An Elementary Approach to a Model Problem of Lagerstrom

Transcription

1 An Elmntay Appoach to a Modl Poblm of Lagstom S. P. Hastings and J. B. McLod Mach 7, 8 Abstact Th quation studid is u + n u + u u = ; with bounday conditions u () = ; u () =. This modl quation has bn studid by many authos sinc it was intoducd in th 95s by P. A. Lagstom. W us an lmntay appoach to show that th is an innit sis solution which is unifomly convgnt on < : Th st fw tms a asily divd, fom which on quickly dducs th inn and out asymptotic xpansions. W also giv a vy shot and lmntay xistnc and global uniqunss poof which covs all > and n. Intoduction Th main poblm is to invstigat th asymptotics as! of th bounday valu poblm u + n u + uu = () with u () = ; u () = : () Ou intst in this poblm, oiginally du to Lagstom in th 95s [3], was stimulatd by two cnt paps by Popovic and Szmolyan [4],[5], who adopt a gomtic appoach to th poblm, and th a many paps which us mthods of matchd asymptotics o multipl scals, with vaying dgs of igo. (Th paps [4] and [5] giv lists of fncs.)

2 Ou point is that it is possibl to giv a compltly igoous and lativly shot answ to th poblm without making any appal ith to gomtic mthods o to matchd asymptotics. W can xpss th solution as an innit sis, unifomly convgnt fo all valus of th indpndnt vaiabl, fom which on can ad o th asymptotics as!. Lagstom cam up with th poblm as a modl of viscous ow, so his wok cntd on n = o 3, but th innit sis can b obtaind fo any al numb n. What n contols is th at of convgnc of th sis. W stat in sction two by showing that th poblm dos indd hav on and only on solution fo any n and any >. This is basd on a simpl shooting agumnt plus a compaison pincipl. Th poof is vy much shot than th on fo small in [4], and givs global uniqunss. Bfo pocding, w mak th futh maks. Th st is that, fo any solution of () () ; u > on [; ). Fo if at any point u = ; thn also u = and so th solution is constant and could not satisfy both bounday conditions. Ou solutions a thfo monoton, with u > ; < u <. Scondly, th is an obvious distinction btwn n > and n. If n >, thn th poblm () () has th solution u = n ; (3) so that th solution with small is psumably som sot of ptubation of this. If n thn th is no such solution. A consqunc is that th convgnc as! is mo subtl whn n thn whn n >. Ou analysis will show that th is littl pospct of discussing th bhavio fo small if n < ; but fotunatly w can handl all n. Finally, ou mthods a not stictd to Lagstom's poblm. Hinch, who givs a cla discussion of th poblm in [], gos on to intoduc a \tibl poblm which h gads as vn mo complicatd than Lagstom's. W show in ou nal sction that ou mthods dal qually wll with that, and vn with gnalizations of it, with littl incas in complxity. W not that it is claimd in many cass that th mthod of matchd asymptotics, whil phaps dcint in igo, is at last cint in xcution, but that claim is dubious fo this last xampl.

3 Existnc and uniqunss Pvious xistnc and uniqunss poofs hav bn valid only fo small ; and giv just local uniqunss. A much stong sult is asily povd. Thom Th xists a uniqu solution to th poblm () () fo any > and n : Poof: W us a shooting mthod, by considing th initial valu poblm u + n u + u u = (4) u () = ; u () =c (5) fo ach c > : It is asy to s that solutions to this poblm a positiv and incasing, and xist on [; ). W wit th quation in th fom so that, on intgation, n u + u n u = ; (6) u () = s n c R s udt ds: (7) Fo n ; (7) implis that ith u () o u () c ( ). Lt = min ; c ( ) Thn, fo > ; u () < c s ds + c (s ) ds: (8) n Fom this it follows that u () is nit. Also, fom (7), u () = c s n R s udt ds c so that u ()! u () xponntially fast. s n R s udt ds; Fom (8) ; w s that u () < if c is sucintly small. Equation (7) futh implis that if u on [; ], thn u () R c ds; and this givs a contadiction s k if c > R s k ds : (9) 3

