Blow-up and Extinction Phenomenon for the Generalized Burgers Equation

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1 Snd Ordrs for Rrins o rrins@bnhamscinca h On Cybrnics & Sysmics Journa On Accss Bow-u and Exincion Phnomnon for h Gnraizd Burgrs Euaion Chn Ning * ian Baodan and Chn Jiian Schoo of Scinc Souhws Univrsiy of Scinc and chnoogy Mianyang 6 China; Souhws Univrsiy of Scinc and chnoogy Mianyang 6 China Absrac: In his ar h auhor considr condiions of ha iniia boundary vau o sudy h dynamic bhavior for Gnraizd Burgrs uaion and giv hs sysm for ha nw condiions of bow-u and Exincion Phnomnon of hs uaion wih in rich using nginring Such as hs horis and mhod can b usd in ohr simiar sysms hy hav a wid rang of aicaions Fina som numrica simuaions ar carrid ou o suor hs nw rsus h auhors xnd dynamic bhavior and h criica vau o coninuum mor rvious work [3 9 3] for in-dh achiv for ay vau Kywords: Bow-u and xincion boundary vau robms gnraizd burgrs uaion noninar sourc INRODUCION AND PRELIMINARIES Lik Burgrs uaion and Schrodingr uaion is on of h mos modrn scinc univrsa uaion as dscrib h noninar wav uaion of h corrsonding ffc of h disrs-on and inracion mod in maria mchanics and fuid mchanics ar ay vau in hysics ara and aso coninuous sochasic rocss in h acua robm in many fids such as hs horis and mhod can b usd in ohr simiar sysms hy hav a wid rang of aicaions Johanns Burgrs firs by using of bow noninar uaion in yar 948 for h u + uu = u x xx ( whr > dno h fuid fow us dscrib ha inracion bwn h convcion and h ffciv sns of fow for h noninar sourc mod Rcny is dynamic im snior bhaviour and is anayica souion and numrica souion wr invsigad o sudy If h burgrs uaion was noninar uaion in h foowing of iniia boundary vau robm u + uu = u a < x < b > x xx u( x = ( x a < x < b u( a = u( b = > ( which > as h viscosiy cofficin (x for a givn funcion Baman gav ou sab of souion Many robms ar basd on h Burgs uaion basic mods o xnsion and aicaion [] manwhi Burgr uaion and Navir- Sok uaion can b hough of as h aroxima uaion [] mak i in h noninar wav gas dynamics h shock wav and coninuous sochasic rocss has a broad aicaion sac c (s [-5] AIkis S rsnov sudy in h foowing of iniia boundary vau robm ( ( u u + g x u u = u+ f (3 formua a condiion guaraning h absnc of h bow-u of a souion and discuss h oimaiy of his condiion [3] In aricuar funcions f ( u = u u u and f ( u = saisfy his condiion E (3 wih f ( u = u ariss in many aicaion (s [4-7] E ( wih g( x u and i is w known ha if f ( u is sur-inar h Phnomnon of h souion bowing u and xincion may occur (s [8-] Concrning h rvniv ffc of h inar gradin rm ( au + b u x ( i= or a s [] Du o h fac ha aramr is ofn sma w wi considr h hfu infunc of h diffusing rm u and u (in xincion; concrning h rvniv ffc of inar and noninar diffusion s [8 ] HE EXINCION PHENOMENON HE BUR- GERS EQUAION n L b a boundd domain of R having sufficiny smooh boundary n b our norma dircion and n u = / x = ( x [ + ha 3 = ( i= i 874-X/4 4 Bnham On

