 Mt hm u V Lr+1 () c gi l nghim dng ca bi toỏn (3.1) nu nú tha ((u, v)) + b(u, u, v) + vi mi v V Lr+1 () f (x, u) ã v dx = g, v S tn ti nht ca nghim dng c phỏt biu nh lớ sau nh lớ 3.3 Di cỏc gi thit v kớ hiu nh trờn, bi toỏn (3.1) cú ớt nht mt nghim dng u tha u + 2à u r+1 Lr+1 g + L1 (3.19) Hn na, nu gi thit sau c tha > K1 + g 2 + 1/4 21 L1 1/2 (3.20) thỡ nghim dng ca bi toỏn (3.1) l nht Chng minh (i) S tn ti u tiờn, ta thy vic chng minh ỏnh giỏ (3.19) cho nghim dng u (nu tn ti) l d dng Tht vy, nu nghim dng u tn ti thỡ phi tha ng thc ((u , u )) + b(u , u , u ) + Footer Page 115 of 123 f (x, u ) ã u dx = g, u 114 Header Page 116 of 123 Do b(u , u , u ) = v f (x, v) ã v à|v|r+1 (x), ta cú u +à |u |r+1 dx L1 + g L1 + g u + u , v t ú ta c ỏnh giỏ (3.19) Bõy gi, ta ch cn chng minh nghim dng ca bi toỏn tn ti Tht vy, V Lr+1 () l tỏch c v V trự mt V Lr+1 () nờn tn ti mt dóy cỏc phn t c lp tuyn tớnh, trc giao {w1 , w2 , } V, y V Lr+1 () Kớ hiu Vm = span{w1 , , wm } Mt nghim dng xp x ca bi toỏn (3.1) l hm cú dng m u m cmi wi = i=1 tha ((um , wi )) + b(um , um , wi ) + f (x, um ) ã wi dx = g, wi (3.21) chng minh s tn ti nghim, ta nh ngha toỏn t Rm : Vm Vm bi ((Rm u, v)) = ((u, v)) + b(u, u, v) + f (x, u) ã v dx g, v vi mi u, v Vm Vi mi u Vm ta cú ((Rm u, u)) = ((u, v)) + b(u, u, u) + u 2+à u u f (x, u) ã u dx g, u r+1 Lr+1 L1 L1 g g u Vỡ vy, nu chn k= g + L1 1/2 , ta c ((Rm u, u)) vi mi u Vm cho u = k v ú, s dng B 1.7, l h qu ca nh lớ im bt ng Brouwer, ta cú: vi mi m tn ti um Vm cho um k v Rm (um ) = Footer Page 116 of 123 115 Header Page 117 of 123 Thay wi (3.21) bi um vi chỳ ý b(um , um , um ) = v f (x, v) ã v à|v|r+1 (x), ta c um + 2à um r+1 Lr+1 g + L1 Do ú, {um } b chn V Lr+1 (), dn n {f (x, um )} cng b chn L(r+1)/r () Vỡ vy, t dóy {um } ta cú th trớch c mt dóy con, kớ hiu l {um } cho um hi t yu V Lr+1 () n u v ta cú th chng minh u chớnh l mt nghim dng yu ca bi toỏn (3.1) (ii) Tớnh nht nghim Gi s u1 v u2 l hai nghim dng ca bi toỏn (3.1) t u = u1 u2 ta cú: vi mi v V Lr+1 () Au , v + b(u1 , u1 , v) b(u2 , u2 , v) + (f (x, u1 ) f (x, u2 )) ã vdx = Chn v = u ta c u = b(u1 , u1 , u ) + b(u2 , u2 , u ) b(u , u , u2 ) 1/4 21 K1 + u (f (x, u1 ) f (x, u2 )) ã u dx fu (x, )u ã u dx u2 + K|u |2 g 2 + 1/4 21 L1 1/2 u (3.22) T (3.20) v (3.22) ta suy tớnh nht ca nghim dng Bõy gi, ta s chng minh nghim dng nht u nhn c trờn l n nh m ton cc nh lớ 3.4 Gi s rng cỏc gi thit nh lớ 3.1 vi g khụng ph thuc vo bin thi gian v iu kin (3.20) c tha Khi ú, nghim u(t) ca bi toỏn (3.1) vi = tha ỏnh giỏ sau u(t) u 1+ 1 et u(0) u , vi l s dng tha (1 + ) + 1/4 g + L1 1/2 + 2K1 < (S tn ti s > nh vy suy t iu kin (3.20)) Footer Page 117 of 123 116 Header Page 118 of 123 Chng minh t w(t) = u(t) u ta cú dw(t) ,v