Một số tính chất định tính của phương trình navier stokes

PP PP t rt tr qts t rsr Pr r r VIN HN LM KHOA HC V CễNG NGH VIT NAM VIN TON HC O QUANG KHI MT S TNH CHT NH TNH CA NGHIM PHNG TRèNH NAVIER-STOKES Chuyờn ngnh: Phng trỡnh vi phõn v tớch phõn Mó ngnh: 62 46 01 03 LUN N TIN S Ngi hng dn khoa hc: GS TSKH Nguyn Minh Trớ H NI 2017 ts t t sr Prssr r sr s t r tt t s ssr r t t rts rs s t t r rrt s t sts rt t rs ssrt t t t Prssrs r tr r r t srt ssrtt r tr strt ts ssts t t sttt t sttt tts r r rt srt trt P st s s ssrtt r t tt tr ss srt t t t tt t P r s srt r s rtt s t r rt rt s r s t t sttt tts t r t srs Pr r r r r tt t rsts rst ts tss r r s sr rr Túm tt Trong lun ỏn ny, chỳng tụi s dng nhng tin b t c lnh vc gii tớch iu hũa mi lm nm gn õy nghiờn cu phng trỡnh Navier-Stokes Chỳng tụi mun núi n vic s dng bin i Fourier v cỏc tớnh cht ca nú, phự hp hn cho vic nghiờn cu cỏc bi toỏn phi tuyn Chng c dnh cho vic nhc li mt s kt qu ó bit v gii tớch iu hũa Trong Chng 2, chỳng tụi xõy dng v nghiờn cu cỏc khụng gian Sobolev sau (khụng gian Sobolev trờn mt khụng gian Banach bt bin vi phộp dch chuyn): - Khụng gian Sobolev khụng thun nht v khụng gian Sobolev thun nht trờn cỏc khụng gian Lebesgue - Khụng gian Sobolev thun nht trờn cỏc khụng gian Fourier-Lorentz - Khụng gian Sobolev thun nht trờn cỏc khụng gian Lorentz - Khụng gian Sobolev thun nht trờn cỏc khụng gian vi chun Lorentz hn hp Trong cỏc khụng gian ny, chỳng tụi chng minh mt s bt ng thc kiu Young cho tớch chp ca hai hm, mt s bt ng thc kiu Holder cho tớch thụng thng gia hai hm v mt s bt ng thc kiu Sobolev Chỳng tụi ỏp dng nhng bt ng thc ny nghiờn cu bi toỏn Cauchy cho phng trỡnh NavierStokes Chỳng tụi xõy dng nghim mm cho phng trỡnh Navier-Stokes nhng khụng gian ny bng nguyờn lý ỏnh x co Picard v ch rng phng trỡnh Navier-Stokes c t chnh cỏc khụng gian ny theo ngha Hadarmard Chỳng tụi chng minh s tn ti ton cc v nht ca nghim mm giỏ tr ban u nh v s tn ti a phng ca nghim mm i vi giỏ tr ban u tựy ý Nhng kt qu thu c cú mt quan h cht ch gia thi gian tn ti v ln ca d liu ban u: Thi gian ln vi d liu ban u nh hoc d liu ban u ln vi thi gian nh Trong Chng 3, s dng phng phỏp ca Foias-Temam, chỳng tụi nghiờn cu s chiu Hausdorff ca hp cỏc im k d theo thi gian ca nghim yu ca phng trỡnh Navier-Stokes trờn hỡnh xuyn chiu strt ts tss s t rrss t r ss r t st t rs t st t rts qts s t ts rr ss rrts rr trsr rr t st t r ts qts tr s t t t r s rsts r ss tr tr st t