Modelling phase transitions via Young measures

We consider the elastic theory of single crystals at constant temperature where the free energy density depends on the local concentration of one or more species of particles in such a way that for a given local concentration vector certain lattice geometries (phases) are preferred. Furthermore we consider possible large deformations of the crystal lattice. After deriving the physical model, we indicate by means of a suitable implicite time discretization an existence result for measure-valued solutions that relies on a new existence theorem for Young measures in infinite settings. This article is an overview of [2].

1. Introduction. We consider the general elastic theory of single crystals at constant temperature where the free energy density depends on the local concentration of one or more species of particles in such a way that for a given local concentration vector certain lattice geometries (phases) are preferred. Large deformations of the crystal lattice are explicitly allowed.
The local concentration of the molecules may change due to diffusion. The time scales typical for diffusion and for elastic deformation are usually significantly different. In good approximation it is admissible to assume that the deformation adjusts infinitely fast to the local situation. In the developed model there is surface energy contributing to the free energy of the crystal. The model allows for m different coexisting macroscopic phases. We assume that the crystal does not possess interstitials and that the time-evolution of the boundary of the domain is known.
After deriving the physical model with the above properties we discuss the existence of solutions to this model by means of an implicit time discretization which results from a Q − Q * formulation, see section 3. In the Langrangian picture we will show for a special class of volumetric free energy densities that in the limit of vanishing time step the time-discrete solutions converge in the sense of Young measures on suitable separable and reflexive Banach spaces. Finally we state an energy inequality for the limit solution.
In this overview of [2] we focus on presenting a general strategy to solve the model equations. We omit all proofs of the stated theorems which can be found in german in [2].
For symbols not defined in the text see the List of Symbols at the end of this paper.
Key words and phrases. Phase transition in crystals, non-linear elasticity, diffusion, implicit time discretization, Q-Q* formulation, Young measures, infinite-dimensional Banach spaces. 30

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2. Derivation of the model. In this section we derive a model of the physical phenomenon outlined in the introduction. We make use of non-equilibrium thermodynamics, see [15], [20], in the context of continuum mechanics, see [16], [9]. In particular we neglect the atomistic structure of the crystal and disregard possible effects of the microstructure. The model is based on the following fundamental considerations: Diffusion is caused by gradients of chemical potentials. The diffusive flux causes a local change of crystal's free energy. The free energy shall depend on particle density, elasticity of the crystal and phase parameter only with a term representing the surface energy of the boundary layers.
In good approximation the system is in mechanical equilibrium. For the analytical treatment, we assume that deformation and phase parameters are global minimisers of the free energy.
Furthermore we assume a constant temperature, no interstials and that the boundary of the crystal is fixed.
A crucial point of the model is that it allows possible large deformations of the crystal lattice that can especially occur at the phase boundaries and for which the assumptions of linear elasiticity theory does not hold. Because of these deformations we also consider surface energies between different phases. Our approach generalizes existing models of solid-solid phase transitions which assume linear elasticity laws, see [18], [29], [30], [23], [8], [28], [14], and it is applicable to materials with non-linear elastic behaviour, see [17], [24]. Remark 1. All theorems presented here hold also for a given boundary evolution if this evolution allows an energy releasing process, see (1.1.38) in [2], p.23.
We describe the crystal by a non-empty, bounded Lipschitz domain Ω ⊂ R 3 . The mechanical deformation is given by a family of mappings {Φ t : Ω → Ω : t ≥ 0} that satisfy for all t ≥ 0 det∇Φ t > 0 a.e. in Ω.
