Sharp interface analysis of a diffuse interface model for cell blebbing with linker dynamics

We investigate the convergence of solutions of a recently proposed diffuse interface/phase field model for cell blebbing by means of matched asymptotic expansions. It is a biological phenomenon that increasingly attracts attention by both experimental and theoretical communities. Key to understanding the process of cell blebbing mechanically are proteins that link the cell cortex and the cell membrane. Another important model component is the bending energy of the cell membrane and cell cortex which accounts for differential equations up to sixth order. Both aspects pose interesting mathematical challenges that will be addressed in this work like showing non‐singularity formation for the pressure at boundary layers, deriving equations for asymptotic series coefficients of uncommonly high order, and dealing with a highly coupled system of equations.


INTRODUCTION
The phenomenon of cell blebbing is connected with various biological processes such as locomotion of primordial germ or cancer cells, the programmed cell death (apoptosis), or cell division.Its importance has been recognised and emphasized in the last decade [6,17,16], and attracts more and more interest.Cell blebbing results from chemical reactions that cause the selection of sites on the cell cortex, which lies underneath the cell membrane, where it contracts.This contraction causes the fluid inside the cell (the cytosol) to be pushed towards the cell membrane, which is then stretched out and moved away from the cell cortex.The cell membrane is pinned to the cell cortex via linker proteins.Only if a sufficient amount of protein bonds can be broken, the membrane can freely develop a protrusion that is called a bleb.
Besides experimental studies [13] there are also many endeavours to understand cell blebbing from a theoretical perspective, cf.[20,14,22,21,3,2,19,26].While all these modelling approaches concentrate on selected aspects of the whole process, a full 3D model that brings together the linker proteins, their surface diffusion, and the fluid-structure interaction has only recently been proposed in [25]: the authors derive a phase field model in which cell cortex and cell membrane are defined by two coupled phase fields, with phase field parameter , that interact with the cytosol.The coupling of the phase fields reflects the linker proteins connecting both surfaces and brings in new interesting mathematical challenges such as well-posedness of equations on evolving 'diffuse manifolds' (the linker protein densities on the cell cortex undergo changes due to surface diffusion and bond 0 1 breaking), developing numerical schemes for solving non-linear, sixth-order phase field equations, and answering the question what model is reached in the limit → 0.
This article is aimed at investigating the last problem and showing that the phase field model of [25] formally approximates a sharp interface model that has also been derived by physical first principles [24].For that we will use the method of formal asymptotic analysis.The techniques we employ are similar to those applied for the asymptotic analysis of related phase field models like [4], the Stokes-Allen-Cahn system in [1], or the Willmore 2 -flow [9].Another related asymptotic analysis is that of [23] for minimisers of the Canham-Helfrich energy.
We start by briefly recalling the phase field model from [25] and show the sharp interface system that is expected in the limit.After we have introduced the notation and gained some understanding of the system of partial differential equations, we introduce foundations of the technique we use to pass to the limit → 0. The major part of this paper follows, which is to plug in series expansions of the solutions of the phase field model in powers of .Via separation of scales, we are able to derive equations for the leading order summands of the series.Using these findings, we can finally pass to the limit in the equations of the phase field model and find the sharp interface system of equations that we initially reviewed.

MODELLING
Besides the numerical advantage of making topological changes such as pinch-offs (like when vesicles form out of the membrane) easy to handle, a phase field approach for modelling cell blebbing is also apt for bio-physical reasons: cell membranes are bilayers of lipid molecules which can be subject to undulations, and so the membrane is not strictly demarcated to the surrounding fluid.Depending on the scale we look at these membranes, the diameter of the lipid molecules involved, and the spacing between them, it may be desirable to model uncertainty in the lipid molecules' position and thus take them to be diffuse layers of some thickness .Another pecularity when considering cell blebbing is experimental evidence [13] that at sites where blebbing occurs, the cell membrane is folded multiple times providing for enough material to be unfolded, and is thus thicker than a typical biological membrane.
Let us assume we observe the process of cell blebbing for a certain time ∈ (0, ∞) in a domain Ω ⊆ ℝ 3 .We consider two evolving diffuse interfaces-the cell membrane and the cell cortex-that can be defined as those subsets of Ω, on which phase fields (modelling the membrane) and (modelling the cell cortex) are close to zero, respectively.Additionally, there is a surrounding fluid with density , velocity , and pressure .Also in the domain, but concentrated on the cell cortex, are linker proteins with mass volume density , .They connect the cell membrane and the cell cortex.The linker proteins behave like springs, but may break if overstretched, so we introduce another density , which gives the mass of linkers per volume that are broken.This is important because 'repairing mechanisms' of the cell take care of reconnecting those broken linkers back to the cell membrane.A scheme in which the aforementioned quantities are all depicted together is given in Figure 1.For deriving the phase field model, Onsager's variational principle [15,18] is combined with a reaction-diffusion-like surface evolution equation for the active and inactive linker proteins.To establish a basic understanding of how a PDE system for cell blebbing can be obtained, let us mention the principle steps in the derivation.

