A short physical derivation of the generalised model is given and global existence and unique- ness of the solution are shown under suitable growth conditions on the non- linearity

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MULTI-COMPONENT ALLEN-CAHN EQUATION FOR ELASTICALLY STRESSED SOLIDS

THOMAS BLESGEN, ULRICH WEIKARD

Abstract. The vector-valued Allen-Cahn equations are combined with elas- ticity where a linear stress-strain relationship is assumed. A short physical derivation of the generalised model is given and global existence and unique- ness of the solution are shown under suitable growth conditions on the non- linearity.

1. Introduction

The Allen-Cahn equation, introduced in [2], provides a well-established frame- work for the mathematical description to free boundary problems for phase transi- tions. Unlike sharp interface models, it postulates a diffuse interface with a small thicknessγ >0. The equations have been the subject of intense mathematical in- vestigations, see for instance [27, 5, 1, 9, 14, 15]; and adequate numerical methods have been developed for their solution, see [25, 23], that contain also references to other numerical work.

The Allen-Cahn equation has been generalised in many directions, see [10, 24]

for a generalisation to the phase field equations; [21], where also the vector-valued system of Allen-Cahn equations is derived; [6], where a statistical framework is considered, and finally [27] for a mixed Allen-Cahn/Cahn-Hilliard formulation.

The physical applications of the Allen-Cahn system are numerous. An overview over the Allen-Cahn and phase field equations is [11], in [20] an overview over the Cahn-Hilliard equation with elasticity is found. Furthermore we mention [26] and [22] with applications to dislocations and lattice instabilities, [3], where droplet motion is described, [5] for the study of travelling waves, [28] for applications to crystallisation, and [7] for diffusion induced segregation phenomena.

In this article we consider a generalisation of the vector-valued system of Allen- Cahn equations to linear elasticity. To this end we will first give a short physical derivation of the complete model, then show existence and uniqueness of a solution to the generalised system.

The existence proof can be roughly split into two parts. Part I, presented in sections 2 to 4, treats the case of polynomial free energy densities that fulfill the mild growth conditions stated in Section 2.3. The second part, starting in Section 6,

2000Mathematics Subject Classification. 35K15, 74B10, 74N20.

Key words and phrases. Linear elasticity; modeling and simulation of multiscale problems.

c

2005 Texas State University - San Marcos.

Submitted November 18, 2004. Published August 15, 2005.

Supported by the German Research Community DFG within priority program 1095.

1

treats the physically relevant case of logarithmic free energy densities and makes use of the results shown in Part I. The employed mathematical methods consist in starting from the time-discrete formulation and finding suitable uniform estimates independent of the time stepτ >0 which by well-known compactness results allow to pass to the limit τ → 0. A former version of this argument can be found in [4] and is classical by now. Our approach will follow closely [17, 18, 19] where the elastic Cahn-Hilliard model is treated.

1.1. Derivation of the Model. Let Ω ⊂ R^D, 1 ≤ D ≤ 3 be a bounded do- main with Lipschitz boundary. We introduce the vector u:= (u₁, . . . , u_n) of non- conserved order parameters. Depending on the physical context, u_i can be either the concentration, or the density, or the volume fraction of thei-th phase.

These quantities fulfill for 1≤i≤n ui≥0, ui ∈H^1,2(Ω),

n

X

i=1

ui= 1.

ByH^m,2(Ω) we denote the Sobolev space ofm-times weakly differentiable functions in the Hilbert space L²(Ω), by H₀^m,2(Ω) the closure of C₀^∞(Ω) w.r.t. k · k_H1,2(Ω). By k · k_H1 we always mean k · k_H1,2(Ω). C₀^∞(Ω) := ∩^∞_m=0C₀^m(Ω) where C₀^m(Ω) is the space ofm-times continuously differentiable functions over Ω with compact support.

In order to describe elastic effects we consider the displacement fieldv(x) which describes the position of a material point xin the undeformed body after defor- mation. We assume that the displacement gradient is small, such that the strain tensor can be approximated by

E =E(v) =Eij(v) := 1

2 ∂_iv_j+∂_jv_i .

We postulate that the system free energy is of the generalised Landau-Ginzburg form

F(u(t),E(v(t))) =F^out(E(v(t)))+

Z

Ω

γ² 2

n

X

i=1

|∇ui(x, t)|²+f(u(x, t),E(v(x, t))) dx.

(1.1) In this formulation, the first term represents energy effects due to applied outer forces,

F^out(E(v)) :=

Z

Ω

W(E(v)).

We assume that there are no external body forces and that the tractions applied to

∂Ω are dead loads and equalS~n, where~nis the unit outer normal to∂Ω. We assume that the symmetric tensorS defined by this property is constant, i.e. independent of time t. The work necessary to transform the undeformed body into the state with corresponding displacement vectorv(t) is therefore

− Z

∂Ω

v·S~n=− Z

Ω

∇v:S=− Z

Ω

E(v) :S

and we find thatW(E(v)) :=−E(v) :Sdescribes the energy density of the applied outer forces.

The term ^γ₂²Pn

i=1|∇ui|² in (1.1) represents the interfacial energy of the tran- sition layers. Here we assume for simplicity that the contributions enter with the same weight for every interface between any two phases.

