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A Kinetic Model for Grain Growth

We provide a well-posedness analysis of a kinetic model for grain growth introduced by Fradkov which is based on the von Neumann-Mullins law. The model consists of an infinite number of transport equations with a tri-diagonal coupling modelling topological changes in the grain configuration. Self-consistency of this kinetic model is achieved by introducing a coupling weight which leads to a nonlinear and nonlocal system of equations. We prove existence of solutions by approximation with finite dimensional systems. Key ingredients in passing to the limit are suitable super-solutions, a bound from below on the total mass, and a tightness estimate which ensures that no mass is transported to infinity in finite time.


Abstract
The subject matter of this thesis is a detailed analysis of the self-consistent kinetic model for grain growth introduced by Fradkov [5]. The model is based on the von Neumann-Mullins law describing the change of area of grains according to their topological class, i.e. the number of edges they have. Topological events are performed by coupling terms between equations for the number densities of different topological classes. The resulting system of transport equations is infinite-dimensional with a tridiagonal coupling structure. Self-consistency of this kinetic model is achieved by introducing a coupling's weight Γ making the equations nonlinear and nonlocal in space.
We start with an introduction in the first chapter. Afterwards in the second chapter we derive Fradkov's model and carry out formal calculations to illustrate self-consistency.
In the third chapter we present a-priori calculations mainly allowing us to bound the nonlinearity Γ. This enables us to prove existence and uniqueness of solutions to finite-dimensional systems in the first part of the fourth chapter. Further bounds on the number densities established in the fifth chapter allow for passing to the limit concerning the number of equations in the second part of the fourth chapter. Therefore we prove existence of solutions to the infinite-dimensional system by a suitable approximation procedure. Uniqueness and continuous dependence on the data is then provided by energy methods. The sixth chapter focusses on long-time behaviour and mainly on stationary solutions of a rescaled system as candidates for self-similar solutions. Finally we prove Lewis' law asymptotically.

Introduction
Most technologically useful materials are polycrystalline aggregates, composed of a huge number of crystallites, called grains, separated by so-called grain boundaries. The application of such materials covers many scales, from steel girders for power poles to base plates of microprocessors. Important material properties like fracture, toughness, or conductivity are determined by the polycrystalline microstructure, i.e. the sizes, shapes, orientation, and arrangement of grains. Unfortunately such materials undergo a temperature controlled aging process leading to a coarsening of the grain structure and therefore inducing changes in mechanical, electrical, optical, and magnetic properties of the material. For further details we refer to the review articles by Fradkov and Udler [6] and Thompson [20].
Different approaches for modelling grain growth in two space dimensions are established in the literature. In Monte-Carlo models [1,2] the microstructure is mapped onto a discrete lattice. The kinetics of the boundary motion are simulated by employing a Monte-Carlo technique for moving these lattice points. An attractive feature of this model is the simple handling of topological events like grain boundary flipping and grain disappearance. Using boundary tracking models based on partial differential equations [12,14] offers an attractive alternative to Monte-Carlo models since they deal with quantities of lower dimension. A disadvantage arises as topological changes require extra treatment. The idea of reducing interfacial energy as driving force in grain growth is also carried on in vertex models where movement of grain boundaries is projected onto the triple-junctions [11,9]. Another class are mean field models for grain growth [20,Section IV]. In the sequel we focus on kinetic models [5,15,4] based on the von Neumann-Mullins law. Such models consider time-dependent distribution functions for the grain areas and the number of sides per grain. Grain areas change ac-

INTRODUCTION
cording to the von Neumann-Mullins law, topological changes are performed by collision operators. Fradkov was the first to develop a model of this type [5] and up to now there is no analytic treatment of such a model. Therefore this thesis establishes a rigorous theory for the arising infinite-dimensional system of transport equations with nonlocal weight making the equations nonlinear.
In Chapter 2 we derive Fradkov's self-consistent model and verify certain natural relations by formal calculations.
Chapter 3 provides us with some necessary a-priori calculations mainly enabling us to bound the solution and the coupling's weight. Furthermore we indicate how to transform the system via the method of characteristics and prove non-negativity of solutions. The key features of this chapter are a (in the continuous variable, the grain area) constant supersolution decaying exponentially with respect to the discrete variable (the topological class) and an argument preventing the total mass from dropping down to zero within finite times based on considerations regarding the maximum annihilation speed for disappearing grains.
We first prove existence of solutions to finite-dimensional systems by a fixed point argument in Chapter 4. Then we pass to the limit proving existence of solutions to the infinite-dimensional system using the bounds on the solution which we obtained Chapter 3 and Chapter 5 and achieving compactness by Arzela-Ascoli. We also prove uniqueness and continuous dependence on the data by energy methods.
In Chapter 5 we prove that no mass runs off at infinity neither with respect to the continuous variable nor to the discrete one. Our main idea is to exploit an interplay between the decay concerning the discrete variable and the one regarding the continuous variable by using a bounding frame that grows in time. Furthermore we verify the natural relations treated in Chapter 2 in the infinite-dimensional case, too.
Chapter 6 is dedicated to the long-time behaviour of solutions. Besides a characterization of stationary solutions we focus our attention on self-similar solutions. We rescale the equations, consider stationary solutions, and solve the resulting system of ordinary differential equations formally. Furthermore we deal with Lewis' law in self-similar variables asymptotically.

Derivation of a consistent kinetic model
The purpose of this chapter is to derive a kinetic model for two-dimensional grain growth proposed by Fradkov [5,8]. Furthermore we carry out formal calculations to illustrate self-consistency of this model.

Preliminaries
We start our considerations with some remarks on a well-established model for grain growth leading to the well-known von Neumann-Mullins law which is the foundation of our kinetic model.

Motion by mean curvature and equilibrium of forces at triple junctions
Mean curvature flow coupled with equilibrium of forces at triple junctions is a widely accepted model for two-dimensional grain growth (cf. [3,12,14]). In the sequel we briefly recall the basic model. Therefore our objects are networks of curves which meet in triple junctions. (We will refer to this as the triple junction condition.) Since we are mainly interested in settings with large numbers of grains and not so much in the influence of boundary conditions, we consider one-periodic spatial networks. Furthermore we restrict ourselves to the case of isotropic surface energies and assume the mobility of the triple junctions to be infinitely fast compared to the mobility of the grain boundaries. (We also assume the mobilities of the grain boundaries to be equal.) Now we can evolve the network due to mean curvature flow as long as no grain boundaries vanish. Furthermore we have to take the Her-ring condition [10] into account that prescribes equilibrium of forces at triple junctions. In the isotropic case this just means that the curves meet in an angle of 2π/3. In general Young's relations imply that the ratios of surface energies and the sine of the opposite angle are equal [17]. Observing that mean curvature flow has a natural interpretation as a gradient flow [19,9], the Herring condition arises as a natural boundary condition (coming up via an integration by parts) when computing the differential of the associated L 2 energy (cf. Appendix B).

von Neumann-Mullins law
Under the assumptions stated above (isotropic surface energy, equal mobility of grain boundaries, and infinite mobility of triple junctions), one can use the concept of a network evolving by mean curvature flow coupled with equilibrium of forces at triple junctions to derive a law of motion for the area of a single grain with n edges [16], known as the von Neumann-Mullins law: M denotes the mobility of the grain boundaries and σ the surface tension. The proof can be done by a direct geometric computation using motion by curvature of the grain boundaries and the prescribed jumps of the outer normal by 2π/3 at triple junctions (cf. Appendix C).
The von Neumann-Mullins law implies that grains with less than six edges shrink, those with more than six grow, and such with exactly six edges retain their area (possibly not their shape).