ss ss r strt s s ss s ss r t s ss s ss r t rrrt ss s ss r t rt ss s ss r t r rt ss ts ss r s rss s qt t r ts t ts s rss rs qt t r ts rt t ts s rss s qt ts qts t st t r r t rts qts strt sts t t rts qts ts ss t Pr trt r s tt rts qts r s ts ss t ss rr r t q st sts t t t s s t st sts r rtrr t rsts str rt t st t t s r t t s t r r t t s t tr s t t s stt t sr s t sr st t sts t t rts qts t trs tts trt Prrs rsts r r ss ttP st s ss tr s t ss rrt ss rt ss rts qts t t rssr tr rt r t rts qts r t t ssrtt sts s ss r strt s ss r strt s strts sts rt ss r rsts t tt rrt t r rtr ts t t rts qts t t t d d 1 q rt ss Hq (Rd ) H qq (Rd ) r d 4, q d ts t t rts qts t t t d qq (Rd ) r d < q d rt ss H ts t t rts qts t t t d qq (Rd ) r d < q rt ss H ss sts ss t rr ts t t rts qts t t t t ps (Rd ) r d 2, p > d , and d s < d ss H p 2p ss sts t rrrt ss rrt ts t t rts qts t t t t d pp,r rt ss H (Rd ) t < p d r < L ts t t rts qts t t t t d pp,r rt ss H (Rd ) t d p < r < L rt ss ts t t rts qts t t t t d H Ld1 1,r (R ) t r < ss sts rt ss rt ss r ss t tt rrt t r rtr ts t t rts qts t t t t rt ss ss sts r rt ss r rt ss r rt ss Lp Lq,r qss trs ss sts t rts qts sr s t st srts r sts t stt t qts sts Lr H sts Lr W 1,q r ss st t trs ts rt t t ssrtt r t s Rd Hqs H qs S S Bqs,p B qs,p Fqs,p F qs,p M p,q M p,q Lp,q H Ls q,r H Ls p,r H s q,r L ds s s ss s ss r strt rt ss s s ss s s ss s rr ss s rr ss s rrt ss s rrt ss rt ss rt ss rrrt ss rt ss tt rts qts u X [x] {x} Ld àD X u , u (X ,X) v v (u) P et Rj [ã, ã]ã r u X t r s tr rt x rt rt x s sr Rd s sr srs st s t r s X t rt u (u) u X rt t sr t R1 u X v v t sr t r t tr t u r rt rtr r s srt rtr t r rr trsr rs rr trsr sr rt s trsrs trt ss t t ss 2(1+2) 21 >2 r t 1d |u(t)|2 + |u(t)|2+1 dt 2(1+2) C()|u(t)| 21 + C(f, ) C(f, )(|u(t)|2 + 1) (1+2) 21 st y(t) = |u(t)|2 + 1, r t s tt y (t) 2C(N, )y r C = 2C(N, ) y(t) (Ca) a= t (1+2) 21 , t Cay a (0) a1 , 0t< , Cay a (0) 21 T = = Cy s stt trt t a1 r (1+2) 21 qs T C(, N ) 2a = a aCy (0) (|u(0)|2 + 1) 21 t t rt s s q t 2y(0) r y(t) 2y(0), t [0, T ] |u(t)|2 2|u(0)|2 + 1, t [0, T ] s t trt t ss t qt s t T |u(t)|2+1 dt 2T C(N, )(2|u(0)|2 + 2) (1+2) 21 + |u(0)|2 s r r ss tt f L (0, T ; V1 ), u0 V , 12 s rr u L (0, T ; L ) u stss rrs qss t s ts t r t tr ( 12 , 32 ) s u C((t1 , t2 ), H (T3 )) u tr s t st tt st u s H (T ) rr (t1 , t2 ) s tt H rrt tr (t1 , t2 ) s r tr H rrt strt t (t1 , t2 ) r r s r tt (t1 , t2 ) s H tr st u t t lim sup |u(t)| = + tt2 