A9117

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1 A9117 hp://znaurforsch.com/ieo/a117.pdf Symmery Reducions and Exac Soluions of he +1)-Dimensional Navier-Sokes Equaions Xiaorui Hu a, Zhongzhou Dong b, Fei Huang c, and Yong Chen a,b a Nonlinear Science Cener and Deparmen of Mahemaics, Ningbo Universiy, Ningbo, 31511, China b Shanghai Key Laboraory of Trusworhy Compuing, Eas China Normal Universiy, Shanghai, 0006, China c Physical Oceanography Laboraory and Ocean-Amosphere Ineracion and Climae Laboraory, Ocean Universiy of China, Qingdao, 66100, China Reprin requess o Y. C.; chenyong@nbu.edu.cn Z. Naurforsch. 65a, ); received May 8, 009 / Sepember 13, 009 By means of he classical symmery mehod, we invesigae he +1)-dimensional Navier-Sokes equaions. The symmery group of Navier-Sokes equaions is sudied and is corresponding group invarian soluions are consruced. Ignoring he discussion of he infinie-dimensional subalgebra, we consruc an opimal sysem of one-dimensional group invarian soluions. Furhermore, using he associaed vecor fields of he obained symmery, we give ou he reducions by one-dimensional and wo-dimensional subalgebras, and some explici soluions of Navier-Sokes equaions are obained. For hree ineresing soluions, he figures are given ou o show heir properies: he soluion of saionary wave of fluid real par) appears as a balance beween fluid advecion nonlinear erm) and fricion parameerized as a horizonal harmonic diffusion of momenum. Key words: Navier-Sokes Equaions; Classical Lie Symmery Mehod; Opimal Sysem; Explici Soluion. 1. Inroducion Symmery group echniques provide one mehod for obaining exac soluions of parial differenial equaions [1 4]. Since Sophus Lie [1] se up he heory of Lie poin symmery group, he sandard mehod had been widely used o find Lie poin symmery algebras and groups for almos all he known differenial sysems. One of he main applicaions of he Lie heory of symmery groups for differenial equaions is o ge group-invarian soluions. Via any subgroup of he symmery group, he original equaion can be reduced o an equaion wih fewer independen variables by solving he characerisic equaion. In general, o each s-parameer subgroup of he full symmery group, here will correspond a family of group-invarian soluions. Since here are almos always an infinie number of such subgroups, i is usually no feasible o lis all possible group-invarian soluions o he sysem. Tha needs an effecive, sysemaic means of classifying hese soluions, leading o an opimal sysem of group-invarian soluions from which every oher such soluion can be derived. Abou he opimal sysems, a lo of excellen work has been done by many famous expers [3 7] and some examples of opimal sysems can also be found in Ibragimov [8]. Up o now, several mehods have been developed o consruc opimal sysems. The adjoin represenaion of a Lie group on is Lie algebra was also known o Lie. Is use in classifying group-invarian soluions appeared in [3] and [4] which are wrien by Ovsiannikov and Olver, respecively. The laer reference conains more deails on how o perform he classificaion of subgroup under he adjoin acion. Here we will use Olver s mehod which only depends on fragmens of he heory of