On a nonlocal viscous phase separation model

Mathematik

On a nonlocal viscous phase separation model

M. Hassan Farshbaf-Shaker

Preprint Nr. 23/2011

M.Hassan Farshbaf-Shaker

Abstract

A nonlocal viscous model of phase separation is presented. It is derived from a minimization of free energy containing a nonlocal part due to particle interaction.

In contrast to the classical Cahn-Hilliard theory with higher order terms this leads to an evolution system of second order parabolic equations for the particle densities, coupled by nonlocal drift and viscosity terms, which allow reasonable bounds for the concentrations. Applying fixed-point arguments and compactness results we prove the existence of variational solutions in standard Hilbert spaces for evolution systems.

Using the free energy as Lyapunov functional the asymptotic state of the system is investigated and characterized by a variational principle.

Key words. Nonlocal phase separation models; viscous phase separation models, Cahn- Hilliard equation; Integrodifferential equations ; Initial value problems; Nonlinear evolution equations.

AMS subject classification. 80A22, 35B40, 35B50, 45K05, 35K20, 35K45, 35K55, 35K65, 47J35

1 Introduction

Phase separation phenomena in material sciences are modeled usually by Cahn-Hillard equation, see [5] and references therein, which is derived from a free energy functional.

Often the classical Ginzburg-Landau free energy which contains gradient terms is used as the free energy functional. These models have been extensively analyzed, see [21] and references therein. But inspecting Van der Waals early works, see [20], and later Cahn and Hillard paper [4] it seems to be reasonable and even more adequate, see [10], to choose an alternative expression for thefree energy functional like

FN L(u) = Z

Ω

F(u)dx, (1.1)

∗Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany

1

whereudenotes the local concentration of a component occupying a spatial domainΩand F(u) =f(u) + ¹₂uw. Here f(u) is a convex function and

w(x) :=

Z

K(|x−y|)(1−2u(y))dy. (1.2) The kernel K of the integral term (1.2) describes nonlocal or long-range interactions [5, 11, 12, 14]. Hence, the difference between local and nonlocal models consists in a different choice of the particle interaction potential in the free energy functional. Moreover the local free energy can be obtained as a formal limit from the nonlocal one, see [17]. In [10]

the above nonlocal free energy functional has been used to derive a nonlocal Cahn-Hillard equation

u_t− ∇ ·(µ∇(f⁰(u) +w)) = 0, where f is the convex (Information) entropy function

f(u) =ulog(u) + (1−u) log(1−u).

Consequently

f⁰(u) = log u

1−u

and u=f⁰⁻¹(v−w) = 1

1 + exp(v−w),

where f⁰⁻¹ is the Fermi-function, whose image is the interval [0,1]. Thus, the nonlocal model naturally satisfies the physical requirement

0≤u(x)≤1, ∀t ≥0.

and the maximum principle is available, which is not true for fourth order equations like in the case of the local Cahn-Hillard equations.

1.1 Nonlocal viscous model

Following [10] our aim is to formulate a general nonlocal model, which also takes into accout viscosity effects, see [19]. In the local theory this was done by adding a rate term to the chemical potential. Now we are going to formulate this additional term in the nonlocal philosophy, so we not only want to get nonlocality in space (1.2) but also nonlocality in time. Hence, the chemical potential in our case is given by

v := δF(u)

δu +ψ, (1.3)

We propose two models:

Model I:

−γ∆ψ_t+ψ =u_t, γ >0. (1.4)

Model II:

−γ∆ψ+ψ =u_t, γ >0.

In both casesγ is a model parameter, which is positive and guarantees the nonlocal struc- ture of the additional term ψ in the chemical potential (1.3). This means in Gurtin’s language that the influence of microforces is nonlocal, but we are not able to postulate a generalized nonlocal balance law for nonlocal microforces similar to the balance law in Gurtin [16]. Settingγ = 0we recover the local viscous model, see [19]. Hence, our model is a real expansion of previous existing models. From mathematical point of view the terms

−γ∆ψ_t respectively −γ∆ψ have regularizing effects. Model I will be analysed in this pa- per. The Anlayis of Model II are left to a forthcoming paper, see [8]. Taking into account (1.3) and (1.4) we end up with the nonlocal viscous Cahn-Hillard equation:

u_t− ∇ ·µ∇v = 0, v =f⁰(u) +w+ψ, w(x) =

Z

Ω

K(|x−y|)(1−2u(y))dy,

−γ∆ψ_t+ψ =u_t, γ >0,

(1.5)

which is complementedby suitable initial and boundary conditions.