r st ( 21 , 32 ), u0 H, f L (0, T ; V1 ) ss tt u s u s H (T ) rr st (0, T ) s t t s s sr Pr u s ts r [0, T ] t H u(t) s r r t = {t [0, T ], u(t) / V } , = {t [0, T ], u(t) V } , O = {t (0, T ), > 0, u(t) C((t , t + ), V )} O u L1 (0, T ; V ) ( 21 , 1] t s r t r t t st (1, ) r t r r t tt u L (0, T ; V ) t s r t ) = s L1 ( ) = s tr st r sttt L ( t t t r qt t t sts t0 \(O ) t r t r t qss tr tr rr s t0 s t t tr H rrt t ts O s \(O ) s st r t t trr L1 ( \(O )) = t tt [0, T ] \ O ( \(O )) t L1 ([0, T ] \ O ) = tr s r t s r s t s y(t) L [0, T ] O t rst tt T > [0, T ] ss tt st O [0, T ] s t S t ss tt L (S) = s tt tr sts t sts t ts y(t) 0, t O, t, t + r t s = + r M M ,T y a (t) a O, t O, r stts sts t t M > 0, a > s à1 as (S) = Pr O = iI (ci , di ), r O I (ci , di ) r t ts (ci , di ) r t s s st r t t st t rtrr iI t tt di + M y a ( ) ,T r tt y s ( ) q , s M a (T ) a s max (di ) a r r (ci , di ) s iI di s Ma t di di t d s (T ) a y s ( )d + ci s a (di ci ) iI t s a ci ci as r y s ( ) + s s Ma (T ) a trt t qt r (di ci )1 a > r st I I s tt M T as s T a y ( )d + as s L1 (S) = t sst tt s (di ci )1 a < , iI\I < + t s tt tr sts (di ci ) < iI\I t s s s tt tr sts tr r m s tt m [0, T ]\ iI (ci , di ) = m j=1 [aj , bj ] = j=1 Bj , r Bj Bj = j j=j t Ij t st i I\I s tt (ci , di ) Bj r r (Bj S) (ci , di ) Bj = iIj r tt diam(Bj ) = bj aj = (di ci ) iIj (di ci ) < iI\I s t qt n n ai , (ai 0, i = 1, n), < 1, i=1 i=1 m m à1 as , diam(Bj ) as (di ci ) = j=1 j=1 as iIj m s j=1 iIj tt ss tt (di ci )1 a < iI\I tt r s (di ci )1 a = à1 as (S) = ( 12 , 23 ), u0 H, f L (0, T ; V1 ) u s st stss t t u Lr (0, T ; V ), r > 0, r(2 1) < tr sts s st à1 r(21) (S ) = S [0, T ] s tt u C([0, T ] \ S ; V ) S = [0, T ] \ O r O s tr t r r r r t L (S ) = r y (t) L (0, T ) r y (t) = , s = 2r |u(t)|2 + t O = O , y (t) = |u(t)|2 + 1, a = 21 s tt r(21) (S ) = s rs t tr Pr t r t r(2 1) t u à0 (S ) = r(2 1) < r s t sst s stss t rr t u s st f s st ts s r st r t r qt t t s ts s sst S rr r t tss r t r(21) s sr sr s q t r r r = 2, = t t t u L2 (0, T ; V1 ) r s rt s t s ss t t sts ts s rr t r rst r r s r(2 1), ( , 1] r t r s st t t u L (0, T ; V ) s rt ts s r rst s t r r(2 1), ( , 1] r rst s t rst t rr r f L (0, T ; V1 ) r r t ts r t [ 32 , ) t s r s r for = ( 21 , 23 ) b1, (u) t t t r r sts Lr W 1,q q2 s t tts u |u| q q2 = : à1 r2 + (S ) = for any > 0, for > : à1 r2 (S ) = r t s t stt r ts st r r q Lq (T3 ) i,l=1 2 L2 (T3 ) i,l,j=1 T3 q = T3 ui q dx, xl = ui xl xj i,j=1 ui xj ui xj q 2 L2 (T3 ) q2 dx [ 32 , ) t t r st t rts qts r st t d u q(q 1) dt q Lq (T3 ) ui ui ui ul xl x2j xj T3 + l,j=1 q |u| q2 q2 = fi T3 j=1 T3 ui ui x2j xj q2 L2 (T3 ) p ui ui xi x2j xj dx + j=1 q2 dx dx Pr rst trt rts t tt d u dt q Lq (T3 ) ui xl =q T3 i,l=1 = q(q 1) T3 i,l=1 T3 i,j,l=1 = q i,j=1 = q i,j=1 = T3 i,j=1 t t trt r T3 ss tt t u(t) ui ui x2l xj xj q1 q2 dx = sign ui dx xj q |u| 2 q L2 (T3 ) ui q2 ui s t ttr l=1 xl x2l 2,q ) L (0, T ; W 1,q ), q [2, 3), u L2 (0, T ; W d u(t) q(q 1) dt ul l,j=1 T3 ui ui ui xl x2j xj q Lp (T3 ) = f L (0, T ; Lq (T3 )) T3 dx + j=1 fi j=1 q |u(t)| q2 r ui dx xj sign t t s st ts st rr stss t qt q2 i dx xj q1 ui dx, t q2 u ui xj ui xj i x2l s t rs t t sr rst p Lq (T3 ) + T3 ui xl xj iqt ui ui x2l xj ui ui sign dx xl t xj q2 u ui xl ui x2l xj T3 q1 ui ui x2j xj T3 L2 (T3 ) p ui ui xi x2j xj q2 dx q2 dx, p(t) Lq (T3 ) C (u, )u(t) Lq (T3 ) tr st stts q , Lq (T3 ) T , K, L u(0) q, sup0tT f (t) Lq (T3 ) s tt u(t) sup 0tT T q Lq (T3 ) K, q |u(t)| L2 (T3 ) dt L Pr t qt u(t) r t q q L3q (T3 ) g Lq (T3 ) = |u(t)| q C |u(t)| L6 (T3 ) L2 (T3 ) t r qt ui ui g(x) xj xj T3 q2 ui ui dx x2j xj q2 L2 (T3 ) u(t) q2 Lq (T3 ) g Lp (T3 ) rr fi T3 ui ui x2j xj p2 ui ui x2j xj q2 dx L2 (T3 ) q u(t) q(2q1) 2q3 Lq (T3 ) |u(t)| + u(t) q2 Lq (T3 ) f (t) Lp (T3 ) L2 (T3 ) + C f (t) q(2q1) 3(q1) Lq (T3 ) rtrr ul ui xl Lq (T3 ) u(t) u(t) L (T3 ) u(t) q+1 Lq (T3 ) Lq (T3 ) q+1 Lq (T3 ) u(t) |u(t)| 3q q q L2 (T3 ) u(t) 3q L3q (T3 ) ul l,j=1 T3 ui ui ui xl x2j xj q2 dx u(t) 2q1 Lq (T3 ) u(t) |u(t)| + L2 (T3 ) q q(2q1) 2q3 Lq (T3 ) q q |u(t)| L2 (T3 ) r s p(t) Lq (T3 ) u(t) q+1 Lq (T3 ) q |u(t)| 3q q L2 (T3 ) s l,j=1 T3 p ui ui xi x2j xj q2 dx u(t) q u(t) q(2q1) 2q3 Lq (T3 ) |u(t)| + 2q1 Lp (T3 ) |u(t)| q q L2 (T3 ) L2 (T3 ) + C f (t) q(2q1) 3(q1) Lq (T3 ) qts s r sr trs t q d dt |u(t)| u(t) q Lq (T3 ) + L2 (T3 ) C u(t) q Lq (T3 ) q(2q1) 2q3 Lq (T3 ) u(t) (2q1) 2q3 +1 + C f (t) q(2q1) 3(q1) Lq (T3 ) s t r tt y(t) = u(t) q Lq (T3 ) +1 r t t s r ss tt f L (0, T ; Lq (T3 )), u0 W 1,q (T3 ), q [2, 3) tr sts stt T u(0) q , q, Lq (T3 ) sup0tT f (t) Lq (T3 ) q str st t sts 2,q ) C(0, T ; W 1,q ) u L2 (0, T ; W Pr r ts tr s sr t t r s t t ts q [2, 3) s tt st u s W 1,q (T3 ) rr (t1 , t2 ) u C((t1 , t2 ), W 1,q (T3 )) s tt W 1,q rrt tr (t1 , t2 ) s r u tr s t st tr W 1,q rrt strt t (t1 , t2 ) r 1,q r s r tt (t1 , t2 ) s W tr st u t lim sup