4 Hnc, in this ang of c, u () >. W thfo know that th a c and c ; with < c < c, such that if c = c thn u () < ; whil if c = c ; thn u () >. Futh, fo any > R > ; u () = u (R) + c If n > th scond tm is boundd by by c R (s R s n R s u(t)dt : c ; whil if n ; it is boundd (n )R n ) ds: Hnc fo any n this tm tnds to zo as R! ; unifomly fo c c c. Sinc u (R) is a continuous function of c; fo any R; it follows that u () also is continuous in c. Hnc th is a c with u () = ; giving a solution to () () : Fo uniqunss w again us th fom (6) ; which w can intgat to obtain n u () = c n u (n ) + s n u ds: () Fom this it follows that if c > c ; thn th cosponding solutions satisfy u > u on (; ). Howv this dos not quit show th dsid uniqunss. Suppos that th a two solutions of () () ; say u and u ; with u () = c > u () = c : Thn fom (), n (u u ) = (c c ) n u u (n ) + s n u u ds: () W hav shown that u! ; u! ; and a assuming that u! ; u! ; all limits bing at an xponntial at. Futh, c > c ; and u > u > on (; ). It follows that as!, th lft sid of () tnds to zo and th ight sid to a positiv limit. This contadiction povs uniqunss. Rmak. Th xistnc thom in [4] has on addd pat. It is shown th that as! ; th solution tnds to a so-calld \singula solution obtaind by taking a fomal limit as! : S [4] fo dtails. This limit sult follows fom ou igoous asymptotic xpansions givn blow. Rmak. Th would sm to b no diculty in xtnding th xistnc poof vn to n < ; but th uniqunss poof dos us ssntially th fact that n : W tun to this point at th nd of sction 5. 4

5 3 Th innit sis (with n ) Stating again with (), and u () = ; w obtain n u = B R u(t)dt () fo som constant B. Sinc u () =, () implis that u () is xponntially small as!. Hnc w can wit () as so that u n u = C R (u )dt ; = C t t R t n (u Stting =, t =, and s = ; w obtain u () = C n n )ds dt: R (u() )d d; (3) wh w us th agumnts and to indicat that w man th scald vsion of u. H C is a constant satisfying = C n n R (u )d d: (4) Sinc fo ach th is a uniqu solution, this dtmins a uniqu C; dpndnt on. Bcaus u < ; w s that th -intgal tm in (4) tnds to innity as! ; so lim! Cn = : By dintiating (3) w a ld to xpct that C is clos to n n >. Sinc both xponnts in (3)hav ngativ xponnts, that if E n () = d; n thn fo small if w can immdiatly say ju () j < C n E n () : (5) 5

6 Fo puposs of futu stimats, w mak th obvious mak that 8 < O ( n ) as! if n > E n () = O (log ) as! if n = : O ( n ) as! : (6) Hnc if n > th is a constant K such that E n () K min n ; n : (7) Th mthod now is to wok fom (3:). As obsvd bfo, sinc u () is xponntially small as!, th intgal tm R (u ) d convgs. Hnc, fo givn > and > ; and any, ( u () = C n (u ) d + ) (u ) d d; n (8) wh th sis in th intgand convgs unifomly fo <. In fact, w will nd to us this sis fo all. Thus w nd to chck its convgnc in this intval. This follows fom (5) and (6) ; which imply that fo any ; if n ; thn (u (s) ) ds < Cn E n (s) ds (9) and n E n (s) ds = o () as! if n > O () as! if n = : Hnc fo n > and any C; th sis in th intgand of (8) convgs unifomly on [; ): Now st = C n E n (s) ds: W not that, if n > ; thn! as! ; whil if n = ; thn! as C!. W pocd to solv (8)by itation. Thus, th st appoximation is, fom (9) ; u () = C n d + O ; n 6

