Boundary Control of the Viscous Generalized Camassa-Holm Equation

Transcription

1 ISSN prin, online Inernaional Journal of Nonlinear Science Vol89 No,pp Bounary Conrol of he Viscous Generalize Camassa-Holm Equaion Yiping Meng 1,, Lixin Tian 1 School of Mahemaics an Physics, Jiangsu Universiy of Science an Technology Nonlinear Scienific Research Cener, Faculy of Science, Jiangsu Universiy Zhenjiang, Jiangsu,113,PRChina Receive April 9, accepe 9 Augus 9 Absrac: This paper invesigaes he bounary conrol of he viscous generalize Camassa- Holm equaion on [, 1] The exisence of is soluion is prove in a shor ime inerval uner he bounary coniion Meanwhile, we prove he global exponenial sabiliy of he soluion in four ifferen spaces, ie in L,H 1,H,H 3 Keywors: he viscous generalize Camassa-Holm equaion; bounary conrol; global exponenial sabiliy 1 Inroucion In [1], Camassa an Holm use Hamilonian mehos o erive a new compleely inegrable ispersive wave equaion for waer waves by reaining wo erms ha are usually neglece in he small ampliue shallow waer limi u + ku x u xx + 3uu x = u x u xx + uu xxx, 11 where u is he flui velociy in he x irecion or equivalenly he heigh of he waer s free surface above a fla boom, k is a consan relae o he criical shallow waer wave spee They showe ha for allk, 11 is inegrable, an fork =, 11 has raveling soluions of he formce x c Afer ha, here has been much research an observaions abou his equaion Zhengrong Liu an Tifei Qian e al [3] invesigae generalize Camassa-Holm enoe GCH equaion, or calle moifie Camassa-Holm or enoe mch equaion, u + ku x u xx + au m u x = u x u xx + uu xxx, 1 where a >,k R, m Nan mis calle he srengh of he nonlineariy They iscusse he peakons an heir bifurcaion in 1 an hey inrouce GCH equaion from he mahemaical poin of view In [-4], Lixin Tian e al suie is peakons an raveling wave soluions In he oher han, more people have pai aenions o bounary conrol of ifferen equaions Byrnes e al suie local sabiliy of Burgers equaion, Vanly e al furher consummae his resul, bu sill local Miroslav Krisic suie global sabiliy of Burgers equaion [5] Biler Rassel an Zhang suie KVB equaion uner perioical bounary coniion [6]-[8] Liu an Krisic suie he sabiliy of KVB equaion in a limie area [9] Wei-jiu Liu e al i some suies on K-S equaion [1, 11] Yiping Meng suie bounary conrol of b-family equaion [1] In his paper, we suy he exisence of he bounary conrol of he viscous generalize Camassa-Holm equaion wih he following bounary u u xx ε u u xx xx + ku x + u 3 x = u x u xx + uu xxx u x, = u x u, = u 1, = u x, = u x 1, = u xx, = u xx 1, = where >, x Ω, Ω = [, 1], u H = L Ω, 13 Corresponing auhor aress: ianlx@ujseucn Copyrigh c Worl Acaemic Press, Worl Acaemic Union IJNS9115/63

