On the Stabilizing Effect of Convection in 3D Incompressible Flows

Transcription

1 On the Stabiliing Effect of Convection in 3D Incompessible Flows Thomas Y. Hou Zhen Lei Febuay 5, 8 Abstact We investigate the stabiliing effect of convection in 3D incompessible Eule and Navie- Stokes equations. The convection tem is the main souce of nonlineaity fo these equations. It is often consideed destabiliing although it conseves enegy due to the incompessibility condition. In this pape, we show that the convection tem togethe with the incompessibility condition actually has a supising stabiliing effect. We demonstate this by constucting a new 3D model which is deived fo axisymmetic flows with swil using a set of new vaiables. This model peseves almost all the popeties of the full 3D Eule o Navie-Stokes equations except fo the convection tem which is neglected in ou model. If we add the convection tem back to ou model, we would ecove the full Navie-Stokes equations. We will pesent numeical evidence which seems to suppot that the 3D model may develop a potential finite time singulaity. We will also analye the mechanism that leads to these singula events in the new 3D model and how the convection tem in the full Eule and Navie-Stokes equations destoys such a mechanism, thus peventing the singulaity fom foming in a finite time. Keywod: Finite time singulaities, 3D Navie-Stokes equations, stabiliing effect of convection. Intoduction The question of whethe a solution of the 3D incompessible Navie-Stokes equations can develop a finite time singulaity fom smooth initial data with finite enegy is one of the most outstanding mathematical open poblems []. A main difficulty in obtaining the global egulaity of the 3D Navie-Stokes equations is due to the pesence of the votex stetching tem, which has a fomal quadatic nonlineaity in voticity. So fa, most egulaity analyses fo the 3D Navie-Stokes equations use enegy estimates and equie some kind of smallness assumption on the initial data [3, 7, 34, 7]. Due to the incompessibility condition, the convection tem does not contibute to the enegy nom of the velocity field o any L p < p ) nom of the voticity field. As a esult, the convection tem has been basically ignoed in the egulaity analysis fo the Navie- Stokes equations. Most of the effots have focused on how to use the diffusion tem to contol the nonlinea votex stetching tem without making use of the convection tem explicitly. Applied and Comput. Math, Caltech, Pasadena, CA hou@acm.caltech.edu. School of Mathematics and Statistics, Notheast Nomal Univesity, Changchun 34, P. R. China; School of Mathematical Sciences, Fudan Univesity, Shanghai 433, P. R. China; Applied and Comput. Math, Caltech, Pasadena, CA leihn@yahoo.com.

2 In this pape, we show that the convection tem has a supising stabiliing effect in the 3D incompessible Eule and Navie-Stokes equations. It plays an essential ole in depleting the votex stetching tem. We demonstate this stabiliing effect of convection by constucting a new 3D model fo axisymmetic flows with swil. This model is fomulated in tems of a set of new vaiables elated to the angula velocity, the angula voticity, and the angula steam function. The only diffeence between ou 3D model and the efomulated Navie-Stokes equations in tems of these new vaiables is that we neglect the convection tem in the model. If we add the convection tem back to the model, we will ecove the full Navie-Stokes equations. This new 3D model peseves almost all the popeties of the full 3D Eule o Navie-Stokes equations. In paticula, the stong solution of the model satisfies an enegy identity simila to that of the full 3D Navie-Stokes equations. We also pove a non-blowup citeion of Beale-Kato-Majda type [] as well as a non-blowup citeion of Podi-Sein type [3, 3] fo the model. In a subsequent pape, we will pove a new patial egulaity esult fo the model [7] which is an analogue of the Caffaelli-Kohn-Nienbeg theoy [] fo the full Navie-Stokes equations. Despite the stiking similaity at the theoetical level between ou model and the Navie-Stokes equations, the fome has a completely diffeent behavio fom the full Navie-Stokes equations. We will pesent numeical evidence which seems to suppot that the model may develop a potential finite time singulaity fom smooth initial data with finite enegy. By exploiting the axisymmetic geomety of the poblem, we obtain a vey efficient adaptive solve with an optimal complexity which povides effective local esolutions of ode fo the viscous model and 89 3 fo the inviscid model. With this level of esolution, we obtain an almost pefect fit fo the asymptotic blowup ate of maximum axial voticity in the inviscid model. If we denote by ω the axial voticity component along the -diection, we find that ω t) CT t) with a logaithmic coection and the potential singulaity appoaches the symmety axis the -axis) as t T. Moeove, ou peliminay study seems to suggest that the potential singulaity is locally self-simila and isotopic. We also pesent numeical evidence which seems to suggest that the viscous model may develop a potential finite time singulaity. The behavio of the nealy singula solution is simila to that of the solution of the inviscid model. We find that the solution of the viscous model expeiences temendous dynamic gowth. The gowth ate is much faste than what has been obseved fo the 3D Navie-Stokes equations. On the othe hand, we obseve that the solution of the viscous model seems to be dominated by the dynamics of the inviscid model duing the time inteval of ou computation. In ode to detemine whethe the 3D model actually develops a finite time singulaity and to study the local scaling popety of the potential singulaity, we need to solve the viscous model much close to the potential singulaity time to captue the viscous effect accuately. This would equie substantially highe numeical esolutions than what we have used in the cuent pape. Depending on the local scaling popety of the nealy singula solution and the balance between the votex stetching tem and the viscous tem, it is still possible that the viscous tem may eventually egulaie the nealy singula solution induced by the nonlinea votex stetching tem. We will investigate this issue futhe in ou futue wok. To undestand the mechanism fo geneating the potential finite time singulaity, we monito closely how the vaiables that contibute to the votex stetching tem inteact dynamically. Nea the suppot of the maximum voticity, which is defined as the egion in which voticity is compaable to its maximum, we find that the physical vaiables that contibute to the votex stetching tem have a stong alignment locally. This seems to be the main mechanism fo geneating the potentially singula solution of the model.