In Section 2 we formulate the problem, general assumptions and the main theorems. The rest of the Sections are devoted to the proof of the corresponding theorems.

2 Assumptions and main results

2.1 Statement of the problems and assumptions

Let be Ω⊂ R³ an open, bounded and smooth domain with boundary Γ = ∂Ω and ν the outer unit normal on Γ. In the sequel, |Ω| denotes the Lebesgue measure of Ω. We denote byL^p(Ω), W^k,p(Ω) for 1≤p≤ ∞ the Lesbegue spaces and Sobolev spaces of functions on Ωwith the usual norms k · k_L^p_(Ω),k · k_W^k,p_(Ω), and we writeH^k(Ω) =W^k,2(Ω), see [7]. For a Banach spaceX we denote its dual by X^∗, the dual pairing between f ∈X^∗, g ∈X will be denoted by hf, gi. If X is a Banach space with the norm k · k_X, we denote for T > 0 by L^p(0, T;X) (1 ≤ p ≤ ∞) the Banach space of all (equivalence classes of) Bochner measurable functions u: (0, T)−→X such that ku(·)kX ∈L^p(0, T). We set R¹₊ = (0,∞) and, as already mentioned, Q_T = (0, T)×Ω,Γ_T = (0, T)×Γ. ”Generic” positive constants are denoted by C and for u∈L¹(Ω) we put

u= 1

|Ω|

Z

Ω

u(x)dx.

Furthermore we define following time dependent Sobolev spaces by V^1,∞(0, T) :={f ∈L^∞(Q_T)| ∇f ∈L^∞(Q_T)},

V^2,∞(0, T) :={f ∈L^∞(Q_T)| ∇f ∈L^∞(Q_T),∆f ∈L^∞(Q_T)}.

So the initial-boundary value problem we want to discuss takes the form:

ut− ∇ ·

=µ∇v

z }| {

(∇u+µ∇(w+ψ)) = 0 in QT, (2.1)

−γ∆ψ_t+ψ =u_t, w=P(1−2u) inQ_T, (2.2)

µν · ∇v =ν· ∇ψ = 0 onΓ_T, (2.3)

ν· ∇ψ₀ = 0, u(0, x) = u₀(x), ψ(0, x) =ψ₀(x) x∈Ω. (2.4) We make the following general assumptions.

(A1) f(u) = ulogu+ (1−u) log(1−u).

(A2) The potential operator P defined by

ρ7→P ρ= Z

Ω

K(|x−y|)ρ(y)dy satisfies

kP ρk_Y ≤r_pkρk_L^p, 1≤p≤ ∞, where the kernel K ∈(R¹+ 7→R¹) is such that

Z

Ω

Z

Ω

|K(|x−y|)|dxdy=m0 <∞, sup

x∈Ω

Z

Ω

|K(|x−y|)|dy =m1 <∞.

(A3) The mobility µ has the form µ(u) = 1

f⁰⁰(u) =u(1−u). (2.5)

(A4) u0(x)∈[0,1] a.e. inΩ and u0 ∈(0,1).

The next assumptions concern different regularity assumptions on the data.

(B1) u₀ ∈L^∞(Ω) or (B1’)u₀ ∈L^∞(Ω)∩H¹(Ω), (B2) ψ₀ ∈H²(Ω) or (B2’)ψ₀ ∈H³(Ω),

(B3) Y :=W^1,p(Ω) or (B3’)Y :=W^2,p(Ω).

Remark 1. The kernel K is chosen to be symmetric. Hence, the potential operator P is symmetric, too. Examples for kernels K satisfying (A2) are Newton potentials, Gauss functions and usual mollifiers, see [10].

Remark 2. A concentration-dependent mobility appeared in the original derivation of the Cahn-Hillard equation, see [4], and a natural and thermodynamically reasonable choice is of the form (2.5) and were considered in [6].

2.2 Main results

Due to different regularity assumptions on the initial data we formulate two different Theorems, which will be proven separately in the next two chapters.