Here, a > 3 can be arbitrary. The conditions (2) and (3) ensure that Φ t are deformations. The condition a > 3 guarantees the integrability of the functional determinant and therefore that positive volumes will be transformed into positive ones. Condition (1) reflects the fact that the initial state is undeformed. The space-and time-dependent particle densities of the n ∈ N different species of molecules are described by ρ(t) : Ω → R n , t ≥ 0, with the natural conditions are postulated for t ≥ 0: ρ 0 is a given initial density-vector with ρ 0 ∈ L 1 (Ω, R n ), ρ 0 ≥ 0 a.e. in Ω, |ρ 0 | 1 ≤ 1 a.e. in Ω and ρ 0 1 < |Ω| Equation (4) 3 ensures the conservation of particles, (5) 2 is due to the fact that the crystal does not possess interstitials and that the number of lattice positions in a volume element is a uniform constant. We assume (7) 2 with strict inequality: This means that we always allow for vacancies in our system in order to permit diffusion. Let m ∈ N denote the number of different possible phases. At each time t ≥ 0 every x ∈ Ω belongs to exactly one phase. The evolution of the phases is described by a family of phase-vectors {χ t : Ω → R m : t ≥ 0} with given χ 0 which have to fulfil Therefore at each time t ≥ 0 there is a well-defined surface area between phase i und j given by S(χ it , χ jt ), where Assuming further that the densities of the surface energy ς ij on the interface between phase i and phase j are positive constants with ς ij = ς ji , the surface energy F s (χ t ) at time t ≥ 0 can be introduced by where σ ij := ςij 2 . Additionally we postulate for all i, j, k ∈ {1, ..., m} Therefore energetically it is not favourable to add a third phase between two other existing phases. We assume that the volumetric free energy density f depends on the phasevector, the gradient of the deformation and the particle densities and that there is a representation where cofA denotes the matrix of co-factors of the quadratic matrix A. By introducing for (χ, A, ρ) ∈ L 1 := L 1 Ω, R m + × L 1 Ω, R 3×3 × L 1 (Ω, R n ) the free volumetric energy at time t ≥ 0 can be computed by F v (χ t , ∇Θ t , ρ(t)). Consequently the free 32 STEFFEN ARNRICH energy of the system at t ≥ 0 is given by F (χ t , ∇Θ t , ρ(t)), where To model the dynamics of the system we use the following notation.
Notation. We define (formally) for quantities J : Ω → R 3·n and µ : Ω → R n : The evolution of the particle densities can be described by a continuity equation where J p t denotes the particle-flux-vector. We suppose there is a linear superposition of the diffusive flux −J t and the mechanical flux or equivalently At a fixed constant temperature the diffusive fluxes are caused by the negative gradients of the chemical potentials which are the thermodynamical forces, [15]. According to Onsager's postulate, [25], [26]), every thermodynamical flux is a linear combination of the thermodynamical forces. Therefore we set for t > 0 in Ω (14) with a symmetric and positive definite matrix L := (L ik ) n i,k=1 and the chemical potential µ k of species k, where the symmetry and positive definiteness of L comes from Onsager's reciprocity relation, [25], [26], [15]. Due to the definition of the chemical potential one has in Ω at each time t > 0 Let us formulate now the aforementioned minimality condition on the free energy. Considering the time-evolution of the deformation of a representative volume element and keeping in mind that the number of particles only changes due to diffusion, we find for two possible deformations Φ 1 t , Φ 2 t the relation where ρ 1 (t), ρ 2 (t) are the densities corresponding to Φ 1 t , Φ 2 t . The minimality condition for any t > 0 reads where Finally our model consists of the equations (1) -(5), (8), (13) - (15) and (17). We refer to this as the Eulerian description or picture.
3. Solution strategy and implicit time discretization. A suitable strategy to solve our equations was developed by Luckhaus and named Q − Q * -ansatz or principle. It has its roots in a special treatment of the heat equation, see section 1.2 in [2]. Up to now this principle is not published. We exemplify it by our equations.
To this end we compute formally the time derivative of the free energy by exploiting the minimality condition (17). The following argument is only heuristic.
If we formally consider the free energy F due to the relation Θ = Φ −1 as a function of the phase-vector χ, the deformation Φ and the particle-density-vector ρ then for t > 0 we find Now we compute ∂ Φ ρ(t)∂ t Φ t from (16) by using to obtain: If we plug (12) and (22) into (21) we find Assuming that the surface terms do not depend on ρ and the normal component of J t vanishes on ∂Ω, cf. (13) and (4) 3 , it follows, see [5] p.58, where Q * is the Fenchel conjugate of Q, see [7], p.49. We call the continuity equation (13) together with (15), (25) and the minimality condition (17) the Q − Q * formulation of our system. A solution to the Q − Q * formulation can be regarded as a weak solution to our original system. This is do to the fact that such a weak solution also fulfils Onsager's Law (14) if we assume (23) to be true. (14) is then an immediate consequence of (25) and of the Fenchel-Young equality, see proposition 3.3.4 in [7], p.51. An advantage of the Q − Q * -formulation is that it provides a natural implicit time discretization of our original system. For similar discretizations see also [29], [21], [22].
