aa r X i v : . [ m a t h . A P ] A p r Nonlinear PDEs and Scale Dependence
Garry Pantelis
Abstract
The properties of nonlinear PDEs that generate filtered solutions are explored with particularattention given to the constraints on the residual term. The analysis is carried out for nonlinearPDEs with an emphasis on evolution problems recast on space-time-scale. We examine the role ofapproximation that allow for the generation of solutions on isolated scale slices.
We are interested in applications, particularly of the evolution type, that are governed by nonlinearPDEs that generate solutions that exhibit fluctuations at all spatial/temporal scales. It is often thecase that one is interested in obtaining a macroscopic solution that is defined as some kind of filteredrepresentation of the dependent variables associated with some scale of resolution. Consequently, themacroscopic equations satisfied by the filtered variables will depend not only on space-time but also ona parameter associated with scale. We start by example and then generalise in the following sections.Let x = ( x , ..., x n ) be the cartesian coordinates of R n ( n = 1 , , t ∈ R + ( R + = the set of positive real numbers excluding 0). In orderto define the filtered variables we introduce the scale parameter η ∈ I on the scale interval I = (0 , η )where 0 < η <<
1. We work on space-time-scale M = R n × R + × I ⊂ R m , m = n + 2, and introducethe linear operator L = ∂∂η − △ (1.1)where △ = ∇ · ∇ and ∇ is the spatial gradient operator. A filtered scalar variable φ : M → R can nowbe defined by L φ = 0 ( x, t, η ) ∈ M . (1.2)Suppose that there exists Φ( x, t ) ( k Φ k L ( R n ) < ∞ ) such that lim η → + φ = Φ( x, t ). If we can also setlim | x |→∞ ∇ φ = 0 then φ is uniquely defined by φ ( x, t, η ) = Z R n G ( x − ˆ x, η )Φ(ˆ x, t ) d ˆ x . . . d ˆ x n (1.3)where G ( x, η ) = (4 πη ) − n/ exp[ −| x | / (4 η )] with the property R R n G ( x, η ) dx . . . dx n = 1. Hence φ ( x, t, η )becomes the Gaussian filter of Φ( x, t ). In this way we can identify η = βδ , where δ is a parameterassociated with the characteristic spatial resolution and β is a constant associated with the filter width.A major sticking point with this approach is the need to assume the existence of a regular limitingsolution, as η → + , associated with the space-time pointwise representation of the dependent variables.Here we shall depart from the approach made in [3] and [4] by attempting to work entirely in M andavoid any reference to a limiting fully resolved solution on space-time.We denote by R ( M, R N ), N ≥
1, the set of smooth vector functions on M , i.e. smooth mappings M → R N . For the case N = 1 members of R ( M, R N ) are smooth scalar functions.Consider now an ideal fluid whose variables are defined on M as follows. Let the fluid velocity v ∈ R ( M, R n ) be a filtered field, defined by L v = 0 ( x, t, η ) ∈ M , (1.4)and satisfy the continuity condition ∇ · v = 0 ( x, t, η ) ∈ M . (1.5)1e now construct the macroscopic equations from (1.4) and (1.5).On each scale slice M | η =const , in a time interval I ⊂ R a fluid element follows a path under the flowcharacterized by a mapping γ : I → M | η =const (integral curve of v ) so that v ◦ γ ( t ) = dx ◦ γ ( t ) dt . (1.6)The acceleration, a , of the same fluid element can be defined along the flow path γ ( t ) by a ◦ γ ( t ) = d x ◦ γ ( t ) dt = dv ◦ γ ( t ) dt . (1.7)Alternatively we may write a ∈ R ( M, R n ) as a = ∂v∂t + v · ∇ v . (1.8)We introduce the symmetric second order tensor σ whose components are given by σ ab = n X c =1 ∂v a ∂x c ∂v b ∂x c ≤ a, b ≤ n . (1.9)We also define s = − ∇ · σ . (1.10)The properties assigned to v above dictate that the acceleration must satisfy the following constraint. Proposition 1.1 If v ∈ R ( M, R n ) satisfies (1.4) and (1.5) then L a = s ( x, t, η ) ∈ M , (1.11) where s is given by (1.10). Proof.