Topological changes
The evolution sketched in Subsection 2.1.1 is well-defined until two vertices on a grain boundary collide, after which topological rearrangements may take place. This happens when either an edge or a whole grain vanishes. In the first case an unstable fourfold vertex is produced, which immediately splits up again, usually in such a way that two new vertices are connected by a new edge. In this case, two neighbouring grains decrease their topological class, whereas the two other grains increase it (Figure 2.1). The second case causing topological rearrangements is grain vanishing. Each grain vanishing is accompanied by disappearance of two vertices and three edges. Due to the von Neumann-Mullins law we only take grains with topological class 2 ≤ n ≤ 5 into account. Grains with n = 2 and n = 3 vanish in a single . For further details on the resulting topological classes associated with adopted topological configurations after vanishing events we refer to the review article by Fradkov and Udler [6].
At this stage it is completely unclear by which mechanism a specific topological configuration is selected within switching or after vanishing events. A natural idea is to assume that this selection process is driven by the same tendency to reduce energy the whole network structure evolves by. One possibility is to compute all possible local configurations and select the one that minimizes energy locally in the best way [9].

One-particle distribution
In common with Fradkov [5,6] we introduce a number density f n (a, t) that measures the number of grains with topological class n and area a at time t. Using the von Neumann-Mullins law (2.1) we can describe the evolution of f by transport equations as long as no topological rearrangements take place. Note that the factors M and σ in (2.1) are constants and therefore were scaled out in the same way as π/3.
To model topological changes we introduce a collision term (Jf ) n on the r.h.s. of (2.2) coupling the equations. We define topological fluxes X + n and X − n denoting the flux from class n to n + 1 and from n to n − 1 respectively.
In the sequel we state the 'gas' approximation of the collision kernel by Fradkov [5]. Each grain of topological class n in a two-dimensional oneperiodic network is bounded by n edges and features n triple junctions on it's boundary. We can identify three possible events causing transitions of the grain from one topological class to another: • switching of an edge (which is part of the grain boundary) leading to a transition from topological class n to (n − 1) • switching of an outgoing edge causing a transition from topological class n to (n + 1) • vanishing of a neighbour grain, which causes a transition from topological class n to (n − 1) Here we ignore that the topological class of a grain is lowered by two if the neighbouring annihilated grain was a lense, i.e. had topological class n = 2. Fradkov and Udler argue [6] that such an event takes place only very rarely as the number of lenses itself is already very small. Now we make a strong assumption by neglecting correlation effects when switching or vanishing events take place, i.e. we assume that the probability for the occurrence of topological changes is only proportional to n (and independent of a and neighbour correlations). Furthermore we assume that the probabilities of switching are equal for all boundaries in the system. This implies that for any given grain the two switching events described above are equally probable. Due to these assumptions the topological fluxes can be expressed as follows: Note that β ∈ (0, 2) is a free parameter in this model describing the ratio between "symmetric" and "asymmetric" topological events. The bounds on β are needed (cf. Lemma 2.3) to estimate the nonlinearity Γ = Γ (f ) which we determine later on (2.5) to achieve self-consistency (cf. Lemma 2.2). Now the collision terms (we will call them coupling terms in the following) read as

Infinite system
The equations stated in the sequel are mainly the same as in the work of Fradkov [5,8,6]. The coupling term (Jf ) 2 differs and we do not neglect .
where β ∈ (0, 2) , n > 2 The coupling's weight Γ making the equations nonlinear (and nonlocal in space) is chosen as which ensures the preservation of the triple junction condition as stated in Lemma 5.4. As boundary conditions we set f n (0, t) = 0 n > 6 (2.6) for 0 < t < ∞ ensuring that no additional mass is transported from the negative half-axis to the positive one. This means no additional grains can be created.

Finite system
Within our proof we will also use the finite-dimensional analogue of (2.3) .
where β ∈ (0, 2) , 2 < n < n 0 (2.8) The coupling defined in (2.8) operates pointwise and has the important zero balance property (2.9) reflecting that no grains can be generated or annihilated by topological rearrangements. n (Jf ) n (a, t) = 0 (2.9) The following choice of Γ : 10) reflects the preservation of the triple junction condition (cf. Lemma 2.2) and will be derived in Subsection 2.3.2.

Bounded and conserved quantities
Within this section we will carry out some formal calculations -assuming a pointwise non-negative solution f of (2.7) exists -to identify certain bounded and conserved quantities. Calculations concerning pointwise non-negative solutions of (2.3) are formally carried out in the same way and are therefore not stated here.
From now on we consider solutions f to the finite system (2.7) which are bounded and continuously differentiable w.r.t. a and t separately. Furthermore the nonlinearity Γ (f (t)) shall be bounded, too.

Total number of grains
If equations (2.7) reflect a coarsening process, it should be clear that the total number of grains decreases in time.
To be more precise we define N (t) by simply counting all grains at time t.

Definition 2.1 (total number of grains) We call
the total number of grains at time t.

Lemma 2.1
If a solution f to non-negative initial data g exists, then we have d dt for all finite times.

Proof of Lemma 2.1
Differentiating N (t) and using (2.7) gives us as f takes zero values at ∞ w.r.t. a due to the exponentially decaying supersolution (cf. Lemma 3.3) and by using the zero balance property (2.9) of n (Jf ) n (a, t). q.e.d.

Triple junction condition
Another important feature that solutions to (2.7) should reflect is the validity of the triple junction condition. Hence we will check whether Euler's polyhedral formula holds during the evolution.

Proposition 2.1 (polyhedral formula) For a periodic polygon Euler's polyhedral formula reads
where V denotes the number of vertices, F the number of facets, and E the number of edges.

Proof of Proposition 2.1
Poincaré's version of the polyhedral formula reads where g is the genus of the surface and the Euler characteristic. Considering a periodic polyhedron means looking at a polyhedron on a torus. A torus has genus g = 1 and therefore the Euler characteristic is χ (g) = 0. q.e.d.
Now we translate the polyhedral formula into our setting.

Proposition 2.2 The polyhedral formula reads
for solutions to (2.7) (under triple junction condition).

Proof of Proposition 2.2
The number of facets F of our periodic network is given by and the number of edges E can be computed via at any time t. The number of vertices V is given by as we constrain that edges shall only meet in triple junctions. Plugging F (t), E (t), and V (t) into (2.12) completes the proof. q.e.d.
We treat the validity of the polyhedral formula as evidence that the triple junction condition holds. This gives rise to the following lemma. is satisfied for all times 0 < t < ∞.