The last termf(u,E(v)) in (1.1) represents the free energy density. The compu- tations in this article are based on the equality

f(u,E(v)) = ¯f(u) +W^el(u,E(v))

with suitable structure and growth conditions on ¯f, see Section 2.3. W^el is the contribution of the elastic energy to f. It was first studied by Eshelby, [16]. By Hooke’s law, a possible ansatz forW^el is

W^el(u,E) := 1

2(E −ε(u)) :C(u)(E −ε(u)). (1.2) We assume the linear relationship (Vegard’s law)

ε(u) :=

n

X

i=1

u_iε_i, (1.3)

where εi := ε(ei) and ei is the i-th basis vector of Rⁿ. This means ε(ei) is the eigenstrain when the system is equal to the ith pure component. C(u) is the elasticity tensor that maps symmetric tensors in R^D×D onto itself. We assume that C is symmetric and positive definite. Instead of (1.2) other forms ofW^el are permitted as long as Assumption (A4) in Section 2.3 remains valid.

We define the time evolution of the unconserved order parameteruas gradient flow of the free energy,

Z

Ω

∂tu=−δ

δuF(u(t),E(v(t))).

Thus for large timet,u(t) tends to a local minimiser ofF.

The mechanical equilibrium is attained on a much faster time scale than the time scale significant for diffusion. Therefore we will assume a quasi-static elastic equilibrium, i.e. the displacementv is obtained by solving the elliptic equation

div(S) = 0 in Ω with the stress tensor

S:=∂εW^el(u,E(v)).

Hence, for a given stop timeT >0 we end up with the following model:

Find for t≥0 a solution pair(u, v)such that inΩT := Ω×(0, T)

∂tu=γ²4u−P(∂uf(u,E(v))), (1.4)

div(S) = 0, (1.5)

S=∂εW^el(u,E(v)), (1.6) with the initial data fort= 0 in Ω

u(·,0) =u0(·) (1.7)

and the boundary conditions fort >0 in∂Ω

u=ud, S·~n=S·~n. (1.8) The projection operatorP in (1.4) is due to algebraic constraints on∂_uf(u,E(v)).

This is explained in the subsequent section.

The boundary conditionS·~n=S·~non∂Ω determinesvonly up to infinitesimal rigid displacements (these are translations and infinitesimal rotations). This fact is well-known for formulations that depend on a linearised strain tensor E. The resulting non-uniqueness in v is of no importance as v only enters through the symmetric termE(v).

2. Preliminaries to existence theory

In this section we discuss the existence theory to the sharp interface model (1.4)- (1.8). We will show that under suitable growth conditions on the free energy density, stated for polynomial f in Section 2.3 and for logarithmic energies in Section 6, discrete solutions to the implicit time discretisation exist. A-priori estimates allow to pass to the limit showing the existence of solutions to the model first with polynomial free energy. This result is then used to generalise to logarithmic free energies.

We will carry out the proof for classical Dirichlet boundary data, i.e. set w.l.o.g.

u_d= 0 in (1.8). Other boundary conditions are shortly discussed in the remark at the end of this section. We begin by collecting general properties of the model and necessary tools that will be needed in the sequel.

The vector of order parameters lies inside the simplex Σ, u∈Σ :=

u⁰ = (u⁰₁, . . . , u⁰_n)∈Rⁿ :

n

X

i=1

u⁰_i = 1 . (2.1) Notice that the condition 0 ≤ ui ≤ 1 in Ω may be violated for polynomial free energies considered in the first part of this section.

If we write (1.4) as ∂tu=w, as a consequence of (2.1), w fulfillsPn

i=1wi = 0.

Thus, withe:= (1, . . . ,1)∈Rⁿ, the right hand sidewsatisfiesw=P(z) for some z∈Rⁿ, where

P(z) :=z− 1 n(z·e)e is the projection ofRⁿ to

TΣ :=n

u⁰ = (u⁰₁, . . . , u⁰_n)∈Rⁿ :

n

X

i=1

u⁰_i = 0o , the tangent space to Σ. Let

X1:={u⁰∈H₀¹(Ω;Rⁿ) :u⁰ ∈Σ almost everywhere in Ω}, X2:={v⁰∈H¹(Ω,R^D) : (v⁰, w)_H1 = 0 for allw∈Xird}, where

Xird={v∈H¹(Ω,R^D) : there existb∈R^D, A∈R^D×D such thatv(x) =Ax+b}

is the space of all infinitesimal rigid displacements.

Since we have (classical) Dirichlet boundary conditions for the equations of con- servation of mass, we consider the space of test functions

Y :=H₀^1,2(Ω;Rⁿ) and its dual

(H₀^1,2(Ω;Rⁿ))⁰=H^−1,2(Ω;Rⁿ).

Remark: If we replace the Dirichlet conditions for u by a Neumann boundary condition or periodic boundary conditions, a (generalised) Poincar´e inequality holds inH^1,2(Ω) and all the results found below continue to hold.

2.1. The weak formulation. A pair (u, v)∈L²(0, T;H₀^1,2(Ω;Rⁿ))×L²(0, T;X2) is called aweak solution of(1.4)-(1.8) if

− Z

ΩT

∂_tξ·(u−u₀) +γ² Z

ΩT

∇u:∇ξ+ Z

ΩT

P(∂_uf(u,E(v)))·ξ= 0 (2.2) for allξ∈L²(0, T;H₀¹(Ω;Rⁿ))∩L^∞(ΩT; Rⁿ) with∂tξ∈L²(ΩT),ξ(T) = 0, and

Z

Ω_T

W^el(u,E(v)) :∇ζ= Z

Ω_T

S:∇ζ. (2.3)

for allζ∈L²(0, T; H¹(Ω,R^D)).