Definition of an energy functional
, , , that is the sum of all kinds of energy of the cell: the ingredients are the kinetic energy of the fluid, the surface and bending energy of the cell cortex and cell membrane, and a potential energy that accounts for the coupling of both membrane and cortex via the linker proteins.

Definition of appropriate boundary conditions (see below).
3. Variation of plus a dissipation functional.With regard to the linker proteins, our process is assumed to be quasi-static, i.e., we assume the linker proteins to be given parameters of although their evolution is given by a reaction-diffusion-like surface equation.

4.
Extending the stationarity condition derived by the previous variation step, the aforementioned surface evolution equations for the linker proteins are added.

Phase field model
Several computations and formulae are the same for the phase field representing the cell membrane and that representing the cell cortex .For those, we always use the symbols ∈ { , }, and Φ ∈ {Γ, Σ} to avoid copious repetition.
In the phase field approach, we approximate two important geometrical quantities known from the sharp interface perspective, namely the normal (everywhere where ≠ 0), and the mean curvature Having the velocity and density of the fluid, we may express the kinetic energy as Let us consider the following energies at a particular point in time ∈ [0, ], so we can ignore the time-dependency for now.The surface energy of the diffuse cell membrane with a surface tension proportional to Γ is given by the Ginzburg-Landau energy A well-established [5,11,7] model for the bending energy of a cell membrane with bending rigidity Γ and spontaneous mean curvature Γ 0 is the phase field version of the Canham-Helfrich energy The spontaneous mean curvature corresponds to an intrinsic bending of the membrane which is typical for biomembranes.The additional term in the energy introduced by that, however, does not introduce new theoretical challenges compared to using a Willmore functional, which is why we will omit it for the sake of a straightforward presentation, i.e, Γ 0 = 0.In this configuration,  ,Γ [ ] is the phase field version of the Willmore energy.We simplify the situation for the cell cortex in that we assume it to be just a stiffer membrane thus employing the same types of energies just with different surface tension and bending rigidity.Both energies associated to membrane and cortex are summarised in the energy functionals For the coupling of cell membrane and cell cortex, we account with a generalised Hookean spring energy.


, , , = where is a spring constant, and , , assigns to points , ∈ Ω the particle-per-volume density of protein linkers connecting in direction − .A possible choice is with being a suitable standard deviation and ̂ an appropriate scaling factor.To outline the idea for this modelling choice, we first point out that can be pictured as a 'smooth Dirac delta function' if is the so-called optimal profile .
Looking at the sharp interface equivalent of the coupling energy, we can identify 1. 2 | − | 2 as a Hookean energy density, which is integrated over the membrane and cortex, and weighted additionally by 2.
The Hookean energy ansatz accounts for the earlier mentioned assumption that the linker proteins behave like springs.Additionally, since linkers might not be distributed homogeneously, we should scale the coupling force by their actual amount, which explains 3. The necessity to consider a weight might not be so obvious: it has not yet been agreed upon in the biological literature how to identify the pairs of points ( , ) ∈ Σ × Γ that are connected by protein linkers.That is why we allow the weight to model a certain probability for this state.An easy way to describe such a probability is in terms of the angle between − and a gauge direction.As this gauge direction, we chose the cortex normal, which enters as the third argument of .
Remark 2.1.It shall be remarked that there are other choices for 'smooth Dirac delta functions' like 1 ( ), which is smoother and easier to handle analytically and numerically.It turns out, however, that for passing to the limit → 0, the latter two choices are not appropriate.The reason for that becomes clear when we compare the right hand side of the momentum balance for the different choices of the integral weight: only for [ ], we have phase field counterparts in for every term we expect in the sharp interface system as derived from physical first principles (cf.[25]).
Summing all potential energies, we obtain the Helmoltz free Energy of the cell as and the inner energy as , , , , = 1 2 Via Onsager's variational principle (cf.[25]), the following system of partial differential equations is then found as stationarity conditions The imposed boundary conditions are where , , , , and In addition, we consider evolution equations for the active and inactive linkers on the diffuse surface of the cell cortex: where The term , is the effective reconnection rate, ≥ 0, of the inactive linkers, and is the effective disconnection rate of the active linkers in relation to the membrane position in space and the orientation of the cortex given by its normal.
For a thorough discussion and further references, the reader may please refer to [24].In the following section, we describe steps one, two and four, but leave out the lengthy calculations involved for step three.For the following discussion, however, we need the concrete expression for all the 2 -gradients of the energies, so we give them here without doing the calculations.Note that these calculations depend on the boundary conditions (1e), (1f), (1g), and (1h): For easier expression of the coupling energy gradients, we introduce Then, Solutions of (1) fulfil an energy inequality, cf.[25].This energy inequality reads (3)