Proof of Lemma 2.2
We carry out the proof by differentiating the polyhedral formula (2.13) and using the zero balance property (2.9) n (Jf ) n (a, t) = 0 to obtain the last equality in the above calculation. Now we will focus our attention on the weighted sum of the coupling terms omitting the arguments a and t of f .
The above computations are done by using n (n − 1) = (n − 1) 2 + (n − 1) and n (n + 1) = (n + 1) 2 − (n + 1) and some index shifts in the resulting sums. By the choice of Γ (f (t)) in equation (2.10) With this knowledge we are able to establish a first estimate on Γ (f (t)).

Lemma 2.3
If f is a solution to (2.7) and its initial data satisfy the polyhedral formula (2.13), then where c > 0 is a uniform constant.

Proof of Lemma 2.3
The numerator of Γ (f (t)) can be bounded from above via as f n (0, t) = 0 for n ≥ 6 due to the boundary conditions (2.11). We will proceed by bounding the denominator of Γ (f (t)) from below by using Lemma 2.2.

Total covered area
With the result of Lemma 2.2 we are able to state the conservation of total covered area A (t) . i.e. total covered area is conserved.

Proof of Lemma 2.4
Differentiating A (t) and using (2.7) gives us via an integration by parts and by using the zero balance property (2.9) of the sum of the coupling terms n (Jf ) n (a, t); f takes zero values at ∞ w.r.t. a (cf. Lemma 3.3).
As the initial data satisfy the polyhedral formula (2.13), Lemma 2.2 gives us the result. q.e.d.
During a coarsening process the average area of grains increases in time. This can also be concluded easily from the above inspection of N (t) and A (t) .

Definition 2.3 (mean grain area) We call
the mean grain area at time t.

Corollary 2.2
If the initial data satisfy the polyhedral formula (2.13), then i.e. the mean grain area of solutions f of (2.7) is increasing in time.

Proof of Corollary 2.2
Differentiating From now on we consider solutions f to the finite system (2.7) which are bounded and continuously differentiable w.r.t. a and t separately. Furthermore the nonlinearity Γ (f (t)) shall be bounded, too.

Characteristics
A first step to construct solutions to (2.7) is to transform the given system of transport equations into a system of integral equations. This will enable us to use a fixed point argument later on. The transformation will be done via the method of characteristics.

Proposition 3.1 (integral equations)
The time-integrated version of the system (2.7) is given by for a ∈ (0, ∞) and t ∈ (0, ∞). We set g n (a) = 0 for a < 0. The coupling (Jf ) n (a, t) is defined by equations (2.8) and the nonlinearity Γ (f (t)) is given by equation Note that we set f n (α, t) = 0 if the argument α is negative in the formulas above.

Proof of Proposition 3.1
For an arbitrary n ∈ {2, ..., n 0 } we set for any a ≥ s (n − 6) and t + s ≥ 0. Differentiating z (s) yieldṡ by using (2.7) for the last equality. Now we observe which completes the proof. q.e.d.

Remark 3.1 (pointwise coupling) As the coupling acts in a pointwise way we shall clarify how to evaluate
are treated in the same way.

Remark 3.2 If f ε is a sufficiently smooth solution to
for a ∈ (0, ∞) and t ∈ (0, ∞), this system can be transformed into for any ε > 0. Nonlinearity, coupling, and boundary conditions are defined in the same way as for f in equations (2.10), (2.8), and (2.11).

Pointwise non-negativity
It is reasonable that a solution f to (2.7) with non-negative initial data should not become negative for any time t. This is stated in the following lemma.

Proof of Lemma 3.1
First we observe f n (0, t) = 0 n > 6 due to the boundary conditions (2.11). The cases can be treated by considering f on R instead of R + and setting f n (a, t) ≡ 0 for a < 0 and n > 6 for all t.
For keeping notation simple we only discuss 0 < a < ∞ from now on.
We regard solutions f ε to system (3.2) as mentioned in Remark 3.2 in the previous Section 3.1 which is similar to our original system (2.7) except that a positive ε is added to the right hand sides. If Γ is bounded, solutions f ε to (3.2) can be constructed as fixed points of (3.3).
We consider the set of triples (n, a, t) where f ε n (a, t) is negative and assume this set to be non-empty. We determine the time τ = τ (ε) after which at least one f ε n (a, t) becomes negative.
We label the indices n and a that belong to τ = τ (ε) by k and α, i.e. ∃ k, α such that we have has a local minimum in a-direction at (k, α, τ ) by construction. Note that neither k nor α can tend to infinity due to Lemmas 3.2 and 3.3.
which is contradictory to the construction of (k, α, τ ). The above calculation is valid for 2 < k < n 0 . Computations are similar in the case k = 2 and also in the case k = n 0 leading to the same desired contradiction. Therefore the set ds + tε and so we have

Supersolutions
Within this section we will present supersolutions to the finite system of transport equations (2.7) introduced in Subsection 2.2.2. The first one will enable us to bound the numerator of Γ (f (t)) from above and is also applicable to the infinite system (2.3).
Note the dependence of Γ = Γ (f (t)). There is no Γ f (t) in the definition above; we consider Γ (t) as a function of time once f (t) is known.
If a function f solves (2.7) andf is a corresponding supersolution in the sense of Definition 3.1 we are able to bound f byf by considering a strict supersolutionf · exp (εt), using a comparison principle, and passing to the limit ε → 0.
To be more precise we state the following proposition.

Proposition 3.2 (comparison principle)
If we can bound the initial data of f strictly by the initial data off , namely

5)
and if the strict inequality holds for all n ∈ {2, ..., N }, 0 < a < ∞, and t > 0, then we can conclude thatf bounds f pointwise for all positive times, namely

Proof of Proposition 3.2
The proof is the same as the one of Lemma 3.1. q.e.d.
The following lemma presents a special supersolution which is constant in time and space. Therefore terms containing partial derivatives ∂ t and ∂ a of f will vanish and we can focus our attention on the effect of the coupling terms.

Lemma 3.2 (constant supersolution)
is a supersolution to (2.7) in the sense of Definition 3.1; c is a constant depending only on sup n,a f n (a, 0) in such a way that (3.5) is satisfied.

Proof of Lemma 3.2
In order to emphasize thatf is exponentially decreasing in n we define γ as and note γ > 0. Nowf reads Asf is constant in a and t we only have to check whether holds for all n. We only consider Γ (f (t)) > 0 as the case Γ (f (t)) = 0 is trivial and Γ (f (t)) < 0 cannot occur due to Lemmas 2.2 and 3.1 as β ∈ (0, 2). With a slight abuse of notation we set c = 1 instead of dividing by c and start examining the coupling terms. n = 2: The choice of γ in (3.9) is sufficient to ensure the desired non-negativity as we only need 1 Again due to (3.9) we have 1 + 1 β ≥ exp (γ) ensuring the non-negativity. 2 < n < n 0 : The necessary non-negativity is again provided by (3.9).
In the following lemma we present an exponentially decaying (w.r.t. a) strict supersolution that allows the use of a comparison principle to show that solutions f of (2.7) must at least decay exponentially. This result is only true for finite n 0 .