2.2. The implicit time discretisation. We fix an M ∈N and seth:= _M^T. For m≥1 and givenu^m−1∈X₁ consider

u^m−u^m−1

h =γ²4u−P(∂uf(u^m,E(v^m))), (2.4)

div(S^m) = 0, (2.5)

S^m=∂εW^el(u^m,E(v^m)). (2.6) 2.3. Structural Assumptions. To establish the existence of weak solutions in the sense of Section 2.1, the following assumptions are made:

(A1) Ω⊂R^D is a bounded domain with Lipschitz boundary.

(A2) The free energy densityf can be written as

f(u⁰,E(v⁰)) =f¹(u⁰) +f²(u⁰) +W^el(u⁰,E(v⁰)) for allu⁰∈Rⁿ, v⁰∈R^D withf¹∈C¹(Rⁿ;R) and convex. Additionally we postulate

(A2.1) f¹≥0.

(A2.2) For allδ >0 there exists a constantCδ >0 such that

|∂uf¹(u⁰)| ≤δf¹(u⁰) +Cδ for allu⁰ ∈Σ.

(A2.3) There exists a constantC₁>0 such that

|∂uf²(u⁰)| ≤C₁(|u⁰|+ 1) for allu⁰∈Σ.

(A3) The initial datum u0 fulfillsf(u0,E(v0))<∞, where v0 is the solution of (2.3).

(A4) The elastic energy densityW^el∈C¹(Rⁿ×R^D×D;R) satisfies

(A4.1)W^el(u⁰,E⁰) only depends on the symmetric part ofE⁰ ∈R^D×D, i.e.

W^el(u⁰,E⁰) =W^el(u⁰,(E⁰)^t) for allu⁰∈Rⁿ andE⁰∈R^D×D.

(A4.2) ∂_εW^el(u⁰,·) is strongly monotone uniformly inu⁰, i.e. there exists ac₁>0 such that for all symmetricE₁⁰,E₂⁰ ∈R^D×D,

(∂εW^el(u⁰,E₂⁰)−∂εW^el(u⁰,E₁⁰)) : (E₂⁰ − E₁⁰)≥c1|E₂⁰ − E₁⁰|².

(A4.3) There exists a constantC1>0 such that for all u⁰ ∈Σ and all symmetric E⁰ ∈R^D×D,

|W^el(u⁰,E⁰)| ≤C₁(|E⁰|²+|u⁰|²+ 1),

|∂uW^el(u⁰,E⁰)| ≤C1(|E⁰|²+|u⁰|²+ 1),

|∂εW^el(u⁰,E⁰)| ≤C₁(|E⁰|+|u⁰|+ 1).

(A5) The energy density of the applied outer forces is given byW(E⁰) =−E⁰ :S whereS is a symmetric constant tensor.

For the rest of this article, we assume without further stating that the assumptions (A1)-(A5) hold.

3. Existence of solutions to the time discrete scheme

For each time step m ≥ 1 in the implicit time discretisation (2.4)-(2.6), given time step sizeh >0, and givenu^m−1∈X1 we define the discrete energy functional

F^m,h(u⁰, v⁰) :=F(u⁰,E(v⁰)) + 1

2hku⁰−u^m−1k²_L2.

Lemma 3.1 (Existence of a minimiser). For givenu^m−1∈X1 and any h >0 the functionalF^m,h possesses a minimiser (u^m, v^m)in X1×X2.

Proof. The proof is an application of the direct method in the calculus of variations.

Combined, (A4.2), (A4.3) imply thatW^el(u⁰,E⁰)≥C(|E⁰|²−|u⁰|²)−Cfor a constant C >0. With Korn’s inequality, see for instance [12], this guarantees the coercivity ofF with respect tov∈X2. Similarly, the termγ²R

Ω

Pn

i=1|∇ui|²in the definition of F guarantees with the Poincar´e inequality the coercivity of F w.r.t. u ∈ X1. Using (A2) on f¹ and f² we thus find that the functional F^m,h is weakly lower semicontinuous and coercive inX₁×X₂ and hence possesses a minimiser.

The following lemma shows that the energy functional F^m,h is the correct one and corresponds to the implicit time discretisation (2.4)-(2.6).

Lemma 3.2 (Euler-Lagrange equations). The minimiser (u^m, v^m)∈ X₁×X₂ of F^m,h fulfills

Z

Ω

u^m−u^m−1 h ·ξ+

Z

Ω

γ²∇u^m:∇ξ+ Z

Ω

P(∂uf(u^m,E(v^m)))·ξ= 0 for allξ∈Y ∩L^∞(Ω;Rⁿ),

(3.1) Z

Ω

∂εW^el(u^m,E(v^m)) :∇ζ= Z

Ω

S:∇ζ for allζ∈H¹(Ω,R^D). (3.2) Proof. We choose directions ξ ∈ Y ∩L^∞(Ω;Rⁿ) with Pn

i=1ξ_i = 0, ζ ∈ X₂ ∩ L^∞(Ω;R^D) and determine variations ofF^m,h(u, v) with respect touandv forξ, ζ.

The variation w.r.t. uis

s→0lim

(F^m,h(u^m+sξ, v^m)−F^m,h(u^m, v^m))s⁻¹

. (3.3)

Sincef¹ is convex, we have

f¹(u^m)≥f¹(u^m+sξ)−s∂_uf¹(u^m+sζ)·ξ.

This implies

f¹(u^m+sξ)≤f¹(u^m) +|s∂uf¹(u^m+sξ)| kξkL^∞

≤f¹(u^m) +|s|f¹(u^m+sξ)kξkL^∞+C|s|.

The last is by Assumption (A2.2) withδ= 1. Hence, for s small enough, we find

f¹(u^m+sξ)−f¹(u^m) s

≤C(f¹(u^m) + 1).