3 To see how convection depletes the mechanism fo geneating a potential finite time singulaity of ou model, we add the convection tem back to the model. We use the solution of the model at a time sufficiently close to the potential singulaity time as the initial condition fo the full 3D Navie-Stokes equations. Supisingly, the solution of the 3D Navie-Stokes equations immediately becomes defocused and smoothe along the symmety axis. As time inceases, the solution develops a thin jet that moves away fom the symmety axis. As we know fom the Caffaelli-Kohn-Nienbeg patial egulaity theoy [] see also [5] fo a simplified poof), the 3D axisymmetic Navie-Stokes equations cannot develop finite time singulaities away fom the symmety axis. The fact that the convection tem foces the most singula pat of the solution to move away fom the symmety axis shows that convection has effectively destoyed the mechanism that leads to a potential finite time blowup obseved in the model. Recent numeical study of the 3D Eule equations by Hou-Li in [5, 6] shows that the convection tem tends to intoduce lage defomation to the local votex stuctue. The suppot of maximum voticity becomes seveely flattened as the votex stetching intensifies in time. This anisotopic collapse of the suppot of maximum voticity seems to play an essential ole in depleting the votex stetching. Some pogess has been made ecently along this diection, see [9]. The esults pesented in this pape may have some impotant implication to the global egulaity of the 3D Navie-Stokes equations. A successful stategy in analying the global egulaity of the 3D Navie-Stokes equations should take advantage of the stabiliing effect of the convection tem in an essential way. So fa most of the egulaity analyses fo the 3D Navie-Stokes equations do not use the stabiliing effect of the convection tem. In many cases, the same esults can be also obtained fo ou model. We have pesented numeical evidence in this pape which shows that the 3D model is much moe singula than the coesponding 3D Navie-Stokes equations. New analytical tools that exploit the local geometic stuctue of the solution and the stabiliing effect of convection may be needed to pove the global egulaity of the 3D Navie-Stokes equations. We also popose a genealied model in any dimension d with d <. The 3D model we discussed ealie coesponds to the special case of d = 5 in the genealied model. Based on the balance between the votex stetching tem and the diffusion tem, we can futhe classify the case of d = 4 as the citical case, d < 4 as the subcitical case, and d > 4 as the supecitical case. The genealied model in the supecitical case shaes many difficulties of the full Navie-Stokes equations. Global egulaity of the genealied model in the supecitical case can be poved only fo small initial data. On the othe hand, we pove the global egulaity of the genealied model in the citical and the subcitical cases. We emak that this stabiliing effect of convection has been studied by Hou and Li in a ecent pape [8]. They showed that convection plays an essential ole in canceling the destabiliing votex stetching in a new D model which can be used to constuct a family of exact solutions of the 3D Eule o Navie-Stokes equations. This obsevation enabled them to obtain a cucial a pioi pointwise estimate fo a high ode nom of solutions in thei model. Using this a pioi estimate, they poved the global egulaity of the 3D Navie-Stokes equations fo a family of lage initial data, whose solutions can lead to lage dynamic gowth, but yet have globally smooth solutions. The stabiliing effect of convection has been also used in deiving localied non-blowup citeia fo the 3D incompessible Eule equations by Deng-Hou-Yu in [, ]. By using a Lagangian fomulation and exploiting the connection between votex stetching and the local geometic egulaity of votex lines, they showed that the latte, even in an extemely localied egion containing the maximum voticity, can lead to depletion of votex stetching, thus avoiding finite time singulaities. Recently, Okamoto and Ohkitani [8] investigated the ole of the convection tem in peventing singulaity 3

4 fomation by studying seveal one-dimensional models and a D model deived fom the D Eule equations. They also discussed othe wok whee the idea has appeaed in some fom o othe, including the wok by Constantin [6] and by De Gegoio [8, 9], among othes. Thee has been some inteesting development in the study of the 3D incompessible Navie- Stokes equations and elated models. In paticula, by exploiting the special stuctue of the govening equations, Cao and Titi [3] poved the global well-posedness of the 3D viscous pimitive equations which model lage scale ocean and atmosphee dynamics. Fo the axisymmetic Navie- Stokes equations, Chen-Stain-Tsai-Yau [4, 5] and Koch-Nadiashvili-Seegin-Sveak [3] ecently poved that if ux,t) C t / whee C is allowed to be lage, then the velocity field u is egula at time eo. It is woth pointing out that the analyses of Chen-Stain-Tsai-Yau and Koch- Nadiashvili-Seegin-Sveak have made use of the convection tem indiectly by woking with the consevative convection diffusion equation satisfied by u θ, whee u θ is the angula velocity. The est of the pape is oganied as follows. In Section, we efomulate the 3D axisymmetic Navie-Stokes equations in tems of some new vaiables. In Section 3, we deive ou 3D model and pove the enegy identity fo this model. In Section 4, we genealie ou model to an abitay space dimension and pove its global egulaity in the subcitical and citical cases. Section 5 is devoted to the analysis of a special D model. In Section 6, we pesent numeical evidence which seems to suggest that ou 3D model may develop a potential finite time singulaity fom some lage smooth initial data with finite enegy. We also analye the mechanism fo geneating the potential finite time singulaity fo the 3D model and demonstate how the convection tem destoys the mechanism that leads to the finite time blowup of the 3D model. In Sections 7 and 8, we pove two non-blowup citeia fo ou model equations. The fist one is an analogue of the well-known Beale-Kato-Majda non-blowup citeion, and the second one is an analogue of the Podi-Sein non-blowup citeion [3, 3]. Refomulation of the 3D axisymmetic Navie-Stokes equations Conside the 3D axisymmetic incompessible Navie-Stokes equations with swil u t + u )u = p + ν u, u =, u t= = u x), x = x,x,). Let x e =, x ),, e θ = x, x ),, e =,,), be the thee othogonal unit vectos along the adial, the angula, and the axial diections espectively, = x + x. We will decompose the velocity field as follows:.) u = u,,t)e + u θ,,t)e θ + u,,t)e,.) whee u, u θ, u ae called the adial, angula and axial velocity espectively. In paticula, u θ is also efeed to as the swil component of the velocity. One can deive the following axisymmetic fom of the Navie-Stokes equations in the cylindical coodinates [7]: t u θ + u u θ + u u θ = ν x ) u θ u u θ, t ω θ + u ω θ + u ω θ = ν x ) ω θ + u θ ) ) + u ω θ,.3) x ) ψ θ = ω θ, 4