Lemma 3.3 (decaying supersolution)
is a strict supersolution to (2.7) for 0 < a < ∞; c is an arbitrary constant depending only on sup n,a f n (a, 0) in such a way that (3.5) holds.

Proof of Lemma 3.3
We check iff is a strict supersolution and observe that the coupling terms Jf n (a, t) were already discussed in

Positivity of total number of grains
The goal of this section is to show that the total number of grains N (t) cannot drop down to zero within finite time. We will prove this by contradiction using the conservation of total covered area A (t) and a lemma concerning the tail of n f n (a, t) w.r.t. a. We will motivate and derive the key lemma mentioned above now.
According to the von Neumann-Mullins law (2.1) incorporated into our system of transport equations (2.7), the highest possible speed by which the area of a grain can shrink is 4. Now consider and observe that this quantity should be non-decreasing in is a characteristic line of the transport operator selected by s with gradient −4 in (a,t)-space. This observation shall be cast into formulas.
We used (2.7) and (2.9) to obtain the equality ( * ). The choice of σ as is sufficient to ensure the desired inequality d dt Q (σ, t) ≥ 0. These calculations give rise to the following lemma.

Proof of Lemma 3.4
From the above calculations (3.12) and by plugging in the choice (3.13) of σ we have for all times t 0 and t 0 + t such that 0 < t 0 < ∞, 0 < t 0 + t < ∞, and for all s ≥ 4t 0 . Mapping s → s + 4t and labeling s 0 = s − 4t 0 leads to the desired inequality (3.14). q.e.d.
We can exploit Lemma 3.4 to illustrate that grains bigger than a given area 4α "survive" at least for a certain time α.
To be more precise, we state the following corollary.
where N (t 1 + t 2 ) denotes the total number of grains at time t 1 + t 2 .

Proof of Corollary 3.1
Set s 0 = 0, t 0 = t 1 , and t = t 2 in Lemma 3.4. q.e.d. Now we will prove by contradiction to the conservation of total covered area that the total number of grains remains positive within finite times.
for all 0 < t < t 0 . This implies for all 0 < t < t 0 . Conservation of total covered area (proved in Lemma 2.4) and again Lemma 3.4 allow for the following estimate which is contradictory to our assumption on N (t 0 ). q.e.d.

Bounding total mass from below
Within the previous Section 3. 4 we have shown that the total number of grains -or total mass -N (t) cannot drop down to zero within finite times. Unfortunately this result is not uniform in f . It turns out that we can achieve a real a-priori estimate not depending on f (t) in exchange for a dependence on n 0 in the case of initial data with finite support. In case of initial data with infinite support we shall have a closer look on Lemma 3.4 and especially on Corollary 3.1 from the previous Section 3.4 to come up with an a-priori estimate depending on the quantiles of the initial data.
As the denominator of Γ (f (t)) can be bounded by N (t) from below, the results of this section provide us with the other ingredient (besides the supersolution (3.8) in Lemma 3.2) to bound Γ (f (t)) itself for finite times.
First we restrict ourselves to initial data with finite support. In that case solutions f to (2.7) have finite support for finite times as well.
With this knowledge we can easily achieve the desired result for initial data with finite support by exploiting conservation of total covered area.
where ω is the length of the support of the initial data.

Proof of Lemma 3.5
According to Remark 3.3 we have where 0 < t < ∞ is an arbitrary finite time.
Proof of Lemma 3. 6 We observe

Bounding the coupling's weight
We now collect results on bounding the coupling's weight Γ (f (t)) in terms of the initial data.

Lemma 3.7
If f is a solution to (2.7) and its initial data satisfy the polyhedral formula (2.13), then we can bound Γ (f (t)) by if the initial data have finite support and by if the initial data have infinite support. The constants c 1 and c 2 only depend on sup n,a f n (a, 0) essentially; ω denotes the length of support of the initial data.

Proof of Lemma 3.7
From Lemma 2.3 we have for any finite t > 0. The numerator is now bounded via Lemma 3.2 and the denominator either by Lemma 3.5 or by Lemma 3.6. q.e.d.

Existence of solutions
In this chapter we prove existence of solutions to the infinite-dimensional system (2.3) by taking a suitable limit of solutions to finite-dimensional systems (2.7). Furthermore we use energy methods to prove uniqueness and continuous dependence on the data of solutions to (2.3).

Function spaces and mild solutions
In order to study the existence and other properties of solutions to (2.7), we introduce an open subset X n 0 of the Banach function space of integrable and bounded continuous vectorial functions x = (x n ) n 0 n=2 of dimension n 0 − 1 with zero boundary condition for n > 6, labelled [L 1 ∩ BC 0,n>6 ] n 0 −1 .

Definition 4.1
For convenience we introduce the phase space P n 0 describing the initial data. [I n (f )] (a, t) =g n (a − (n − 6) t)

s) ds
Note we set f n (α, t) = 0 if the argument α is negative in the formulas above.
Our aim is to construct solutions to (2.7) as fixed points of I on Y n 0 . This gives rise to the following definition of a mild solution.

Definition 4.5 (mild solution)
We call a function f ∈ Y n 0 satisfying f n (a, t) =g n (a − (n − 6) t)

Existence for short times
We will construct short-time solutions f in a certain neighbourhood of the initial data g with heavy restrictions on the possible maximal time T .

Definition 4.7 We consider
where C 0 > 0 is given by and β ∈ (0, 2) is a free parameter.
and therefore depends continuously on the initial data.

Proof of Theorem 4.1
We intend to apply Banach's fixed point theorem to the operator I on Y n 0 . Therefore we have to check if I maps Y n 0 M 1,∞,Γ to itself and if I is contractive. The continuity of f w.r.t. a follows from Proposition 4.1 in Subsection 4.1.4. This allows us to evaluate f pointwise in a especially at the boundary a = 0 to compute the numerator of Γ (f (t)). Now we observe that the boundary conditions f n (0, t) = 0 for n > 6 are preserved by I as we set f n (a, t) = 0 for a < 0.
The self mapping property of I on Y n 0 M 1,∞,Γ can be easily verified for any

REINER HENSELER
Before examining |Γ ([I (f )] (t))| we carry out an estimate on a part of the denominator of Γ ([I (f )] (t)) depending on the initial data g to recover C 0 where β ∈ (0, 2). We used the polyhedral formula (2.13) to obtain the first inequality.

So we have
Before we start to verify the contraction property w.r.t. sup t · 1 +sup t · ∞ we carry out an auxiliary calculation on |Γ (u) − Γ (v)| by decomposing Γ (f ) into it's numerator N (f ) and denominator D (f ) for all T ≤ T c . We now combine (4.2) with (4.3) and use (4.1) to achieve The continuous dependence of T on the initial data g is clear as the components the minimum is taken from are all continuous in g. q.e.d.