Theorem 1. Suppose that the assumptions (A1)-(A4) and (B1)-(B3) hold. Then there exists a unique triple of functions (u, w, ψ) such that u(0) =u₀, ψ(0) =ψ₀ and

1. u∈L²(0, T;H¹(Ω)), 0≤u(t, x)≤1 a.e. in Q_T, 2. u_t∈L²(0, T;H¹(Ω)^∗),

3. w∈ V^1,∞(0, T), 4. ψ ∈L²(0, T;L²(Ω)), 5. ∇ψ ∈L^∞(0, T;H¹(Ω)), 6. ∇ψ_t ∈L²(0, T;H¹(Ω)^∗),

which satify (2.1)-(2.4) in the following sense:

T

Z

0

hu_t, ϕidt+

T

Z

0

Z

Ω

(∇u+µ∇(w+ψ))∇ϕdxdt= 0, ∀ϕ∈L²(0, T;H¹(Ω)), (2.6)

γ

T

Z

0

h∇ψ_t,∇φidt+

T

Z

0

Z

Ω

ψφdxdt=

T

Z

0

hu_t, φidt, ∀φ∈L²(0, T;H¹(Ω)), (2.7)

w=P(1−2u) a.e. in Q_T. (2.8)

Theorem 2. Suppose that the assumptions (A1)-(A4) and (B1’)-(B3’) hold. Then there exists a unique triple of functions (u, w, ψ) such that u(0) =u₀, ψ(0) =ψ₀ and

1. u∈L²(0, T;H²(Ω)), 0≤u(t, x)≤1 a.e. in Q_T, 2. u_t∈L²(0, T;L²(Ω)),

3. w∈ V^2,∞(0, T), 4. ψ ∈L²(0, T;L²(Ω)), 5. ∇ψ ∈L^∞(0, T;H²(Ω)), 6. ∇ψ_t ∈L²(0, T;L²(Ω)),

which satisfy (2.1)-(2.4) in the following sense:

T

Z

0

Z

Ω

u_tϕdxdt+

T

Z

0

Z

Ω

(∇u+µ∇(w+ψ))∇ϕdxdt= 0, ∀ϕ∈L²(0, T;H¹(Ω)), (2.9)

γ

T

Z

0

Z

Ω

∇ψt· ∇φdt+

T

Z

0

Z

Ω

ψφdxdt=

T

Z

0

Z

Ω

utφdxdt, , ∀φ∈L²(0, T;H¹(Ω)), (2.10)

w=P(1−2u) a.e. in Q_T. (2.11)

Remark 3. Note that using the testfunctionsϕ= 1 and φ = 1 in (2.6)-(2.7) we get u(t, x) =u₀ a.e. in [0, T],

T

Z

0

Z

Ω

ψ(t, x)dxdt= 0. (2.12) Under the assumptions of Theorem 2 we can state the following

Theorem 3. Supposef⁰(u0)∈L^∞(Ω). Then f⁰(u)∈L^∞(QT).

Remark 4. We get from Theorem 3 that0< u(t, x)<1 a.e. in Q_T, provided 0< u₀(x)<

1 a.e. in Ω.

The main tool for studing the global behaviour of the solution (2.9)-(2.11) for T → ∞ is the energy estimate. Because of Theorem 3 and (A2) the function f⁰(u) +w+ψ =:v is in L^∞(Q_T) and an admissible testfunction in (2.9) and gives the global energy estimate

γ 2sup

t≥0

Z

Ω

|∇ψ(t)|²dx+

∞

Z

0

Z

Ω

|ψ|²dxdt+

∞

Z

0

Z

Ω

µ(u)|∇v|²dxdt≤C₅₅<∞. (2.13) Thus we can state the following Theorem which can be proven exploiting (2.13), see [9].