2 94 h On Cybrnics & Sysmics Journa 4 Voum 8 Ning a W considr Burgrs uaion wih highr-ordr rm as foows n u + g( x u u = u+ f( u Q = ( R ( ( i u x n = x i = / ( ( ( u x + u x / n = ( x ( ux ( = u( x x > whr ( ( = + ( + ( g x u a x u a x u b x ( = f u cu c u c u Dfiniion: If hr is ux ( of robm (-(4 saisfy: for souion ( ( [ u( x x u x x ( (3 (4 ( [ ] hn h robm (-(4 wih souion ux ( b cad xincion in fini im W wi giv h horm o xnd som rsus [9] w hav h foowing rsus as mma u x W L + hn for Lmma [] If > xiss C > such ha hods inuaiy: k + ( ( + ( + u C u u N ( + 4( + whr 4+ N ( sac L ( k = N = n (5 imis norm of Proof From Gagiardo-Nirnbrg inuaiy [9] and Hodr inuaiy ha w asy g h concusion horm 3 Assum ha = min a ( x > min a ( x > max ( a a = > hr xiss and saisfis foowing inuaiy: + + a u u x dx + a u u x dx u dx max { } + ( ( max a a u x dx+ u x dx hn u ( x ( L ( L L 4 u x (6 h souion ux ( of robm (-(4 wi b wih xincion in fini im and wih foow dcay sima: ( k k ( * u x dx u x dx C ( k [ u ( x dx = [ (7 Proof By u( x muiis h boh sid of ( and ingraing on for x o combin Grn s formua and boundary condiions w hav: u x dx + a u + u( x dx+ a u + u( x dx + = u dx+ u x dx By (8 and (6 w hav d d u ( x dx { } + + max (8 max a a u x dx u dx Uiiz mma w obain max sc suiing aram- k = ( n { } 4 n( max { } + r * C w obain ha bow inuaiy: d * k udx C ( u ( x dx d By ingra abov inuaiy for and noic iniia condi- u x = u x I is (7 hrfor h souion of ion (-(4 is xincion in fini im and has h rsu of simas ha coms h roof of horm 3 3 HE BLOW-UP PHENOMENON FOR HE BUR- GERS EQUAION Considr h foowing uaion of [3] u + g( x u u = u+ f ( u n in Q = ( R (3 Coud wih iniia and boundary condiions u x = x for ( x Q \{( x : x } Hr > and ar consans ( = (3 g = g g g g = g x u i = n n i i Assum ha g a( x u b( x a( x = a > min Q = + and k

3 Bow-u and Exincion Phnomnon for h Gnraizd Burgrs Euaion h On Cybrnics & Sysmics Journa 4 Voum 8 95 whr ab C( Q and h osiiv consans ar such ha y Rfor any y R If h souion is nonngaiv h as assumion is unncssary Concrning h noninar sourc f ( u w assum ha f f for a and such ha (33 L us formua our rsu Dno b = b( x min m = max u hr is h araboic boundary of { } Q Q i = Q \ x : x Wihou oss of gnraiy suos ha h domain is ying in h sri < x < W wi rov h goba (i for arbirary > cassica sovabiiy of robm (3 (3 undr h foowing assumion: hr xiss a consan M m such ha f M + bm a M + Lmma 3[3] Assum ha ( ( ( C g x u C Q i M M C ( (34 f u M M If condiion (33 and (34 ar saisfid hn for an arbirary > hr xiss a uniu cassica souion of robm (3 (3 such ha u x M (35 Morovr considr dynamic acion in arg im wih comound rm of gnraizd Burgrs uaion Considr h auxiiary uaion ( ( ( ( u+ a x u + a x u + b x u = u+ u u u + in( W sudy h mos yica cas: w omi subscri in x and : ( ( ( ( u+ a x u + a x u + b x u u ( ( = u + u u u + in xx Coud wih iniia and boundary condiions ( = ( x u x for ( x ( ( ( {( x x( } \ : x (36 (37 whr a a bc ( ( and h osiiv consans ar such ha y y R for any y R Undr h basd of corrsonding condiion and combin mma 3 w wi rov h goba cassica sovabiiy of robm (36 (37 undr h foowing assumion: hr xiss a consan M msuch ha + min { } + M + bm a + a M W wi suy h foowing horm: horm 3 Assum ha C ( (38 ( au + au + b C (( ( ( M M + ( + u u u C ( M M + If condiion (33 and (38 ar saisfid hn for an arbirary > hr xiss a uniu cassica souion of robm (36 (37 such ha u x M whr (39 ( ( \{( x : x } = max { } Proof L a ( x = a a ( x = a b( x = b = M M M ( = ( + f u u u u + ( + u + au + a u + b u = u + u u + u in x xx ( ( (3 W assum hr is ha u + and u + ar dfind ohrwis w ak u uand u u I is known ha hr xiss a goba souion of robm (3 and (37 wihou smanss rsricions on iniia daa for + min + whn and zro boundary condiions morovr if > + whn = (or { } = ; (or > max + hn a fini bow u occurs if h iniia daa is sufficiny arg L us ay our Lmma3 o E (39 If w considr wo cas (i and (ii: { } (i < + whn = hn condiion (38 is aways fufid wih M m b a a < + ( = ( + { } = max su / + + < min + whn ; Simiar cas (i hr wih (ii { }