Continuous dependence on the data Lemma 4.1
The mild solution f in the sense of Definition 4.5 depends continuously on its initial data g.

Proof of Lemma 4.1
The continuous dependence on the data is a direct consequence of the structure of I and can be verified by an application of Gronwall's lemma.
Let f 1 and f 2 be mild solutions to the initial data g 1 and g 2 . Now we define from a computation similar to the one leading to (4.4) in the proof of Theorem 4.1. f 1 and f 2 are continuous functions in a and t; therefore z is continuous, too. We apply Gronwall's lemma and get implying the desired continuous dependence on the data. q.e.d.

Regularity of mild solutions Proposition 4.1
The mild solution f to initial data g ∈ P n 0 is continuous in a and t.

Proof of Proposition 4.1
Looking at Definition 4.5 we observe that all components involved are continuous functions: The initial data g are continuous by assumption, Γ (f (t)) is continuous, the coupling (Jf ) n (a, t) is continuous in a and t if the f n (a, t) involved are continuous, and the integration mapping is continuous. q.e.d.

Corollary 4.1
If the initial data g ∈ P n 0 are continuously differentiable, then a mild solution f is continuously differentiable, too.

Proof of Corollary 4.1
We can construct mild solutions u = ∂ a f to initial data v = ∂ a g as fixed points of I on Y n 0 in the same way as f to initial data g. This is possible as Γ (f (t)) only depends on n ∞ 0 f n (a, t) da and f n (0, t) such that we have a set of first-order equations in a. We apply Proposition 4.1 to v = ∂ a g and u = ∂ a f to identify a candidate for ∂ a f formally. Now we consider the difference quotient h and use Definition 4.5 to achieve the following bound allowing us to take the limit h → 0. Finally we have to verify and by Gronwall's lemma we have q.e.d.

Corollary 4.2
If the initial data g ∈ P n 0 are continuously differentiable, then a mild solution f is continuously differentiable in t.

Proof of Corollary 4.2
We repeat the proof of Corollary 4.1 and apply Proposition 4.1 on ∂ a g. Then we differentiate (3.1) w.r.t. t and deduce boundedness of ∂ t f n (a, t). q.e.d.

Remark 4.1 Corollaries 4.1 and 4.2 imply that a mild solution to continu-
ously differentiable initial data is a strong solution of (2.7).

Mild and admissible solutions
Proposition 4. 2 We can approximate mild solutions in Y n 0 with initial data in P n 0 by solutions in

Proof of Proposition 4.2
The function space C 1 is dense in C 0 . The regularity results (Proposition 4.1, Corollary 4.1) on mild solutions f and their continuous dependence on the initial data g (Lemma 4.1) together with the exponentially decaying supersolution in Lemma 3.3 allow for an approximation of mild solutions in Y n 0 by mild solutions in q.e.d.

Corollary 4.3
The mild solution with initial data g ∈ P n 0 is an admissible solution for all times t ∈ [0, T ], i.e. f ∈ C 0 ([0, T ] ; P n 0 ).

Proof of Corollary 4.3
We approximate the solution in Y n 0 by solutions in

Existence for arbitrary finite times
Theorem 4.2 For given initial data g ∈ P n 0 there exists a unique admissible solution f ∈ C 0 ([0, T ] ; P n 0 ) where T < ∞ is arbitrary.

Proof of Theorem 4.2
The strategy of the proof is to divide (0, T ] into appropriate subintervals (T j−1 , T j ], 0 = T 0 < T 1 < · · · < T k−1 < T k = T , and to apply Theorem 4.1 on each of these subintervals. Therefore we have to ensure that the constants M 1 , M ∞ , and M Γ do not need to be changed throughout the whole sequence of subintervals ((T j−1 , T j ]) j .
• M 1 is defined to be strictly bigger than g 1 and as the total number of grains decreases in time (Corollary 2.1) we have g 1 ≥ f 1 for all times.
• M ∞ is defined to be strictly bigger than g ∞ and due to the supersolution in Lemma 3.2, which is constant in a, we have g ∞ ≥ f ∞ for all times by using a comparison principle.
• M Γ depends on M ∞ (which remains the same) and the number of grains N (T j−1 ) at the beginning of each subinterval.
In the case of finite support of the initial data we have where ω denotes the length of the support of the initial data.
If the initial data have infinite support we know

Infinite system
The goal of this section is to construct mild solutions of (2.3) by taking a limit of admissible solutions of the finite-dimensional system (2.7). Further properties of solutions to (2.3) will be studied in Chapter 5.

Appropriate spaces and mild solutions
We now introduce the Banach sequence space X as the infinite-dimensional analogue to X n 0 (cf. Definition 4.1). Again the natural norm used is Definition 4.8 Furthermore we define Y to describe the space of solutions f to (2.3) by considering the supremum w.r.t. time t of elements of X using the natural norm sup t f 1 + sup t f ∞ .

Remark 4.2
We consider elements of X n 0 as elements of X (and elements of Y n 0 as elements of Y ) by setting x n = 0 (and y n = 0) for n > n 0 .
We extend our definition of mild solutions to (2.7) (cf. Definition 4.5) to the infinite-dimensional case.

Definition 4.10
We call a function f ∈ Y satisfying f n (a, t) =g n (a − (n − 6) t) for all n ≥ 2, 0 < a < ∞, and t > 0 a mild solution to (2.3). We set f n (α, t) = 0 if the argument α is negative.

EXISTENCE OF SOLUTIONS
We will prove existence (cf.

Proof of Theorem 4.3
The main idea of the proof is to construct mild solutions to (2.3) as a limit of admissible solutions to (2.7) where the largest possible n = n 0 is increasing. We label the final n, namely n 0 , by k from now on to keep notations simple. Furthermore we introduce the following notation to finite-dimensional systems.
Note that elements f k of the sequence f k k only make sense for k > 6 due to (2.13). Before we start to identify weak limits of this sequence in Y we carry out some calculations bounding the spatial and the time derivatives of the f k n (a, t) by the initial data g. As these bounds will be independent of k we omit the upper index k and write f n (a, t) instead.