5 whee u = ψ θ, u = ψ θ )..4) The incompessible constaint in cylindical coodinates is given by u + u + u = o u ) + u ) =,.5) which is tivially satisfied in view of.4). In [6], Liu and Wang showed that if u is a smooth velocity field, then u θ, ω θ and ψ θ must satisfy the following compatibility condition at = : u θ = = ω θ = = ψ θ = =. The voticity can be epesented in cylindical coodinates as follows: Note that the axial voticity component has the fom ω = u θ ) e + ω θ e θ + uθ ) e..6) ω = uθ ) = uθ + u θ..7) The last two tems on the ight hand side have the same asymptotic limit as since u θ = =. Thus the vaiable uθ chaacteies the axial voticity nea =. In [8], Hou and Li intoduced the following new vaiables: u = uθ, ω = ωθ, ψ = ψθ,.8) and deived the following equivalent system that govens the dynamics of u, ω and ψ as follows: t u + u u + u u = ν ) u + ψ u, t ω + u ω + u ω = ν ) ω + u ) ),.9) ) ψ = ω, whee u = ψ ), u = ψ )..) Note that in the new system.9)-.), the convection tem has absobed one of the votex stetching tems, u ω θ, which oiginally appeas in the second equation of.3). In some sense, the convection tem has aleady stabilied one of the potentially destabilied votex stetching tem in the above efomulation. Finally, we ecall the basic enegy identity fo the Navie-Stokes equations.): d u dx + ν u dx =..) dt R 3 R 3 Natually, the system.3)-.4) and the new system.9)-.) enjoy the the same enegy identity.). 5

6 3 A new 3D model and its popeties In this section, we intoduce ou 3D model fo axisymmetic flows with swil. The pupose of intoducing this model is to study the stabiliing effect of the convection tem in the 3D incompessible Eule o Navie-Stokes equations. Ou model shaes many popeties with the 3D Eule o Navie- Stokes equations. Fist of all, it has the same nonlinea votex stetching tem. Secondly, it has the same type of a pioi enegy identity. Thidly, almost all the existing non-blowup citeia fo the 3D Eule o Navie-Stokes equations ae also valid fo ou model. A 3D model that satisfies all these popeties seems had to find in geneal. But in tems of the equations fo the new vaiables, u, ω, and ψ, we can get ou 3D model equations by simply dopping the convective tem fom.9): t u = ν ) u + ψ u, t ω = ν ) ω + u ), 3.) ) ψ = ω. Note that 3.) is aleady a closed system. The main diffeence between ou 3D model and the oiginal Navie-Stokes equations is that we neglect the convection tem in ou model. If we add the convection tem back to ou 3D model, we will ecove the Navie-Stokes equations. Below we will deive some impotant popeties of the model equations 3.). Fist of all, we note that thee is an intinsic incompessible stuctue in the 3D model equations 3.). To see this, we define the velocity field as with It is easy to check that u = u e + u θ e θ + u e 3.) u = ψ ), u θ = u, u = ψ ). 3.3) u = u + u + u =, 3.4) which is the same incompessibility condition fo the oiginal incompessible Eule o Navie-Stokes equations. Next, we will pove a compatibility condition fo the solution of ou 3D model. This compatibility condition was fist obtained by Liu and Wang in [6] fo the 3D axisymmetic Eule o Navie-Stokes equations. Poposition 3.. Any smooth solution u,ω,ψ ) to the 3D viscous model equations 3.) on t < T with ν > satisfies the following compatibility conditions: k u θ t,,) = = k ω θ t,,) = = k ψ θ t,,) = =, 3.5) fo all even intege k and t < T, whee u θ = u, ω θ = ω, ψ θ = ψ. 3.6) 6

7 Poof. Let u,ω,ψ ) be a smooth solution of 3.) fo t < T. It is easy to see that u θ,ω θ,ψ θ ) satisfies t u θ = ν + + ) u θ + ψθ u θ, t ω θ = ν + + ) ω θ u + θ ), + + ) ψ θ = ω θ, fo >. Multiplying each equations in 3.7) by yields t u θ = ν + + ) u θ + ψ θ u θ, t ω θ = ν + + ) ω θ + [u θ ) ], + + ) ψ θ = ω θ. Since u,ω,ψ ) is smooth, so is u θ,ω θ,ψ θ ). By letting in 3.7), we pove the compatibility condition 3.5) fo k =. Next we diffeentiate 3.7) with espect to twice, then let. We can see that 3.5) is tue fo k =. By using an induction agument, one can pove that 3.5) is tue fo all even k >. Remak 3.. We emak that the above compatibility condition is necessay if we equie that the econstucted 3D velocity field using 3.)-3.3) be smooth at =. This is due to the fact that the unit vectos e and e θ ae singula at =, but k+ e and k+ e θ ae egula at =. Thus u θ, ω θ and ψ θ must have asymptotic expansions in tems of the odd powes of nea =. This would give ise to the compatibility condition 3.5), even in the case of ν =. Notations. To make it easie to compae the 3D model equations 3.) with the axisymmetic Navie-Stokes equations.9)-.), we ecast ou 3D model equations in R R 5 and intoduce some notations. We will denote by x = x,x,) a point in R 3 and y = y,y,y 3,y 4,) a point in R 5. The time deivative is denoted by t. The space deivative with espect to x o y is denoted by x = x, x, ) T in R 3 o by y = y, y, y3, y4, ) T in R 5, espectively. Similaly, we will use x = x + x +, y = y + y + y 3 + y 4 +. Thoughout this pape, if the function is axisymmetic, we will denote its space vaiable by,), whee -axis is the symmety axis and = x + x in R3 and = y + y + y 3 + y 4 in R5, espectively. In paticula, the Laplacians using cylindical coodinates ae 3.7) x = + +, y = ) Now we state the impotant enegy identity of the 3D model equations 3.), which is equivalent to.) of the axisymmetic Navie-Stokes equations. Poposition 3.3. The solution of the 3D model equations 3.) satisfies the following enegy identity: d dt u + y ψ ) 3 dd + ν y u + y ψ ) 3 dd =. 3.9) 7

8 Moeove, we have u + y ψ ) 3 dd = u u θ ) dd, 3.) = y u + y ψ ) 3 dd 3.) x u u θ + u θ + uθ )) dd, whee u is defined in 3.)-3.3). Futhemoe, the enegy identity 3.9) is equivalent to that of the Navie-Stokes equations in the sense that u dd x u dd u + y ψ ) 3 dd y u + y ψ ) 3 dd u dd, 3.) x u dd. 3.3) Poof. The enegy identity 3.9) follows by the standad enegy method. Obseve that the diffusion opeato in the 3D model equations, which is given by + 3 +, is actually the Laplacian opeato in R 5, y. Thus, it would be easie to pove the enegy identity by pefoming enegy estimates in R 5. Multiplying the fist equation of 3.) by u and then integating ove R 5 yields d u dt dy + ν y u dy = ψ u dy. 3.4) R 5 R 5 R 5 Multiplying the second equation in 3.) by ψ and then integating ove R 5 yields t ω ψ dy ν ψ y ω dy = ψ u )dy. R 5 R 5 R 5 Using the thid equation in 3.) and integation by pats, one has d y ψ dy + ν y ψ dy = ψ u dy. 3.5) dt R 5 R 5 R 5 Multiplying 3.5) by and adding the esulting equation to 3.4) gives 3.9). Next, we pove 3.)-3.). The equality 3.) follows by the following staightfowad calculation: u + y ψ ) 3 dd = = = = u + ψ + ψ ) 3 dd u θ + ψ θ ψθ + u ) dd u θ + ψ θ + ψθ + u ) dd u θ + u + u ) dd, 8