7 and w obtain th scond appoximation by substituting this back in (8) : Rpating this, w ach (C n u = C n E n + (C n ) R ) 3 R + (Cn ) 3 R n R R R E n n d d R E n n d d s n s R s E n dt ds dd + O ( 4 ) ; as!. To obtain C; w nd to b abl to valuat ach of ths tms fo small (in paticula, fo = ), and this is a matt of intgation by pats. Thus, fo non-intgal n, E n () = d = n n n + n d n () = n n + n E n ; () and this can b patd to giv E n as a sum of tms of th fom c k k and E n p ; until < n p <. Thn E n p = d n p = (p + n) n p d d n p and w can thn continu to intgat by pats as fa as w lik. (If n is an intg, w will ach R d, which intoducs a logaithm.) Thus E n () can b xpssd as a sum of tms of th fom c k k, and so obviously th sam is tu of En, with in plac of. Also, E n () d = d d n = d j n + d n = E n E n ; () so that R E n d can b xpssd as th sam typ of sum. Hnc th scond tm in () givs a sum of tms of th fom E k () and th thid and fouth tms a sum involving E k (3). 7

8 W now cay th pocss though in th most intsting cass, n = ; 3. 4 Th cas n = Whn n = w a intstd in E () = d = log + = log + log d log d log d = log (log ) + O log ; fo small ; = log + + O log : (3) (S, fo xampl, [], Chapt.) Also, fo futu puposs, using () w obtain E () = E () (4) = + log + ( ) + O log as! : (5) Looking now at () ; with = ; w s that as! ; C log! and C = log + O Hnc th sis in () is in pows of : log log! : Also, w will wok ou appoximations (in od to compa th sults with thos of Hinch) to od, so that (fo xampl) log ( ) u = a () + b () + O log 3 log log 8

9 fo any xd valu of ( of od ). This, as w shall s, ncssitats nding ( C = + A B + + O log log log ) 3 ; and quis us of all th tms in () : log With this in mind, w look at th scond tm of (). Thus fom () ; so that th scond tm is C E d = C Fom (3), th scond tm is thfo as!. E d = E + ; (6) E d = C E j d d = C E () + E () : (7) C (log + log + O ()) = C ( log log + O ()) (8) In th thid and fouth tms of () w nd only th lading tms, i.. w can igno th quivalnt of + log in (8). Using (6) th thid tm bcoms C3 E d = C3 (log + O ()) as! : (9) Finally, in th fouth tm, th intgand in th -intgal is just th scond tm, (as a function of ), so that fom (6) ; th fouth tm is M = C 3 E () + E () d d: (3) 9

10 It is sn fom (6) that fo any ; R E () d convgs. Hnc w can wit th inn intgal abov in th fom R R ; and it follows that M = C 3 fk + ()g d wh K is a constant, is boundd and () = O ( log ) as!. follows that M = C 3 (KE () + O ()) as! : W can valuat K using (6) and (3): E () d = E (u) dv = ; E () d = E It futh d = lim f E () + E ()g = lim (log log ) = log : (3)!! Hnc, fom (3), th fouth tm of () is C 3 fe () (log ) + O ()g = C 3 f(log ) log + O ()g as! : (3) Now stting = and using (8) ; (9), and (3), w obtain that as!. log and wh = C ( log + O ()) + C ( log log + O ()) + C3 (log + O ()) C 3 f(log ) log + O ()g Hnc, = C C = log log! + C A log! + log 3 +C 3 log + B log 3 + O log 4 + A = ; log +O log 4 ;! ; (33) B A A + ( + log ) + 3 log = :

11 Hnc, A = + B = + + log : Thus, fo xd ; of od ; w hav, with = log ; u = ( +! ) log (log + log + ) 3 + ( + ) + ( log log log ) + 3 log (log + log ) + O 4 ; 3 so that, aft cancllation, u = log + log + O 3 : This is th \inn xpansion. Fo th \out xpansion, i.. xd ; of od ; w hav u = E () E () E () + O 3 : Ths sults a in accodanc with thos of Hinch and of oths on this poblm. 5 Th cas n = 3 H w a intstd in (fom (3) and (4)) E () = E () = + log + ( ) + O log as! : Thus, th st tm on th ight of () valuatd at = is C + log + ( ) + O as! :