Lie algebras o consruc he opimal sysem of Navier-Sokes equaions. One of he mos imporan open problems in fluid is he exisence and smoohness problem of he Navier- Sokes equaions, which has been recognized as he basic equaion and he very saring poin of all problems in fluid physics [9 10]. Therefore solving Navier- Sokes equaions becomes very imporan and valuable bu difficul. Here, by means of he classical Lie sym / 10 / $ c 010 Verlag der Zeischrif für Naurforschung, Tübingen hp://znaurforsch.com

2 X.Hue al. Symmery Reducions in +1)-Dimensional Navier-Sokes Equaions mery mehod, we invesigae he +1)-dimensional Navier-Sokes equaions: ω = ψ xx + ψ yy, 1) ω + ψ x ω y ψ y ω x γω xx + ω yy )=0. ) Since he iniial derivaion of 1) and ), many auhors have been sudying hem [11 14]. Subsiuing 1) ino ), we can ge ψ xx + ψ yy + ψ x ψ xxy + ψ x ψ yyy ψ y ψ xxx 3) ψ y ψ xyy γψ xxxx + ψ xxyy + ψ yyyy )=0. So we can invesigae 3) insead of Navier-Sokes equaions 1) and ) in he following secions. This paper is arranged as follows: In Secion, by using he classical Lie symmery mehod, we ge he vecor fields of he +1)-dimensional Navier-Sokes equaion 3). Then he ransformaions leaving he soluions invarian, i. e. is symmery groups are obained. In Secion 3, afer an opimal sysem of onedimensional symmery group of 3) is consruced, he corresponding one-parameer and some wo-parameer reducions are given ou. Thanks o he Maple, we can obain some exac soluions [15 17] of 3). Finally, some conclusions and discussions are given in Secion 4.. Symmery Group of Navier-Sokes Equaions To 3), by applying he classical Lie symmery mehod, we consider he one-parameer group of infiniesimal ransformaions in x,y,,ψ) given by x = x + εξx,y,,ψ)+oε ), y = y + εηx,y,,ψ)+oε ), = + ετx,y,,ψ)+oε ), ψ = ψ + εψx,y,,ψ)+oε ), 4) where ε is he group parameer. I is required ha he se of equaions in 3) be invarian under he ransformaions 4), and his yields a sysem of overdeermined, linear equaions for he infiniesimals ξ, η, τ, and Ψ. Solving hese equaions, one can have ξ = c 1x c 3y c 4 y + f ), η = c 1y + c 3x + c 4 x + g), τ = c 1 + c, Ψ = g )x f )y + h)+ c 3x + y ), where c i i = 1,,3,4) are arbirary consans and f ), g), andh) are arbirary funcions of. Andheassociaed vecor fields for he one-parameer Lie group of infiniesimal ransformaions are v 1,v,,v 7 given by v 1 = x x + y y +, v =, v 3 = y x + x y + x + y ψ, v 4 = y x + x y, v 5 = f ) x f )y ψ, v 6 = g) y + g )x ψ, v 7 = h) ψ. 5) Equaions 5) show ha he following ransformaions given by expεv i ),i = 1,,,7) of variables x,y,,ψ) leave he soluions of 3) invarian: expεv 1 ) : x,y,,ψ) xe ε,ye ε,e ε,ψ), expεv ) : x,y,,ψ) x,y, + ε,ψ), expεv 3 ) : x,y,,ψ) xcosε) ysinε),xsinε) + ycosε),,ψ + x + y ) ε, expεv 4 ) : x,y,,ψ) 6) xcosε) ysinε),xsinε)+ycosε),,ψ), expεv 5 ) : x,y,,ψ) x + f )ε,y,,ψ f )yε), expεv 6 ) : x,y,,ψ) x,y + g)ε,,ψ + g )xε), expεv 7 ) : x,y,,ψ) x,y,,ψ + h)ε). And he following heorem holds: Theorem 1: If ψ = px,y,) is a soluion of 3), so are he funcions: ) ψ 1) = p xe ε,ye ε,e ε, ψ ) = px,y, ε), ψ 3) = p xcosε)+ysinε), ) xsinε)+ycosε), + x + y ε, ) ψ 4) = p xcosε)+ysinε), xsinε)+ycosε),, ψ 5) = px f )ε,y,) f )yε, ψ 6) = px,y g)ε,)+g )xε, ψ 7) = px,y,)+h)ε.