Sharp interface model
We introduce two evolving, two-dimensional manifolds Γ = (Γ ( )) ∈[0, ] for the cell membrane, and Σ = (Σ ( )) ∈[0, ] for the cell cortex.These evolving manifolds can also be described as the level sets Γ ( ) = −1 ( , 0) and Σ ( ) The cell we consider is swimming in a fluid with pressure and velocity .Additionally, we have the density ∶ Σ → ℝ of linker proteins connecting cell membrane and cell cortex, which we call active linkers.Another density ∶ Σ → ℝ is introduced to model the density of the disconnected or broken proteins, called inactive linkers; these no longer couple cell membrane and cell cortex, but may be reconnected due to healing mechanisms inside the cell.

FORMAL ASYMPTOTIC ANALYSIS
Having outlined the physical principles, we are going to analyse the sharp interface limit of the phase field model.Let us now turn to the main result of this paper: we will demonstrate, using the method of formal asymptotic expansions, that classical solutions of the system (1) converge, for → 0, to solutions of (4).For a thorough theoretical introduction into the subject of formal asymptotic expansions, we refer to [8], whereas a more application-oriented perspective is taken in [12].

Interfacial coordinates
For the following analysis, we will need a coordinate transformation typical for asymptotic analysis of phase field equations for which boundary layers are expected in the regions where the phase fields are close to zero.Let us denote a tubular neighbourhood of a smooth, orientable hypersurface ⊆ ℝ 3 by ( ).We require that ∈ (0, ∞) is small enough such that (Γ ( )) ∩ (Σ ( )) = ∅ for all ∈ [0, ].The local boundary layer coordinates, or interfacial coordinates (as they are most often termed in this context), with respect to are defined by the map For two evolving manifolds Γ , Σ , we extent this definition to and then set for ( ) ∈ {Γ ( ) , Σ ( )}.We always consider small enough such that the interfacial coordinate transformations are well-defined.
Generally, for a function on The function ̂ depends on three arguments: The first is time, the second a point on one of the manifolds Γ ( ) or Σ ( ), and the third a real number from − , .The latter is occasionally referred to as 'fast variable' and derivatives with respect to this variable are denoted by (⋅) ′ ; derivatives with respect to the first variable are denoted by (⋅).
The following (standard) formulae will be important later.

Assumptions on the solution
Typically, formal asymptotic theories rely on non-trivial properties on the solution of the system under investigation, (1) in our case.A rigorous justification requires treatment of its own and is not in the scope of this work.We shall restrict ourselves to clearly formulating the properties we need in form of assumptions, and rather focus on the relation of the quantities of a solution of (1) that assure a sensible behaviour in the limit.These assumptions can serve as a hint what needs to be investigated when a mathematical proof is to be given.
6.The species densities' evolution is irrelevant outside the diffuse layers around Γ , Σ .We thus consider them to be asymptotically constant in time away from the diffuse layers: For every ⋐ Ω 0 , it holds 7. The components of every classical solution to (1) shall have a regular asymptotic expansion in ( ), ∈ {Γ ( ) , Σ ( )}, after transformation into local coordinates: For all ∈ { , , , , , , , }, it holds for , ∈ ℕ 0 , where all ̂ shall be integrable in and as smooth as .We call these series inner expansions of .
9. For the species density , , we additionally require that blow-ups are of order at most −1, i.e., ̂ , = 0 for all ∈ {− , … , −2}.The reason why we cannot naturally expect boundedness here is that , does not give the volume fraction, but the number of particles per volume of the active linkers.
Let us further exercise some smaller expansions.
A common principle, which we will make use of in the following multiple times, is summarised in the following for some ∈ [0, ∞) and ∈ (0, ∞), ∈ (Ω), and for all sequences →0 ←→ , it holds ( ) After the preliminaries are fixed, we shall proceed by analysing the asymptotic behaviour of the solution of (1).