9 whee we have used Poposition 3. and the identity Using 3.), we can easily obtain 3.). ψ θ ψ θ dd =. It emains to pove 3.) and 3.3). Fist, we compute x u as follows: x u = e + e θ θ + e ) u e + u θ e θ + u e ) 3.6) = u e e + u θ e e θ + u e e uθ e θ e + u e θ e θ + u e e + u θ e e θ + u e e. Thus, by the definitions of u and u in 3.3) and the incompessibility constaint 3.4), one has x u = u θ + u θ + uθ 3.7) + u + u + u + u + u = u θ + u θ + uθ + u u u + u + u = u θ + u θ + uθ ) + ψ θ + ψ θ + ψ θ + [ ψ θ )] [ ψ θ )] + + ψ θ ψ θ ) + ψ θ ψ θ ). On the othe hand, Poposition 3. and a simple computation give Note that y u 3 dd = u θ + u θ + uθ ) dd. 3.8) y ψ = = 4 ijψ + ψ + ψ i,j= 4 i,j= ijψ + ψ θ + ψ θ + [ ψ θ )] 4 ψ θ ψ θ ) ). 9

10 A staightfowad calculation gives 4 ijψ = i,j= Theefoe, we obtain = = 4 i,j= 4 i,j= 4 i,j= yj ) i ψ δ ij ψ y iy j 3 ψ + y iy j ψ δ ij ψ 3y iy j 3 = ψθ + 3 ψ ) + ψ + y iy j 3 ψ θ 4 i,j= 3y iy j δ ij ψ ) + y iy j δ ij ψ ψ θ 3y iy j y i y j ψ ψ θ) = ψθ + 7 ψ 4 ψ ψθ). y ψ = ψθ + ψθ + ψθ + [ ψ θ )] 3.9) + 7 [ ψ θ )] ψ θ ) 4 ψ θ ψ θ ) 4 ψ θ). Combining 3.8) and 3.9), we have By compaing 3.7) with 3.), we have y u + y ψ ) 3 dd 3.) = [ u θ + u θ + uθ + ψ θ + ψ θ + ψ θ + [ ψ θ )] + 7 [ ψ θ )] ψ θ ) 4 ψ θ ψ θ ) 4 ψ θ)] dd. = + Thus to pove 3.), it suffices to show that y u + y ψ ) 3 dd 3.) x u u θ + u θ + uθ )) dd [ ψ θ )] ψ θ ) ψ θ ψ θ ) ψ θ) dd. [ ψ θ )] ψ θ ) ψ θ ψ θ ) ψ θ) dd =. 3.)

11 Indeed, diect calculations give = = = = [ ψ θ )] ψ θ ) ψ θ ψ θ ) ψ θ) dd 3.3) [ ψ θ )] ψ θ + ψθ [ ψ θ + ψ θ ]) dd [ ψ θ ) )] ψ θ ψ θ ] ) dd + ψ θ ψθ ψ θ) d =+ ) ψ θ dd + ψ θ ) ψ θ d =, =+ ψ θ ψθ ψ θ) =+ d whee f =+ = lim + f). This poves 3.). The second inequality of 3.3) follows immediately fom 3.). It emains to pove the fist inequality of 3.3). Using 3.), 3.7), and an agument simila to that of poving 3.3), we obtain y u + y ψ ) 3 dd x u dd = ψ θ + ψ θ + ψ θ + [ ψ θ )] + 3 [ ψ θ )] ψ θ ) ψ θ ψ θ ) ψ θ) dd = ψθ + ψθ + ψθ + [ ψ θ )] [ ψ θ )] dd +3. This completes the poof of Poposition Genealied model equations in an abitay space dimension In this section, we genealie ou 3D model 3.) to an abitay space dimension. On one hand, the genealied model impoves ou undestanding of the oiginal 3D model. On the othe hand, the esult we obtain fo the genealied model is of independent inteest in itself. Let n be an intege and denote y = y,y,,y n+,) R n+. The genealied model is given by the following system of equations: t u = ν n+ u + ψ u, t ω = ν n+ ω + u ) ), 4.) n+ ψ = ω,

12 whee n+ = y + + y n+ +. The 3D model we intoduced in the pevious section coesponds to the case of n = 3, which can be also consideed as a model in five space dimensions. It is easy to check that 3.4) and 3.5) ae still valid fo the genealied system 4.). Thus we have the same enegy identity d u dt + y ψ ) dy + ν y u + y ψ ) dy =. 4.) R n+ R n+ The initial condition fo the genealied model 4.) is of the fom: u,y) = u y), ω,y) = ω y), ψ,y) = ψ y). 4.3) Based on the balance between the diffusion tem and the nonlinea votex stetching tem, we can classify the genealied model as subcitical if n <, citical if n =, and supecitical if n >. Like the 3D Navie-Stokes equations, ou 3D model, which coesponds to n = 3, is supecitical. In the following, we pove the global egulaity of the genealied model fo the subcitical and the citical cases. Theoem 4.. Assume that n and u,ω,ψ ) satisfies ω = y ψ, u, ψ H R n+ ). Then thee exists a globally smooth solution to system 4.) with the initial data 4.3). Poof. We fist conside the citical case of n =. Applying y to the fist equation in 4.) and then taking the L inne poduct of the esulting equation with y u, we have d y u dy + ν y u dy = ψ u y dt u dy 4.4) R 4 R 4 R 4 ν yu L R 4 ) + ν u L 4 R 4 ) ψ L 4 R 4 ) ν y u L R 4 ) + C ν yu L R 4 ) y ψ L R 4 ) ν y u L R 4 ) + C ) y u 4 L ν R 4 ) + ω 4 L R 4 ), whee we have used the Sobolev embedding inequality: u L 4 R 4 ) C y u L R 4 ). Similaly, taking the L inne poduct of the second equation in 4.) with ω and pefoming integation by pats, we have d ω dy + ν y ω dy dt R 4 R 4 ν yω L R 4 ) + ν u 4 L 4 R 4 ) 4.5) ν yω L R 4 ) + C ν yu 4 L R 4 ). Combining 4.4)-4.5) with the basic enegy identity 4.), one aives at the a pioi estimate y u + ω ) t dy + ν y u + y ω ) dyds 4.6) R 4 R 4 y u + ω ) { C t dy exp y u + ω ) } dydt R 4 ν R 4 y u + ω ) { C dy exp u R 4 ν, y ψ ) ) } L R 4 ).