u(t) W 1,q (T3 ) = + t tt2 r st t (0, T ) q [2, 3), u0 H, f L (0, T ; Lp (T3 )) ss tt u s 1,q t L (0, T ; W ) u s W 1,q rr st t s t s s sr r ss tt q [2, 3), u0 H, f L (0, T ; Lp (T3 )) u s st t stss t t u Lr (0, T ; W 1,q ), tr sts s st Sq [0, T ] r(2q 3) < 2q s tt u C([0, T ] \ Sq ; W 1,q ) q Lq (T3 ) + à1 r(2q3) (Sq ) = 2q Pr t tss t tr t r s= t r ,a q = r(2q3) 2q r y(t) L q (0, T ) r y(t) = u(t) t t sr rst 2q3 < r s t sst s r(2q3) t u stss t rr t 2q ts s à0 (Sq ) = u s st f s st r r t tss r t r sr r r(2q3) 2q s s s q t r L2 (0, T ; V1 ) r s rt s t s ss t t sts r = 2, q = t t t u ts s rr t r rst r r s r r 2, q = t t u L (0, T ; V ) s s rt s t s r t t u L (0, T ; V1 ) q r(2q 3) r rst s t r r t rs rs s tt t t u Lr (0, T ; V ) t = r u Lr (0, T ; W 1,q ) u Lr (0, T ; H) r u Lr (0, T ; Lq ) rst s rqrts s st rr t t rs rs s t rrs tr This chapter was written on the basis of the paper r t sr s t sr st t r sts t t sttr rts qts trst r tts r ss ts tss strt sts t t rts qts s (q > 1, d s < d ) t Pr trt r r t ss H q q q t t st sts t ss L [0, T ]; H qs (Rd ) t rtrr t H qs (Rd ) t s rt s (q > 1, s = dq 1) t t st d sts t ss L ([0, ); H qq (Rd )) t r t t s s s rt s t ss d rt rrrt ss rt ss s s= d q H Lpp,r (Rd ), (r 1, p < ) H Ls q,r (Rd ), (s 0, q > 1, r 1, dq s < dq ) t rt m < trs q = (q1 , q2 , , qd ), r = (r1 , r2 , , rd ) r < qi < , ri , i d tr st r m , stt t st qss sts t t rt ss H L p m , ) r p > 2, T > rts qts t ss Q := QT = L ([0, T ]; H L d d ã t t s t t ss I = {u0 (S (R )) , div(u0 ) = : e u0 Q < } r q r q r rsts str rt t st t t s r t t s t r r t t s t p + d i=1 qi m = t s I t s t T = rt s ,p p m s t t s s s B Lq,r stt t sr s t ss sr st t sts t t rts qts t tr s trs r s rrt ts rrs t rsts t tr rt t rrt ts rrs t t t sr s t sr st t st t trs ts rt t t ssrtt r sss r t rts qts t t rt ss r ss sss r t rts qts t t t ss t t t s r sss r t rts qts t t rrrt ss r tt ss ts r t t r r t rts q ts t t t t rt s ss r tt s t rst r t sr s t sr st t r sts t t sttr rts qts trst r tts r ts r rt ss t t t r r t rts qts r tt ss ts trs tr rt rs r r str t sts r t r ts qts trt r t qts t t r r t rts qts t t t t t ss rrt r r st rts str sts r rts qts sstt ss rrt r r st st rts str L (|x| dx) L (|x| dx) ss rrt sts t rts qts r rsts t ssrtt rst t P tts r sttt tts P tts r sttt tts t P tts r sttt tts t r tts rt qts ts t sttt r s s rss st r str trt s rrr r Pộ sss t rts qts rt s t r q t sr sr s ss s Prột rstộ Prs r r ộst s ss