We have L a = L ( ∂v∂t + v · ∇ v ) = ∂∂t ( L v ) + L ( v · ∇ v ) . (1.12)By a straightforward calculation L ( v · ∇ v ) = ( L v ) · ∇ v + v · ∇ ( L v ) + s (1.13)where we make use of the fact that v is divergent free to express the source term s in the divergent form(1.10). The required result follows from (1.12) and (1.13) and the fact that v is a filtered field .We can always introduce a filtered scalar function p ∈ R ( M, R ), L p = 0 ( x, t, η ) ∈ M , (1.14)and write a = −∇ p + r , (1.15)for some r ∈ R ( M, R n ). Noting that L a = −L∇ p + L r = −∇L p + L r = L r , we have constructed from(1.4) and (1.5) the system of macroscopic equations: ∇ · v = 0 ( x, t, η ) ∈ M , (1.16) ∂v∂t + v · ∇ v + ∇ p = r ( x, t, η ) ∈ M , (1.17) L r = s ( x, t, η ) ∈ M , (1.18)where s is given by (1.10).We can think of the residual r as incorporating the effects of the large scale viscous and stress/strainsof the fluid that are manifestations of the filtering process. In the case that we can assume the existenceof a limiting pointwise representation of the filtered variables we can impose, under some appropriatenorm k · k , the condition that lim η → + k r − ν △ v k = 0, for some constant ν . In this way we recover, atleast in the generalised sense, the Navier-Stokes equation from the macroscopic equations (1.16)-(1.17)as a limiting formulation. Since the notion of viscosity is itself meaningful only in some locally averagedsense, under the continuum fluid model the inviscid limit, lim η → + k r k = 0, is more natural.2 Macroscopic Equations
We now attempt to generalise avoiding application specific issues. The system of nonlinear PDEs con-sidered will be of first order but it should be apparent that the analysis can incorporate higher ordersystems. As in the previous section, let x = ( x , ..., x n ) be the cartesian coordinates of R n ( n = 1 , , t ∈ R + ( R + = the set of positive realnumbers excluding 0). Let M = R n × R + × I ⊂ R m , m = n + 2, where I = (0 , η ) is the scale interval, η ∈ I is the scale parameter and 0 < η <<
1. We shall write ξ = ( x, t, η ) = ( ξ , . . . , ξ m ), where we haveset ξ a = x a (1 ≤ a ≤ n ), ξ n +1 = t and ξ m = ξ n +2 = η is the scale parameter.For any D ⊆ R N , N >
0, we denote by R ( D, R N ), N >
0, the set of smooth ( C ∞ ) maps u : D → R N . For u = ( u α ) = ( u , . . . , u N ) ∈ R ( M, R N ) we shall denote u i = ∂ i u = ( u αi ), u ij = ∂ i ∂ j u = ( u αij ), . . . , with the shorthand notation ∂ i = ∂ /∂ξ i . Throughout, unless otherwise stated, we use the range1 ≤ i, j, k ≤ m for lower case Latin indices and the range 1 ≤ α, β, γ ≤ N for the lower case Greekindices. For easier identification of certain important operators we use the indices t instead of n + 1 and η instead of m = n + 2, e.g. we write ∂ t instead of ∂ n +1 and ∂ η instead of ∂ n +2 = ∂ m .We shall express our macroscopic equations in terms of a core function of u and u i that is forminvariant with respect to scale and a residual term. Let F ∈ R N such that F ( u, u i ) is a smooth vectorfunction with respect to its indicated arguments. Given any u ∈ R ( M, R N ) there exists an r ∈ R ( M, R N )such that F ( u, u i ) = r ξ ∈ M . (2.1)Alternatively, we may regard (2.1) as a system of N first order PDEs for the unknown dependent vectorfunction u ( ξ ) if r ( ξ ) is known. We shall henceforth refer to r as the residual and F ( u, u i ) as the corefunction. The core function may be regarded as retaining the structure of the N PDEs that are associatedwith the fully resolved system as shall be outlined below. The construction of the core function in awider context is application specific and follows along the lines discussed in the previous section. Weshall return to this application in a later section. Our first task is to find constraints on r ( ξ ) such thatthe solutions of (2.1) have well defined filter properties.We