Theorem 4. Let (u, w, ψ) be a solution of (2.9)-(2.11). Then there exist a sequence {t_k : k = 1,2,· · · }with t_k → ∞for k→ ∞ and a triplet (u^∗, w^∗, ψ^∗)such that u_k=u(t_k), w_k = w(t_k), ψ_k =ψ(t_k) satisfy

u_k → u^∗ strongly in L² and weakly in H¹, w_k → w^∗ strongly in H¹,

ψ_k → 0 strongly in L²,

∇ψ_k → ∇ψ^∗ strongly in L²,

(2.14)

and

arctan(e^−vk/2)→arctan(e^−v∗/2) strongly in H¹, v^∗ = const. . (2.15) Moreover, the following relations hold:

w^∗ = Z

Ω

K(|x−y|)(1−2u^∗(y))dy, u^∗ =u₀, (2.16)

u^∗ = 1

1 + exp(w^∗−v^∗), v^∗ =const. (2.17)

3 Proof of Theorem 1

3.1 Existence

The idea of existence proof is as follows: we construct regularized problems with non- degenerate mobility functions. These regularized problems then are approximated by semi- discrete problems, which we solve by applying the Schauder’s fixed-point principle. After constructing suitable a priori estimates and compactness we can converge from the semi- discrete approximation to the regularized problem. The similar procedure we repeat for regularized problem to get uniform a priori estimates and compactness results, which finaly give convergence to the original problem. We devide our existence proof into a sequence of steps.

3.1.1 Regularized problems

At first we modify the moblity. We introduce a non-degenerate positive mobility µ_ε as µ_ε(u) :=







µ(ε) for u≤ε,

µ(u) for ε < u≤1−ε, µ(1−ε) for u >1−ε.

(3.1) This means that we symmetrically cut the mobility and constantly extend it to whole R. Simillarly we regularizef⁰⁰(u) and f⁰(u), see [9]. Furthermore we introduce the truncation

Πu:=







1 for u≥1, u for 0< u <1, 0 for u≤0,

(3.2) which is necessary to be able to apply the Schauder’s fixed-point principle. Hence, we get the truncated regularized system:

T

Z

0

hu_t, ϕidt+

T

Z

0

Z

Ω

(∇u+µ_ε∇(w+ψ))· ∇ϕdxdt= 0, ∀ϕ∈L²(0, T;H¹(Ω)), (3.3)

γ

T

Z

0

h∇ψ_t,∇φidt+

T

Z

0

Z

Ω

ψφdxdt=

T

Z

0

hu_t, φidt, ∀φ∈L²(0, T;H¹(Ω)), (3.4)

w=P(1−2Πu) a.e. in Q_T. (3.5)

Remark 5. We have by (A2) and (A4)

kwk²_H1(Ω) ≤r₂²k1−2Πuk_L²_(Ω) ≤r₂²|Ω|. (3.6) Remark 6. ∃ε0 :=ε0(w)so that ∀ε∈(0, ε0]

F_{N L,ε}(u) :=

Z

Ω

f_ε(u) + 1 2uw

dx≥ −C_F, where C_F >0.

Proof. Using (A1), (3.1) and (3.5) we see that it depends on the choice of ε to ensure that f_ε(u) dominates ¹₂uw. Thus, there exists an ε₀ = ε₀(w) so that ∀ε ∈ (0, ε₀] this is

true. 2

Existence result for the regularized problems We will denote the solution to the regularized system (3.3)-(3.5) by (u_ε, w_ε, ψ_ε). Let ∀ε ∈(0, ε₀] be fixed but arbitrary. The strategy of constructing solutions to (3.3)-(3.5) is to employ a semi-discrete approximation. To this end, letM ∈N be given andh:=T /M. In the sequel, we will denote byC_i, i∈N,positive constants that may depend on Ω, T and the initial data, but not onM orm ∈ {1, . . . , M}.

For 1≤ m ≤M, we consider the semi-discrete problem on the time level t :=mh for the unknown functions u^m, w^m, ψ^m : Ω→R given by

1 h

Z

Ω

(u^m−u^m−1)ϕdx+

+ Z

Ω

∇u^m+µ_ε(u^m)∇

w^m+w^m−1 2 +ψ^m

· ∇ϕdx= 0, ∀ϕ∈H¹(Ω), (3.7) γ

h Z

Ω

∇(ψ^m−ψ^m−1)· ∇φdx+ Z

Ω

ψ^mφdx= 1 h

Z

Ω

(u^m−u^m−1)φdx, ∀φ ∈H¹(Ω), (3.8)

w^m =P(1−2Πu^m)a.e. in Ω. (3.9)

For 1 ≤ m ≤ M the system (3.7)-(3.9) is a nonlinear elliptic system. Note that u⁰ = u₀, ψ⁰ =ψ₀.