Due to Lemma 3.2 we have a supersolution
for f k independent of k. It is very easy to compute that we also have a subsolution f n (a, t) = −f n (a, t) < 0 , 2 ≤ n ≤ k of the same structure. Note that the coupling's weight Γ (f (t)) does not depend on a directly but only on f n (0, t) and f n (a, t) da. Therefore the spatial derivative of a solution satisfies an equation analogue to (2.7) where Γ = Γ (t) is known. Hence we can bound the spatial derivative ∂ a f k n by the supersolutionf n (a, t) and the subsolution −f n (a, t) depending only on the spatial derivative ∂ a g of the initial data.
We can also bound the time derivative of solutions to (2.7) via due to the previous considerations and Lemma 3.7 (bounding sup t Γ). The constants c 1 and c 2 only depend on the initial data and their spatial derivative, c 0 also depends on T (via n ∞ 4T g n (a) da).
We have that f n (a, t), |∂ a f n (a, t)|, and |∂ t f n (a, t)| are uniformly bounded by sup n≥2 c n+6 n−1 exp (−γn) on (0, ∞) × (0, T ] for all n ≥ 2 (independent of a and t). Therefore f k k is equicontinuous. Now consider an arbitrary, but fixed n. By Arzela-Ascoli we know that there exists a subsequence (f kν n ) kν such that f kν n → f n uniformly on a compact subset of (0, ∞) × (0, T ] as ν → ∞. Furthermore f n is continuous w.r.t. both variables a and t. We exhaust (0, ∞) × (0, T ] (and especially (0, ∞)) with compact subsets. On each of these subsets we apply the above argument, i.e. there exists a subsequence (f kα n ) kα such that f kα n → f n pointwise where α denotes an indexing of the compact subsets (e.g. with length increment 1). By choosing an appropriate diagonal subsequence we deduce that f kµ n → f n pointwise, f n continuous, on (0, ∞) × (0, T ] as µ → ∞. Note f kµ n ≡ 0 for n > k µ .
We start executing the procedure described above for n = 2 and achieve that f The sequence (f kµ 3 ) kµ is also bounded and equicontinuous. This impliesagain by Arzela-Ascoli -that there exists a further subsequence (f k λ 3 ) k λ which converges pointwise to a continuous f 3 . This convergence is again uniform on compact subsets of (0, ∞) × (0, T ]. Proceeding as above implies that we can choose a suitable diagonal subsequence (f kκ ) kκ such that f kκ n → f n pointwise, f n continuous, for all n as κ → ∞. Now we observe that our bounds on Γ f kκ (t) (cf. Lemmas 2.3 and 3.7) are independent of k κ as we omit the term k κ β ∞ 0 f kκ kκ (a, t) da while estimating the denominator of Γ. It remains to show that Γ f kκ (t) → Γ (f (t)) pointwise in time in the limit κ → ∞: The convergence of the numerator is clear as it contains only weighted contributions f n (0, t) for n = 2, . . . , 5 due to the boundary conditions (2.6). A-priori calculations stated in Lemma 5.3 (Chapter 5) imply convergence of the denominator's essential part n n ∞ 0 f n (a, t) da and its last term n 0 β ∞ 0 f n 0 (a, t) da in the limit process f kκ → f as κ → ∞. Convergence of the term −2 (β + 1) ∞ 0 f 2 (a, t) da is trivial. Therefore the limit Γ (f (t)) is given by (2.5).
With this knowledge we are able to pass to the limit in the integral formulation f kκ n (a, t) = g n (a − (n − 6) t) and achieve a mild solution to (2.3) in the sense of Definition 4.10. This solution is non-negative and continuous in time (and space).
In order to prove differentiability of solutions w.r.t. a and t we differentiate the integral formulation (4.5) and bound the modulus of the r.h.s. indepen-dently of k κ . We first investigate using Lemma 3.7 to bound Γ and the supersolution (and subsolution) to control the derivatives within the integrand. Therefore differentiating (4.5) w.r.t. a leads to ∂ a f kκ n (a, t) ≤ |∂ a g n (a − (n − 6) t)| + c (f (·, 0) , t) t exp (−γn) and we can pass to the limit κ → ∞ as the initial data are smooth. Differentiating (4.5) w.r.t. t gives us using the Leibniz integral rule. Again we can pass to the limit κ → ∞ as the initial data are smooth and finite w.r.t. n.
Conservation of the polyhedral formula is stated in Lemma 5.4 under the assumptions of this theorem.
Corollary 5.1 in Section 5.2 implies that total covered area is conserved as we can plug (f kκ ) kκ into the proof of Lemma 5.2, set α = k κ , and pass to the limit (w.r.t. κ → ∞). Note that the appearing sums are cut off at k κ by definition of f kκ . q.e.d.

Energy methods and uniqueness
Within this subsection we will show that solutions to (2.3) in the sense of Theorem 4.3 are unique and depend continuously on the initial data. The proof is carried out using energy methods. We start our considerations with a weighted L 2 -energy aiming at getting (almost) a sign for the coupling terms.

Definition 4.12
We define for solutions in the sense of Theorem 4.3 where c = c (f (·, 0) , T ) is a constant.

Proof of Lemma 4.2
First we investigate how the coupling terms behave when they are multiplied by nf n . These computations are inspired by an integration by parts with a continuous variable n but carried out purely discrete by index shifts and using binomial formulas (cf. Appendix A).
The proof is carried out by differentiating the energy w.r.t. t and using (2.3).
We can also establish a linear growth rate of E s (t).
for solutions in the sense of Theorem 4.3 where C = C (f (·, 0)) is a constant.

Proof of Lemma 4.3
We repeat our calculations within the proof of Lemma 4.2, decompose Γ into it's numerator and denominator, and use Hölder's inequality to proceed.
The last inequality is obtained by using the supersolution (3.8). q.e.d.
Considering the difference of two solutions it turns out that E s (t) is insufficient to get a result as in Lemma 4.2. Therefore we introduce an "energy" E (t) with a slightly modified integrand and an additional term.

Definition 4.13
We define We could also use exp (−k a) instead of exp (−a) for any k > 0 within the definition of E (t) and still acquire the same results that are presented below.

Remark 4.3 For E (t) we can achieve the same results (Lemmas 4.2 and 4.3) as for E s (t).
The only technical differences are an additional integration by parts w.r.t. a and the use of Cauchy's inequality with ε in order to absorb (Ṅ ) 2 into the negative boundary term of the partial integration mentioned before. ment of E (t) instead of f .
for any two solutions u and v to (2.3) in the sense of Theorem 4.3.

Proof of Lemma 4.4
Again will carry out the proof by differentiating E (u − v) (t) and using (2.3). For simplicity we omit the arguments (a, t) of u and v wherever possible.
The first term on the r.h.s. of (4.7) is treated via an integration by parts:

REINER HENSELER
The second term on the r.h.s. of (4.7) has to be split into several parts in order to overcome the nonlinearity Γ = N /D (which we also split into it's numerator N and denominator D. We now treat the first term on the r.h.s. of (4.9) by a computation similar to (4.6) within the proof of Lemma 4.2.
The second term on the r.h.s. of (4.9) is estimated by using Young's inequality and (4.10): Here ε has to be chosen again in such a way that the first term of the last inequality above is less than half of the modulus of the second term of the last inequality within (4.8).
The third term on the r.h.s. of (4.9) can be treated by using Cauchy's inequality and again (4.10): Recall that N is a notation for the total number of grains in contrast to N which is the numerator of Γ.
Now we present the computation (4.10) that was used above: The estimates (4.10) are mainly achieved by using the supersolution (3.8) and Hölder's inequality twice.
The third term on the r.h.s. of (4.7) can be handled by Cauchy's inequality with ε and by using the supersolution to estimate the intermediate terms of (Ṅ (u) −Ṅ (v)) 2 . Here ε has to be chosen in such a way that the second term of the last inequality within (4.11) is less than half of the modulus of the second term of the last inequality within (4.8).