2 174 Inernaional Journal of Nonlinear Science,Vol89,No,pp We will iscuss he global well-poseness of 13 I will be shown ha for given T >,u H 3, 13 amis a unique soluion u C, T, H 3, 1 C, T, H, 1 Main heorem Denoe A = Δ, where Δ is he Laplace operaor Le v = u + Au Denoe bi-linear iem B u, v = uv x an C u = ku x Eq13 can be enoe as v + εav + B v, u + B u, v + C u + u 3 x 3u x = u = ux, u, = u1, = u xx, = u xxx 1, = u x 1, = u x, where A is a self-ajoin posiive operaor wih compac inverse Theorem 1 Wih u H 3, Eq4 has a global soluion u C,, H 3, 1 C,, H, 1, 1 an saisfies he following L, H 1, H, H 3 sabiliy: 1 u L + u H 1 u L + u H 1 exp ελ 1, λ 1 is a Poincare coefficien u H 1 + u H u H 1 + u expb 1 ελ, 3 H where ε c λ, + an b 1 is a nonnegaive consan, λ is a Poincare coefficien 3 u H + u H 3 u H + u H 3 exp b ελ 3, 4 where ε b λ 3, + an b is a nonnegaive consan, λ 3 is a Poincare coefficien 3 Proof of Theorem Firs, we prove he global exisence of he soluion We use he Galerkin proceure o prove global exisence Le {φ j } j=1 be an orhonormal basis of Hconsising of eigenfuncions of he operaor A For m N efine he iscree ansaz space by H m = {φ 1, φ,, φ m }, an P m is he orhogonal projecion from H o H m Performing he Galerkin Proceure for Eq1, hen 1 is he orinary ifferenial sysem, { v m, + εav m + P m B v m, u m + P m B u m, v m + P m C u m + u 3 m x 3u m x =, 31 u m x, = p m u m, x where v m = u m u m,xx Since he nonlinear erm is quaraic in u m, hen by he classical heory of orinary ifferenial equaion, he sysem 31 has a unique soluion for a shor inerval of he ime, T m Our purpose is o show he global exisence of he soluion uner given bounary conrol Taking inner prouc in 31 o be u m onω, we have u m L + u m H 1 + ε u m H 1 + Au m L = 3 By Poincare inequaliy an Young s inequaliy, we can consruc u m L + u m H 1 + ελ 1 u m L + u m H 1, where λ 1 is a Poincare coefficien Then, we can ge u m L + u m H 1 u m L + u m H 1 exp ελ 1 u m L + u m H 1 Δr 1, 33 IJNS for conribuion: eior@nonlinearscienceorguk

3 Y Meng, L Tian: Bounary Conrol of he Viscous Generalize Camassa-Holm Equaion 175 [, T ], where r 1 is posiive consan Le r be a posiive consan, inegraing 3 in he inerval[, + r] [, T ], we have +r Taking inner prouc in 31 o be Au m onω, we obain u m H 1 + Au m L + ε u m s H 1 + Au m s L s r 1 ε 34 Au m L + Au m H 1 + P m B u m, v m, Au m +P m B v m, u m, Au m + P m C u m, Au m + u 3 m x, Au m 3u m x, Au m = By compuing, we have u 3 m x, Au m = 3 u m u mx 3 x, 3u m x, Au m = 3 u mx 3 x P m B u m, v m, Au m + P m B v m, u m, Au m + P m C u m, Au m 3 P m B u m, u m, Au m + P m B Au m, u m, Au m + P m B u m, Au m, Au m Accoring o Agmon inequaliy when n = 1, hen we have P m B u m, u m, Au m u m,x L Ω u m H 1 C 1 u m 5 L, 35 P m B Au m, u m, Au m u m,x L Ω Au m L C u m 1 H 1 Au m 5 L, P m B u m, Au m, Au m u m,x L Ω Au m L C 3 u m 1 H 1 Au m 5 L, u mu mx 3 x u m L Ω u m,x L Ω u m H 1 C 4 u m 1 L u m 1 H 1C 5 u m 1 L u m H 1 C 4 C 5 C 1 6 C 1 7 Au m 1 L u m 5 H 1 C 8 Au m 1 L u m 5 H 1 where C 8 = C 4 C 5 C 1 6 C 1 7 an u m L C 6, u m H 1 C 7 u m,x 3 x u m,x L Ω u m H 1 C 9 u m 5 L From above inequaliies, we can obain Pm B u m, v m, Au m + P m B v m, u m, Au m + P m C u m, Au m + u 3 m x, Au m 3u m x, Au m, 3C 1 + 3C C 9 u m 5 L + C + C 3 u m 1 H 1 Au m 5 L C 1 u m 1 L u m H 1 + Au m L, where C 1 = max { 3C 1 + 3C C } 9, C + C 3 Then, we have u m H 1 + Au m L + ε Au m L + Au m H 1 C 1 u m 1 L u m H 1 + Au m L + 1 u m H 1 + Au m L 36 Applying Young s inequaliy an Poincare inequaliy, we have u m H 1 + Au m L + ελ u m H 1 + Au m L ελ 1 u m H 1 + Au m L IJNS homepage:hp://wwwnonlinearscienceorguk/