Remark 7. Forϕ= 1 and φ= 1 we get from (3.7)-(3.9) u^m =u₀ and

Z

Ω

ψ^m = 0, ∀m ∈ {1, . . . , M}.

We prove existence of approximate solutions step by step via Schauder’s fixed-point prin- ciple.

Lemma 1. Suppose that the assumptions (A1)-(A4) and (B1)-(B3) hold. Then for every m ∈ {1, ..., M} there exists a triple of functions (u^m, w^m, ψ^m)∈ H¹(Ω)×H¹(Ω)×H²(Ω) satifying (3.7)-(3.9).

Proof. 1. Our proof is based on the application of Schauder’s fixed-point principle. Let m ∈ {1, ..., M} be fixed but arbitrary, and assume that the data (u^m−1, ψ^m−1) are known and given. Now for a given u^m ∈L² we consider theauxiliary linear problems

Z

Ω

∇(T_mu^m)· ∇ϕdx+ 1 h

Z

Ω

T_mu^mϕdx= Z

Ω

g₂ϕdx, ∀ϕ∈H¹(Ω), (3.10) Z

Ω

∇ψ^m· ∇φdx+ h γ

Z

Ω

ψ^mφdx= Z

Ω

g1φdx, ∀φ ∈H¹(Ω), (3.11) where

Z

Ω

g1φdx := 1 γ

Z

Ω

(u^m−u^m−1)φdx+ Z

Ω

∇ψ^m−1· ∇φdx, ∀φ ∈H¹(Ω), (3.12) Z

Ω

g₂ϕdx:=

Z

Ω

µ_ε(u^m)∇

w^m+w^m−1 2 +ψ^m

· ∇ϕdx+ 1 h

Z

Ω

Tm−1u^m−1ϕdx,∀ϕ∈H¹(Ω). (3.13)

The existence and uniqueness theory of (3.10)-(3.11) is standard and can be found in [7]. The strategy is to convert the integral expression in (3.10)-(3.13) into linear- and bilinearforms and use the Lax-Milgram Theorem. From [15], Corollary 2.2.2.4, respectively, we find that for a given u^m ∈L² and consequently a given g₁ ∈L²(Ω) in (3.12) the linear equation (3.11) admits a unique solution ψ^m ∈ H²(Ω). Setting w^m = P(1−2Πu^m) ∈ H¹(Ω), we find (3.9). Finally again from [15], Corollary 2.2.2.4, respectively, we conclude that for a given g₂ ∈L²(Ω) (3.10) admits a unique solutionT_mu^m ∈H¹(Ω).

2. Thus, we have properly defined a fixed-point operator T_m : L²(Ω) −→ L²(Ω). We can apply Schauder’s theorem, if we are able to prove, that T_m : L²(Ω) −→ L²(Ω) is completely continuous and T_m[B]⊂ B hold true for a closed ball B ⊂L²(Ω) with a radius depending only on the data of the problem.

3. Let ψ^m ∈ H²(Ω) be a solution of (3.11). We obtain using ψ^m as a testfunction in (3.11)

Z

Ω

|∇ψ^m|²dx+h γ

Z

Ω

|ψ^m|²dx≤ 1 γ

Z

Ω

(u^m−u^m−1)ψ^mdx+ Z

Ω

∇ψ^m−1· ∇ψ^mdx.

Applying Young’s inequality in the form 1

γ Z

Ω

(u^m−u^m−1)ψ^mdx≤ 2γ

Z

Ω

|u^m−u^m−1|²dx+ 1 2γ

Z

Ω

|ψ^m|²dx, (3.14)

and using the Poincaré inequality for the last term in (3.14) by choosing = 2c_p/γ, where c_p is the Poincaré constant, we finally conclude

Z

Ω

|∇ψ^m|²dx+ 4h γ

Z

Ω

|ψ^m|²dx ≤ 4c_p γ²

Z

Ω

|u^m−u^m−1|²dx+ 2 Z

Ω

|∇ψ^m−1|²dx. (3.15)

4. Let T_mu^m ∈ H¹(Ω) be a solution of (3.10). Applying the admissible test function ϕ=T_mu^m ∈H¹(Ω) in (3.10) and Young’s inequality we get

