On the Benney Lin and Kawahara Equations

Transcription

1 JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS 11, ARTICLE NO AY On he BenneyLin and Kawahara Equaion A Biagioni* Deparmen of Mahemaic, UNICAMP, , Campina, Brazil and F Linare IMPA, Erada Dona Caorina 11, 64-3, Rio de Janeiro, Brazil Submied by Colin Roger Received March 6, 1996 We eablih global well-poedne for he iniial value problem Ž IVP aociaed o he o-called BenneyLin equaion Thi model i a Korewegde Vrie equaion perurbed by diipaive and diperive erm which appear in fluid dynamic We alo udy he limiing behaviour of oluion o hi IVP when he parameer of he perurbed erm approach 1997 Academic Pre 1 INTRODUCTION In hi paper conideraion i given o he udy of he iniial value problem aociaed o he BenneyLin equaion, ha i, u uu x u x Ž u u u x ½ už x, Ž x, Ž 11 where, The equaion above wa fir derived by Benney 1 and laer by Lin 7 Žee alo 11 I decribe one-dimenional evoluion of mall bu finie ampliude long wave in variou problem in fluid dynamic Thi model include conervaive diperive effec and nonconervaive diipaive *Parially uppored by CNPq X97 $5 Copyrigh 1997 by Academic Pre All righ of reproducion in any form reerved

2 13 BIAGIONI AND LINARES one repreened by he erm Ž u u and Ž u u x x, repec- ively Thi alo can be een a an hybrid of he well known fifh order Korewegde Vrie Ž KdV equaion or Kawahara equaion Ž Žee 9 and reference herein and he derivaive Korewegde Vrie KuramooSivahinky Ž KdV-KS equaion Ž Žee Our purpoe here i o eablih he global well-poedne of problem Ž 11 for daa in Sobolev pace, Ž Before geing ino he deail of our reul we fir make ome commen Thi work i moivaed by he reul obained by Bona, Biagioni, Iorio, and Scialom and he review made in 8 by Roiner of ome perurbed KdV equaion In he iniial value problem aociaed o he KdV-KS equaion, ie, 1 u Ž u u Ž u u x x, x,, Ž 1 už x, Ž x wih, aifying, wa udied and he global well poedne of Ž 1 for daa in Ž, 1, eablihed Alo he limi behavior of oluion of Ž 1 wa udied The heory in e ome analyic fac ha had appeared previouly a concluion in he numerical and heoreical work of Ercolani, McLaughlin, and Roiner 3 and Roiner 9 where he main concern wa he udy of raveling wave aracor for he model in Ž 1 A we commened above in 8 variou perurbed KdV equaion were conidered including our model of udy, problem Ž 11 The main queion raied wa abou he effec produced when he perurbaion end o zero We will ry o give ome anwer when he IVP Ž 11 i conidered in he enire line We can how ha IVP Ž 11 i in fac globally well-poed in Ž, I will be done by mean of a erie of a priori eimae ha allow u o exend local oluion previouly obained for any ime T In addiion, we udy he limiing behavior of oluion of he IVP Ž 11 when he parameer and approach zero So, in paricular, we how ha oluion of Ž 11 converge o oluion of he Kawahara equaion when end o zero in Ž When we le approach zero, he oluion of he problem Ž 11 hen converge o oluion of he derivaive 1 of he KdV-KS equaion in Ž, In he cae ha boh parameer and end o zero, oluion of he BenneyLin equaion will 5 converge o oluion of he KdV equaion in Ž The plan of hi paper i a follow Local well-poedne for he IVP Ž 11 i eablihed in Secion In Secion 3 we will find everal a priori eimae needed o exend he local oluion o global one and o udy