Outer expansion
We start with investigating the solutions' behaviour away from the boundary layer, i.e. on a set Ω = Ω ⧵ (Γ) ∪ (Σ) for some > 0. Let ∈ { , } for the following considerations.
Further, the sufficiently fast decay of the species densities' time derivative, see Assumption 6 imply ˆΩ [ ] ( ) From (1c) and (1d), we also obtain It thus follows from (3) that .
Hence, it must hold For the initial data of , we have (cf.Assumption 2) 0 (0, ⋅) = −1 in Ω − and 1 in Ω + , ∈ {Γ, Σ}, so that we can argue by continuity in time that which is the essential result of this paragraph.

Inner expansion
As there is no danger of confusion, we drop the subscript on the physical quantities.Let us first not that the result of the previous paragraph can be combined with the principle of asymtptotic matching on the phase fields such that we obtain for ∈ (Γ) ∩ Ω + Γ , i.e., Γ ( ) > 0. Analogously, for ∈ (Γ) ∩ Ω − Γ and mutatis mutandis for .Remark 3.5.An immediate consequence of the matching principle and the assumption that = 0 for all ∈ {− , … , −1} of the outer expansion is lim of the inner expansion.This also holds for all derivatives as long as they exist.
Repeating the arguments of the previous paragraph, we may from now on assume w.l.o.g.̂ = 0 for all ≤ −3 and ̂ = 0 for all ≤ −4.
At order −4 , we have an additional right hand side term Multiplying again by ̄ and noting that due to the previous considerations hence, ̂ ′′ −2 = 0 and we may conclude ̂ −2 = 0 as before.We cannot go further now.However, in Section 3.6 we show that actually ∇ 2  ∈ −2 and in Section 3.7 that ∈ (1)-using only the results on velocity and pressure we've derived here-, which gives ̂ ′ −3 • = 0, and further

Optimal profiles of ̂ and ̂ to leading order
Leading order of ∇ 2  and ∇ 2  is at most −2 .We consider the evolution law (1d): The left hand side is at most of order −2 (since the velocity is at most of order −1 , see the previous Section 3.5).So requiring ≤ 2, the leading order terms of Δ ∇ 2  are of order −3 and must be zero, which is equivalent to the equation We pose the additional condition ̂ 0 ( , , 0) = 0 (otherwise, we had infinitely many solutions by shifting along the abscissa).Further, we set ∶= ̂ ′′ 0 − ′ ̂ 0 ′′ − ̂ ′′ 0 − ′ ̂ 0 ′′ ̂ 0 and observe that thanks to the counterparts of ( 21), ( 22 we further conclude with the counterparts of ( 21), (22) for that ̂ 0 ( ) = tanh √
The very same argument applies verbatim for .

Properties of ̂ and ̂ to leading order
The following analysis is conducted on the example of ̂ , but the arguments are the same for ̂ .We consider the equation (1i) on (Σ): Using (16), the results from Section 3.5, (27), the optimal profile found for in Section 3.6 together with (18), we have so to leading order only the terms at − −3 of the diffusion term matter: 21), (22).Simultaneously, ̂ ′ − decays as | | → ∞, see (23).Thus, it must even hold and so ̂ ′ − ( , ) = 0 for all , .Consequently, ̂ − is constant in .Leveraging (23) again, it follows ̂ − = 0. We may repeat this argument and find Finally, we have to leading order: and conclude