3 X. Hu e al. Symmery Reducions in +1)-Dimensional Navier-Sokes Equaions 3 In [18], Clarkson and Kruskal CK) inroduced a direc mehod o derive symmery reducions of a nonlinear sysem wihou using any group heory. For many ypes of nonlinear sysems, he mehod can be used o find all he possible similariy reducions. Then Lou and Ma modified CK s direc mehod [19 ] o find ou he generalized Lie and non-lie symmery groups of differenial equaions by an ansaz reading ux,y,)=αx,y,)+β x,y,)uξ,η,τ), 7) where ξ,η,τ are all funcions of x,y,. 7) also poins ha if Ux,y,) is a soluion of he original differenial equaion, so is ux,y,). Acually, insead of he ansaz 7), he general one-parameer group of symmeries can be obained by considering linear combinaion c 1 v 1 +c v +c 3 v 3 +c 4 v 4 +c 5 v 5 +c 6 v 6 +c 7 v 7 of he given vecor fields. Bu he explici formulae for he above ransformaions are very complicaed. Facually, i can be represened uniquely in he form g = expε 1 v 1 )expε v )expε 3 v 3 )expε 4 v 4 ) 8) expε 5 v 5 )expε 6 v 6 )expε 7 v 7 ). Thus making use of group ransformaions 8), he mos general soluion obainable from a given soluion px,y,) is in he form for simpliciy, one can do i by compuer algebra): ψ = a 4 x + y ) a 4 a 5 f )+a 6 g ))x + a 4 a 6 g)+a 5 f ))y 1 a 4a 5 f ) 1 a 4a 6 g) + a 5 a 6 f )g) a 7 h) + px,y,t), X = a 1 [cosa 4 ) 1 a 3 a 3 sina 4 ))x sina 4 ) 1 a 3 a 3 cosa 4 ))y + 1 a 3 a 5 f )cosa 4 ) a 6 g)sina 4 )) ] a 3 a 5 f )sina 4 )+a 6 g)cosa 4 )), Y = a 1 [sina 4 ) 1 a 3 + a 3 cosa 4 ))x +cosa 4 ) 1 a 3 a 3 sina 4 ))y + 1 a 3 a 5 f )sina 4 )+a 6 g)cosa 4 )) ] + a 3 a 5 f )cosa 4 ) a 6 g)sina 4 )), T = a 1 + a ), where a 1,a,,a 6 are arbirary consans. 3. Reducions and Soluions of Navier-Sokes Equaions By exploiing he generaors v i of he Lie-poin ransformaions in 5), one can build up exac soluions of 3) via he symmery reducion approach. This allows one o lower he number of independen variables of he sysem of differenial equaions under consideraion using he invarians associaed wih a given subgroup of he symmery group. In he following we presen some reducions leading o exac soluions of he Navier-Sokes equaions of possible physical ineres. Firsly, we consruc an opimal sysem o classify he group-invarian soluions of 3). As i is said in [4], he problem of classifying group-invarian soluions reduces o he problem of classifying subgroups of he full symmery group under conjugaion. And he problem of finding an opimal of subgroups is equivalen o ha of finding an opimal sysem of subalgebras. Here, by using he mehod presened in [3 4], we will consruc an opimal sysem of one-dimensional subalgebras of 3). From 5), ignoring he discussion of he infiniedimensional subalgebra, one can ge he following four operaors: v 1 = x x + y y +, v =, v 3 = y x + x y + x + y ψ, v 4 = y x + x y. Applying he commuaor operaors [v m,v n ]=v m v n v n v m, we ge he following able he enry in row i and he column j represening [v i,v j ]): Therefore, here is v 1 v v 3 v 4 v 1 0 v v 3 0 v v 0 v 4 0 v 3 v 3 v v Proposiion 1: The operaors v i i = 1,,3,4) form a Lie algebra, which is a four-dimensional symmery algebra.