introduce the vector field operators V i defined by V i = ∂ i + u αi ˙ ∂ α + u αij ˙ ∂ jα + u αijk ˙ ∂ jkα + . . . (2.2)with the notation ∂ i = ∂∂ξ i , ˙ ∂ α = ∂∂u α , ˙ ∂ iα = ∂∂u αi , ˙ ∂ ijα = ∂∂u αij , . . . (2.3)Here and throughout we use the summation convention of repeated upper and lower indices over theassigned ranges for the given indice type given above. For u ∈ R ( M, R N ) the operator V i acting onany F ∈ R N using the more general representation F ( ξ, u, u i , u ij , . . . ) generalises the partial derivativeoperator ∂ i acting on members of R ( M, R N ). ( V i F is sometimes referred to as the total derivative of F with respect to ξ i ). The action of V i on any w ∈ R ( M, R N ), w = w ( ξ ), gives the partial derivative of w with respect ξ i , i.e. V i w = ∂ i w . The vector field operators V i may be truncated up to terms associatedwith the largest order of the partial derivatives of u present in the expressions on which V i is operating.In truncated form the vector fields V i have special geometric significance in the general theory of PDEs[1],[2].We introduce the spatial Laplacian operator △ = ∇ · ∇ , (2.4)where ∇ = ( ∂ , · · · , ∂ n ) is the spatial gradient operator. The associated operator acting on vectorfunctions F ∈ R N using the general representation F ( ξ, u, u i , u ij , . . . ) is defined by L = n X b =1 V b V b . (2.5)The set of filter maps is defined as follows: Definition. P ( M, R N ) is the set of vector functions u ∈ R ( M, R N ) such that( ∂ η − △ ) u = 0 ξ ∈ M . (2.6)3e can assume that P ( M, R N ) is sufficiently populated that it is of interest to the analysis thatfollows. Suppose for the moment that we can assume that there exists e u ∈ R ( R n × R + , R N ) satisfyingthe first order PDEs F ( e u, ∂ i e u ) = 0 ( x, t ) ∈ R n × R + (2.7)for some F ∈ R N such that F ( u, u i ) is smooth with respect to its indicated arguments. If for some u ∈P ( M, K ) we have u | η =0 + = e u and lim | x |→∞ ∇ u = 0 then u ( x, t, η ) = R R n G ( x − ˆ x, η ) e u (ˆ x, t ) d ˆ x . . . d ˆ x n ,where G ( x, η ) = (4 πη ) − n/ exp[ −| x | / (4 η )]. Hence u ( ξ ) = u ( x, t, η ) is the Gaussian spatial filter of e u ( x, t ).The PDEs (2.7) represent the fully resolved system. It should be mentioned at this point that inmany applications of interest F will not explicitly depend on terms u η (see application below). Underour assigned range for the lower case Latin indices the representation F ( u, u i ) will indicate a possibleexplicit dependence on u η . The more general representation F ( u, u i ) will not effect the calculationsbelow and is used to avoid introducing more notation for the indices and their ranges.Since for u ∈ P ( M, R N ) we cannot expect that in general F ( u, u i ) will vanish identically on M wemust introduce a residual term r ∈ R ( M, R N ) as in (2.1). The system (2.7) is recovered from (2.1) byrequiring that lim η → + r = 0.In many applications one cannot gaurantee the existence of regular solutions e u ∈ R ( R n × R + , R N )for the fully resolved system and it is at this point that we depart from the assumptions made in [3],[4].Our focus is on nonlinear problems that generate solutions that exhibit fluctuations at all space and timescales. We propose that such systems can be defined in a meaningful way enirely within space-time-scaleavoiding any reference to a fully resolved system.We shall make extensive use of the vector field operator W = ∂ η + ( △ u α ) ˙ ∂ α + ( △ u αi ) ˙ ∂ iα + ( △ u αij ) ˙ ∂ ijα + . . . (2.8)We are now in a position to establish the constraints on the residual for filter maps. Proposition 2.1 If u ∈ P ( M, R N ) and r ∈ R ( M, R N ) such that F ( u, u i ) − r = 0 , ξ ∈ M , where F ∈ R N and F ( u, u i ) is smooth with respect to its indicated arguments, then ( ∂ η − △ ) r = s , ξ ∈ M ,where s = ( W − L ) F ( u, u i ) . Proof.