Further properties of ̂ and ̂
The expansion of the Willmore-Energy gradient in interfacial coordinates shall be and in original coordinates We are going to show that ̂ −3 = ̂ −2 = ̂ −1 = 0 by dint of the energy inequality.Afterwards, we are going to see that important properties of ̂ 0 , ̂ 1 , and ̂ 2 follow from these equations that we will use when passing to the limit in the next Section 4. Before going on, let us calculate Thanks to the optimal profiles at leading order for both phase fields, cf.Section 3.6, we have ∇ 2  [ ] ∈ −2 , and also 1).We note further which follows from the optimal profile of and to leading order combined with Lemma 3.4.(The optimal profiles allow for showing the decaying condition that is the main prerequisite of Lemma 3.4.)For we have to additionally consider (57), and note (58), as well as (again, leveraging Lemma 3.4 and using the optimal profile of and to verify the prerequisites).The energy inequality (3) gives us additionally .
Note that we can restrict to ( ) since the energies and their 2 -gradients are zero outside to leading order.Applying the Poincaré-Wirtinger inequality, we deduce, , which implies, using the reversed triangle inequality for ‖⋅‖ 2 ( ( )) , for ≤ 2. By applying Young's inequality, we can deduce further ˆ which in turn implies so with the co-area formula, it follows From (31), the expansion of the integrand follows directly.Equation (34) then requires This in turn gives ∇  ∈ −1 , so ffl ( ) ∇  d 3 ∈ (1), which we insert into (32), and choose < 1 to obtain ˆ so that ̂ −1 = 0. Now that we have found equations ̂ −1 = 0, ̂ −2 = 0, and ̂ −3 = 0, we may derive information on ̂ from them.

Expansion of the 2 -gradient of the Willmore energy
We recall that First, we expand the chemical potential, by expanding the Laplacian term: and the double well potential's first derivative: The expansion of the chemical potential then reads 1 . (37)

SHARP INTERFACE LIMIT
By inserting the expansions in interfacial coordinates of the components of the solution of (1) into the systems' equations, we have managed to • eliminate the velocity expansion's summands up to (and including) order −1 , • eliminate the pressure expansion's summands up to (and including) order −3 , • show that both phase fields assume the optimal profile at leading order, • show that 1 = 0, • and derive (48).
Before we can make use of these findings and pass to the limit → 0, we compute the expansions of the remaining terms in (see the right hand side of (1a)).

Expansion of ∇ 2  and remaining force terms
We compute the asymptotic expansions of

Expansion of ∇ 2 
We recall from (2d) that with We further expand and note and thus .
In order to expand , we first compute Therefore, using ( 16), we find Multiplication with − ∇ gives

Expansion of ∇ 2 
We recall (2e), with Before we start expanding these terms, we prove the following formulae: Proof.Ad (55): Due to ̂ 0 being the optimal profile and it thus being independent of the tangential variable , and considering (43), we have This observation brings the claimed expansion for the product of ∇ and | | ∇ | | −1 using (19).
Ad (56): We again use the optimal profile and (43) to conclude , and the claim follows.
is expanded just like : We continue by expanding : First note Then we observe ∇ ∇ = ( ) , so where the last equality is justified by (30).Then, At last, we turn to and compute On the first term, we use ( 56) and (43) (for the expansion of the double well potential) to obtain Second, we calculate Third, We further compute and using (56), we find Finally, .

Expansion of
As before, we employ ( 16), (55) to obtain We now show, using the results of the analysis in the previous sections, that classical solutions of (1) converge formally to solutions of (4) for ↘ 0. In the following, we will often use that is integrable, and we will abbreviate We also partition all integrals over Ω into an interal over ( ) and one over Ω = Ω ⧵ ( ).In the latter region, the outer expansions hold, and thus the integrands are all of lower order vanishing in the limit, so we can neglect them.

Momentum balance and mass conservation 4.2.1 Outer region
At order 0 , we find with the results of Section 3.3 (causing all the energy gradient terms on the right to vanish) and for the incompressibility condition This gives (4a) and (4b).
Plugging further the inner expansion into (1a) and using (50), (51), we find where = ̂ • with ̂ ∈ (1).For understanding the limit of this equation, let us consider its variational formulation with test functions ∈ [ 1 (Ω)] 3 .The left hand side then reads We can rewrite the integral of − ̂ ̂ ′ 1 + ∇Φ 0 ̂ ̂ + ̂ ⊗ ̂ ′ 1 ̂ in by looking at the expansions of ∇ ̄ and of ∇ ̄ in interfacial coordinates: and respectively (following with (11) and ( 51)).With the matching conditions for ∇ ̄ and ∇ ̄ , we further obtain The computations go analogously for the limits ↗ 0 and ↘ −∞.Together, both limits form a jump ⋅ .Now we pass ↘ 0 in (59) and insert the matching condition for the pressure and (60).This reveals where ∈ { Γ , Σ } depending on what surface the integrand is to be understood.
The right hand side of (1a) in variational form is . (62) Note that we can rewrite and which we use in the following as abbreviation.
To pass (62) to the limit, we treat the gradients of the energies separately.The gradients of  and  have the same structure for both and , and can therefore be treated verbatim.For  we distinguish the derivatives w.r.t. and .Let us start our analysis with ∇ 2 .