Proof of Corollary 4.4
Uniqueness follows directly from Lemma 4.4 by setting u (·, 0) = v (·, 0). Continuous dependence of solutions on the initial data is also a direct consequence of Lemma 4.4. q.e.d.

Properties of solutions to the infinite system
In Section 4.2 of the previous chapter we have proven existence of strong solutions to (2.3). Now we focus on verifying that properties which hold for the finite system (2.7) (e.g. conservation of total covered area) also to be true in the infinite-dimensional case.

No runoff at infinity
Within this section we carry out some a-priori calculations concerning the infinite system (2.3). From Lemma 3.2 we know that the amount of mass in f n (a, t) is exponentially small w.r.t. n. Now we will show that the amount of mass in f n (a, t) is also exponentially small with increasing a, which means that there is no runoff at infinity.

Definition 5.1
For any given ν > 6 and α > 0 we define as the non-essential part or "quasi-complement" of N (t) .
The idea is now to choose ν (t) and α (t) as monotonically increasing functions of time to control N ⊥ (t). A first step is to understand how N ⊥ evolves in time if ν and α grow. (2.3). Let µ (t) > 6 and α (t) > 0 be monotonically increasing functions with d dt α (t) ≥ µ − 6. Then we have

Lemma 5.1 Suppose f is a solution to
exp (−γµ (s)) α (s) ds (5.1) for all finite times t; c is an arbitrary constant depending on sup n,a f n (a, 0) (in such a way that (3.5) holds) and on sup 0≤t≤T Γ (f (t)). Furthermore γ = log(1 + 1 β ) is a constant, too.

Proof of Lemma 5.1
We define as the integer part of µ (t) and denote the jump of ν (t) by [[ν]].
In the case [[ν]] = 0 we observe that and so we have as an additional part in the r.h.s. of (5. q.e.d.

Remark 5.1
We can assume d dt α (t) ≥ µ instead of d dt α (t) ≥ µ − 6 within Lemma 5.1 to keep notation as simple as possible in the following.
In order to exploit Lemma 5.1 we have to choose the functions µ (t) and α (t) depending on each other via d dt α (t) ≥ µ (t) in a clever way; α should grow at least linearly in ν to compensate for the transport in a along characteristic lines. On the other hand ν should grow exponentially controlling the diffusion in n.
For simplicity we use the notation if the third argument of N ⊥ is given by rounding down the second one. Theorem 5.1 Consider the functions α (t) = α 0 exp (t) and µ (α (t)) = α (t). Furthermore denote by ν (t) = µ (α (t)) the integer part of µ. If f is a solution to (2.3) we have for all finite times t > 0 and all finite α 0 > 0.

Proof of Theorem 5.1
Instead of (5.1) we consider an equation for the supersolution N ⊥ of N ⊥ , namely for any arbitrary α 0 > 0. Now we plug in our ansatz for α and µ: Changing variables a = α 0 exp (s) leads to and by lengthening the domain of integration (and using α 0 = α exp (−t)) we have finally. Carrying out the integration completes the proof. q.e.d.

Conservation of total covered area
The purpose of this section is to show that total covered area A (t) of solutions f to (2.3) is a conserved quantity for suitable initial data g.

Proof of Lemma 5.2
We have via an integration by parts. Theorem 5.1 implies We recall the definition of A (t) (cf. Definition 2.2) in the infinite-dimensional case and prove the announced assertion. q.e.d.

Validity of triple junction condition
Within this section we want to argue that the polyhedral formula (cf. Subsection 2.3.2, Proposition 2.2) is preserved in the infinite-dimensional case, too. A first step is to understand why a key quantity within the computations for the finite-dimensional casen n ∞ 0 f n (a, t) da -is also bounded for solutions of (2.3).

Lemma 5.3 For a solution
for all times t with 0 < t < T < ∞. The constant c 0 depends on T , β ∈ (0, 2), and sup n,a f n (a, 0).

Proof of Lemma 5.3
We split the integration at n for the weighted sum of coupling terms. Passing to the limit κ → ∞ leads to the desired result by the choice of Γ (f (t)) in (2.5) and the zero balance property (2.9) of the coupling, i.e. n (Jf ) n (a, t) = 0. Note that

Decrease of total number of grains
The last fact which we can expect to verify is that total number of grains (cf. Subsection 2.3.1, Definition 2.1) is a decreasing quantity in the infinite-dimensional case, too.
Proof of Lemma 5.5 Theorem 5.1 implies that the boundary values at a = 0 for n = 2, . . . , 5 are the only "drainage" (for suitable initial data) -no mass can be lost at infinity w.r.t. a or n. Therefore the proof is essentially the same as in Lemma 2.1 and Corollary 2.1.
We consider a suitable diagonal sequence (f kκ ) of solutions to the finitedimensional system (2.7) as in the proof of Theorem 4.3. For each f kκ we can carry out the proof of Lemma 2.1, but we integrate only up to k κ instead of ∞ (w.r.t. a). We also sum up to k κ only.
Bounding ∂ t f kκ n (a, t) in the same way as within the proof of Theorem 4.3 allows us to differentiate w.r.t. t. We use the zero balance property (2.9) and pass to the limit κ → ∞ with f kκ n (a, t). Convergence of the appearing terms is ensured by Theorem 5.1. q.e.d.
as f s n (0) = 0 for n > 6 due to the boundary conditions (2.6) and the contribution from the term n = 6 is zero due to the prefactor (n − 6). Furthermore we have f s n (0) ≥ 0 for all 2 ≤ n ≤ n 0 . This implies Γ (f s ) = 0 due to the definition of Γ in (2.5). Now the criterion for stationary solutions reduces to (n − 6) ∂ a f s n (a) = 0 for all n ≥ 2 and 0 < a < ∞. Together with (6.1) and the boundary conditions (2.6) this completes the proof. q.e.d.

Lemma 6.2
A nontrivial stationary solution as described in Lemma 6.1 is not attractive as slightly perturbed data lead to a positive Γ (f (t)) for some times t and are therefore affected by the coupling operator (Jf ) n (a, t).

Proof of Lemma 6.2
Assume there exists a finite time t such that we have f 6 (a, t) da = 1 − ε, n>6 (n − 6) f n (a, t) da = ε/2, and Γ (f (t)) = 0. Then (2.13) implies n<6 (6 − n) f n (a, t) da = ε/2. This gives us an upper bound on the time τ after which Γ (f (t)) is positive: τ = min a {f n (a, t * ) > 0, 2 ≤ n ≤ 5}. At that time two things happen: 1 st the total number of grains decreases, and 2 nd the discrete diffusion is active and shuffles mass away from f 6 . Then we can repeat our estimate on τ . q.e.d.
We are not able to prove that the trivial stationary solution is attractive for nonstationary initial data, i.e. ∃k ≥ 2, k = 6, : g k (a) da > 0, but we show that the total number of grains N (t) is strictly decreasing for most finite times t.