4 176 Inernaional Journal of Nonlinear Science,Vol89,No,pp u m H 1 + Au m L + C 1 ελ 1 u m H 1 Au m L u m H 1 + Au m L C 11 u m H 1 + Au m L, where C 11 = C 1 ελ 1 is a consan, { } λ is a Poincare coefficien an ε > max 1 λ, 1 Le y = u m H 1 + Au m L,g = C 11 u m H 1 + Au m L Then, we can erive +r y ss r ε, +r r g ss C ε 11, for r > Accoring o uniform Gronwall inequaliy, we have exp u m H 1 + Au m L r εr Inegraing 36 in he inerval [, + r], we can ge C 11 r ε u m H 1 + Au m L Δr 3 +r ε Au m L + Au m H s C 1r r 3 + C 1r + r 3 4ε ε + r ε Δr 4 37 Taking inner prouc in 31 o be A u m on Ω, we have Au m L + Au m H 1 + ε Au m H 1 + A u m L + P m B um, v m, A u m +P m B vm, u m, A u m + Pm C um, A u m + u 3 m x, A u m 3u m x, A u m = By compuing, we have u 3 m x, A u m = 3 3u m x, A u m = 3 u m u m,xu m,xxx + u mu m,xx u m,xxx x = 15 u m,xu m,xxx + u m u m,xx u m,xxx x = 15 Accoring o Agmon inequaliy when n = 1, hen we have Pm B um, u m, A u m um,x L Ω Au m L u m u m,x u m,xxx, u m,x u m,xxx C 1 u m 1 L Au m L + Au m H 1, Pm B Aum, u m, A u m um,x L Ω Au m H 1 C 13 u m 1 L Au m L + Au m H 1, P m B um, Au m, A u m u m,x L Ω Au m H 1 C 14 u m 1 L Au m L + Au m H 1, u m u m,x u m,xxx u m L Ω u m,x L Ω Au m L C 15 u m 1 L u m 1 H 1C 16 Au m 1 L Au m L + Au m H 1 C 17 u m 1 L Au m L + Au m H 1,whereC 17 = C 15 C 16 C 1 6 C 1 7, u m,x u m,xxx u m,x L Ω Au m L C 18 u m 1 L Au m L + Au m H 1 From above inequaliies, we can obain Pm B um, v m, A u m + Pm B vm, u m, A u m + Pm C um, A u m +u 3 m x, A u m 3u m x, A u m IJNS for conribuion: eior@nonlinearscienceorguk

5 Y Meng, L Tian: Bounary Conrol of he Viscous Generalize Camassa-Holm Equaion 177 3C 1 + C C 14 + C C C 18 u m 1 L Au m L + Au m H 1 Then, we have Au m L + Au m H 1 C 19 u m 1 L C 19 u m 1 L Au m L + Au m H 1 + ε Au m H 1 + A u m Au m L + Au m H Applying Young s inequaliy an Poincare inequaliy, we can obain L Au m H 1 + A u m L 38 Au m L + Au m H 1 C u m H 1 Au m L Au m L + Au m H 1, 39 { } where C = C 19 λ 3 ε 1 is consan, ε > max 1 λ 3, 1 an λ 3 is a Poincare coefficien We know ha +r C u m H 1 Au m L s C +r u, m H 1 + Au m L s C r ε 31 Nex, by uniform Growall inequaliy, i follows ha Au m L + Au m H 1 r4 C r exp Δr εr ε We inegrae 38 in he inerval[, + r], an have ε 1 +r Au m H 1 + A u m s C 19r 3 r 4 L 4ε } { where ε > max 1 λ 3, 1 Taking he inner proucof 31 o wih A 3 u m in Ω, we can ge + C 19r 4 ε + r 5 Δr 6, 31 Au m H 1 + A u m L r from he meho iscusse above an uniform Gronwall lemma similarly Now, we have obain ha u m L, u m H 1, Au m L, Au m H 1an A u L m are boune Then v m L, v m H 1 an Av m L are boune Hence we can conclue ha u m, L an v m, L are boune By Aubin s Compacness Theorem, we conclue ha here is a subsequenceu m, such hau m u, or equivalenly v m v Le us replace u m an v m by u m anv m Now we prove u, v saisfy equaion 1 Le ω D A We know ω L is boune from he above iscussion From orinary ifferenial equaions 31, we ge v m, w+ε + Av m s, P m ws+ Cu m, P m ωs + B v m s, u m s, p m w s+ u 3 m x, P m ωs 3 Now, i is clear ha lim p mω ω =, lim p maω Aω =, an so m + m + Av m, ω Av, ω, Av m s, P m ωs T Av s, ωs,asm +, C u m s, P m ω s B u m s, v m s, p m w s u m u m, x, P m ωs = v m, w C u s, ω s IJNS homepage:hp://wwwnonlinearscienceorguk/