1 2h

Z

Ω

|T_mu^m|²dx ≤ 1 4

Z

Ω

µ_ε(u^m)∇

w^m+w^m−1 2 +ψ^m

2

dx+ 1 2h

Z

Ω

|Tm−1u^m−1|²dx

≤ 2 4³

Z

Ω

∇

w^m+w^m−1 2

2

dx+ 2 4³

Z

Ω

|∇ψ^m|²dx

+ 1 2h

Z

Ω

|T_m−1u^m−1|²dx. (3.16)

Using (3.6) we obtain by the estimates (3.15), (3.16) kT_mu^mk²_L2(Ω) ≤hr²₂|Ω|

4² +2h

4²k∇ψ^m−1k²_L2(Ω)+kTm−1u^m−1k²_L2(Ω)

+2h

4²ku^m−1k²_L2(Ω)+2h

4²ku^mk²_L2(Ω).

That means, we have kT_mu^mk²_L2(Ω) ≤ λ² for all u^m ∈ L²(Ω), if we choose h so that 1−h/8 =: 1/β >0 and fix radiusλ >0 by

λ² ≡hβr₂²|Ω|

4² + 2hβ

4² k∇ψ^m−1k²_L2(Ω)+βkTm−1u^m−1k²_L2(Ω)+ 2hβ

4² ku^m−1k²_L2(Ω). Hence, we get T_m[B]⊂ B for a closed ballB :={u^m ∈L²(Ω) :ku^mk_L²_(Ω) ≤λ}.

5. To show the continuity of T_m, let {u^m_i }i∈N ⊂ L²(Ω) be a sequence such that limi→∞ku^m_i −u^mk_L²_(Ω) = 0. For every i ∈ N there exists a uniquely determined solu- tion T_mu^m_i ∈ H¹(Ω) of the problem (3.10)-(3.13). Because T_mu^m ∈ H¹(Ω) is a solution of

the problem (3.10), for every i∈N it follows Z

Ω

∇(T_mu^m_i − T_mu^m)· ∇ϕdx+

+ 1 h

Z

Ω

(T_mu^m_i − T_mu^m)ϕdx= Z

Ω

(g_i,2−g₂)ϕdx, ∀ϕ∈H¹(Ω), (3.17) Z

Ω

∇(ψ^m_i −ψ^m)· ∇φdx+ h γ

Z

Ω

(ψ^m_i −ψ^m)φdx = Z

Ω

(g_1,i−g₁)φdx, ∀φ∈H¹(Ω), (3.18) where

Z

Ω

(g_i,1−g₁)φdx := 1 γ

Z

Ω

(u^m_i −u^m)φdx− 1 γ

Z

Ω

(u^m−1_i −u^m−1)φdx

+ Z

Ω

∇(ψ^m−1_i −ψ^m−1)· ∇φdx, ∀ϕ∈H¹(Ω), (3.19) and

Z

Ω

(g_i,2−g₂)ϕdx :=

Z

Ω

(µ_ε(u^m)−µ_ε(u^m_i ))∇

w^m+w^m−1 2 +ψ^m

· ∇ϕdx

+ Z

Ω

µ_ε(u^m_i )∇(w_i^m−w^m)· ∇ϕdx

+ Z

Ω

µ_ε(u^m_i )∇(w_i^m−1−w^m−1)· ∇ϕdx

+ Z

Ω

µ_ε(u^m_i )∇(ψ_i^m−ψ^m)· ∇ϕdx

+1 h

Z

Ω

(T_m−1u^m−1_i − T_m−1u^m−1)ϕdx, ∀ϕ∈H¹(Ω). (3.20)

Using in (3.18) the testfunction φ= (ψ_i^m−ψ^m)∈H²(Ω) we find that Z

Ω

|∇(ψ_i^m−ψ^m)|²dx+h γ

Z

Ω

|ψ_i^m−ψ^m|²dx= Z

Ω

(g_i,1−g₁)(ψ_i^m−ψ^m)dx.

Similar calculations like in (3.15) give k∇(ψ_i^m−ψ^m)k²_L2(Ω)≤8c_p

γ² ku^m_i −u^mk²_L2(Ω)+8c_p

γ² ku^m−1_i −u^m−1k²_L2(Ω) (3.21) + 2k∇(ψ_i^m−1−ψ^m−1)k²_L2(Ω). (3.22)

where c_p is the Poincaré constant.