For any u ∈ R ( M, R N ) there will always exist an r ∈ R ( M, R N ) such that F ( u, u i ) − r vanishesidentically on M . We have ( V η − L )( F ( u, u i ) − r ) = 0 . (2.9)Manipulating the left hand side we obtain( V η − L )( F ( u, u i ) − r ) = ( V η − L ) F ( u, u i ) − ( ∂ η − △ ) r = ( V η − W ) F ( u, u i ) + ( W − L ) F ( u, u i ) − ( ∂ η − △ ) r = ( V η − W ) F ( u, u i ) + s − ( ∂ η − △ ) r , (2.10)where we take r ∈ R ( M, R N ) to mean that it has the representation r ( ξ ) so that V i r = ∂ i r . It isstraightforwad to see that if u ∈ P ( M, R N ) then V η = W and hence the first term in this last expressionof (2.10) vanishes. Thus ( V η − L )( F ( u, u i ) − r ) = − ( ∂ η − △ ) r + s . (2.11)The desired result follows from (2.9) and (2.11) .We can now write the full system of macroscopic equations for the filtered variables u ∈ P ( M, R N )as F ( u, u i ) = r ξ ∈ M , (2.12)( ∂ η − △ ) r = s ξ ∈ M , (2.13)where s = ( W − L ) F ( u, u i ) . (2.14)Given any core function F ∈ R N such that F ( u, u i ) is smooth with respect to its arguments, forany u ∈ P ( M, R N ) there will always exist an r ∈ R ( M, R N ) in (2.12) satisfying (2.13)-(2.14). From the4ractical viewpoint, we must consider as a separate issue the task of developing algorithms that extractfrom P ( M, R N ) those members that can be regarded as meaningful, in some formal sense, to the specificapplication under consideration.We now establish an important link between the filter error, ψ , and the residual equation error, e ,as defined in the following. Proposition 2.2
Let u, r ∈ R ( M, R N ) such that F ( u, u i ) − r = 0 , ξ ∈ M , where F ∈ R N and F ( u, u i ) is smooth with respect to its indicated arguments. Let ψ ∈ R ( M, R N ) be defined by ψ ( ξ ) = ( ∂ η − △ ) u ( ξ ) . (2.15) It follows that ψ satisfies the linear system of first order PDEs C αiβ ∂ i ψ β + C αβ ψ β = e α ξ ∈ M , (2.16) where the coefficients are given by C αiβ = ˙ ∂ iβ F α ( u, u j ) , C αβ = ˙ ∂ β F α ( u, u i ) , and e = ( ∂ η − △ ) r − s , (2.17) where s = ( W − L ) F ( u, u i ) . Proof.
We follow a similar argument to that of the proof of the previous proposition but note that now r need not satisfy (2.13)-(2.14) and hence u is not necessarily a member of P ( M, R N ). Since F ( u, u i ) − r vanishes identically on M we have ( V η − L )( F ( u, u i ) − r ) = 0 . (2.18)The left hand side becomes( V η − L )( F ( u, u i ) − r ) = ( V η − L ) F ( u, u i ) − ( ∂ η − △ ) r = ( V η − W ) F ( u, u i ) + ( W − L ) F ( u, u i ) − ( ∂ η − △ ) r = ( V η − W ) F ( u, u i ) + s − ( ∂ η − △ ) r = ( V η − W ) F ( u, u i ) − e , (2.19)where we take r ∈ R ( M, R N ) to mean that it has the representation r ( ξ ) so that V i r = ∂ i r . The firstterm in the last expression will not in general vanish because u is not necessarily a member of P ( M, R N ).Using the definitions of the vector field operators V η and W , we have the explicit representation V η − W = ψ α ˙ ∂ α + ( ∂ i ψ α ) ˙ ∂ iα + ( ∂ i ∂ j ψ α ) ˙ ∂ ijα + . . . (2.20)The identity (2.16) then follows from (2.18)-(2.20) .If the residual r satisfies (2.13), where the source term is given by (2.14), then e vanishes identicallyon M and (2.16) admits the trivial solution from which it follows that u ∈ P ( M, R N ).The system of PDEs (2.12)-(2.13) are the exact equations satisfied by the filtered variables u ∈P ( M, R N ) on space-time-scale M . They are of little use in applications since the presence of the ∂ η term in the residual equation (2.13) means that the solution of the system can only be obtained by scaleintegration from some arbitrary scale slice of M on which u is prescribed. To overcome this problem weresort to approximation by replacing the residual equation (2.13) with a constraint on the residual thatallows us to generate regular solutions u independently on scale slices M | η =const such that u is a goodapproximation of some member of P ( M, R N ) on that scale slice. An example of a residual approximation that has been found to be useful in applications is one in whichthe exact residual equation (2.13) is replaced by the constraint (see [3],[4]) △ r − rη + s = 0 , (3.1)5here the source term s is given