dt dw(t) , v dt + ((w(t), v)) + + b(u(t), u(t), v) b(u , u , v) + (f (x, u(t)) f (x, u ), v) = Chn v = et w(t) (vi l mt s dng bt kỡ, giỏ tr s c chn sau), ta cú d t e |w| + et w et (|w|2 + w ) + et w + et b(u, u, w) dt et b(u , u , w) + et (f (x, u) f (x, u ), w) = Vỡ vy, d t e |w| + et w dt et |w|2 + w et w t e (1 2 w + w + ) + 2 + 2b(w, w, u ) w 1/4 1/4 + w g 2 + f (x, )w ã wdx u + 2K|w|2 1/2 L1 + 2K1 Do (3.20), ta cú 1/4 + g 2 + L1 1/2 + 2K1 < Vỡ vy, ta cú th chn > nh cho (1 + ) + v thu c 1/4 g 2 + L1 1/2 + 2K1 < 0, d t (e [w(t)]22 ) dt Vỡ vy, [u(t) u ]22 et [u(0) u ]22 Chỳ ý rng u [u]2 2 ta c iu phi chng minh Footer Page 118 of 123 + u u V, w 117 Header Page 119 of 123 KT LUN CHNG Trong chng ny, chỳng tụi nghiờn cu h Kelvin-Voigt-Brinkman-Forchheimer ba chiu khụng nht thit b chn nhng tha bt ng thc Poincarộ Cỏc kt qu t c bao gm: 1) Chng minh c s tn ti v nht ca nghim yu (nh lớ 3.1) 2) Chng minh c s tn ti hỳt lựi ca quỏ trỡnh sinh bi nghim yu ca bi toỏn ngoi lc g cú th ph thuc vo bin thi gian (nh lớ 3.2) 3) Chng minh c s tn ti nht v tớnh n nh ca nghim dng yu ca bi toỏn ngoi lc g khụng ph thuc vo bin thi gian v " nh" (nh lớ 3.3) Footer Page 119 of 123 118 Header Page 120 of 123 KT LUN CC KT QU T C Trong lun ỏn ny, chỳng tụi nghiờn cu h Navier-Stokes-Voigt v h KelvinVoigt-Brinkman-Forchheimer ba chiu trng hp ngoi lc ph thuc thi gian (trng hp khụng ụtụnụm) v xột h khụng nht thit b chn m ch cn tha bt ng thc Poincarộ Cỏc kt qu t c bao gm: i vi h Navier-Stokes-Voigt: Chng minh c s tn ti nht nghim yu, s tn ti v ỏnh giỏ s chiu fractal ca hỳt lựi, tớnh trn ca hỳt lựi, tớnh na liờn tc trờn ca hỳt lựi trng hp hai chiu i vi h Kelvin-Voigt-Brinkman-Forchheimer: Chng minh c s tn ti nht nghim yu, s tn ti ca hỳt lựi, s tn ti nht v tớnh n nh ca nghim dng KIN NGH MT S VN NGHIấN CU TIP THEO Tip tc nghiờn cu cỏc tớnh cht ca hỳt lựi ca quỏ trỡnh sinh bi h Kelvin-Voigt-Brinkman-Forchheimer: tớnh trn, ỏnh giỏ s chiu ca hỳt, s ph thuc liờn tc ca hỳt theo cỏc tham s Nghiờn cu dỏng iu tim cn nghim ca h Navier-Stokes-Voigt v h Kelvin-Voigt-Brinkman-Forchheimer trng hp ngoi lc ph thuc vo tr hoc cha nhiu ngu nhiờn Footer Page 120 of 123 119 Header Page 121 of 123 DANH MC CễNG TRèNH KHOA HC CA TC GI 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Navier-Stokes equations) [42] Trong [6], Cai... trng hp hai chiu, tc l so sỏnh hỳt ca h Navier-Stokes- Voigt vi hỳt ca h Navier-Stokes gii hn tng ng õy ch xột c trng hp hai chiu vỡ tớnh t ỳng ton cc ca h Navier-Stokes ba chiu l m rt ln Khú khn... trỡnh Navier-Stokes- Voigt mụ t chuyn ng ca cỏc cht lng loi Kelvin-Voigt, khụng nộn c, nht, n hi (vi tham s c trng cho tớnh n hi l ) (xem [38]) Chỳ ý rng = h Navier-Stokes- Voigt tr thnh h Navier-Stokes