Replacing (13), (23), (25) in use of (20) by their time-discrete versions for timestep h > 0 and t ≥ 0: (31) is automatically satisfied and there holds for all η ∈ C 1 0 (Ω, R 3·n ) where ρ(t + h) is defined by (28). Hence Therefore for given time step h > 0 we suggest the following algorithm to compute a time-discrete solution: and compute ρ(t + h), µ t+h by (28), (33). Motivated by these considerations our strategy is first to solve the time-discrete system and then to show that the time-discrete solutions converge for vanishing time step h to an object that can be interpreted as a physically meaningful solution of the model equations.
The next step is to give the precise definition of the time-discrete solution and its analysis. 4. The time-discrete model. To define the notion of a time-discrete solution we have to clarify first what we will understand by the divergence of a L 2 -mapping. This is done in a way adapted to the equations such that the conservation of particles (4) 3 and the implication (23)⇒(24) holds. The deeper reasons why we need a definition for all L 2 -mappings are that Q is defined on L 2 Ω, R 3·n and that we want to apply the direct method of variational analysis to show existence of a time-discrete solution (see below).
Remark 2. Take notice of the following properties of the divergence, see Bemerkung 2.1.3, p.41 in [2].

STEFFEN ARNRICH
For fixed time step h > 0 we call every mapping a time-discrete solution if it satisfies and for t ≥ 0 where for t ≥ 0 Crucial to prove existence of a time-discrete solution is to show that the minimization problem F ĥ ρ → min, (χ, Θ, J ) ∈ M has a solution. We try to applicate the direct method. To this end we need additional mathematical conditions on the volumetric free energies f j , especially convexity, lower semicontinuity and growth conditions. They are specified in the following theorem. Using the abbreviations  (4), (5) where ρ 0 is replaced byρ.
The first assumption i., this is polyconvexity and lower semicontinuity of the volumetric free energy densities, is mainly needed to apply the direct method. ii. assures that there is a sequence (Θ ι ) ι∈N such that Θ ι ι → ∞ which implies H = cof∇Θ, ∆ = det∇Θ, see theorem 7.6-1, p.365 in [9]. It also guarantees the same for the deformations Φ ι due to To satisfy (4), (5) we need iii. while iv. yields inf M F ĥ ρ < ∞. Beside of this the proof relies on the following statements • Sobolev embeddings, Mazur's lemma and Fatou's lemma.
• For a minimizing sequence there holds J ι • Q is weakly sequentially lower semicontinuous.
• F s is strongly lower semicontinuous on P and it holds m j=1 Ω |∇χ j | ≤ CF s (χ) for some fixed constant C > 0 and all χ ∈ P .
By means of this theorem we can construct to every h > 0 a mapping γ h : [0, ∞[→ Π e which fulfils all restrictions of definition 4.2 but (45) and (46). Beside the existence of vacancies (7) 2 we need some extra informations about the behavior of the f j at their effective domains to assure this relations. The reason for this is that the argument (32) doesn't work in general, because we cannot exclude concentrations of (∇Θ t , ρ(t)) near or at  and for A ∈ R 3×3 with detA > 0 If f satisfies 1. To every (A, ρ) ∈ ∂Z o and every χ ∈ C n there exists ∇ ρ f(χ, A, ρ), 2. To every (A, ρ) ∈ ∂Z u there is z ∈ R n such that for all χ ∈ C n there holds lim then under the assumptions of theorem 4.3 there holds where we used the abbreviation (54).