4 96 h On Cybrnics & Sysmics Journa 4 Voum 8 Ning a M + su{ b } = m a + a min { } ( min { } max{ } max < + = + Combin h wo cas and as a consunc for arbirary daa hr xiss a goba souion saisfying h sima u max M M { } Nx considr anohr wo cas wih invrs form: (i Suos ha > + = Condiion (38 bcoms a smanss rsricion In fac rwri (38 in h form M M m such ha a + a Obviousy if + { b} max = a + a m + = max{ b} hn (38 is fufid wih M = m and hr xiss a goba souion such ha u m > max + whn Simiar cas (i hr M m ha is (ii L ha { } + { } m a + a /max b = max min { } { } hn (38 is fufid wih M = m and hr xiss a goba souion such ha u m Now considr h wo criica cas Cas (i = + whn = : + ( u + ( a + a u + b u = u + ( u u x xx For h funcion hav v u (3 = wih μ / ( a a ( μ v + a + a v + bv v = v + + xx ( μ c v = + w x (3 v= for x = { = x } < x=± { } Cas (ii = { } + whn ; min max{ } min { } u + ( a + a u + b u max{ } min { } = u + u u xx Simiar cas (i w hav x μ v + a + a v + bv v = u + + ( c v { } { } μ = max min xx v= for ( x = { = x } { < x=± } x (33 (34 Condiion (38 for E (3 ak h form M m μ such ha 4 ( μ + c M + μ M a + a M μ + Obviousy his condiion fufid wih { } su 4 ( μ + c μ M = max m a + a μ = / ( a + a and h sima u( x M μ is obaind As a consunc h horm guarans h goba sovabiiy of robm (3 (37 and h souion saisfis u x M μ x h sima Simiar cas (i hr M M > w hav M m μ and M m μ such ha 4 ( μ + c M + μ M ( max { } + max { } a a μ + M + Obviousy his condiion fufid wih { μ + c μ} su 4 M = max m a + a ( a + a = ( { } μ = / max + and h sima u( x M is obaind As a consunc h horm guarans h goba sovabiiy of μ