by (2.14). The approximate macroscopic system (2.12) and (3.1) canbe solved independently on any scale slice M | η =const by updating the residual via (3.1) at each timestepas soon as u is available. If we can assume that lim η → + | r | = 0 it follows that | e | = (cid:12)(cid:12)(cid:12)(cid:12) ∂r∂η − rη (cid:12)(cid:12)(cid:12)(cid:12) ≤ η M (cid:12)(cid:12)(cid:12)(cid:12) ∂ r∂η (cid:12)(cid:12)(cid:12)(cid:12) . (3.2)Motivated by a discussion of residual approximations in general, we shall avoid any focus on specificformulations such as (3.1).In practical application space-time solutions are generated from the system of PDEs (2.12) on somescale slice M | η =const while using a residual that is not exact, i.e. one in which e does not vanish identicallyon that scale slice. If we enforce (2.12) with a residual that is not exact then it follows from Proposition2.1 that u cannot be a member of P ( M, R N ).Our objective here is to investigate how well members of R ( M, R N ) serve as approximations to mem-bers of P ( M, R N ). More precisely, suppose that we employ a residual that is based on an approximationof the residual equation (2.13) for which we know that a regular solution u ∈ R ( M, R N ) for (2.12)exists. We are interested in how u will deviate from some member of P ( M, R N ) as the scale parameteris increased if the two solutions agree on some arbitrary scale slice. Proposition 3.1
For some ε ∈ (0 , η ) , let M ε = M | η ∈ ( ε,η ) . Let u ∈ R ( M ε , R N ) and ¯ u ∈ P ( M, R N ) such that lim η → ε + ( u − ¯ u ) = 0 . Suppose further that ψ ∈ R ( M ε , R N ) , defined by ψ ( ξ ) = ( ∂ η − △ ) u ( ξ ) , isbounded on M ε and that lim | x |→∞ ∇ ¯ u, ∇ u = 0 . It follows that ( u − ¯ u )( x, t, η ) = Z ηε Z R n G ( x − ˆ x, η − ˆ η ) ψ (ˆ x, t, ˆ η ) d ˆ x . . . d ˆ x n d ˆ η ( x, t, η ) ∈ M ε , (3.3) where G ( x, η ) = (4 πη ) − n/ exp[ −| x | / (4 η )] . Proof.
Setting ω = u − ¯ u we have ( ∂ η − △ ) ω = ψ ξ ∈ M ε . (3.4)We also have ω | η = ε + = 0 ; lim | x |→∞ ∇ ω = 0 . (3.5)The solution of (3.4)-(3.5) is unique and is given by (3.3) .Given that R R n G ( x, η ) dx . . . dx n = 1 we obtain from (3.3) the bound, for ξ ∈ M ε , | u − ¯ u | ≤ η sup ξ ∈ M ε | ψ | . (3.6) We return to the example of an ideal fluid. We set u = ( v, p ), where v = ( v , . . . , v n ) is the filteredfluid velocity vector and u n +1 = p is the filtered fluid pressure. In this case N = n + 1 and we set F = ( F ( v ) , F ( p ) ), where F ( v ) = ( F , . . . , F n ) and F ( p ) = F n +1 . Motivated by Section 1 we define thecore function by F ( v ) = v t + v · ∇ v + ∇ p , F ( p ) = ∇ · v . (4.1)Note that the core function is not explicitly dependent on ξ and u η .Since we insist that admissible solutions have well defined filter properties, i.e. u ∈ P ( M, R N ),Proposition 2.1 demands that the residual satisfy (2.13). Here the source term s is obtained by applyingthe identity (2.14) to the explicit representation of the core function given in (4.1): s a = − n X b =1 n X c =1 v bc v abc , (1 ≤ a ≤ n ) , s n +1 = 0 , (4.2)6sing the notation u i = ( v i , p i ), u ij = ( v ij , p ij ). We set r = ( r ( v ) , r ( p ) ), where r ( v ) = ( r , · · · , r n ) and r ( p ) = r n +1 , and similarly s = ( s ( v ) , s ( p ) ), where s ( v ) = ( s , · · · , s n ) and s ( p ) = s n +1 . If we can assumethat r ( p ) → η → + then, in light of the residual equation (2.13) and the fact that the componentof the source term s ( p ) = s n +1 vanishes identically on M , r ( p ) also vanishes identically on M . It followsthat the continuity equation ∇ · v = 0 is form invariant with repsect to scale. We can now write s ( v ) = − ∇ · σ , s ( p ) = 0 , (4.3)where σ is a symmetric second order tensor whose components are given by (1.9). Thus (4.3) is inagreement with the results presented in the first section. We note that the source term in the residualequation also does not contain any explicit dependence on terms involving ∂ η and ∂ t .For u ∈ P ( M, R N ) the macroscopic equations (2.12) then take the explicit form ∂v∂t + v · ∇ v + ∇ p = r ( v ) , ∇ · v = 0 . (4.4)In practice we wish to solve (4.4) (subject to suitable initial conditions) independently