Lebesgue’s dominated convergence theorem and Assumption (A2.3) imply

s→0lim 1 s

Z

Ω

(f¹+f²)(u^m+sξ)−(f¹+f²)(u^m)

= Z

Ω

(∂_uf¹+∂_uf²)(u^m)·ξ.

With the help of (A4.3) we find

s→0lim Z

Ω

s⁻¹

W^el(u^m+sξ,E(v^m+sζ))−W^el(u^m,E(v^m))

= Z

Ω

∂uW^el(u^m,E(v^m))ξ+∂εW^el(u^m,E(v^m)) :∇ζ . The variation of the quadratic formu7→ _2h¹ ku^m−u^m−1k²_L2 yields

s→0lim

s⁻¹(2h)⁻¹ ku^m+sξ−u^m−1k²_L2− ku^m−u^m−1k²_L2

= u^m−u^m−1 h , ξ

L². Taking into account that u^m−u^m−1 as well as∇u^m : ∇ξ for every ξ lie on TΣ, this finally yields (3.1). To derive (3.2) we varyF^m,h with respect tov. From the

symmetry of∂_εW^el andS we find (3.2).

3.1. Uniform estimates. In the preceding section we proved the existence of a discrete solution (u^m, v^m) for 1 ≤m ≤M and arbitraryM ∈ N. We define the piecewise constant extension (uM, vM) of (u^m, v^m)_1≤m≤M by

(uM(t), vM(t)) := (u^m_M, v_M^m) := (u^m, v^m) fort∈((m−1)h, mh]

withu_M(0) =u₀, andv_M(0) given by Equation (2.6).

The piecewise linear extension (u_M, v_M) for t= (βm+ (1−β)(m−1))hwith appropriateβ∈[0,1] is given by the interpolation

(uM, vM)(t) :=β(u^m_M, v_M^m) + (1−β)(u^m−1_M , v_M^m−1).

Lemma 3.3 (A-priori estimates). The following a-priori estimates are valid.

(a) For allM ∈Nand allt∈[0, T]we have the dissipation inequality F(u_M,E(v_M))(t) +1

2 Z

Ωt

|∂_tu_M|²≤F(u₀,E(v₀)).

(b) There exists a constantC >0 such that sup

0≤t≤T

nkuM(t)k_H1+kvM(t)k_H1

o≤C, (3.4)

sup

0≤t≤T

Z

Ω

f¹(u_M(t)) +k∂tu_MkL²(Ω_T)≤C. (3.5)

Proof. Since (u^m, v^m) is a minimiser of F^m,h, it holds for everym≥1 F(u^m,E(v^m)) + 1

2hku^m−u^m−1k²_L2≤F(u^m−1,E(v^m−1)). (3.6) After writingu^m−u^m−1 as a time derivative, iterating (3.6) yields

F(u^m_M,E(v_M^m)) +1 2

Z mh

0

k∂tuMk²_L2 dτ ≤F(u0,E(v0)).

Using the assumptions (A2)-(A4) and with the help of the inequalities of Poincar´e

and Korn, this proves the lemma.

For the linear interpolationuM ofu^m_M, the Euler-Lagrange equation (3.1) can be rewritten as

Z

Ω

∂tuM(t)·ξ+ Z

Ω

γ²∇uM(t) :∇ξ+ Z

Ω

P(∂uf(uM(t),E(vM(t))))·ξ= 0 (3.7) for all ξ ∈Y ∩L^∞(Ω;Rⁿ), which holds for almost all t ∈(0, T). Equation (3.7) controls the variation of uM in time and, together with the uniform estimates of Lemma 3.3, allows to show compactness in time.

The following theorem is the first main result as it can also be used to proof convergence of numerical solution schemes. In the next part we will show that this limit is in fact a solution to (1.4)-(1.8).

Theorem 3.4 (Compactness of (uM, vM)). There exists a constant C >0 such that for all t1, t2∈[0, T]

ku_M(t₂)−u_M(t₁)k_L2 ≤C|t₂−t₁|^1/4.

Furthermore, there are subsequences (u_M)_M∈N and (v_M)_M∈N with N ⊂ N and there are u∈L^∞(0, T;H₀¹(Ω)) andv∈L^∞(0, T;H¹(Ω)) such that

(i) u_M → u in C^0,α([0, T]; L²(Ω;Rⁿ))for allα∈(0,¹₄), (ii) u_M → u in L^∞(0, T;L²(Ω;Rⁿ)),

(iii) uM → u almost everywhere in ΩT, (iv) u_M * u^∗ in L^∞(0, T;H₀¹(Ω;Rⁿ)), (v) v_M → v in L²(0, T;H¹(Ω)), (vi) ∂_uf^k(u_M)→ ∂_uf^k(u) in L¹(Ω_T)fork= 1,2 asM ∈ N tends to infinity.

Proof. For chosen constantL >0 let PL(u⁰) :=

(u⁰ if|u⁰| ≤L,

u⁰

|u⁰|L if|u⁰|> L. (3.8) In (3.7) we test withξ:=PL(uM(t2)−uM(t1)), where t1, t2∈[0, T] with t1 < t2. After integration in time fromt1 tot2we obtain

ku_M(t₂)−u_M(t₁)k²_L2+ Z t2

t1

Z

Ω

γ²∇u_M(t) :∇(u_M(t₂)−u_M(t₁))dt +

Z t2

t₁

Z

Ω

P(∂_uf(u_M(t),E(v_M(t))))P_L(u_M(t₂)−u_M(t₁))dt= 0.