The idea of proof is to approximate the f j by suitable smooth and convex functions from below and to show by using (32) that the solution of the corresponding variational problem fulfils (45), (46). Then one shows that this solutions converge to the solution of the original problem where in the limit Onsager's relation remains valid. This is done by a subgradient argument where 1.-4. enters. Theorem 4.3 and corollary 1 imply togehter with their assumptions and together with the inequality F ĥ ρ(t) (χ t+h , Θ t+h , J t+h ) ≤ F (χ t , ∇Θ t , ρ(t)) the main theorem for the time-discrete system, see Korollar 2.2.3, p.56 and Korollar 2.2.4, p.63 in [2]. It deals mainly with estimates on the time-discrete solutions.
where for t ≥ 0 (64) in combination with Jensen's inequality and Kolmogorov's criterion allows to prove a statement about the convergence of the Lagrangian particle densitiesρ as h → 0, see Korollar 2.2.5, p.65 in [2]. Corollary 2. Let (h k > 0) k∈N with lim k→∞ h k = 0 and let γ k be a time-discrete solution to time step h k for k ∈ N. Then to every φ ∈ C 1 0 (R 3 ) and T > 0 there is a subsequence of (γ k ) k∈N also labeled by k andρ ∈ L 2 ([0, T ] × Ω, R n ) such that there holds Note that the folding is in space and understood to be componentwise.

5.
The time-continuous model. The next step according to our strategy is to give an answer to the question: Can we find a sequence of time steps (h k ) k∈N with h k ↓ 0, k → ∞ and a corresponding sequence (γ k ) k∈N of time-discrete solutions which converges (in some sense) to a physically meaningful solution of the timecontinuous problem as k → ∞?
We restrict ourselfes to finite time intervalls [0, T ], T > 0. At first one would try to show convergence to a weak solution in the sense of Sobolev. Due to the nonlinearities of the model and the fact that the equations do not provide estimates on χ t , Θ t in time there is no hope to find a weak solution like that.
An inspiration to get a useful notation of solution gives the original Young measure theorem, see [31]: Theorem 5.1. Let I ⊂ R be an interval, p ∈ N and let (u k : I → R p ) k∈N be a sequence of measurable mappings with sup k∈N sup τ ∈I |u k (τ )| < ∞. Then there is a subsequence (u k l : I → R p ) l∈N and for almost all τ ∈ I there is a probability measure P τ on R p such that for all f ∈ C(R p ) and ψ ∈ L 1 (I) there holds If we consider the time-discrete solutions as mappings γ h : [0, T ] → Π e , see defintion 4.2, having in mind the estimates of theorem 4.4 and suppose that there is a generalisation of theorem 5.1 to a function space X with Π e ⊂ X then by (65) it would be natural to look for a formulation of our time-continuous model in terms of functionals according to the right-hand side of (65) where in (66) R p has to be replaced by X. This is the underlying idea of the following notion of a measure-valued solution, see Definition 3.1.1, p.71 in [2].
A weak formulation of our problem belongs to the following class of weak diffusion problems.
Definition 5.2. Let T > 0, X, I, J be sets, (N t ) t∈[0,T ] be a family of subsets of X and (D i ) i∈I , (F i ) i∈I , (C j ) j∈J , be families of functionals on [0, T ] × X with values in [−∞, ∞]. By a weak diffusion problem with respect to T , X, (N t ) t∈[0,T ] , (D i ) i∈I , (F i ) i∈I , (C j ) j∈J we understand the following task.
Find a mapping γ : [0, T ] → X such that for all ϑ ∈ C ∞ 0 (]0, T [) and all t ∈ [0, T ] there holds T 0θ (τ )D i (τ, γ(τ )) + ϑ(τ )F i (τ, γ(τ )) dτ = 0 for all i ∈ I, (67) is an abstract diffusion equation, (68) can be regarded as a coupling of the different (physical) quantities represented by γ and (69) is an abstract formulation of the side conditions. "Averaging over the values of γ(t)" leads to the notion of a measure-valued solution for weak diffusion problems.. Definition 5.3. We say there is a measure-valued solution to (67) -(69), if there is an σ−Algebra Σ on X and to almost every t ∈ [0, T ] a probability measure P t on Σ such that for all ϑ ∈ C ∞ 0 (]0, T [) and almost all t ∈ [0, T ] there holds We will denote a measure-valued solution by (X, Σ, P t ) t∈[0,T ] .
As mentioned below theorem 5.1 our hope to find a measure-valued solution to our problem is based on a generalisation of Young's theorem to function spaces X with Π e ⊂ X. Fortunately this is possible as the following theorem and its corollary shows, see Theorem 3.2.8, p.105 and Theorem 3.2.11 p.106 in [2].