Lemma 6.3
The total number of grains N (t) to nonstationary initial data, i.e. ∃k ≥ 2, k = 6, : g k (a) da > 0, decreases strictly for most finite times.
Proof of Lemma 6.3 Lemma 2.1 implies that the total number of grains N (t) decreases in time. Due to (2.13) and the leftwards transport for n < 6 we observe thatṄ < 0 for most times. We will elaborate on this. Assume there is a time t * such that f n (0, t) = 0 for all n at t = t * . The polyhedral formula (2.13) implies 5 n=2 f n (a, t) da > 0. We have an upper bound on the time τ after which at least one f n (0, t), 2 ≤ n ≤ 5, is strictly positive implyingṄ < 0 at least at that time τ = min a {f n (a, t * ) > 0, 2 ≤ n ≤ 5}. q.e.d.

Self-similar scaling
Self-similar scaling behaviour, also called normal grain growth, often occurs in experiments. To observe this in our model we start our analysis by rescaling equations (2.3) (cf. Subsection 2.2.1) and considering stationary solutions of the rescaled system. The resulting system of ordinary differential equations will be examined afterwards.

Natural rescaling
Following an ansatz by Fradkov [5] we introduce the relative quantity  which is important to compute the rescaled Γ (f (t)). We have for the terms on the r.h.s. of (2.3). The derivatives on the l.h.s. change via and by using aN = ξ within the second term of the time derivative. Collecting terms and dividing by N 3 leads to as the rescaled version of (2.3). Using Lemma 2.1 (or better Lemma 5.5) we observė and by considering stationary solutions of (6.2) and plugging in our definition of α (6.3) we immediately get (n − 6 − αξ) ∂ ξ ϕ n = G (ϕ) (Jϕ) n + 2αϕ n (6.4) with positive boundary conditions for ϕ n (0) where 2 ≤ n ≤ 5 and zero boundary conditions elsewhere. Note that the factor α in (6.4) can be scaled out mainly by using a different G (ϕ) instead of G (ϕ).

Simple rescaling
We can also rescale (2.3) directly by and consider stationary solutions to achieve a simpler form of (6.4): (n − 6 − ξ) ∂ ξ ϕ n = G (ϕ) (Jϕ) n + 2ϕ n , n ≥ 2 (6.5) These equations are also subject to the boundary conditions ϕ n (0) = 0 (6.6) for n > 6. The nonlinearity takes the following form: This simple scaling also ensures conservation of total covered area (before considering stationary solutions) and furthermore we can recover some information on ϕ n dξ by integrating (6.5) and summing up: Here we used the zero balance property (2.9) of the coupling (Jϕ) n .
We do not prove existence of solutions to (6.5) as two major difficulties arise: First there are no a-priori bounds available (like supersolutions) as in the time-dependent case and second the zeros of n − 6 − ξ in front of the derivatives ∂ ξ ϕ n (ξ) are not that easy to overcome due to the alternating influence of the ϕ n by the coupling (Jϕ) n .
The remaining task is to select a self-similar solution that is "physically reasonable", i.e. covers the same total area A as the solution to (2.3) in the time-dependent case. We observe that ξϕ n (ξ) dξ depends continuously on the weighted sum of the initial values 5 n=2 (6 − n) α n .

Remark 6.2 It is unclear if we can reach ∞
n=2 ∞ 0 ξ ϕ n (ξ) dξ = A for any given finite A by a choice of finite initial values α n , 2 ≤ n ≤ 5.
To achieve consistency with experimental results [7] we can demand as an additional selection criterion, e.g. by prescribing ϕ 2 (ξ) dξ = εN for any given 0 < ε 1 where ε should be extracted from experimental data.
We are able to compute the topological classes distribution for suitable sets of input parameters. The initial values ϕ 2 (0) , . . . , ϕ 5 (0) are chosen such that (n − 6) ϕ n (ξ) dξ = 0 and (6 − n) ϕ n (0) = ϕ n (ξ) dξ hold and also all φ n are non-negative. This choice is not unique and we can also adjust the ratios of the φ n , n < 6, a bit in this way. β = 1/2 is chosen according to [8] and in common with [7] we set φ 2 = 10 −3 as starting value for the iteration. The condition ξϕ n (ξ) dξ = A is satisfied by a suitable rescaling of the ϕ n (0). The iteration is carried out up to n = 1000. Besides a typical shape  Fig. 1] and also with simulation results of a completely different model [9, Fig. 9] we treated earlier. The parameter β seems to influence the ratio between φ 5 and φ 6 and also the decay rate of the φ n .

Lewis' law
A natural question concerning grain growth is to ask whether there is a correlation between the topological class and the area of a grain. Lewis observed a linear relationship [13] examining cellular structures arising in biology. In common with Flyvbjerg [4] this so-called Lewis' law reads ξ n = a (n − 6) + b (6.14) concerning our model (in scale invariant, dimensionless variables). Here ξ n = ξϕ n (ξ) dξ/ ϕ n (ξ) dξ denotes the mean value of ϕ n (ξ). It is unclear if this phenomenological law is really applicable for grain growth. Rivier and Lissowski derived Lewis' law by maximum entropy arguments applied to cell distributions [18].
Concerning our model, arguments given by Flyvbjerg are reasonable that we expect Lewis' law as a consequence of von Neumann-Mullins law, as we describe the fundamental dynamics of grain growth through one of it's consequences [4], namely the von Neumann-Mullins law (2.1). Furthermore we observe that Lewis' law cannot be true for small n and arbitrary a, b ≥ 0. In the sequel we will show that Lewis' law is valid for asymptotically large n. Similar results are achieved by Flyvbjerg [4]. for the topological class distributions φ n = ϕ n (ξ) dξ.

Lemma 6.4
Assume there exists a smooth solution to (6.5). Let the asymptotic expansion for φ n in Proposition 6.1 be accurate. Then we have with a = 1/ (G + 1) and b = a ((2β + 1) − 6G) for large n.

Conclusion [The curtain falls.]
We have established a rigorous existence theory for a nonlinear system of transport equations with nonlocal weight that arose from a model for grain growth almost twenty years ago. Now the field is open for further studies concerning self-similar behaviour that we started partially in the last chapter. At this stage it is unclear whether one can succeed with rigorous analytic treatment or if detailed numerical simulations might provide deeper insight.
Besides the use of standard analytic results it was necessary to develop problem specific techniques. Key ideas are the supersolution in Lemma 3.2, the quantile considerations in Lemma 3.4, the "energy" used in Lemma 4.4, and the behaviour of the "quasi-complement" computed in Lemma 5.1.
Using the "quasi-complement" of total mass as a bounding frame growing in time (cf. Theorem 5.1) seems to be -to our knowledge -a new idea in the analysis of infinite-dimensional systems.
energy on a single grain boundary first