Theu^m_M are uniformly bounded inH¹(Ω;Rⁿ), therefore the linear interpolantsuM

are uniformly bounded inL^∞(0, T;H¹(Ω;Rⁿ)). Thus we obtain kuM(t2)−uM(t1)k²_L2

≤CkuMk_L^∞_(H1)

Z t₂

t1

γ²k∇uM(t)k_L2+k(∂uf¹+∂uf²)(uM(t))k_L2

dt +CkPLuMk_L∞(Ω_T)

Z t2

t1

k∂uW^el(uM(t), vM(t))k_L1 dt

≤CkuMkL^∞(H¹)(t₂−t₁)^1/2

k∇ukL²(Ω_T)+k(∂uf¹+∂_uf²)(u)kL²(Ω_T)

+CkPLuMkL^∞(Ω_T)(t2−t1)k∂uW^el(u,v)kL^∞(L¹).

Employing the a-priori estimate (3.4) and with the help of (A2.2), (A2.3) and (A4.3) we have proved

kuM(t2)−uM(t1)k_L2 ≤C|t2−t1|^1/4 for allt1, t2∈[0, T] for a positive constantC. This is the equicontinuity of (uM)M∈N.

The boundedness of (uM) inL^∞(0, T; H¹(Ω)) together with the fact thatH¹is compactly embedded inL² yields with the Arzel`a-Ascoli theorem statement (i).

The claims (ii), (iii) and (iv) are shown as follows. Choose for t∈[0, T] values m ∈ {1, . . . , M} andβ ∈ [0,1] such that t = (βm+ (1−β)(m−1))h. From the definition ofuwe get at once

kuM(t)−u_M(t)kL² =kβu^m_M+ (1−β)u^m−1_M −u^m_MkL²

= (1−β)ku^m_M −u^m−1_M kL² ≤Ch^1/4.

This tends to zero as M becomes infinite. With the help of (i), this proves (ii).

Since for a subsequence we have convergence almost everywhere, (iii) is proved, too.

Claim (iv) is a direct consequence of Estimate (3.4) which gives the boundedness ofu_M inL^∞(0, T;H¹(Ω;Rⁿ)).

The proof of (v) is contained in [17, Lemma 3.5].

To prove (vi), we first notice that by Assumption (A2), ∂_uf¹ is a continuous function. Hence, by (iii),

∂_uf¹(u_M)→∂_uf¹(u) almost everywhere in Ω_T.

The growth condition of Assumption (A2.2) on f¹ now yields that for arbitrary δ >0 and all measurable E⊂Ω

Z

E

|∂uf¹(uM)| ≤δ Z

E

f¹(uM) +Cδ|E| ≤δC+Cδ|E|.

Therefore,R

E|∂uf¹(uM)| →0 as|E| →0 uniformly in M and by Vitali’s theorem we find∂uf¹(uM)→∂uf¹(u) inL¹(ΩT) asM ∈ N tends to infinity.

Assumption (A2.3) yields with Lebesgue’s dominated convergence theorem ac- cordingly

∂uf²(uM)→∂uf²(u).

4. Global existence of solutions I

Theorem 4.1 (Global existence of solutions for polynomial free energy). Let the assumptions of Section 2.3 hold. Then there exists a weak solution (u, v)of (1.4)- (1.8) in the sense of Section 2.1 such that

(i) u∈C^0,14([0, T]; L²(Ω;Rⁿ)), (ii) ∂tu∈L²(ΩT; Rⁿ),

(iii) v∈L²(0, T;H¹(Ω)).

Proof. We are going to prove that (u, v) introduced in Theorem 3.4 is the desired weak solution in the sense of Section 2.1. From Equation (3.7) we learn

− Z

Ω_T

∂_tξ·(u_M−u₀) + Z

Ω_T

γ²∇uM :∇ξ+ Z

Ω_T

P(∂_uf(u_M,E(vM)))·ξ= 0 for all ξ ∈ L²(0, T; Y) with ∂_tξ ∈ L²(Ω_T) and ξ(T) = 0. In this equation we pass to the limitM → ∞ and exploit Theorem 3.4. The convergence of the linear expressions is clear. The convergence

Z

ΩT

∂uf(uM,E(vM))·ξ→ Z

ΩT

∂uf(u,E(v))·ξ

follows similar to the proof of Theorem 3.4 with Vitali’s theorem by using the growth condition (A2.2) onf¹, (A2.3), Estimate (3.5), the almost everywhere convergence ofuM and the boundedness ofξ. The generalised Lebesgue convergence theorem, the growth condition (A4.3), and the strong convergence of∇vM anduM inL²(Ω) yield that we can pass to the limit inR

Ω∂uW^el(uM,E(vM))·ξ. This implies (2.2).

Similarly we can pass to the limit in (3.2) and obtain (2.3). This is done in the same way as before by using once more growth condition (A4.3) and the strong

convergence of∇vM anduM in L²(Ω).

5. Uniqueness of the solution

We show uniqueness of a solution to (1.4)-(1.6) under the simplifying assumption that

W^el(u⁰,E⁰) =1

2(E⁰−ε(u⁰)) :C(E⁰−ε(u⁰)), (5.1) with a symmetric constant positive definite tensor C and with ε(u⁰) defined by (1.3).

The proof of the following theorem is straightforward and uses an integration in time method and a Gronwall argument.

Theorem 5.1(Uniqueness of solutions to the elastic Allen-Cahn system). LetW^el be given by (5.1). Then the solution pair(u, v)obtained in Theorem 4.1 is unique in the spaces stated in this theorem.