5 Bow-u and Exincion Phnomnon for h Gnraizd Burgrs Euaion ab h On Cybrnics & Sysmics Journa 4 Voum 8 Aking h aramrs of Burgr-E (36 a a b 8 ab 97 c c aking h aramrs of Burgr-E (36 a a b 8 robm (33 (37 and h souion saisfis h sima u ( x M Undr (37 w can show h rnd of dynamics bhavour of Burgr-E (36 wih aking h aramrs in ab (S Fig By (i and (ii w obain sima u ( x {M M } μ = / ( a + a According o cas ab W com his roof of horm CONCLUSION his aric mainy sudis h gnra dynamics bhaviour of Burgr-uaion of basing and Exinguishing hnomnon o discuss hs cas hrough crain aramrs vaus Such uaions aways xhibi a rich hnomnoogy aracs many anion in nginring mchanics maria mchanics and fuid mchanics wih aicaion vau h auhors xnd dynamic bhaviour and h criica vau o coninuum mor rvious work [ ] for in-dh achiv for ay vau CONFLIC OF INERES Fig ( h souion found by numrica ingraion of Burgr-E (36 dscriion aking h aramrs in ab h auhors confirm ha his aric conn has no confic of inrs ACKNOWLEDGEMENS his work is suord by h Naura Scinc Foundaion (NoZB9 of Sichuan Educaion Burau and h ky rogram of Scinc and chnoogy Foundaion (NoZd7 of Souhws Univrsiy of Scinc and chnoogy Fig ( h souion found by numrica ingraion of Burgr-E (36 dscriion aking h aramrs in ab 4 NUMERICAL SIMULAIONS In ordr o iusra our horica anaysis numrica simuaions ar aso incudd in aid of MALAB 7 Mos of aramrs ar aking foow wih from abs and According o cas ab : Figur dscriion aking h aramrs (S ab Undr (37 w can show h rnd of dynamics bhavour of Burgr-E (36 wih aking h aramrs in ab (S Fig Figur dscriion aking h aramrs (S ab REFERENCES [] [] [3] [4] [5] [6] [7] Fchr C A Burgrs uaion: a mod for a rasons in: Noy (d Numrica Souion of Paria Diffrnia Euaions NorhHoand Amsrdam 98 Karman V I Noninar Wavs in Disrsiv Mdia Prgamon Prss Nw York 975 AIkis S rsnov A condiion guaraning h absnc of h bow-u hnomnon for h gnraizd Burgrs uaion Noninar Anaysis No Gurs A Van Bn H Simifid hydrodynamic anaysis of surfuid urbuncin H II: inrna dynamics of inhomognous vorx ang PhysRvB Rao Ch S Sachdv P L Ramaswamy M Sf-simiar souions of a gnraizd Burgrs uaion wih noninar daming Noninar Ana RWA No Sun X Ward M J Masabiiy for a gnraizd Burgrs uaion wih aicaions o roagaing fam frons Euroan J A Mah rsnov AIS On h gnraizd Burgrs uaion NoDEA noninar diffrnia uaions A

6 98 h On Cybrnics & Sysmics Journa 4 Voum 8 Ning a [8] Samarskii A A Gaakionov V A Kurdjumov S P Mikhaiov A P Bow-u in Quasiinar araboic uaions In: d Gruyr Exosiios Mah Vo9 (d Gruyr Brnin [9] CHEN N Exincion of souion for a cass of Ginzbur-Landau racion-diffusion uaion in ouaion robms Jorna of Biomahmaics [] Ladyznska O A Soonnkav V A ra-sva N N Linar and uasi-inar uaion of araboic y Amr Mah Soc Proridn R I 968 [] rsnov AIS h Dirich robm for scond ordr smiinar iic and araboic uaion Diffr Eu A [] rsnov A I S A rmark on h orous mdium wih sourc A Mah L [3] Chn N ian B D and Chn J Q Bow-u of souions for som a cass of gnraizd noninar Schrodingr uaions A Mah Inf Sci6-3S [4] Chn N Chn JQ and ian B D Fixd Poin horms of Mui-vau ma And H s Variaion Iraion A Mah Inf Sci 6-3S [5] Liu XJ Zhou YH ZhangL & Wang JZ Wav souions of Burgrs uaion wih high Rynods numbrs Scinc China (chnoogica Scincs Juy Rcivd: Smbr 6 4 Rvisd: Dcmbr 3 4 Accd: Dcmbr 3 4 Ning a; Licns Bnham On his is an on accss aric icnsd undr h rms of h (hs://craivcommonsorg/icnss/by/4/gacod which rmis unrsricd noncommrcia us disribuion and rroducion in any mdium rovidd h work is rory cid

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