Applying ϕ=T_mu^m_i − T_mu^m ∈H¹(Ω) as a testfunction in (3.17) we get Z

Ω

|∇(T_mu^m_i − T_mu^m)|²dx+ 1 h

Z

Ω

|T_mu^m_i − T_mu^m|²dx

= Z

Ω

(gi,2−g2)(Tmu^m_i − Tmu^m)dx.

Young’s inequality gives 1

h Z

Ω

|T_mu^m_i − T_mu^m|²dx≤4 Z

Ω

(µ_ε(u^m)−µ_ε(u^m_i ))∇

w^m+w^m−1 2 +ψ^m

2

dx

+ 8 Z

Ω

|µ_ε(u^m_i )∇(w_i^m−w^m)|²dx

+ 4² Z

Ω

|µε(u^m_i )∇(w^m−1_i −w^m−1)|²dx

+ 4² Z

Ω

|µ_ε(u^m_i )∇(ψ_i^m−ψ^m)|²dx

=:I₁+I₂+I₃+I₄.

Each summand will be analyzed separately: Because of the continuity of Π we get I₂ ≤8kw_i^m−w^mk²_H1(Ω) ≤2kP(Πu^m_i −Πu^m)k²_H1(Ω) ≤2Cr₂²ku^m_i −u^mk²_L2(Ω).

The summandI3can be trated in a similar way likeI2. ForI4we use the estimate (3.22). In the limit processi−→ ∞the expression(µ_ε(u^m)−µ_ε(u^m_i ))tends pointwise to zero, because of the Lipschitz continuity ofu^m 7→µ_ε(u^m)and the convergencelimi→∞ku^m_i −u^mk_L²_(Ω) = 0.

Hence applying Lebesgue’s theoremI1 tends to zero. The convergence ofI2-I4 follows from the boundedness of µ_ε(u^m_i ) and the convergencelimi→∞ku^m_i −u^mk_L²_(Ω) = 0.

6. Bacause of T_m[L²(Ω)]∈H¹(Ω) and the completely continuous embedding of H¹(Ω) into L²(Ω), the fixed-point map Tm : L²(Ω) −→ L²(Ω) is completely continuous. Having in mind the first step of the proof, Schauder’s fixed-point theorem yields a solution u^m ∈ H¹(Ω)∩ Bof the equation Tu^m =u^m. Settingw^m =P(1−Πu^m)∈H¹(Ω), we have found a solution (u^m, w^m, ψ^m)∈H¹(Ω)×H¹(Ω)×H²(Ω) of the problem (3.7)-(3.9). 2 In order to prove convergence of the semi-discrete problem (3.7)-(3.9) to the regularized problems (3.3)-(3.5) we need to derive (uniformly inm) a priori estimates. The key estimate is the following energy estimate in its discrete form

Lemma 2. (Discrete energy estimate) Let(u^m, w^m, ψ^m)be solution of (3.7)-(3.9) for every

m∈ {1, ..., M}. Then γ

2

M

X

m=1

k∇(ψ^m−ψ^m−1)k²_L2(Ω)+ γ 2 max

1≤k≤Mk∇ψ^kk²_L2(Ω)+

M

X

m=1

hkψ^mk²_L2(Ω) (3.23)

+

M

X

m=1

h Z

Ω

µ_ε(u^m)|∇v^m|²dx≤C₁ (3.24) and

M

X

m=1

h Z

Ω

|∇u^m|²dx ≤C₂(T). (3.25)

Proof. 1. Because of Lemma 1 the testfunctionϕ=v^m =f_ε⁰(u^m) +^wm+w₂^m−1+ψ^m ∈H¹(Ω) is admissible in (3.7) and we find

1 h

Z

Ω

(u^m−u^m−1)

f_ε⁰(u^m) + w^m+w^m−1 2 +ψ^m

dx+

Z

Ω

µ_ε(u^m)|∇v^m|²dx= 0.

We will estimate the first summand term by term.

2. The first term can be estimated as follows 1

h Z

Ω

(u^m−u^m−1)f_ε⁰(u^m)dx≥ 1 h

Z

Ω

f_ε(u^m)−f_ε(u^m−1)dx, where we have used the convexity off_ε(u), see (A1).