3 BENNEYLIN AND KAWAARA EQUATIONS 133 he limiing behavior of oluion of problem Ž 11 Some echnical reul ued in hi ecion will be included in he Appendix Secion 4 will be devoed o how he global well-poedne of problem Ž 11 The limiing behavior of oluion of he problem Ž 11 when and approach zero will be udied in Secion 5 and 6 Before leaving hi ecion we will inroduce ome noaion Noaion We will ue beide andard noaion in parial differenial equaion he following: If X and Y are Banach pace, BŽ X,Y will denoe he Banach pace of bounded linear mapping from X ino Y wih he uual operaor norm We denoe by Ž, he inner produc in L The norm in will be denoed by and will denoe he L -norm The Fourier ranform will be denoed eiher by F or by ˆ LOCAL RESULTS In hi ecion we will prove ha IVP Ž 11 i locally well-poed in Ž, Before doing o we ae ome properie of oluion of he linear problem aociaed o Ž 11, ha i, u u x u x Ž u u, x, ½ 1 už x, Ž x Le Ž 4 Q exp i 3 i 5 4,,,, and define Ž, F E Ž f Ž Q Ž Ž FfŽ PROPOSITION 1 Le, be gien Then Ž Ž Ž 1 E B,, for all and and aifie 4 4 E, Ž C e 1 Ž Ž 8Ž1 e 14, '

4 134 BIAGIONI AND LINARES for all Ž, where C i a conan depending only on Moreoer, he map Ž, E Ž, i coninuou wih repec o he opology of Ž Ž E Ž BŽL 1 Ž,, for all and and aifie 4 Ž 1 8 1, L E Ž f C e Ž f 3 for all f L 1 Ž, where C i a conan depending only on The map Ž, E Ž f i coninuou wih repec o he opology of Ž, Ž o 3 For any, he map, E, f define a C -emigroup in aifying 4 E Ž e,, where he norm i ha of BŽ Ž, Ž In paricular, if hen Ž, E Ž i he unique oluion of 1 in he cla, 1 5 C, : C, : Proof The proof of hi propoiion follow he ame line a he proof of, Propoiion and 3 o i will be omied The following reul concern he local well-poedne of he IVP Ž 11 in Ž,, ha i TEOREM Le,, and be fixed If hen here exi T TŽ,, and a unique funcion u CŽ, T : aifying he inegral equaion už, E Ž E Ž u už, d 4 Ž wih E defined a aboe,,, x Proof We eparae wo cae: 5 and 5 Conider 5 Le 4 X Ž T C,T : Ž : Ž E Ž M,,T and d Ž u, up u Ž Ž Noice ha X Ž T, T i a complee meric pace We will how ha for T ufficienly mall he map, E E d i a conracion in X T,, x

5 BENNEYLIN AND KAWAARA EQUATIONS 135 Le X T ; Propoiion 1 implie Ž E Ž, 1 E Ž Ž Ž, x d Ž T4 Ž58 C up Ž Ž e 1 T 5,T Ž T C M e CŽ T, 5 where CT i he erm in bracke Oberve ha CT ato we can chooe T ufficienly mall o ha he righ hand ide of 5 o i le han M I how ha : X Ž T X Ž T o o To ee ha i a conracion ake u, X Ž T So 1 Ž už Ž up u 1CŽ T L o,to 1 up už Ž už Ž CŽ T,To o T o 4 up už Ž Me CŽ T,T o o Ž T 1 4 Thu if T T i uch ha M e CŽ T 1 o 1 1, we have ha i a conracion in X Ž T 1 provided 5 A argumen andard of uniquene hen implie he reul in CŽ, T : Ž 1 The cae 5 follow by uing a boorapping argumen Moreover, a furher boorapping argumen how ha for any and, Ž Ž 5 uc,t : and u C, T : 1 1 and u aifie he equaion in Ž 11 LEMMA 31 3 A PRIORI ESTIMATES Conider he iniial alue problem Ž 11 wih Ž,,, Le u be a oluion of Ž 11 in CŽ, T : for

6 136 BIAGIONI AND LINARES ome T Then he following eimae hold, for T If i i he cae ha ue T4 Ž 31 1 up, if, Ž 3 ož / CTŽu u e if, Ž 33 ž / ž / ž / ž / 1 1 u P, TP, x 1 Ž u P, TP, 3 4, Ž 35 where he P are nondecreaing funcion of heir argumen j If we hae ct ux 1 1 TP5Ž,, e ctž1 e ct u e P Ž,T T 6 T 1, Ž 37 6 where P5 a Proof Muliply Ž 11 by u and inegrae over o obain 1 d u Ž u,u Ž u,u, d where he inner produc i ha of L Inegraion by par and he CauchySchwarz inequaliy hen imply d u u u u u d Inegraing he la relaion over Gronwall lemma we obain Ž 31,, where T, and applying