4 4 X.Hue al. Symmery Reducions in +1)-Dimensional Navier-Sokes Equaions To compue he adjoin represenaion, we use he Lie series in conjuncion wih he above commuaor able. Applying he formula Adexpεv))v 0 = v 0 ε[v,v 0 ]+ 1 ε [v,[v,v 0 ]], we can consruc he following able: Ad v 1 v v 3 v 4 v 1 v 1 expε)v exp ε)v 3 v 4 v v 1 εv v v 3 εv 4 v 4 v 3 v 1 + εv 3 v + εv 4 v 3 v 4 v 4 v 1 v v 3 v 4 wih he i, j)-h enry indicaing Adexpεv i ))v j. Following Ovsiannikov [3], one calls wo subalgebras v and v 1 of a given Lie algebra equivalen if one can find an elemen g in he Lie group so ha Adgv 1 )=v,whereadg is he adjoin represenaion of g on v. Given a nonzero vecor v = a 1 v 1 + a v + a 3 v 3 + a 4 v 4, our ask is o simplify as many of he coefficiens a i as possible hough judicious applicaions of adjoin maps o v. In his way, omiing he deailed compuaion, one can ge he following heorem by he complicaed compuaion: Theorem : The operaors generae an opimal sysem S a) v 1 + a 4 v 4,a 1 0; b1) v 3,a 1 = 0,a 3 0; b) v 3 + v,a 1 = 0,a 3 0; b3) v 3 v,a 1 = 0,a 3 0; c) v,a 1 = a 3 = 0,a 0; d) v 4,a 1 = a = a 3 = 0. Making use of Theorem, we will discuss he reducions and soluions of 3) Reducions by One-Dimensional Subalgebras For case a), from v 1 + a 4 v 4 )ψ)=0, i. e. x ψ x + y ψ y + ψ + a 4 yψ x + xψ y )=0, one can ge ψ = Fξ,η), whereξ = sina 4 ln)) x cosa 4 ln)) y,andη = cosa 4 ln)) x + sina 4 ln)) y.then 3) is reduced o γf ξξ + F ηη ) ξξ + γf ξξ + F ηη ) ηη + ξ F ξξ + F ηη ) ξ + ηf ξξ + F ηη ) η + a 4 ξ F ξξ + F ηη ) η a 4 ηf ξξ + F ηη ) ξ + F ξξ + F ηη ) F ξ F ξξ + F ηη ) η + F η F ξξ + F ηη ) ξ = 0. By solving he above equaion, one can obain Fξ,η)=Fξ ± ηi), where i = 1 andf is an arbirary funcions of he corresponding variable. In case b1), solving yψ x + xψ y x + y = 0, i follows ψ = x + y arcan ) x + Fξ,η), y where ξ = x + y and η =. Subsiuing hem ino 3), and inegraing he reduced equaion once abou ξ, one can have 4γξηξ F ξξξ + F ξξ )+ξ F ξξ ξηf ξη F = 0. In case b) and b3), solving yψ x + xψ y x + y + εψ = 0, i follows ψ = ε x + y )+Fξ,η), where ξ = x + y, ε = ±1 andη = ε arcan x y )+. And he reduced equaion is 8ε 5 γf ηηηη + 8ε 4 ξ F ξ F ηηη ξ F η F ξηη + F η F ηη ) + 8ε 3 γξ F ξξηη + F ηη ) + 8ε ξ ξ F ξ F ξξη ξ F η F ξξξ + F ξ F ξη F η F ξξ ) + 8εγξ ξ F ξξξξ + 4ξ F ξξξ + F ξξ ) ξ = 0. For case c), from ψ = 0, one can ge ψ = Fx,y), which indicae a saionary fluid. Then 3) is cas ino he reduced form F x F xx + F yy ) y F y F xx + F yy ) x γf xx + F yy ) xx γf xx + F yy ) yy = 0.