12 Th scond tm is (C) Fom (4) w s that whil fom () ; = (C) = (C) E In all, th scond tm is E () E () E d d E () d E () d + + E d Ed : E d = + log + O (log ) as! ; E d = E E = + log + + O ( log ) ; E d = log + + (C) log + + O log : + O log : It is adily vid that th thid and fouth tms in ()giv O o nc 3 3 log ; which is ngligibl. Thus, valuating () at =, w hav = C + log + + (C) log + + O C 3 log ; so that C = log ( + ) + O log : Thn, fo xd ; of od ; w hav u = ( log ( + )) + (log + log ) + + log + log + + O log ;

13 u = + O log : log log + log + ( ) Fo xd ; of od ; w hav u = ( log ( + )) E () + E () E () E () Ed + O 3 : (34) Again, ths sults a in agmnt with thos of Hinch, and oths, although (34) givs on tm futh. Rmak 3. It is of intst to consid what happns whn n < ; sinc, at last fo n, th still xists a uniqu solution. Th quation () is still valid at =, but sinc E n () is no long singula at = fo n <, () with = bcoms mly an implicit quation fo C n. This tlls us that C!, sinc n!, but w no long gt an asymptotic xpansion. In paticula, it is no long obvious that C is uniqu. Of cous, w know this fom Thom if n : Fo n <, this uniqunss may fail. 6 Hinch's tibl poblm In [], Hinch intoducs a futh xtnsion of Lagstom's poblm. This is with u + n u + u + u u = ; (35) u () = ; u () = : Existnc fo any n and any > can b povd as bfo. W focus h on th asymptotics. In Hinch's wok it is sn that th mthod of matchd asymptotic xpansions is mo complicatd in this cas than in th standad Lagstom modl. W can in fact tat a gnalization which causs no futh dicultis, u + n u + f (u) u + u u = ; (36) 3

14 with th sam bounday conditions. As makd in th intoduction to th pap, th solution will ncssaily hav u > so that conditions on f (u) a ncssay only fo u. W qui only that f b continuous and positiv in this intval. Thn (36) can b wittn as ( n u ) n u + f (u) u + u = ; so that log n u = F (u) udt + A fo som constant A; wh This bcoms o, on intgation, wh G (u) F (u) = F (u) u = G () = C G (u) = u f (s) ds: C R n (u )ds ; u t t R t n (u F (u) dv: )ds dt; In od to kp th manipulations simpl and ct compaisons, w will consid th cas considd by Hinch, wh f (u) = ; F (u) = u; G (u) = u : Thn, with =, t =, w hav and witing w gt u = C n u = C n n u = u u (u ) ; n R (u )d d; (37) R u u (u )d d: (38) 4

15 As in sction 3, w can intgat by pats, and sinc u in u < ; u w will dvlop a convgnt sis as bfo. To gt th st th tms (ncssay to giv Hinch's accuacy whn n = ), w hav fom (38) that u = C n n ( u u (u ) d + u u (u ) d + ) d: As bfo, sinc u! xponntially fast as! ; th sis in th intgand convgs unifomly fo lag, so that (39) is valid fo lag : But again w nd to xtnd it down to =. Fom (37)w hav u C n E n () and so th convgnc poof is th sam as that pcding () : Bfo pocding futh with n =, w mak a coupl of maks about th simpl cas n >. Thn, as w saw in subsction 5, only two tms a ncssay to giv th quid accuacy, and thn (39) givs u u = C n u n u (u ) d + and sinc u appas in what is alady th highst od tm, w can plac it u by its limit as u! ; i.. : Thus w gt, to th quid od, C u = C n n E n n ( u ) dd: This, apat fom th facto ; is th sam quation as w dalt with in sction 5 (with u in plac of u ), and th solution can b wittn down fom th. (If w had a gnal function f in plac of ; w would gt F (u) F () = C n E n C n F () f () n F (u) F () d d:) Tuning now to th cas n = ; and F (u) = u; w nd th tms on th ight of (39) : Thus, u u = (39) (u ) + O ( u ) as u! : (4) 5