13 Based on the above a pioi estimate, we can easily pove the global egulaity of the solution in any high ode nom fo all t >. The case of n = can be poved similaly. The a pioi estimate now eads y u + ω ) t dy + ν y u + y ω ) dyds 4.7) R 3 R 3 y u + ω ) { Ct ) } dy exp u R 3 ν 3/, y ψ ) L R 3 ). The cases of n = and n = ae staightfowad. We omit the poof hee. 5 A special D model It is also inteesting to conside the special case when the solution develops a lage gadient along the -diection and the deivative along the -diection is elatively smooth. In this case, the Laplacian opeato becomes -dominated and can be appoximated locally by the one-dimensional Laplacian along the -diection. This gives ise to the following D model, which coesponds to the case of n = in ou genealied system: t u = ν u + ψ u, 5.) t ω = ν ω + u ) ), 5.) ψ = ω. 5.3) Futhe we intoduce a new vaiable v = ψ. By integating equation 5.) with espect to, we obtain an evolution equation fo v as follows: t v = ν v u + ct), 5.4) whee ct) is an integation constant. If we conside peiodic bounday condition in with peiod, we can detemine c explicitly by enfocing v d = since v = ψ. This gives ct) = u t,)d. Then we obtain an equivalent system fo u and v as follows: { t u = ν u + u v, t v = ν v u + ct), 5.5) whee ct) = u t,)d. It is easy to show that we have d u dt + v ) = ν u u + v v ) + ct)v. 5.6) Using 5.6) and the fact that v t,)d =, we have u t,)+v t,)) d = ν t u ) + v ) ddt+ u,)+v,)) d. 5.7) This a pioi estimate is sufficient to establish the global egulaity of the D model fo initial data u,), v,) L [,]. 3

14 An inteesting featue of this D model is that even without any diffusion, its solution still has global egulaity. To see this, conside the inviscid model with ν =. Combining 5.6) with 5.7), we get d u dt + v ) = 4ct)v u + v ) + By the Gonwall inequality, we obtain the following a pioi estimate: u + v u,) + v,)) e t + e t ) u,) + v,) ) ) d. ) u,) + v,)) d. 5.8) Using the above a pioi estimate, we can pove global egulaity in any high ode nom. Let m be any intege. We have d dt m u + m v ) d = [ m u m u v ) m v m u ) ] d C ) u L + v L m u + m v ) d. Using 5.8), one obtains m u + m v ) d 5.9) exp { t C ) } u L + v L ds m u,) + m v,) ) d exp exp{c t}) m u,) + m v,) ) d, whee C is an absolute positive constant depending only on the L -nom and the L -nom of the initial data. This poves the global egulaity of the D inviscid model. It is inteesting to obseve that the gowth ate of the high ode nom is double exponential in time. As we will see in next section, when the solution of the oiginal 3D model develops a stong laye along the -diection, the solution does not become singula in a finite time. The analysis of the D model povides us with an impotant guidance in ou computational study of possible finite time singulaities. 6 The nealy singula behavio of the 3D model In this section, we will pesent numeical esults to demonstate the nea singula behavio of the 3D model. We will also illustate the mechanism that leads to the fomation of a potential finite time singulaity of the model and how the convection tem destoys such a mechanism, peventing the fomation of the potential finite time singulaity in the full Navie-Stokes equations. We caution that we need to compute the solution of the viscous model much close to the potential singulaity time with esolutions much highe than what we have used in the cuent pape to detemine whethe the viscosity would eventually egulaie the nealy singula solution of the viscous model. 4

15 6. The setup of the poblem We solve the 3D model equation ove a cylindical domain: Ω = {,) /, }. Note that ψ is detemined only up to a constant. To define ψ uniquely, we may assume that ψ,,t)d = fo all times. We use peiodic bounday conditions along the -diection with peiod / and no-slip, no-flow bounday conditions at =, i.e. u = =, which is equivalent to u = u θ = u =, i.e. ψ = u = ψ + ψ =, on =. 6.) Since ψ,,t)d =, the above condition is equivalent to u = =, ψ = =, ψ = =. 6.) Since ou 3D model is fomulated in the steam function-voticity setting, we need to deive a bounday condition fo ω. Recall that Thus the no-slip, no-flow bounday condition 6.) implies that ω = ψ ψ 3 ψ. 6.3) ω = = ψ =. 6.4) Let us discetie the domain Ω by a N N gid. Denote j = jh, j = jh, with h = /N ) and h = /N. Futhe, we denote ψ ) i,j ψ i, j ), u ) i,j u i, j ), and ω ) i,j ω i, j ). We will discetie the 3D model in space by a second ode centeed diffeence method and discetie the 3D model in time by using the classical fouth ode Runge-Kutta method. To deive a discete bounday condition fo ω at =, we appoximate ψ ) = by a second ode finite diffeence method. This gives ω i,) ψ ) i,n+ ψ ) i,n + ψ ) i,n h. 6.5) Using the bounday condition 6.) fo the steam function at =, we conclude that ψ ) i,n =. Moeove, we can appoximate the second bounday condition ψ ) = = by using a second ode centeed diffeence appoximation to ψ ) =. This gives ψ ) i,n+ ψ ) i,n h =, o equivalently ψ ) i,n+ = ψ ) i,n. 6.6) Substituting 6.6) to 6.5), we obtain ou discete bounday condition fo ω at = : ω ) i,n = ψ ) i,n /h. 6.7) This is the well-known Thom s bounday condition [9]. In [], Hou an Wetton poved that the second ode finite diffeence appoximation of the Navie-Stokes equations using the Thom s bounday condition conveges with a second ode accuacy. 5