7 BENNEYLIN AND KAWAARA EQUATIONS 137 We urn o he eimaion of u for The inner produc in of Ž 11 wih u give 1 d u Ž u,u Ž u,u Ž uu x, u d 1 ux u Ž u,ux Ž u u u u u u u x 1 1 cu Ž u Ž1Ž u u cu Ž9Ž u Ž3Ž c Ž c c u u ž / c Ž4185 u, Ž 39 Ž435 where 1, are arbirary; chooing 1 14c, c, and uing Ž 31 we obain Ž 3 We urn o he eimaion of u for Equaliy Ž 38 implie ogeher wih an inequaliy due o Kao Žee 6 1 d u c u u u u u d cuu u 4 Inegraion over, and Gronwall lemma imply Ž 3 In order o prove Ž 34 and Ž 35, ue i made of he amilonian of he Kawahara equaion namely u uu x u x u x Ž 31 ž / Ž u ux u u dx Ž Equaion 11 can be wrien a u u u u, 31 x

8 138 BIAGIONI AND LINARES where 1 Ž u u u u Ž 313 i he Gaeaux derivaive of Muliplying 31 by u, inegraing over, and uing he fac ha we obain Ž u, Ž u, x ž / 1 Ž u u, u u u,u u u u x,ux ux u,u u,u u u x Inegraing over,, we ee ha ž / x 6 6 u u u dx dx Ž u x,ux ux d Ž u,u Ž u,u d x u u d Ž 314 Subiuing inequaliie Ž 71 Ž 77 from he Appendix ino Ž 314 lead o: Ž 1 In he cae, x 3 3 Ž u,u u x x x d Ž u,u Ž u,u d u u x d u u u dx dx

9 BENNEYLIN AND KAWAARA EQUATIONS Ž u 7 1 x 1 c Ž u d T Ž Le 4, 1, and,, be uch ha Ž for inance, 1, 9 Then, replacing i in Ž give u c1 T which implie 35 Inequaliy 34 follow from 35 and 76 Ž In he cae we have, from 314, 1 1 u u x x x u,u u d u,u u,u d x u u d u c Ž Ž 3 4 u d T Ž Ž

10 14 BIAGIONI AND LINARES Le, 1, and,, aifying From 316 we have u c TŽ and Ž 34, Ž 35 follow, a well To prove Ž 36 we ue he amilonian of KdV equaion, 3 1 u Ž x 3 u u dx I follow ha 11 can be wrien a where u x Ž u Ž uu u x, Ž 317 u Ž u u Ž 318 Muliplying Ž 317 by Ž u, inegraing over, and uing he fac ha we ge Ž u, Ž u x Ž u Ž u, u Ž u,u u Ž u,u x u u u,uu u,u ž ž x / / Ž u,u Ž u,u Ž u,u u x x x Ž u,u x Ž 319

11 BENNEYLIN AND KAWAARA EQUATIONS 141 Inegraing 319 from o i follow ha 3 3 u ux dx Ž u, u Ž u, u d 3 3 Ž u,u u d u,u d Ž 3 x x x x Now ubiue inequaliie Ž 85 Ž 88 from and Ž 78 ino Ž 3 o derive Ž 1 u 1 x Ž 1 3 x 3 3 u u u u u u u d 4 x 4 x x x 5 x u u d u d Ž 31 Seing 1, 3, 13, 1 we have u u x x u 3 u 7 u 14 4 u u d x 13 TP,, 4 u d Gronwall lemma implie Ž 36 Now we eimae u uing he nex conerved quaniy of he KdV equaion: Muliplying 11 by ž / u u 5uu x 3u dx Ž u u 5u 1uu 6u 3 4 x x x

12 14 BIAGIONI AND LINARES and inegraing over give Ž u Ž u,ž u u u 4 4 x 5 3 x Ž x ž / u 5u 1uu 3 6u, u u u Ž 33 Replacing Ž 89 Ž 815 from and Ž 79 Ž 711 ino Ž 33, we ge Ž u Ž u u u u u Ž u u Ž 1 u u Ž Ž 678 u Take , 5 16, and hen ct 4Ž u e ct Ž 1 e u u By inegraion from o and ubiuion of Ž 4, we have ž / 5 u 4 5uu 3u x dx dx ct Te 1 18 ct 1 e u d Replacing now Ž 816 Ž 817 from reul u u u u u ct cte ct 1 e u d Taking 14 and Gronwall lemma imply 37 1