5 X. Hu e al. Symmery Reducions in +1)-Dimensional Navier-Sokes Equaions 5 Fig.. Fig. 1. The above equaion has he soluion Fx,y)=C 3 +C 4 anhc 1 +C x +C yi) + C 5 anh C 1 +C x +C yi), where i = 1, and C i i = 1,,3,4,5) are arbirary consans. When he growh rae of fluid imagine par of he soluion) ends o be zero when C 5 = C 4 E 1 E +1) + E cosc y) ), E = e C 1+C x, he soluion E 4 1 of saionary wave of fluid real par) appears as a balance beween fluid advecion nonlinear erm) and fricion parameerized as a horizonal harmonic diffusion of momenum wih coefficien γ. Figure 1 shows a saionary inerior ocean circulaion wih C 1 = 0.8, C = 0.15, C 3 = C 4 = 1, which looks like an anicyclonic subropical gyre in a closed ocean basin, wo cyclonic ropical and subpolar lows a he norh and souh, respecively [3]. In case d), solving yψ x + xψ y = 0, we obain ψ = Fξ,η), whereξ = x + y and η =. Subsiuing i ino 3) and inegraing he reduced equaion wice abou ξ, one can ge F η 4γξ F ξ ) ξ = 0. Solving he above equaion, we have he soluion of 3) ψ = exp4c 1 γ)c BesselJ0, C 1 x + y )) 9) +C 3 BesselY0, C 1 x + y ))), 3.. Reducions by Two-Dimensional Subalgebras Case 1: {v 1,v }.From x ψ x + y ψ y + ψ = 0and ψ = 0, we have ψ = F x y ). Subsiuing i ino 3), one can ge [ γ ξ + 1) F ξξξξ + 1ξ ξ + 1)F ξξξ ] + 13ξ + 1)F ξξ + 4ξ F ξ + ξ + 1)F ξ F ξξ + 4ξ F ξ = 0, where ξ = x y. Case : {v 1,v 3 }. Solving x ψ x + y ψ y + ψ = 0 and yψ x + xψ y x +y = 0, i follows ψ = x +y arcan x y )+F x +y ). Subsiuing i ino 3), i follows 4γξ F ξξξξ +4ξ F ξξξ +F ξξ )+ξ F ξξξ +5ξ F ξξ = 0. Case 3: {v 1,v 4 }.From x ψ x + y ψ y + ψ = 0and yψ x + xψ y = 0, one can ge ψ = F x +y ). Subsiuing i ino 3), we have F Z)+8γF Z)+3ZF Z)+16γZF Z) + Z F Z)+4γZ F Z)=0, where C i i = 1,,3) are arbirary consans. Figure exhibis he plo of ψ in 9) wih γ = 1, C 1 = 1, C = 10, C 3 = 0, and he ime = 1. Fig. 3.

6 6 X.Hue al. Symmery Reducions in +1)-Dimensional Navier-Sokes Equaions where Z = x +y. Solving he above equaion, i follows ψ = C 1 +C lnz +C 3 Ei 1, Z ) 4γ ) ) 3 3C 4 Γ Ze Z3 4γ 1 Z3 Γ, dz 3 3 4γ ) + 1C 4 γ 3 Γ ln Z C4 πγ 1 3 Ze Z3 8γ 1 WhiakerM 3, 5 ) 6, Z C4 πγ 4 3 Z e Z3 8γ 4γ 4 WhiakerM 3, 5 ) ) 6, Z3 + 1C 4 γ 3 Γ lnz, 4γ 3 where C i i = 1,,3,4) are arbirary consans. Figure 3 exhibis he plo of ψ in 10) wih γ = 1, C 1 = 0, C = C 3 = 1, C 4 = 0, and he ime = 0, appearing an amospheric subropical high or monopole ani-cyclonic blocking in he Norhern Hemisphere. Case 4: {v,v 4 }. The soluion of ψ = 0and yψ x + xψ y = 0hasheformψ = Fx +y ). Then he reduced equaion of 3) is ξ F ξξξξ + 4ξ F ξξξ + F ξξ = 0, which has he soluion F = C 1 +C ξ +C 3 lnξ )+C 4 ξ lnξ ), where C i i = 1,,3,4) are arbirary consans and ξ = x + y. 4. Conclusions In summary, we invesigae he symmery of he Navier-Sokes equaions by means of he classical Lie symmery mehod. The symmery algebras and groups of 3) are