Proof. If there are two pairs of solutions (u¹, v¹), (u², v²) to the equations (1.4)- (1.6), it holds fork= 1,2

∂tu^k =γ²4u^k−P(∂uf¹(u^k) +∂uf²(u^k))−P((εi:C(ε(v^k)−ε(u^k)))_1≤i≤n), 0 = div(C(E(v^k)−ε(u^k))).

(5.2)

Letu:=u²−u¹ andv:=v²−v¹. Then (u, v) solves the weak equation Z

Ω_T

∂_tu·ξ=− Z

Ω_T

γ²∇u:∇ξ− Z

Ω_T

(∂_u(f¹+f²)(u²)−∂_u(f¹+f²)(u¹))·P ξ

− Z

Ω_T

(εi:C(E(v)−ε(u)))_1≤i≤n·P ξ

(5.3) for every ξ ∈ L²(0, T; Y)∩L^∞(Ω_T;Rⁿ) with ∂_tξ ∈ L²(Ω_T) and ξ(T) = 0. Let t₀∈(0, T). We chooseP_L(u²−u¹)X(0,t₀) as a test function in the difference of the weak formulations of (5.2), where L > 0 and P_L(u) is defined as in (3.8). In the limitL→ ∞the terms withP_L(u) are replaced by uand we find

Z

Ωt0

C(E(v)−ε(u)) :E(v) = 0. (5.4) Similarly we chooseξ:=PL(u²−u¹)X_(0,t0)as test function in (5.3) and in the limit L→ ∞we obtain with the help of (5.4)

1 2

Z

Ωt0

d

dt|u|²=− Z

Ωt0

γ²∇u:∇u− Z

Ωt0

(E(v)−ε(u)) :C(E(v)−ε(u))

− Z

Ω_t0

(∂u(f¹+f²)(u²)−∂u(f¹+f²)(u¹))·(u²−u¹).

The convexity off¹ yields

∂u(f¹(u²)−f¹(u¹))·(u²−u¹)≥0 and due tou(t= 0) = 0 we end up with

1 2

Z

Ω_t0

d

dt|u|²= 1

2ku(t0)k²_L2≤ Z

Ω

(∂uf²(u²)−∂uf²(u¹))·u.

With Gronwall’s inequality, asf²is Lipschitz continuous, and sincet0was arbitrary, we findu≡0 in ΩT which leads to

Z

Ω_T

E(v) :CE(v) = 0.

With Korn’s inequality this yieldsv≡0 in the whole of ΩT. 6. Logarithmic free energy

In the upcoming three sections we are going to extend Theorem 4.1 to logarithmic free energies. The results will in particular be valid for the free energy functional,

f(u⁰,E(v⁰)) =k_Bθ

n

X

j=1

u⁰_jlnu⁰_j+1

2u⁰·Au⁰+W^el(u⁰,E(v⁰)) (6.1) whereθ denotes the (fixed) temperature andk_B the Boltzmann constant. We will exploit this particular structure off in the sequel.

As is well known the mathematical discussion is much more subtle, f becomes singular as oneuj approaches 0. To show that 0< uj<1 for everyj, we approxi- mate f forδ >0 by somef^δ that fulfills the requirements of Section 2.3 and find suitable a-priori estimates that allow to pass to the limitδ→0.

Despite of the mathematical difficulties, the logarithmic free energy guarantees that the vectoruof order parameters lies in the transformed Gibbs simplex

G:= Σ∩

u⁰∈Rⁿ:u⁰_j≥0 for 1≤j≤nand

n

X

i=1

u⁰_i= 1 and is therefore physically meaningful.

The assumptions (A2) and (A3) of Section 2.3 are replaced by the following assumptions:

(A2’) f is of the form (6.1), where A ∈ R^n×n is a symmetric positive definite matrix andθ >0 the constant temperature.

(A3’) The initial valueu0= (u01, . . . , u0n)∈X1fulfillsu0∈Galmost everywhere and

Z

Ω

u0j>0 for 1≤j≤n.

The other assumptions are unchanged and continue to hold.

To proceed, we define for d ∈ R and given δ > 0 the regularised free energy functional

ψ^δ(d) :=

(dlnd ford≥δ, dlnδ−^δ₂+^d_2δ² ford < δ.

The regularised free energy functional is defined in such a way that ψ^δ ∈C² and the derivative (ψ^δ)⁰ is monotone. This definition goes back to the work [13] by Elliott and Luckhaus.

Due to Assumption (A2’), this leads to

f^δ(u,E(v⁰)) =f^1,δ(u) +f²(u) +W^el(u⁰,E(v⁰)), (6.2) f^1,δ(u⁰) :=kBθ

n

X

j=1

ψ^δ(u⁰_j), (6.3)

f²(u⁰) :=1

2u⁰·Au⁰. (6.4)

As can be easily checked,f^1,δ, f²fulfill the assumptions of Section 2.3.

6.1. Uniform estimates. The following lemma was first stated and proved in [13]

for logarithmic free energies typical for the Cahn-Hilliard system. The proof of Elliott and Luckhaus can be directly transferred to the situation considered here with the regularised free energy defined by (6.1).

Lemma 6.1 (Uniform bound from below on f^δ). There exists a δ0 > 0 and a K >0 such that for allδ∈(0, δ0)

f^1,δ(u) +f²(u)≥ −K for all u∈Σ.

Now we summarise the results for the regularised problem proved in Lemma 3.3 and Theorem 3.4. Lemma 6.2 also states the boundedness and convergence of the numerical solutions asδ&0.