13 BENNEYLIN AND KAWAARA EQUATIONS GLOBAL RESULTS In hi ecion global well-poedne of he IVP Ž 11 will be eablihed for daa in Ž, The local reul obained previouly combined wih he a priori eimae in Secion 3 will allow u o how he following reul: TEOREM 41 The IVP Ž 11 i globally well-poed in Ž,, Proof Le and le F be he map which applie ino he unique oluion of he IVP u CŽ, T : given by Theorem We hall how ha F i a coninuou funcion of in Ž Le, Ž, ufž, and FŽ If wu i follow ha 1 d 1 w Ž Ž u w x,w Ž w,ww d cw 54 w 34 Ž u w w w 53 cž w w Ž u w w for every So chooing ŽcŽŽu 35 he inequaliy above become d w Ž cž u w d Thu Gronwall lemma implie w exp T Ž for T For 5, he inegral equaion 4 and Propoiion 1, 3 imply,, 1 Ž4 Ž 3 8 e ½ e Ž Ž 5 wž, E Ž E Ž Ž u Ž d Ž u Ž 1 d T e C Ž T, up Ž u 1,T Ž 3 8 Ž wž, d

14 144 BIAGIONI AND LINARES Ž A generalizaion of Gronwall lemma ee 4, implie hen T4 wž, e E Ž, Ž 4 Ž58 Ž n where E z Ý z Ž n1 Ž denoe he gamma funcion n and / up Ž už, k Ž, k C1Ž T, ž 8,T If 5, he inegral equaion 4 and Propoiion 1, give,, x wž, E Ž E Ž Ž u Ž d x 1 e u ½ Ž Ž4 14 e 1 e ' ŽŽ 8Ž1 14Ž wž, T e up Ž u CŽ T, d, 14,T Ž Ž 43 5 d where T 14 T8Ž18 Ž T 4T e Ž T 1 e CŽ T, 14 ' We apply once again he generalizaion of Gronwall lemma menioned above o obain wih T4 wž, e E Ž Ž up už, Ž, C Ž T, Ž 34,T k k 43 which prove our aerion Thi, ogeher wih Theorem and he a priori eimae Ž 31, Ž 3 prove he global well poedne in Ž, of he IVP Ž 11

15 BENNEYLIN AND KAWAARA EQUATIONS CONVERGENCE TO SOLUTIONS OF TE DERIVATIVE OF TE KdV-KS EQUATION In hi ecion we will how ha for fixed he oluion of he BenneyLin equaion converge o oluion of he derivaive of he KdV-KS equaion when, ha i, Ž 1 TEOREM 51 Le Conider u C, T : Ž,, 1, he oluion of IVP 11 correponding o Ž Then Ž 1 lim u u exi in C, T : o and define a oluion of he IVP Ž 11 wih We ae a lemma, he proof of which may be found in 1 : LEMMA 5 Le here be gien,, and Then 4 1 Ž Ž4Ž1' 14 e C 1 e Ž 51 Ž for all, where C i a conan depending only on Moreoer, i follow from 51 ha 4 Ž 1 e d ž / 3 1 Ž e Ž Ž Proof of Theorem 51 Le and conider u1 and u wo oluion of he IVP Ž 11 wih equal o 1 and, repecively Uing ha u, u aify he inegral equaion we obain for w u u 1 1 Ž,, wž, E Ž E Ž 1 1 Ž,, xž 1 1 E E u d 1, x 1 x E Ž Ž u Ž u d Ž 53