obained. Specially, he mos general oneparameer group of symmeries is given ou as he composiion of ransforms in he seven various onesubgroups expεv 1 ),expεv ),,expεv 7 ) and he mos general soluion obainable from a given soluion px,y,) is gained. Nex we have classified onedimensional subalgebras of a Lie algebra of 3). Then he reducions and some soluions of Navier-Sokes equaions by using he associaed vecor fields of he obained symmery are given ou. By one-dimensional subalgebras, 3) is reduced o some 1+1)-dimensional equaions and by wo-dimensional subalgebras, 3) is reduced o some ordinary equaions. For hree ineresing explici soluions of 3), we also give ou figures o show heir properies. Acknowledgemens We would like o hank Prof. Senyue Lou for his enhusiasic guidance and helpful discussions. The work is suppored by he Naional Naural Science Foundaion of China Gran No and ), Program for New Cenury Excellen Talens in Universiy NCET ), Shanghai Leading Academic Discipline Projec No. B41), Program for Changjiang Scholars and Innovaive Research Team in Universiy IRT0734) and K. C. Wong Magna Fund in Ningbo Universiy. [1] S.Lie,Arch.Mah.6, ). [] G. W. Bluman and S. C. Anco, Symmery and Inegraion Mehods for Differenial Equaions, Springer, New York 00. [3] L. V. Ovsiannikov, Group Analysis of Differenial Equaions, Academic, New York 198. [4] P. J. Olver, Applicaions of Lie Groups o Differenial Equaions, Springer-Verlag, New York [5] S. V. Coggeshall and J. Meyer-er-Vehn, J. Mah. Phys. 33, ). [6] J. Paera, P. Winerniz, and H. Zassenhaus, J. Mah. Phys. 16, ). [7] K. S. Chou, G. X. Li, and C. Z. Qu, J. Mah. Anal. Appl. 61, ). [8] N. H. Ibragimov, Lie Group Analysis of Differenial Equaions, Boca Raon [9] D. Sundkvis, V. Krasnoselskikh, P. K. Shukla, A. Vaivads, M. Andr, S. Bucher, and H. Rme, Naure London) 436, ); G. Pedrizzei, Phys. Rev. Le. 94, ). [10] C. L. Fefferman, hp:// Navier-Sokes Equaions/Official Problem Descripion.pdf. [11] S. P. Lloyd, Aca Mechanica, 38, ). [1] C. Y. Wang, Appl. Mech. Rev. 4, S ). [13] W. Fushchych, W. M. Shelen, and S. L. Slavusky, J. Phys. A: Mah. Gen. 4, ). [14] K. Fakhar, Z. C. Chen, and R. Shagufa, IJDEA, 7, ). [15] J. S. Yang and S. Y. Lou, Z. Naurforsch. 54a, ). [16] L. L. Chen and S. Y. Lou, Aca Sinica Physica, 8, S ). [17] Z. Z. Dong and Y. Chen, Symmery Reducion, Ex-

7 X. Hu e al. Symmery Reducions in +1)-Dimensional Navier-Sokes Equaions 7 ac Soluions and Conservaion Laws of he +1)- Dimensional Dispersive Long Wave Equaions, Z. Naurforsch. A. in press). [18] P. A. Clarkson and M. Kruskal, J. Mah. Phys. 30, ). [19] S. Y. Lou, J. Phys. A: Mah. Gen. 38, ). [0] H. C. Ma and S. Y. Lou, Z. Naurforsch. 60a, 9 005). [1] H. C. Hu and S. Y. Lou, Z. Naurforsch. 59a, ). [] Y. Chen and X. R. Hu, Z. Naurforsch. 63a, 1 008). [3] J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York 1979.