6 178 Inernaional Journal of Nonlinear Science,Vol89,No,pp C u m s, P m ω ω s + C u m s C u s, ω s u m H 1 P m ω ω L s + u m u H 1 ω L s,as m +, B v m s, u m s, P m ω s B v s, u s, ω s I1 m + I m + I 3 where I 1 m = I m = I m 3 = B v m s, u m s, P m ω ω s B v m s, u m s P m ω ωs, B v s, u m s u s, ω s v s L u m s u s H 1 ω L s, B v m s v s, u m s, ω s v m s v s L u m s H 1 ω L s as m + Accoring o above iscussion an Lebesgue Conrol Convergence Theorem, we have lim m B v m s, u m s, P m ωs = B v s, u s, ωs Similarly, B u m s, v m s, P m ω s B u s, v s, ω s 4 I m + I m 5 + I 6 where I 4 m, I 5 m,i 6 m as n Accoring o above iscussion an Lebesgue Conrol Convergence Theorem, we have lim m We can also ge u 3 m x, P m ωs B u m s, v m s, P m ωs = B u s, v s, ωs u 3 x, ωs u 3 m x, P m ω ωs + u 3 m x u 3 x, ωs u 3 H m 1 P m ω ω L s + u 3 m u 3 H 1 ω L s u 3 H m 1 P m ω ω L s + um uu m + u m u + u H 1 ω L s, as m +, 3 u m x, P m ωs 3 u x, ωs 3 u m x, P m ω ωs + 3 u m x 3 u x, ωs 3 u m H 1 P m ω ω L s + 3 u m uu m + u H 1 ω L s, as m + Finally, for all ω D A, we can conclue v, ωs + ε + Av s, ωs + u 3 x, ωs 3 u x, ωs + B v s, u s, ωs + Above all, we can conclue ha he exisence of global soluion o Eq1 Uniqueness of he soluion is an immeiae consequencesee [13] Secon, we will prove he global exponenial sabiliy of he soluion m, m, B u s, v s, ωs C u s, ωs = v, ω 314 IJNS for conribuion: eior@nonlinearscienceorguk

7 Y Meng, L Tian: Bounary Conrol of he Viscous Generalize Camassa-Holm Equaion 179 We ake he inner prouc of 1 wih u in, 1 o obain, u L + u H 1 + ε u H 1 + Au L = 315 By Poincare inequaliy, we have u H 1 λ 1 u L, Au L λ 1 u H 1, u L + u H 1 + ελ 1 u L + u H 1, u L + u H 1 ελ 1 u L + u H 1 Thus we ge u L + u H 1 u L + u H 1 exp ελ 1, where λ 1 is a Poincare coefficien Now, ake he inner prouc of 1 wih Au in, 1 o obain, By compuing, we have u H 1 + Au L + ε Au L + Au H 1 + B u, v, Au + B v, u, Au u 3 x, Au = 3u x, Au = + C u, Au + u 3 x, Au 3u x, Au = C u, Au =, 3 u u x u xx x 3 u L u H 1 Au L a 1 3 uu x u xx x 3 u L u H 1 Au L a B u, v, Au + B v, u, Au u H 1 + Au L, u H 1 + Au L, = 3 B u, u, Au + B Au, u, Au + B u, Au, Au, B u, u, Au u L u H 1 Au L a 3 u H 1 + Au L, B Au, u, Au u H 1 Au L a 4 u H 1 + Au L, B u, Au, Au a 5 Au L a 5 u H 1 + Au L So we have u H 1 + Au L a 1 + a + a 3 + a 4 + a 5 ελ u H 1 + Au L, u H 1 + Au L u H 1 + Au L exp a 1 + a + a 3 + a 4 + a 5 ελ Δ u H 1 + Au L expb 1 ελ, where ε b1 λ, + an b 1 is a nonnegaive consan, λ is a Poincare coefficien Then we ge 3 Take he inner prouc of 1 wih A u in, 1 o obain Au L + Au H 1 + ε Au H 1 + A u L + B u, v, A u + B v, u, A u + C u, A u + u 3 x, A u 3u x, A u =, C u, A u =, 316 IJNS homepage:hp://wwwnonlinearscienceorguk/