16 To summaie, the discete no-slip no-flow bounday at = is given by: u ) i,n =, ψ ) i,n =, ω ) i,n = ψ ) i,n /h. 6.8) The peiodic bounday condition along the -diection and the bounday condition 6.8) at = plus the initial condition completely detemine the dynamic evolution of the discetied system of ou 3D model. The initial condition we conside in ou numeical computations is given by u,,) = + sin4π)) ).) 3, 6.9) ψ,,) =, 6.) ω,,) =. 6.) Note that the compatibility condition 3.5) implies that u, ψ and ω must be even functions of. The above initial condition satisfies this constaint and the no-slip no-flow bounday condition. We choose the initial condition in such a way that its suppot is centeed aound = and the solution is nealy flat away fom =. Since the initial value fo ψ is identically equal to eo, the velocity fields along the adial and the axial diections ae all eo. Thus, the motion is induced by the swiling component of the velocity field. Futhe, we note that the swiling component of the velocity field is always non-negative since its initial value is non-negative. We have used both a unifom mesh and an adaptive mesh in ou computations. Since the solution eventually becomes singula at = and is vey smooth and nealy flat away fom =, we use the following coodinate tansfomation along the -diection to achieve the adaptivity: = fα) α.9sinπα)/π. 6.) This tansfomation gives a change of vaiables fom to α via = fα). Note that f maps the unit inteval [,] to itself and is a smooth tansfomation. Moeove, the deivative of the map is given by f α =.9cosπα).. The smallest value of f α is achieved at α = with f α ) =.. If we use a unifom mesh in α with a mesh sie h, then the above tansfomation will poduce an adaptive mesh in the oiginal space with the smallest mesh sie concentated nea =. The mesh sie nea = is popotional to h /. Thus, we gain an adaptivity facto of / nea =. The advantage of using this mesh adaptivity is that the mesh map is vey smooth and is time independent. We do not intoduce mesh adaptivity along the -diection because we would like to use the Fouie tansfom along it fo the discete elliptic equation fo the steam function. Afte applying the Fouie tansfom along the -diection, the discete system fo ψ becomes a ti-diagonal system fo each wave numbe, which can be inveted diectly. This gives ise to a vey efficient elliptic solve fo the steam function, ψ. We have pefomed a detailed esolution study using both the unifom mesh and the above adaptive mesh. We find that the above adaptive mesh woks extemely well. Since the poblem is thee-dimensional, the adaptivity along the -diection gives a facto of savings. In the inviscid computation, which is moe singula than the viscous case, the lagest esolution we use is N = 89 and N = 8 with a time step t =.5 7. This povides an effective esolution of 89 3 fo the coesponding thee dimensional poblem with a unifom mesh. The total numbe of 6

17 time steps we un is ove a hunded thousands. Without taking advantage of the axisymmety of the solution and the mesh adaptivity, we would not have been able to affod to pefom computations using this high esolution. 6. The dange of unde-esolution in poducing a faux singulaity Befoe we pesent ou numeical evidence which suppots the fomation of a potential finite time singulaity in ou model, we fist demonstate the impotance of sufficient numeical esolution in esolving the nealy singula behavio of the 3D Eule o Navie-Stokes equations. Unde-esolution could lead to the conclusion of a faux singulaity in a finite time when the solution is in fact smooth. The guidance fom ou analysis in Section 5 also plays a citical ole in excluding such an atificial singulaity esulting fom an unde-esolved computation. The initial condition we conside is designed in such a way that it has a maximal pobability of poducing a finite time singulaity fo the coesponding inviscid model. A key popety of this initial condition is that u is non-negative and vanishes at =.375. Fom the model equation fo u in 3.) in the inviscid case ν =, we can see that u emains non-negative and vanishes at =.375 fo all times. Fo this paticula initial condition, the solution poduces two lage focusing centes dynamically, one to the left of =.375, and anothe to the ight of =.375. As time inceases, the axial velocity field nea = geneated by this initial condition becomes lage and positive fo <.375 and negative fo >.375. As a esult, the two focusing centes ae attacted towad the plane of =.375 and may expeience a head-on collision in a finite time. The potential collision of the two focusing centes is a pefect candidate fo geneating a finite time singulaity at =.375. Indeed, we find fom ou computations that the two focusing centes ae attacted to the plane of =.375 and develop a thin laye paallel to the -axis. By the time t =.77, the two focusing centes almost collide, and the solution becomes seveely squeeed along the -diection, see Figue. It is difficult to esolve the thin laye stuctue of the solution of the inviscid model aound t =.77. We had oiginally used a unifom mesh to solve the 3D model equation up to N N = ove the physical domain Ω = [,] [,]. This level of esolution is close to the limit of ou computing esouces without going to a massively paallel cluste. Even with this lage esolution, we find that the solution of the inviscid 3D model is still not completely well esolved nea the potential singulaity time. This motivates us to switch to the adaptive mesh and concentates ou computational mesh nea the most singula egion of the solution, i.e. the egion close to the -axis. The use of adaptive mesh enables us to incease ou local effective esolution substantially. With the inceasing numeical esolution, we can esolve the nealy singula behavio of the solution at t =.78. It is woth emphasiing that numeical unde-esolution may lead to the wong conclusion that the 3D model develops a finite time singulaity aound t =.78 when the solution becomes nealy singula. In fact, ou computations indicate that the highe esolution we use with N = 56, 5, 4, the faste the gowth the solution expeiences aound t =.78, see Figue. This seems to be a stong evidence fo a finite time singulaity. Howeve, with even highe esolution N 48), we can esolve this nealy singula behavio of the solution, see Figue 8. As we discussed ealie, the solution becomes seveely flattened and -dominant locally nea the egion of maximum of u. In the egion whee u is compaable to the maximum of u, the 3D model can be appoximated to the leading ode by the coesponding D model along the -diection. As we showed in Section 5, the solution of the D model cannot blow up in a finite time, even in 7