, and (γ k : [0, T ] → X) k∈N be a sequence of mappings. If there are monoton increasing sequences of compact sets (K 1λ ⊂⊂ (X 1 , · X1 )) λ∈N , (K 2λ ⊂⊂ (X 2 , w X2 )) λ∈N such that there holds then there exists a subsequence (γ k l ) l∈N and a mapping P : [0, T ] → R(X) with P t ≥ 0, P t (X) = 1 for almost all t ∈ [0, T ] and  Additional to the assumptions stated above let there exist a q ≥ 0 such that for all k, λ ∈ N (73) Let f be bounded from below and let f be either lower sequentially semicontinuous or lower sequentially semicontinuous with respect to the second argument and satisfy a uniform continuity in time, i.e. for any λ ∈ N and given > 0 there exists a δ(λ, ) If one of these two conditions is met, it follows A sequentially continuous function f that is not necessarily bounded from below and satisfies (74) fulfils 2. For our model we need only the case where ν k = λ T , k ∈ N. 3. A semi-constructive method to compute the measures P t by means of cylindric functions is given in section 3.2 in [2].
Theorem 5.4, corollary 3 and the estimates stated in theorem 4.4 restrict the choice of the function space X. On one hand one has to guarantee the existence of a sequence of compact subsets such that (70) holds and on the other hand the functionals that corresponds to D i , F i , C j in definition 5.2 have to be (lower semi-)continuous and to fulfil growth conditions like (74). Obviously these are contrarily requirements and so it turns out that 5.4 and corollary 3 are not directly applicable to our system. The reason lies in the fact that we cannot prove the weak sequential continuity of the functional which corresponds to the continuity resp. diffusion equation (13), because there enters a term cof∇Φ(J •Φ), see p.119 in [2]. This is due to the fact that we can find only sequences of weakly compact sets in W 1,a (Ω, R 3 ) resp. in L 2 (Ω, R 3·n ) for the mappings Θ k resp. J k , k ∈ N, which correspond to (K 2λ ) λ∈N in theorem 5.4. Now one can try to use the Lagrangian picture instead of the Eulerian one, i.e. one uses for t ≥ 0 the quantities Φ t and where the Langrangian volume energy density is given bŷ Note that the transformation f →f preserves all presumed properties of f. In this picture the continuity equation reads as ∂ tρ (t) = divĴ t but because of the integrability and differentiability properties of the Φ t we cannot prove existence of the corresponding compact sets with (70). For instance there is no integral estimate for t →Ĵ kt like (60) andχ t loses the BV-property.
A way to overcome these problems is to use a mixture of the Langrangian and the Eulerian description and splitting up the model equations by introducing extra variables. In addition to the Langrangian variables we use the following Eulerian and extra variables for t ≥ 0: χ t , Θ t , J t , µ t , J inft , µ inft and Remark 6. We need the quantities indicated by inf to prove an energy inequality, see (64). Now we model the time-continous problem by this quantities and define functionals according to definition (5.2) to give a weak formulation.
To give a formulation in the sense of definition 5.2 of the above stated model we use List of symbols. Let p, q ∈ N and r ∈ N ∪ {∞}.
euclidian scalar product Id Ω Identity on Ω r ≤ s means s − r ∈ R q + for r, s ∈ R q r < s means s − r ∈ R q + \ {0} for r, s ∈ R q |r| 1 q k=1 |r i | for r ∈ R q |A| spectral norm of A ∈ R q×q TrA q k=1 A kk for A ∈ R q×q · q q-norm on some given L q space · p,q Sobolev-norm on some given W p,q space · BV natural norm on BV(Ω, R m ) C r 0 (U ) set of all C r -functions with compact support in the open set U ⊂ R q X denotes weak convergence in the Banach space X * denotes convergence in the sense of Radon measures X * norm-dual of the Banach space X w X weak topology on the Banach space X w q weak topology on some given L q space R(X) space of Radon measures on the topological space X C b (Y ) space of continuous and bounded functions on the topological space Y B(Y, ω) Borel σ-algebra on the topological space (Y, ω) ri(V ) relative interior of V ⊂ R q