16 146 BIAGIONI AND LINARES Now we eimae each erm of he righ hand ide We begin wih he fir one The mean value heorem combined wih Lemma 5 yield where Ž,, E Ž E Ž Ž 8 ˆ 1 1 e d 8 Ž 4 up e C 1 e Ž ž / Ž4Ž1 116 CŽ T,, Ž Ž T 1 ŽT4Ž1 116T CŽ T, C e Uing a imilar argumen we find Ž E Ž E Ž Ž u,, x ŽŽ 1 Ž ˆ1 1 e u d 4 1 ŽŽ u up e 1 4 u 1 1 1C 1 ž 5 Ž / ' ŽŽ 4Ž1 116Ž e 1 u 4 CŽ T, Ž ' Ž Ž 5 4ŽT T 16T CT, C T 1 e Then eimae 3 implie 1,, E Ž E Ž x Ž u C T, 56 ' '

17 BENNEYLIN AND KAWAARA EQUATIONS 147 Finally, eimae 5 give u, x 1 x E u u 1 E u u, 1 Ž4 Ž 3 8 Ž 1 1 L C e u u Ž Ž4 C e Ž u u wž, Ž 3 8 CŽ T, Ž wž, Ž 57 Combining Ž 53, Ž 55, Ž 56, and Ž 57 we have for 5 and T, 1 Ž 1 1 w, C T, Thu Gronwall lemma implie Ž 3 8 CŽ T, Ž wž, d wž, CŽ T,, E Ž, Ž 58 E i defined in, formula 54 So, we have ha 1 1 Ž58 and C T, 8 ž / up wž,, a, Ž 59 T 1 Thu he limi lim u exi in CŽ, T : Ž, 5 For 5 we only need o ue Lemma 5 in Ž 57 and Gronwall lemma o ge Ž 59 Thu we have proved ha he limi u exi in Ž Moreover, i follow ha u converge weakly o u 1 uniformly over, T In paricular, u Ž, i weakly coninuou and uniformly bounded o

18 148 BIAGIONI AND LINARES Since u i a oluion of Ž 11 i aifie už, už, o ½ 5 u u u u d x u x x x x o So combining he above obervaion wih he fac ha he map, T 3 Ž 4 4 u u u u x o x o x o x o i weakly coninuou we obain ½ 5 u, u, u u u d x uo 3 4 o o o x o x o x o o Ž 4 Ž 1 and herefore u AC, T : L, T : Ž o ence uo aifie IVP Ž 11 wih for almo every From he local exience reul for we have ha he IVP Ž 11 ha a unique oluion in Ž 1 C, T : correponding o daa Therefore uo coincide wih he rong oluion of IVP Ž 11, ha i, he derivaive of he KdV-KS equaion 6 CONVERGENCE TO SOLUTIONS OF TE KAWAARA EQUATION In hi ecion our purpoe i o udy he limiing behavior a of oluion of he IVP Ž 11 for fixed and if I come up ha in he fir cae oluion of he BenneyLin equaion converge o oluion of he o-called Kawahara equaion A before we follow cloely he argumen ued by Bona e al in Our reul read a TEOREM 61 Le and be gien If i he oluion of he IVP Ž 11 correponding o iniial daa, hen he limi lim exi in CŽ, T : and i he unique oluion of Ž 11 wih Proof Le and and be oluion of he IVP Ž 11 1 correponding o daa and differen value of 1 and of he parameer We conider w 1 ; hen i aifie w w ww w w w 1 x x x x 1 Ž 61 1

19 BENNEYLIN AND KAWAARA EQUATIONS 149 Afer muliplying 61 by w and inegraing wih repec o x we have 1 d wž, Ž w, Ž 1 x x 1Ž w,ww d Ž Ž w, Ž Ž Gronwall lemma implie hen 1 Ž c w c w 1 1 w Ž w Ž 1 4 c w w, ct up 1 1 T ½ Ž 1 5 exp ct up Ž 6 T for T Uing 31, 34, and 35 we have ha he -norm of 1 and remain bounded by a funcion of T and, independenly of 1 So, if 1, hen 6 implie ha lim exi in L, uniformly in, T To complee he proof we need o how ha verifie Eq Ž 11 o wih and ha CŽ, T : Ž o The fir par follow uing an argumen imilar o he one employed in he proof of Theorem 51 To how ha CŽ, T : we fir noice ha o lim inf Ž, hu he map, T Ž, o o i coninuou a wih repec o he -opology The coninuiy a and he uniquene of he oluion of he IVP imply righ coninuiy a Ž, T The lef coninuiy follow hen conidering he change of variable Ž, x Ž,x and he fac ha he reul above are ill rue under hi change of variable Thi complee he proof of he heorem We noice ha he above reul can be proved analogouly o he cae Ž, Now we preen a reul concerning he limiing behavior of oluion Ž 11 when boh and approach zero In hi cae we have convergence o oluion of he KdV equaion in 5 More preciely, 5 PROPOSITION 6 Le be gien If i he oluion of he IVP Ž 11 correponding o iniial daa, hen he limi lim, exi Ž 5 in C, T : and i he unique oluion of Ž 11 wih

