Tính hầu tuần hoàn, hầu tự đồng hình và dáng điệu tiệm cận của một số luồng thủy khí trên toàn trục thời gian

BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC BÁCH KHOA HÀ NỘI LÊ THẾ SẮC TÍNH HẦU TUẦN HỒN, HẦU TỰ ĐỒNG HÌNH VÀ DÁNG ĐIỆU TIỆM CẬN CỦA MỘT SỐ LUỒNG THỦY KHÍ TRÊN TỒN TRỤC THỜI GIAN LUẬN ÁN TIẾN SĨ TOÁN HỌC Hà Nội - 2022 BỘ GIÁO DỤC VÀ ĐÀO TẠO TRƯỜNG ĐẠI HỌC BÁCH KHOA HÀ NỘI LÊ THẾ SẮC TÍNH HẦU TUẦN HỒN, HẦU TỰ ĐỒNG HÌNH VÀ DÁNG ĐIỆU TIỆM CẬN CỦA MỘT SỐ LUỒNG THỦY KHÍ TRÊN TỒN TRỤC THỜI GIAN Ngành : Toán học Mã số : 9460101 LUẬN ÁN TIẾN SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC: PGS TSKH Nguyễn Thiệu Huy Hà Nội - 2022 L˝I CAM OAN Tỉi xin cam oan c¡c k‚t qu£ nghi¶n cøu lu“n ¡n T‰nh hƒu tuƒn ho n, hƒu tü çng h…nh v d¡ng i»u ti»m c“n cıa mºt sŁ luỗng thy kh trản to n trửc thới gian l cỉng tr…nh nghi¶n cøu cıa tỉi, ho n th nh dữợi sỹ hữợng dÔn ca PGS.TSKH Nguyn Thiằu Huy CĂc k‚t qu£ lu“n ¡n l ho n to n trung thüc v ch÷a tłng ÷ỉc t¡c gi£ kh¡c cỉng b bĐt ký mt cổng trnh nghiản cứu n o CĂc nguỗn t i liằu tham khÊo ữổc trch dÔn y theo úng quy nh H Ni, ng y 08 thĂng 01 nôm 2022 Ngữới hữợng dÔn Nghiản cøu sinh PGS TSKH Nguy„n Thi»u Huy L¶ Th‚ S›c i L˝IC MÌN Lu“n ¡n n y ÷ỉc thüc hi»n ti Trữớng i hồc BĂch khoa H Ni dữợi sỹ hữợng dÔn ca PGS.TSKH Nguyn Thiằu Huy Thy khổng ch l mt nh khoa hồc m cặn l mt ngữới vổ mÔu mỹc cổng viằc cụng nhữ cuc sng Thy  tn tnh ch bÊo, hữợng dÔn gióp tỉi ho n th nh lu“n ¡n Tỉi xin b y tọ lặng bit ỡn c biằt sƠu sc tợi thy Tổi cụng xin gòi lới cÊm ỡn sƠu sc tợi TS Phm Trữớng XuƠn, ngữới  hữợng dÔn, çng h nh v t“n t…nh gióp ï tỉi suŁt qu¡ tr…nh nghi¶n cøu v ho n th nh lu“n ¡n Trong suŁt thíi gian l m nghi¶n cøu sinh t⁄i Tr÷íng ⁄i håc B¡ch Khoa H Nºi, tỉi  nhn ữổc nhiãu tnh cÊm cụng nhữ sỹ giúp ï cıa c¡c thƒy cỉ bº mỉn To¡n Cì b£n, c¡c thƒy cỉ Vi»n To¡n Ùng dưng v Tin hồc c biằt, tổi  nhn ữổc nhng õng gâp, chia s·, ºng vi¶n cıa c¡c th nh vi¶n nhâm seminar D¡ng i»u ti»m c“n nghi»m cıa ph÷ìng trnh vi phƠn v ứng dửng ti Trữớng i hồc B¡ch Khoa H Nºi PGS.TSKH Nguy„n Thi»u Huy i•u h nh Tổi xin gòi lới cÊm ỡn chƠn th nh ‚n c¡c thƒy cỉ v c¡c th nh vi¶n nhâm seminar Nh¥n dàp n y, tỉi cơng b y tọ sỹ cÊm ỡn chƠn th nh tợi Ban GiĂm hiằu, cĂc Phặng, Ban liản quan, Khoa Cổng nghằ thỉng tin v Bº mỉn To¡n håc thuºc tr÷íng ⁄i hồc Thy lổi  to iãu kiằn thun lổi cho tỉi håc t“p v nghi¶n cøu CuŁi cịng, tỉi xin b y tä lỈng bi‚t ìn ‚n gia …nh v to n th” b⁄n b– ¢ ln khuy‚n kh‰ch, ºng viản chia sà nhng khõ khôn cuc sng, giúp tổi vng tƠm hồc v nghiản cứu Nghiản cứu sinh ii MệC LệC MáT Să K HI U DềNG TRONG LU N N M— U TŒng quan v• hữợng nghiản cứu v lỵ chồn ã t i Möc ch, i tữổng v phm vi nghiản cứu Phữỡng phĂp nghiản