Lemma 6.2 (A-priori and compactness results for regularised problem).

(a) For allδ∈(0, δ₀)there exists a weak solution(u^δ, v^δ)of (1.4)-(1.8) with a loga- rithmic free energy that satisfies (A2’), (A3’), (A4)-(A6) in the sense of Section 2.1.

(b) There exists a constantC >0independent of δ such that for allδ∈(0, δ0) sup

t∈[0,T]

ku^δ(t)kH¹+kv^δ(t)kH¹ ≤C, sup

t∈[0,T]

Z

Ω

f^1,δ(u^δ(t)) +k∂_tu^δk_L2(Ω_T)≤C,

ku^δ(t2)−u^δ(t1)kL² ≤C|t2−t1|^1/4 for allt1, t2∈[0, T].

(c) One can extract a subsequence (u^δ)_δ∈R, where R ⊂ (0, δ₀) is a countable set with zero as the only accumulation point such that

(i) u^δ→uin C^0,α([0, T]; L²(Ω;Rⁿ))for allα∈(0,¹₄), (ii) u^δ→uin L^∞(0, T;L²(Ω;Rⁿ)),

(iii) u^δ→ualmost everywhere in Ω_T, (iv) u^δ* u^∗ in L^∞(0, T;H₀¹(Ω;Rⁿ)),

(v) v^δ→v inL²(0, T;H¹(Ω)) asδ∈ R tends to zero.

Proof. Using Lemma 6.1, the regularised problem satisfies the assumptions of Sec- tion 2.3 and by Theorem 4.1, a weak solution for fixed δ ∈ (0, δ0) exists. This proves (a). Lemma 3.3 and Theorem 3.4 imply directly (b). From Lemma 3.3 it follows that F^δ(u₀,E(v0)) does not depend onδ, hence the constant on the right hand side does not depend onδ. Theorem 3.4 leads to Assertion (c).

7. Higher integrability for the logarithmic free energy Sinceϕ^δ:= (ψ^δ)⁰ will be singular asδ→0 we introduce forr >0

ϕ^δ_r(d) :=

(ϕ^δ(d)|ϕ^δ(d)|^r−1 ifϕ^δ(d)6= 0,

0 ifϕ^δ(d) = 0.

By definition,ϕ^δ_r∈C⁰(R).

For 0 < r < 1, ϕ^δ_r is not differentiable at the zero point of ϕ^δ. To overcome this difficulty, foru >0 we introduce the functionϕ^δ,%_r withϕ^δ,%_r =ϕ^δ_rin R\[0,1]

and define ϕ^δ,%_r in [0,1] such that ϕ^δ,%_r is a C¹ function, monotone increasing and ϕ^δ,%_r →ϕ^δ_r in C⁰(R) asu&0.

First we need a regularity result on the strain tensor. The following Lemma is taken from [17] where it is also proved.

Lemma 7.1(Higher integrability of the strain tensor). Suppose thatu∈L^σ(Ω,Rⁿ) for a σ > 2. Then there exists a p ∈ (2, σ] independent of u such that for all v∈H¹(Ω,Rⁿ)which fulfill for all ζ∈H¹(Ω,Rⁿ)the identity

Z

Ω

∂uW^el(u,E(v)) :∇ζ= Z

Ω

S :∇ζ the integrability property ∇u∈L^p(Ω,R^D×D) holds. In particular,

k∇vk_Lp(Ω,R^D×D)≤C k∇vk_L2(Ω,R^D×D)+kuk_Lp(Ω,Rⁿ)+ 1 independent ofu.

Even though by construction 0< uj<1 almost everywhere, it might still happen that for the limit the sets{x∈Ω|uj(x) = 0}and{x∈Ω|uj(x) = 1}have non-zero Lebesgue measure and that the entropic terms in the free energy density become singular. To show that this is not the case we need the following

Lemma 7.2(Integrability of the regularised free energy). There exists aq >1and a constant C >0 such that for all δ∈(0, δ0)

kϕ^δ(u^δ_j)k_Lq(ΩT)≤C for all1≤j≤n. (7.1) Proof. Starting point is the weak formulation (2.2)

Z

Ω_T

kBθP(ϕ^δ(u^δ_i))1≤i≤n·ξ

=− Z

ΩT

∂_tu^δ·ξ− Z

ΩT

γ²∇u^δ :∇ξ− Z

ΩT

P Au^δ·ξ− Z

ΩT

P ∂_uW^el(u^δ,E(v^δ))·ξ (7.2) which holds for all ξ ∈ L²(0, T;H₀¹(Ω;Rⁿ)) with ∂tξ ∈ L²(ΩT), ξ(T) = 0. We want to use Lemma 7.1 and notice that due to the Sobolev embedding theorem u^δ ∈ L^∞(0, T; L^s(Ω)), where s = _D−2^2D if D ≥ 3 and s ∈ [1,∞) if D = 2 and u^δ ∈L^∞(Ω_T) forD = 1. So we find∇u^δ ∈L^∞(0, T; L^p(Ω)) for some p >2. We choosepsuch thatp∈(2,4] and such thatp∈(2,_D−2^2D ) ifD≥3. This means that also test functionsξ∈L²(0, T; H¹(Ω,R^D))∩L^p−2p (Ω_T,Rⁿ) are allowed. So we can test (7.2) withξ:= [ϕ^δ,%_r (u^δ_j)]_1≤j≤n for 0< r≤1. A reformulation of the left hand side yields

kBθP(ϕ^δ(u^δ_i)_1≤i≤n)·(ϕ^δ,%_r (u^δ_i)_1≤i≤n)