20 15 BIAGIONI AND LINARES Proof Wihou lo of generaliy we aume max, 4 min, Conider he oluion and of he IVP Ž 11 1 wih daa correponding o value of he parameer 1, 1 and,, repecively Le w1 ; hen w aifie w w Ž ww w Ž Ž 1 x x x 1 x 1 Ž w w Ž Ž Ž Ž Nex muliplying by w and inegraing wih repec o x i follow 1 d wž, d ž / Ž 1 x w, x w,w w 1 Ž Ž w, w, x Ž c w w w 4 w Ž c w So Gronwall lemma implie wž, ct½ up Ž 1 1 T up Ž T ½ Ž exp ct up Ž 64 T 5 Uing 33, 36, and 37 we have ha he -norm of 1 and remain bounded by a funcion of T and 5 independenly of So, if, hen Ž 64 1 and he ame argumen ued previouly imply he reul

21 BENNEYLIN AND KAWAARA EQUATIONS APPENDIX ere we prove everal eimae ued in Secion 3, making ue of he GagliardoNirenberg and Young inequaliie: udx u u cu u x 17 u u 7 Ž 71 Ž u,u u u cu u x x x x 1 u u Ž 7 u,u u 3 u 1 u cu 198 u 58 L x 1611 u u 3811 Ž 73 u,u u u u u 3 u 1 u x c u u u u 37 Ž u c u u u u Ž 75 x 1 u c u u u u Ž 76 x u c u u u u Ž 77 x Ž u,u Ž Ž u uu, u x x x L L x x x x u u u u u u x x x x c u u u u u u u c u u u u 14 Ž 78 x 3 u,u 3uu,u u u u u x L L x x x 3 1 u u u u x c u u u u 1 Ž 79 L u,u Ž u u,u u u u x x x x 1 1 u u u u x x c u u u u 18 Ž 71

22 15 BIAGIONI AND LINARES Ž uu, u x Ž uu, u Ž u u,u x x u u u u u u x L L x Ž x x x c u u 1 u 3 u u 1 u 1 u c u u u u 18 Ž 711 REFERENCES 1 D J Benney, Long wave on liquid film, J Mah Phy 45 Ž 1966, J L Bona, A Biagioni, R Iorio, Jr, and M Scialom, On he Koreweg-de Vrie- Kuramoo-Sivahinky equaion, Ad Differenial Equaion 1, No 1 Ž 1996, 1 3 N M Ercolani, D W McLaughlin, and Roiner, Aracor and ranien for a perurbed periodic KdV equaion: A nonlinear pecra analyi, J Nonlinear Sci 3, No 4 Ž 1993, D enry, Geomeric heory of emilinear parabolic equaion, in, Lecure Noe in Mahemaic, vol 84, Springer-Verlag, Berlin, T Kao, On he Koreweg-de Vrie equaion, Manucripa Mah 8 Ž 1979, T Kao, On he Cauchy problem for he Ž generalized Koreweg-de Vrie equaion, Sud Appl Mah Ad Mah Suppl Sud 8 Ž 1983, S P Lin, Finie ampliude ide-band abiliy of a vicou film, J Fluid Mech 63 Ž 1974, Roiner, Applicaion of he Invere Specral Tranform o a Koreweg-de Vrie Equaion wih a Kuramoo-Sivahinky-Type Perurbaion, PhD Thei, Univeriy of Arizona, Roiner, A numerical diagnoic ool for perurbed KdV equaion, preprin 1 E Tadmor, The well-poedne of he Kuramoo-Sivahinky equaion, SIAM J Mah Anal 17 Ž 1986, J Topper and T Kawahara, Approximae equaion for long nonlinear wave on a vicou fluid, J Phy Soc Japan 44, Ž 1978,