cứu K‚t qu£ cıa lu“n ¡n C§u tróc cıa lu“n ¡n Ch÷ìng KI N THÙC CHU N BÀ 1.1 Nßa nhâm 1.1.1 1.1.2 1.2 Khæng gian h m, khỉng gian nºi 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 Ch÷ìng SÜ T˙N T I V T NH ˚N ÀNH CếA MáT Să LP NGHI M CếA PHìèNG TR NH TI N H´A TR N KH˘NG GIAN N¸I SUY 2.1 Tnh chĐt nghiằm ca phữỡng tr 2.1.1 2.1.2 2.2 Tnh chĐt nghiằm ca phữỡng tr 2.2.1 2.2.2 iii 2.3 Mt sŁ øng döng 2.3.1 2.3.2 2.3.3 2.3.4 Chữỡng MáT Să LP NGHI M CÕA PH×ÌNG TR NH NAVIERSTOKES TR N KH˘NG GIAN LORENTZ C´ TR¯NG MUCKENHOUPT p q 3.1 C¡c ¡nh gi¡ L L giœa c¡c khæng gian enhoupt 3.2 Phữỡng trnh tuyn tnh trản khổng gia enhoupt 3.3 Phữỡng trnh nòa tuyn tnh tr¶n khỉ Muckenhoupt Chữỡng MáT Să LP NGHI M CÕA PH×ÌNG TR NH BOUSSINESQ TRONG MI N KH˘NG BÀ CH N 4.1 D⁄ng ma tr“n cıa h» ph÷ìng tr…nh Bous 4.2 L Tnh chĐt nghiằm ca phữỡng trnh tu p 4.2.1 4.2.2 4.3 Sỹ tỗn ti v tnh n ành nghi»m cıa ph÷ 4.3.1 4.3.2 K TLU NV KI NNGH Nhng kt quÊ Â t ữổc ã xuĐt mt s hữợng nghiản cứu tip theo DANH MệC C C CNG TR NHCNG Bă CếA LU N N T ILI UTHAMKH O iv MáT Să K HI U DÒNG TRONG LU N N N R R+ X L(X) AP(R; X) AA(R; X) p S AA(R; X) n R+ p L () L () p Lloc ( ) W k;p () k W ;1 () C(R; X) k 1 := fu L ( ) : D u L ( ); vỵi j j vỵi chu'n kukk;1 := max kD uk1: kg jjk := u : R ! X li¶n tưc := u : R ! X li¶n tưc v BC(R; X) BC(R+; X) := u : R+ ! X li¶n tưc v kuk1 = sup ku(t)k < t2R+ : U U PAA0(R; ) WPAP(R;X) := R WPAA( p ; R W S AA( K(t; x) (X ; X ) ;q (X0; X1) ;1 vỵi x := x kk (X0;X1) ;1 (X0; X1) C () C0 ( ) C0 ; ( ) p;q L () x (X ; X ) ;1 : t!0+ := : Khæng gian c¡c h m kh£ vi cĐp vổ hn trản : Khổng gian cĂc h m kh£ vi vỉ h⁄n vỵi gi¡ compact := fv C0 ( ) : divv = := u Lloc vỵi kukp;q = p L w( ) vỵi u := u p; M— U Tng quan vã hữợng nghiản cứu v lỵ chồn ã t i Nghiản cứu nghiằm tun ho n, hƒu tuƒn ho n v sü kh¡i qu¡t cıa chúng i vợi phữỡng trnh tin hõa l mt hữợng nghiản cứu quan trồng liản quan n t nh chĐt nghi»m cıa ph÷ìng tr…nh ti‚n hâa theo thíi gian Łi vợi trữớng hổp nghiằm tun ho n, mt s phữỡng phĂp thữớng ữổc sò dửng nhữ nguyản lỵ Massera [1, 2], nguyản lỵ im bĐt ng ca Tikhonov [3] hay h m Lyapunov [4] ÷ỉc ¡p dưng cho mºt sŁ lợp phữỡng trnh vi phƠn cử th CĂc phữỡng phĂp ph bin nhĐt cho viằc chứng minh sỹ tỗn ti nghi»m tuƒn ho n l t‰nh bà ch°n cıa nghi»m v t‰nh compact cıa ¡nh x⁄ Poincar† thæng qua c¡c php nhúng compact [3, 4, 5, 6] Tuy nhiản, vợi trữớng hổp phữỡng trnh o h m riảng cĂc miãn khổng b chn hay cĂc phữỡng trnh cõ nghiằm khỉng bà ch°n th… c¡c ph†p nhóng compact n y khổng cặn úng na v õ sỹ tỗn ti nghiằm b chn s khõ t ữổc iãu n y l c¡c i•u ki»n ban ƒu phị hỉp ” £m b£o t‰nh bà ch°n cıa nghi»m khæng d„ d ng t…m ÷ỉc Mºt ph÷ìng ph¡p ” gi£i quy‚t nhœng khõ khôn n y l sò dửng