8 18 Inernaional Journal of Nonlinear Science,Vol89,No,pp u 3 x, A u = 15 uu x u xxx 15 u L Ω u x L Ω Au L 15a 6 u 1 L u 1 H 1a 7 u 1 H 1 Au 1 L Au L + Au H 1 a 8 u 1 H 1 Au 1 L Au L + Au H 1 3u x, A u = 15 1 u x u xxx 15 u L Ω Au L a 9 u 1 H 1 Au 1 L Au L + Au H 1, B u, v, A u + B v, u, A u = 3Bu, u, A u + Bu, Au, A u + BAu, u, A u, B u, u, A u ux L Au L a 1 u 1 H 1 Au 1 L Au L + Au H 1, B Au, u, A u u x L Au H 1 a 11 u 1 H 1 Au 1 L Au L + Au H 1, B u, Au, A u u x L Au H 1 a 1 u 1 H 1 Au 1 L Au L + Au H 1 By 316, we ge u H 1 a 13, Au H 1 a 14 From above inequaliies, we have Au L + Au H 1 + ε Au H 1 + A u L a 8 + a 9 + 3a 1 + a 11 + a 1 u 1 H 1 Au 1 L Au L + Au H 1, a 8 + a 9 + 3a 1 + a 11 + a 1 a 1 13 a 1 14 Au Δ= L + Au H 1 b Au L + Au H 1 where b is a nonnegaive consan By Poincare inequaliy, we have Au L + Au H 1 b ελ 3 Au L + Au H 1, Au L + Au H 1 Au L + Au H 1 exp b ελ 3, where ε b λ 3, + an b is a nonnegaive consan, λ 3 is a Poincare coefficien We ge 4 of he heorem The proof of Theorem is complee References [1] Camassa R, Holm D An inegrable shallow waer equaion wih peake solion Phys Rev Le71: [] Yiqing Li, Lixin Tian, Yuhai Wu: On he bifurcaion of raveling wave soluion of generalize Camassa- Holm equaion Inernaional Journal of nonlinear Science61: [3] Lixin Tian, Chunyu Shen: Opimal conrol of he b-family equaion Inernaional Journal of nonlinear Science41:3-97 [4] Lixin Tian,Xiuying Song: New peake soliary wave soluions of he generalize Camassa Holm equaion Chaos, Solions an Fracals 193: [5] Miroslav KrsicOn global sabiliion of Burgers equaion by bounary conrol, Sysems Conrol Leers [6] PBiler: Large-ime behavior of perioic soluions o issipaive equaions of KV-B ype Bullein of he Polish Acaemy of Sciences, Mah 37-8: [7] DLRussel, BYZhang: Sabilizaion of he Koreweg-e Vries equaions on a perioic omain IMA Mah Appl Springer, New York7: IJNS for conribuion: eior@nonlinearscienceorguk

9 Y Meng, L Tian: Bounary Conrol of he Viscous Generalize Camassa-Holm Equaion 181 [8] DL Russel, BY Zhang: Exac conrollabiliy an sabilizabiliy of he Koreweg-e Vries equaionstransamermahsoc3489: [9] BY Zhang: Bounary sabilizaion of he Koreweg-e Vries equaionsser Numer Mah 118: [1] Wei-jiu Liu,Miroslav Krsic: Sabiliy enhancemen by bounary conrol of he Kuramoo- Sivashinsky equaion Nonlinear Analysis43: [11] Haixia Chao, Dianchen Lu, Lixin Tian: Bounary conrol of he Kuramoo-Sivashinsky equaion wih an exeral exciaion Inernaional Journal of nonlinear Science 1: [1] Yiping Meng: Bounary conrol on he moifie b-family equaion Inernaional Journal of nonlinear Science 6: [13] HRDullin, GAGowal, DDHolm, Camassa-Holm: Koreweg-e Veris-5 an oher asympoically equivalen equaions for shallow waer waves Flui Dynamics Research IJNS homepage:hp://wwwnonlinearscienceorguk/