on scale slices M | η =const . Although the source term (4.3) is suitably defined, the residual equation (2.13) is still notsuitable for this purpose due to the presence of the ∂ η term on the left hand side. For this reason weresort to an approximation of the residual based on (2.13)-(2.14) with the aim of obtaining a memberof R ( M, R N ) that approximates some member of P ( M, R N ).In a similarly fashion as above, let us set e = ( e ( v ) , e ( p ) ) and ψ = ( ψ ( v ) , ψ ( p ) ). Because r ( p ) and s ( p ) vanish indentically on M it follows that e ( p ) also vanishes identically on M . Using the prescription ofthe core function (4.1) to obtain the coefficients, the system of PDEs (2.16) reduce to ∂ t ψ ( v ) + v · ∇ ψ ( v ) + ψ ( v ) · ∇ v + ∇ ψ ( p ) = e ( v ) , ∇ · ψ ( v ) = 0 . (4.5)for some v ∈ R ( M, R n ) that is a solution of (4.4). Clearly the long time behaviour of (4.5) is of interestand will depend crucially on the specific residual approximation used. If one can supply reliable estimatesof the residual equation error e on the scale slice on which the space-time solution is being sought thenthere is the opportunity of resorting to computation. Since the left hand sides of (4.5) contain no partialderivatives with respect to scale we may, from the computational view, obtain numerical estimates of ψ by coupling (4.5) with the system (4.4) using the same algorithms. In this way one may obtain estimatesof the error bound (3.6) on the scale slice of interest as the computations proceed.Before concluding it is important to note that for any u ∈ P ( M, R N ) satisfying the continuityconstraint there will always exist an r ( v ) ∈ P ( M, R N ) satisfying (2.13) with source (4.3) such that (4.4)will hold. This does not entirely resolve the existence problem for applications since many solutionswill have very little interest in this respect. As already mentioned, the problem of extracting from thisset those solutions that are of interest to the application under study is a separate issue and must beexamined in the context of the prescribed initial conditions and, to some extent, the solution method. In avoiding reference to a fully resolved solution we do not insist that regular fully resolved solutions donot exist, only that we do not insist that our space-time-scale solutions uniformly converge to one in thelimit as η → + . In doing so we have the expectation that meaningful solutions to applications may beobtained entirely within space-time-scale.We have already seen that for any member of P ( M, R N ) there will exist an r ∈ R ( M, R N ) satisfying(2.12) and (2.13) where the source term s is given by (2.14). But we have yet to formally establish that P ( M, R N ) is sufficiently populated that it contains members that have any useful meaning for specificapplications. Allied to this is the construction of algorithms that allow us to extract from P ( M, R N )such solutions.Some progress has been made in this direction through numerical solution methods. For nonlinearsystems the introduction of discretisation dictates that one is seeking an approximation of the filteredrepresentation of the dependent variables associated with the scale of resolution rather than an approx-imation of the fully resolved dependent variables. In this context there arises the associated problem ofhow one defines the core function. We can no longer simply associate the core function with the PDEs7hat are to be assumed to govern the fully resolved system. In Section 1 an attempt was made to rederivethe core function of an ideal fluid application without reliance on the well known governing equationsassumed for the fully resolved system. At this stage, the problem of defining the core function is verymuch application specific and needs to be investigated further in a more general context. References [1] D.G.B. Edelen, Alternatives to the Cartan-Kahler Theorem,
Nonlinear Analysis, Theory, Methodsand Applications , , No. 8, (1990), 765-786.[2] D.G.B. Edelen and J. Wang, Transformation Methods for Nonlinear Partial Differential Equations ,(World Scientific, 1992).[3] G. Pantelis, Modelling nonlinear systems by scale considerations,