nguyản lỵ dng Massera, nghắa l nu mt phữỡng trnh vi phƠn cõ nghi»m bà ch°n th… nâ câ nghi»m tuƒn ho n Thỹc t, viằc kt hổp gia nguyản lỵ dng Massera v khổng gian ni suy  ữổc sò dửng chứng minh sỹ tỗn ti nghiằm tun ho n ca cĂc phữỡng trnh cỡ hồc chĐt lọng (cĂc dặng thy kh) v cĂc phữỡng trnh truyãn nhiằt vợi hằ s thỉ, ph÷ìng tr…nh Ornstein - Uhlenbeck [7, 8] Trong c¡c cỉng tr…nh n y, c¡c h m tß nºi suy ữổc sò dửng kt hổp vợi phữỡng phĂp Ergodic [8] i vợi trữớng hổp cĂc dặng thy kh, sỹ tỗn t⁄i cıa nghi»m tuƒn ho n cıa ph÷ìng tr…nh Navier-Stokes v cĂc phữỡng trnh dng Navier-Stokes tr th nh hữợng nghiản cứu quan trồng Trong miãn b chn, Serrin  sß dưng t‰nh Œn ành cıa nghi»m bà ch°n ” ch sỹ tỗn ti nghiằm tun ho n ca phữỡng tr nh Navier-Stokes [9] Sau õ, sỹ tỗn ti, nh§t, t‰nh Œn ành v n d¡ng i»u ti»m c“n nghi»m tuƒn ho n tr¶n to n khỉng gian R , trản miãn khổng n b chn R v trản to n trửc thới gian R ữổc m rng nghiản cứu Nghiản cứu sỹ tỗn ti cıa c¡c nghi»m ti»m c“n hƒu tuƒn ho n v tiằm cn hu tỹ ỗng hnh ca phữỡng trnh Navier-Stokes miãn khổng b chn Nghiản cứu tnh nhĐt to n cửc cho nghiằm ca phữỡng trnh Navier-Stokes trản miãn khổng b chn Nghiản cứu cĂc loi nghiằm ca phữỡng trnh Navier-Stokes v phữỡng trnh Boussineq trản khổng gian Mourey y‚u v c¡c khæng gian nºi suy kh¡c 91 DANH MÖC C C C˘NG TR NH N N CNG Bă CếALU Nguyen Thieu Huy, Le The Sac and Pham Truong Xuan (2018), On Al-most Automorphic Solutions to Incompressible Viscous Fluid Flow Prob-lems , International Journal of Evolution Equations, Volume 11, Number 3, pp 501-516 Nguyen Thieu Huy, Vu Thi Ngoc Ha, Le The Sac and Pham Truong Xuan (2021), Weighted Stepanov-Like Pseudo Almost Automorphic Solutions for Evolution Equations and Applications , Acta Math Vietnam 46, 103 122 (ESCI/SCOPUS) Nguyen Thieu Huy, Pham Truong Xuan, Le The Sac and Vu Thi Ngoc Ha (2021), Periodic and Almost Periodic Solutions to NavierStokes Equa-tions in Interpolation Spaces with Muckenhoupt Weight , Accepted for Publication in Ukrainian Mathematical Journal (SCIE) Vu Thi Ngoc Ha, Nguyen Thieu Huy, Le The Sac and Pham Truong Xuan (2021), Interpolation Spaces and Weighted Pseudo Almost Automorphic Solutions to Parabolic Equations and Application to Fluid Dynamics , Ac-cepted for Publication in Czechoslovak Mathematical Journal (SCIE) Thieu Huy Nguyen, Truong Xuan Pham, Thi Ngoc Ha Vu and Le The Sac, Existence and Stability of Periodic and Almost Periodic Solutions to Boussinesq Systems in Unbounded Domains , Submitted 92 T ILI UTHAMKH O [1] J Massera (1950), The existence of periodic solutions of systems of differ-ential equations , Duke Math J., 17, 457-475 [2] 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