18 Closeup of the u contou at t=.77, N =N =48, ν=, 3D model Figue : Closeup view of u contou at t =.77 using a unifom mesh, N = 48, N = 48, t = 6, ν =. the inviscid case. Afte t =.78, the two focusing centes move away fom each othe and the maximum of u deceases fo a shot time befoe it expeiences anothe peiod of apid gowth. The theoetical esult povides us with a citical guidance which enables us to compute beyond this faux singulaity aound t = The nealy singula behavio of the viscous model We now pesent numeical esults which show that the solution of the viscous model becomes nealy singula. We choose the viscous coefficient to be ν =. and pefom a seies of esolution studies using the adaptive method. We have used both unifom mesh and adaptive mesh with N anging fom 56 to 496. Below we pesent the computational esults obtained by using the adaptive mesh with the highest esolution N = 496, N = 4, and t =.5 7. We will also pefom a esolution study to demonstate that ou computations ae well-esolved Numeical evidence fo a potential finite time singulaity Fom ou analytical study of the 3D model, it follows by using a standad enegy estimate that if u is bounded, then the solution of the viscous 3D model cannot blow up in a finite time. Thus it is sufficient to monito the gowth of u in time. We will pesent numeical evidence which seems to suppot that u may develop a potential finite time singulaity fo the initial condition we conside. The natue of this potential singulaity and the mechanism fo geneating this potential singulaity will be analyed in a late subsection. In Figue 3, we plot the maximum of u in time ove the time inteval [,.] using the adaptive mesh method with N = 496 and N = 4. The time step is chosen to be t =

19 9 8 Maximum of u in time, N =56 geen), N =5 ed), N =4 blue), ν= N =N =56 N =N =5 N =N =4 7 6 u time Figue : Maximum u in time computed by the unifom mesh ove [,] [,], N = N = 56 geen), N = N = 5 ed), and N = N = 4 blue), ν =. The time steps used ae t = 5 6,.5 6 and 6 espectively. This sequence of elatively low esolution computations seems to suggest that the solution would blow up aound t =.78. But highe esolution computations show that this is not the case. This is also suppoted by ou analysis of the educed D model. We can see that u expeiences a vey apid gowth in time afte t =.. In the ight plot of Figue 3, we also plot loglog u )) as a function of time. We can see clealy that u gows much faste than double exponential in time, which implies that the solution of ou model may develop a finite time singulaity. We will pesent moe caeful analysis of this potentially singula behavio late. In Figues 4-5, we show a sequence of contou plots fo u fom t =.4 to t =.. At ealy times, we obseve that the solution foms two lage focusing centes of u which appoach each othe. As this occus, these athe localied egions ae squeeed and fom a thin laye paallel to the -axis and with lage gadients along the -diection. The two focusing centes become the closest aound t =.7, see also Figue fo the contou of u at t =.77 fo the inviscid 3D model. As these egions appoach each othe and develop a thin laye paallel to the -axis, the solution becomes locally -dominant nea the egion whee u achieves its maximum. In this egion, the 3D model can be appoximated to the leading ode by the coesponding D model along the -diection which we intoduced in the pevious section. As we have shown befoe, the solution of the D model cannot blow up. Afte t =.7, the maximum of u stats to decease. The two focusing centes move away fom each othe and thei suppots become moe isotopic. As time inceases, we obseve that thee is a stong nonlinea inteaction between u and ψ ), which is induced by the ovelap between the suppot of maximum of u and the suppot of maximum of ψ ). By the suppot of maximum of u, we mean the egion in which u is compaable to its maximum. The stong alignment between u and ψ ) nea the suppot of maximum of u leads to a apid gowth of the solution which may become singula in a finite time. Anothe impotant obsevation is that as time inceases, the position at which u achieves its 9

20 9 x 4 Maximum u in time, N =496, N =4, t=.5 7, ν=..5 Loglog u )) in time, N =496, N =4, t=.5 7, ν= u loglog u )) Time Time Figue 3: Left figue: u as a function of time ove the inteval [,.]. The ight figue: loglog u )) as a function of time ove the same inteval. The solution is computed by the adaptive mesh with N = 496, N = 4, t =.5 7, ν =.. maximum also moves towad the symmety axis. This suggests that the potential singulaity will be along the symmety axis at the singulaity time. It is easy to see fom.7) that lim + u =.5lim + ω. Thus, the blowup of u chaacteies the blowup of the axial voticity, ω. Next, we pefom a detailed study fo the 3D model and push ou computations vey close to the potential singulaity time. We use a sequence of esolutions using both unifom and adaptive mesh. Fo the unifom mesh, we use esolutions fo N N anging fom to with time steps anging fom t = 5 6 to 5 7. Fo the adaptive mesh, we use N N = 48 56, N N = and N N = espectively. The coesponding time steps fo these computations ae t = 6, t = 5 7, and t =.5 7 espectively. With N N = 496 4, we achieve an effective esolution of 4 4 nea the egion of = whee the solution is most singula. To obtain futhe evidence fo a potential finite time singulaity, we use a systematic singulaity fom fit pocedue to obtain a good fit fo the possible singulaity of the solution. The pocedue of ou fom fit is as follows. We look fo a finite time singulaity of the fom: u C T t) α. 6.3) We have tied seveal ways to detemine the fitting paametes T, C and α. At the end, we find that the best way is to study the invese of u as a function of time using a sequence of numeical esolutions. This appoach has been used successfully befoe by van Dommelen and Shen in thei study of the spontaneous geneation of the singulaity in a sepaating lamina bounday laye [35] see also []). Fo each esolution, we find that the invese of u is almost a pefect linea function of time, see Figs. 6 and 8. By using a least squae fit of the invese of u, we find that α = gives the best fit. The same least squae fit also detemines the potential singulaity time T and the constant C. We emak that the O/T t)) blowup ate of u, which coesponds to the blowup ate of the axial voticity, is consistent with the non-blowup citeion of Beale-Kato-Majda type, see Section 7. To confim that the above pocedue indeed gives a good fit fo the potential singulaity, we plot u as a function of time in the left plot of Figue 6. We can see that the ageement between

21 Contou of u at t=.4, N =496, N =4, t =.5 7, ν=., 3D model Contou of u at t=.6, N =496, N =4, t =.5 7, ν=., 3D model Contou of u at t=.8, N =496, N =4, t=.5 7, ν=., 3D model Contou of u at t=., N =496, N =4, t=.5 7, ν=., 3D model Figue 4: Contou plots of u at t =.4 top left),.6 top ight),.8 bottom left), and. bottom ight), computed by the adaptive mesh with N = 496, N = 4, t =.5 7, ν =.. Contou of u at t=., N =496, N =4, t=.5 7, ν=., 3D model Closeup view of u contou at t=., N =496, N =4, t=.5 7, ν=. x Figue 5: The contou of u at t =. left figue) and its closeup view ight figue) computed by the adaptive mesh with N = 496, N = 4, t =.5 7, ν =..