=kBθ

n

X

j=1

ϕ^δ(u^δ_j)− 1 n

n

X

i=1

ϕ^δ(u^δ_i)

ϕ^δ,%_r (u^δ_j)

=k_Bθ 1 n

n

X

i,j=1

ϕ^δ(u^δ_i)−ϕ^δ(u^δ_j)

ϕ^δ,%_r (u^δ_i)

=k_Bθ 1 n

hXⁿ

i<j

ϕ^δ(u^δ_i)−ϕ^δ(u^δ_j)

ϕ^δ,%_r (u^δ_i) +

n

X

i>j

ϕ^δ(u^δ_i)−ϕ^δ(u^δ_j)

ϕ^δ,%_r (u^δ_j)i

=kBθ 1 n

n

X

i<j

ϕ^δ(u^δ_i)−ϕ^δ(u^δ_j)

ϕ^δ,%_r (u^δ_i)−ϕ^δ(u^δ_j) . Due to (ϕ^δ,%_r )⁰≥0 we furthermore find

−γ² Z

ΩT

n

X

i=1

∇u^δ_i · ∇ϕ^δ,%_r (u^δ_i)≤0.

Using the H¨older inequality, (7.2) implies kBθ1

n

n

X

i<j

ϕ^δ(u^δ_i)−ϕ^δ(u^δ_j)

ϕ^δ,%_r (u^δ_i)−ϕ^δ(u^δ_j)

≤C

k∂tu^δkL²(Ω_T)+ku^δkL²(Ω_T)

max

1≤i≤nkϕ^δ,%_r (u^δ_i)kL²(Ω_T)

+CZ

Ω_T

|∂uW^el(u^δ,E(v^δ))|^p/22_p

1≤i≤nmax Z

Ω_T

|ϕ^δ,%_r (u^δ_i)|^p−2p 1−²_p

. Now we let%&0 and employ the estimates of Lemma 6.2 and the regularity result of Lemma 7.1. With Young’s inequality we can deduce for anyα >0 the existence

of a constantCα with k_Bθ 1

n

n

X

i<j

ϕ^δ(u^δ_i)−ϕ^δ(u^δ_j)

ϕ^δ(u^δ_i)−ϕ^δ(u^δ_j)

≤α

1≤i≤nmax Z

ΩT

|ϕ^δ(u^δ_i)|^p−2p +C_α.

(7.3) A direct computation exploiting the monotonicity ofϕ^δ andϕ^δ,%_r finally yields

1 2 max

1≤i≤n|ϕ^δ(u^δ_i)|^1+r≤CX

i<j

ϕ^δ(u^δ_i)−ϕ^δ(u^δ_j)

ϕ^δ,%_r (u^δ_i)−ϕ^δ,%_r (u^δ_j) . This last result in combination with (7.3), after choosingα sufficiently small and

settingr= ^p−2_p ends the proof.

8. Global existence of solutions II

Theorem 8.1 (Global existence of solutions for logarithmic free energy). Let the assumptions of Section 6 hold. Then there exists a weak solution (u, v) in the sense of Section 2.1 of the sharp interface equations (1.4)-(1.8) with logarithmic free energy such that

(i) u∈C^0,14([0, T]; L²(Ω;Rⁿ)), (ii) ∂_tu∈L²(Ω_T; Rⁿ),

(iii) v∈L^∞(0, T; H¹(Ω,R^D)),

(iv) lnuj ∈L¹(ΩT)for1≤j ≤n and0< uj<1 almost everywhere.

Proof. We pass to the limit δ & 0 in the weak formulation (2.2), (2.3) with f defined by (6.2) and have to show that (u, v) found in Lemma 6.2 is a solution.

The limit for (2.3) can be justified in the same way as in the proof of Theorem 4.1.

It remains to control the limitδ&0 in (2.2),

− Z

ΩT

∂tξ·(u^δ−u0) +γ² Z

ΩT

∇u^δ:∇ξ+ Z

ΩT

kBθP(ϕ^δ(u^δ_i)_1≤i≤n)·ξ +

Z

Ω_T

P(Au^δ+∂_uW^el(u^δ,E(v^δ)))·ξ= 0.

The arguments of Theorem 4.1 can be reused except forkBθP(ϕ^δ(u^δ_i)1≤i≤n)·ξ.

Now we will show that ϕ^δ(u^δ_k) converges to ϕ(uk) almost everywhere in ΩT. From the almost everywhere convergence of u^δ_k to u_k, (7.1) and the Lemma of Fatou we find

Z

Ω_T

lim inf

δ&0 |ϕ^δ(u^δ_k)|^q ≤lim inf

δ&0

Z

Ω_T

|ϕ^δ(u^δ_k)| ≤C.

Next we show that

δ&0limϕ^δ(u^δ_k) =

(ϕ(uk) if lim_δ&0u^δ_k =uk∈(0,1),

∞ if lim_δ&0u^δ_k =u_k∈/(0,1) (8.1) almost everywhere in ΩT. For a point (x, t)∈ΩT with lim_δ&0u^δ_k(x, t) =uk(x, t) we obtain fromϕ^δ(d) =ϕ(d) ford≥δthatϕ^δ(u^δ(x, t))→ϕ(u(x, t)) asδ&0. In the second case of a point (x, t)∈Ω_T with lim_δ&0u^δ_k(x, t) =u_k(x, t)≤0, we have that forδsmall enough,

|ϕ^δ(u^δ_k(x, t))| ≥ϕ(max{δ, u^δ_k(x, t)})→ ∞ forδ&0.

This proves (8.1).