16 W follow th mthod usd just bfo () and obtain fom (39) that u = C n E n + Cn E n n d d + C n 3 E n n d d + C n 3 E n n d d C n 3 s n s s E n n dt ds dd + O 4 = C n E n + F + F + F 3 + F 4 + O 4 ; say. As bfo, if n > thn this is valid fo any C as! ; unifomly in ; whil if n = ; it is valid as C! : Fo n = w can continu to follow th agumnt in sction 4. Thus, as! ; F = C ( log log + O ()) F 3 = C3 (log + O ()) F 4 = C3 [(log ) log + O ()] : Th tm F did not appa bfo. xpansion and this is Only th highst od tm is ndd fo ou C3 d Ed: Now Thus, Ed = E + = E d E d = log ; fom (3) : F = C3 E log + O log = C3 ((log ) log + O ()) ; 6

17 and, valuating at = ; w hav = C (log + + O ()) C (log + + log + O ()) + C3 log ( log + log + ) + O C 3 ; log = C + C3! log 3 4 log + O C C log! + log log C + O log C 3 + O : log Hnc if thn C = + log A log + B log 3 + O log 4 ; B A + ( ) ( ) + A ( + log ) ( ) = ; A = ( + ) ; A ( ) + ( )3 (3 4 log ) = : W can of cous calculat B; but in fact its valu will b b ilvant to th lvl of appoximation that w tak. Thn, fo xd ( of od ), w hav, with l = log ; u = ( ) l + + B= ( ) + (log + log + ) l l ( 3 + ( ) l + + ) ( log log log ) l 3 + ( ) 3 (3 4 log ) (log + log ) + O l 3 l 3 ( ) log ( ) log = O l 3 : (4) l l (Not that th dnitions of A and B w such that u = at = up to and including od l ; so that to that od th can b only tms in log ; not 7

18 constant tms. W do not nd th xplicit valu of B.) To obtain u; w hav to invt, so that u = log + ( ) log + log + O l 3 : l l Fo xd ; of od ; w hav u = l Thus + + ( ) E ()+ E l l () E () +O l 3 : u = (u ) (u ) + = + + E () + l l ( ) ( ) (E () E ()) l E () l + O l 3 : (4) Again, ths sults a consistnt with thos of Hinch, xcpt that Hinch has an algbaic mistak which in (4) placs + by +. 7 Rmak on \switchback Stating with Lagstom, th tms involving log in th inn xpansions hav bn considd stang, and dicult to xplain. Thy a oftn calld \switchback tms, bcaus, stating with an xpansion in pows of, on nds inconsistnt sults which a only solvd by adding tms of low od, that is, pows of log. Th cnt appoach to th poblm by gomtic ptubation thoy xplains this by fnc to a \sonanc phnomnon, which is too complicatd fo us to dscib h [4],[5]. In ou wok, th ncssity fo such tms is sn alady fom th quation (3) and th sulting xpansion (8) : ( u () = C n (u ) d + ) (u ) d d: n 8

19 In th xistnc poof it was sn in (9) that C = O () as!. On th ight of (8) th st tm is simply C n E n () ; and th simpl xpansions givn fo E and E show immdiatly th nd fo th logaithmic tms. Th is no \switchback, bcaus th pocdu dos not stat with any assumption about th natu of th xpansion. Rfncs [] Edlyi, A. t. al., High Tanscndtal Functions, Vol., McGaw Hill, 953. [] Hinch, E. J., Ptubation Mthods, Cambidg Univsity Pss, 99. [3] Lagstom, P. A. and R. G. Castn, Basic Concpts in Singula Ptubation Tchniqus, Siam Rviw 4 (97), 63-. [4] Popovic, N. and P. Szmolyan, A gomtic analysis of th Lagstom modl poblm, J. Dintial quations 99 (4), [5] Popovic, N. and P. Szmolyan, Rigoous asymptotic xpansions fo Lagstom's modl quations, a gomtic appoach, Nonlina Analysis 59 (4),