22 .4 x 4 The invese of u in time with N=496 vs asymptotic fit, ν=.. Asymptotic fit N=496 x u blue) vs C/T t) ed) in time, T=.9, C=8.348, N=496, ν=. 4 u, N=496 9 Fitted solution u.6 u Time Time Figue 6: The left plot: The invese of u blue) vesus the asymptotic fit ed). The ight plot: u blue) vesus the asymptotic fit ed). The asymptotic fit is of the fom: u T t) C with T =.9 and C = The solution is computed by adaptive mesh with N = 496, N = 4, t =.5 7. ν =.. the computed solution with N N = and the fitted solution is almost pefect. In the ight box of Figue 6, we plot u computed by ou adaptive method against the fom fit C/T t) with T =.9 and C = The two cuves ae almost indistinguishable duing the final stage of the computation fom t =.8 to t =.. In Figue 7, we also plot ψ as a function of time. We obseve that ψ decays almost linealy in time with a logaithmic coection. The ageement between the computed solution with N N = and the fitted solution is excellent. On the ight plot, we compae ψ with the fitted solution. Again, we obseve a vey good ageement between the two cuves. We futhe investigate the potential singula behavio of the solution by using a sequence of esolutions to study the limiting behavio of the computed solution as we efine ou esolutions. The space esolutions we use ae N N = 4 8, 48 56, and espectively. The coesponding time steps ae t = 6, 5 7, and.5 7 espectively. Fo each esolution, we obtain an optimal least squae fit of the singulaity of the fom u T t)/c. The esults ae summaied in Table. Based on the fitted paametes T and C fom the thee lagest esolutions, we constuct a second ode polynomial that intepolates T and C though these thee data points. We then use the polynomial to extapolate the values of T and C to the infinite esolution limit. The extapolated values at h = ae T =.83 and C = 8.9 espectively. In Fig. 8, we plot the invese of u as a function of time using fou diffeent esolutions. We can see that as we efine the esolution, the computed solution conveges to the extapolated singulaity limiting pofile. To illustate the natue of the nealy singula solution, we show the 3D view of u as a function of and in Figues 9 and. We also show the 3D view of w as a function of and in Figue. While u is symmetic with espect to =.375, w is anti-symmetic with espect to =.375. We can see that the suppot of the solution u in the most singula egion is isotopic and appeas to be locally self-simila. We will futhe investigate the local scaling popety of the solution in Section

23 x 4 The invese of ψ blue) vs asymptotic fit ed), N=496, ν=. N=496 Asymptotic fit ψ 4 x blue) in time vs asymptotic fit ed), N=496, ν=. 4 N=496 Asymptotic fit ψ ψ Time Time Figue 7: The left plot: The invese of ψ blue) vesus the asymptotic fit ed). The ight plot: ψ blue) vesus the asymptotic fit ed). The asymptotic fit is of the fom: ψ Clog/T t))) / T t) with T =.9 and C =.535. The solution is computed by adaptive mesh with N = 496, N = 4, t =.5 7. ν =...5 x 4 The invese of u in time, N=4, 48, 37, 496, ν=. N=4 N=48 N=37 N=496 Asymptotic fit u Time Figue 8: The invese of u in time. The solution is computed by adaptive mesh with N = 4, 48, 37 and 496 espectively odeing fom top to bottom in the figue), t = 6, 5 7, , and.5 7 espectively. The last cuve is the singulaity fit by extapolating the computational esults obtained by N = 48, 37 and 496 to infinite esolution N =. The fitted cuve is of the fom u T t)/c, with T =.83 and C = 8.9. ν =.. 3

24 h T C / / / / Extapolation to h = Table : Resolution study of paametes T and C in the asymptotic fit: u C using diffeent esolutions h = /N ). The esolutions we use in ou adaptive computations ae N N = 4 8, 48 56, and espectively. The coesponding time steps ae t = 6, 5 7, and.5 7 espectively. The last ow is obtained by extapolating the second ode polynomial that intepolates the data obtained using h = /496, /644 and /89. T t) Figue 9: The 3D view of u at t =. computed by the adaptive mesh with N = 496, N = 4, t =.5 7, ν =.. 4

25 Figue : The 3D view of u at t =. computed by the adaptive mesh with N = 496, N = 4, t =.5 7, ν =.. Figue : The 3D view of ω at t =. computed by the adaptive mesh with N = 496, N = 4, t =.5 7, ν =.. 5

26 x 4 Convegence study of u in time, N =48, 37, 496, ν=. N =496 x 4 Convegence study of u in time, N =48, 37, 496, ν=. N =496 9 N =37 N =48 9 N =37 N = u 5 4 u Time Time Figue : Convegence study fo u in time with thee esolutions: N N = 48 56, t = 5 7 geen), N N = 37 38, t = ed), N N = 496 4, t =.5 7 blue). The left figue is ove the time inteval [,.], while the ight figue is a closeup view ove the time inteval [.,.]. ν =.. Remak 6. While the numeical esults pesented in this subsection seem to suppot that the solution of the viscous model develops a potential finite time singulaity, we caution that this evidence is not yet conclusive. Based on the balance between the nonlinea votex stetching tem and the viscous tem, we find that the solution of the viscous model seems to be dominated by the dynamics of the inviscid model duing the time inteval of ou computation. In ode to detemine whethe the 3D model actually develops a finite time singulaity, we need to compute much close to the potential singulaity time with esolutions much highe than what we have used in the cuent pape in ode to captue the viscous effect accuately. Depending on the local scaling popety of the nealy singula solution and the balance between the votex stetching tem and the viscous tem, it is still possible that the viscous tem may eventually egulaie the nealy singula solution induced by the nonlinea votex stetching tem. We plan to investigate this issue futhe in ou futue wok Resolution study Finally, we pefom a esolution study fo ou computations by compaing the computation obtained by thee diffeent esolutions, which ae N N = 48 56, N N = 37 38, and N N = In Figue, we plot u as a function of time using these thee esolutions N N = geen), N N = ed), and N N = blue) ove the time inteval [,.]. We can see that while the computation with N = 48 unde-esolves the solution nea the end of the computation, the solution obtained by using N = 37 gives an excellent ageement with that obtained by using N = 496. We also compae the solution of u at = using thee diffeent esolutions. Using the patial egulaity theoy fo ou 3D model, which we pove in [7], any singulaity of ou 3D model must lie on the symmety axis, =. Thus it makes sense to pefom a esolution study fo the solution along the symmety axis which is the most singula egion of the solution. In the left box of Figue 6