s ốs r t r rrt ss r t rts qts t Rn s t ss ộtrs t ộqts r ts Prs rt srts s rr s t r rr Prt rrt st sts Pr t rts qts Pr t rú t s rts qts tts Prrts t rts rt tr Prs rt tr t rts qts t rr P t sttr rts qts t tr r q r ttP st t rts qts t r ss ts r s t rss rts qts rr rr s tt s sr str rqs sr st r sstố rts rss t t stss sts t rts q ts t r Prr trs Ps s s r t r r t rts p qts t t L r t r P rsst rr t s L3 (R3 ) t trs P rs r ts t t t t Ps ss ts r rts st t rr rr rr tt s sr rr rt ss ttP r t t t s r rs tts Pr s t tr rrts t sts t rts qts t Prs r P t r sts t t rts qts r r t ss r t P stts stt r sts t t rts qts st rr r ts sr r qts sts t rts sst r q ts r t Lr Lp rrt t rts t r rt rrr r qts r rt rs t ửrr r rt rt rtrs r r r ĩr srt ỹr rs r t r s t rt rrt t r ts qts t Ps t r tr trt rs rr s ts t Prs Prt rt q ts trt r ts t t r t Pr t t t sttr rts sst t P t t t rts t r r t r t qts t rts q ts t strts t ss s t t P t tr Lp sts t rts qts t sts t t Rm t ts t tr sts t rts qts rr ss rs t t P rts qts t t t t t s ts tr sss r t rts qts t P s tt s s r Prss r s t t r t sttr r r t rts sst t t rs Pr t s tt tr ss rss r r r ts rtrs r ts tts r rst Prss r r t rrr tr Pr t P rsst t ts t rts Pr sr ts tts t r ts rss ộqts tộrs ộrs t qqs r ốs q s rq t Prs t r r t q sq sst s t t r ss sr s ts q sq sst s t t sts rts qts ts Pr tts s s t s t sr st sts t t rts qts t Ps Pr tr tỏ r q rts t t Pr t t sss t r r sst s ss t ts s r t rss r r s rssst r ụ s sts s qs Prốs s ộr tr ss r ộrr r Ptr ts s s t rs P ts s t rtt sttq r s qts rts tr ss Ptq r r sr sr t rts qts t Ps r r sr s r rts qts tr ts tts r rr t trr rrt sts t rts qts r t rr t r r t rts qts r Prs Pr s s s ss Prss s s r ss rr ss ts t rts qts tr t qts P rts qts r r ss rt str rts qts r t ss t r rs tts t r str tts P P t t r r trs rts qt ttrs r rts qts r rsr ts t r ssr rts t r Lp r t sss r t rts qts rt s ss t B ,q ... NavierStokes Chỳng tụi xõy dng nghim mm cho phng trỡnh Navier-Stokes nhng khụng gian ny bng nguyờn lý ỏnh x co Picard v ch rng phng trỡnh Navier-Stokes c t chnh cỏc khụng gian ny theo ngha Hadarmard... ny, chỳng tụi s dng nhng tin b t c lnh vc gii tớch iu hũa mi lm nm gn õy nghiờn cu phng trỡnh Navier-Stokes Chỳng tụi mun núi n vic s dng bin i Fourier v cỏc tớnh cht ca nú, phự hp hn cho vic... HN LM KHOA HC V CễNG NGH VIT NAM VIN TON HC O QUANG KHI MT S TNH CHT NH TNH CA NGHIM PHNG TRèNH NAVIER-STOKES Chuyờn ngnh: Phng trỡnh vi phõn v tớch phõn Mó ngnh: 62 46 01 03 LUN N TIN S Ngi hng