Force terms of ∇ 2 
In this section we show the following lemma: Lemma 4.2.Let ∈ {Γ, Σ} and ∈ { , } such that is the boundary layer for .The following limit holds true: for a constant .
The strategy of the proof is as follows: In Section 3.8, we have expanded the gradient ∇ 2  and concluded from the energy inequality that all its terms up to order −1 must equal zero.The terms remaining on order 0 are shifted to order −1 by multiplication with ∇ ⋅ , which is just the right scaling for obtaining the claimed limit using Lemma 3.4.

Proof of 4.2.
Collect all the terms of ∇ 2 , ∇ ⋅ 2 ( (Φ)) on order −1 : The terms on the first and second line are from the expansion of the chemical potential and the double well potential.The left summand on line three stems from (44), the right one from (46).The left summand on line four is taken from (40) (note (42)); the right summand is from (44) (note (45)).First, we substitute some expressions using Lemma 3.1 and (43): Second, we transform the integral using the co-area formula.We denote ( , ) = ( Φ ( ) , ).
With integration by parts, it follows directly that Exploiting this property and the independence of ̂ 0 of the first argument of , we obtain Thanks to Lemma 3.4, we know that this difference converges to zero as ↘ 0. The same reasoning applies for The remaining terms are treated as follows: We recall ̂ ′ ∈ ( ) and ̂ ′′ ∈ 2 (cf.(41), (42)) and ) with 1 = 0).Thus, the following expansion holds: This way we see with where we used To treat ), we take a global parametrisation ∶ ℝ 2 → Φ (this is w.l.o.g.since in case there is no global parametrisation, we make all the following calculations locally and patch the integrals together afterwards), and define ( ) = ( ) + ( ( )).With the area formula, we obtain Note that ( ( ), ) = ( ( ), ).We further integrate by parts We observe and we have ℙ With Jacobi's formula for derivatives of determinants, it can be seen that the derivative of the Jacobian w.r.t is also in ( ).

Force terms of ∇ 2 
A small calculation reveals and from (36) we have with ̂ ′′ 0 • − ′ 0 = 0 and ( 43) So convergence under the integral follows by Lemma 3.4:

Limit of ̄
Using the expansion we computed before, we obtain

Deriving the jump conditions
We now observe that all the previously computed limits have equal, corresponding terms in the right hand sides of (4f) and (4g).Note that these appear with a minus on the right hand side of the momentum balance, so we negate them here accordingly.We

Species subsystem
Like the phase field equations, the species subsystem (1i) (1j) is meaningless in the outer region.For reasons of symmetry, it suffices to conduct the asymptotic analysis for the equation of : it carries over to verbatim.We start our analysis by expanding the term ∇ ⋅ [ ] ∇ .W.l.o.g., we assume that ̂ ′ = 0. First we derive from (7) and using (30) With ( 16) and thanks to (43) and ̂ 0 being the optimal profile, we have also

Figure 1
Figure 1 Illustration of the relationship of the two diffuse layers.The dotted lines indicate the centers of the transition layers of and .In the white region, both and take values close to 1.In the light orange region, takes values close to 1, but has values close to −1.In the dark orange region, both and have values close to −1.
) .(Bringing to the right, all leading order terms of Δ ∇ 2  Γ [ ] + ∇ 2  from order −3 to −1 have no match on the right hand side and thus have to be zero following the separation of scales argument.Using that the Neumann boundary conditions (1g) do not depend on , we can thus conclude that all these terms are of order zero.)Comparing the right hand side of (1a) with its left hand side, we conclude ˆΩ ( ) , ( , ) v ( , ) , , , ) ⋅ [ ] , d 3 ( ) d 3 ( ) ∈ (1) .