3. In order to estimate the second term we use the symmetry of P, see Remark 1.

1 h

Z

Ω

(u^m−u^m−1)

w^m+w^m−1 2

dx= 1 h

Z

Ω

1

4{(u^m+u^m−1)(w^m−w^m−1) + (u^m−u^m−1)(w^m+w^m−1)}dx= 1

h Z

Ω

1

2{u^mw^m−u^m−1w^m−1}dx.

4. For the third term we use the testfunction φ=ψ^m ∈H²(Ω) in (3.8) to get 1

h Z

Ω

(u^m−u^m−1)ψ^mdx= γ h

Z

Ω

∇(ψ^m−ψ^m−1)· ∇ψ^mdx+ Z

Ω

|ψ^m|²dx.

The above estimates give γ

2hk∇(ψ^m−ψ^m−1)k²_L2(Ω)+ γ

2hk∇ψ^mk²_L2(Ω)− γ

2hk∇ψ^m−1k²_L2(Ω)+kψ^mk²_L2(Ω)

+ Z

Ω

µ_ε(u^m)|∇v^m|²dx+ 1 h

F_{N L,ε}(u^m)−F_{N L,ε}(u^m−1)

≤0. (3.26)

We multiply (3.26) by h and sum both sides from m = 1 to m = k, where 1 ≤ k ≤ M. Using Remark 6 we conclude

γ 2

M

X

m=1

k∇(ψ^m−ψ^m−1)k²_L2(Ω)+γ 2 max

1≤k≤Mk∇ψ^kk²_L2(Ω)+

M

X

m=1

hkψ^mk²_L2(Ω)

+

M

X

m=1

h Z

Ω

µ_ε(u^m)|∇v^m|²dx≤C₁, where C₁ :=F_{N L}(u₀)−F_{N L,ε}(u^M) + ^γ₂k∇ψ⁰k²_L2(Ω).

Defining w˜^m := ^wm+w₂^m−1 +ψ^m we have the following estimate Z

Ω

µ_ε(u^m)|∇(f_ε⁰(u^m) + ˜w^m)|²dx= Z

Ω

f_ε⁰⁰(u^m)|∇u^m|²+ 2∇u^m· ∇w˜^m+|∇w˜^m|² f_ε⁰⁰(u^m)

dx

≥ Z

Ω

f_ε⁰⁰(u^m)

2 |∇u^m|² −|∇w˜^m|² f_ε⁰⁰(u^m)

dx

≥ Z

Ω

2|∇u^m|²− 1

4|∇w˜^m|²

dx,

(3.27)

where we have used Young’s inequality and the fact that f_ε⁰⁰(u^m)≥4. We multiply (3.27) byh and sum both sides from m= 1 to m=k, where 1≤k≤M, to get

C₁ ≥

k

X

m=1

h Z

Ω

2|∇u^m|²− 1

4|∇w˜^m|²

dx,

whereC₁ is the constant in (3.23). The definition ofw˜^m, (A2), (B4) and (3.23), (3.6) give

M

X

m=1

h Z

Ω

|∇u^m|²dx ≤C₂(T),

where C₂(T) :=T

r₂²|Ω|+ 4r²₂+ ¹₄ max

1≤k≤Mk∇ψ^kk²_L2(Ω)

+ ^C₂¹. 2

Lemma 3. Let (u^m, w^m, ψ^m) be solution of (3.7)-(3.9) for every m ∈ {1, ..., M}. Then

1≤k≤Mmax k∆ψ^kk²_L2(Ω)≤C₃. (3.28)

Proof. 1. Because of Lemma 1 ∆ψ^m ∈L²(Ω), thus an admissible testfunction in (3.8) γ

h Z

Ω

∆(ψ^m−ψ^m−1)∆ψ^mdx+ Z

Ω

|∇ψ^m|²dx+ 1 h

Z

Ω

(u^m−u^m−1)∆ψ^mdx= 0.

2. Applying the testfunction −u^m/γ in (3.8) we have 1

h Z

Ω

∆(ψ^m−ψ^m−1)u^mdx− 1 γ

Z

Ω

ψ^mu^mdx+ 1 γh

Z

Ω

(u^m−u^m−1)u^mdx= 0.

3. We use the identity Z

Ω

(u^m−u^m−1)∆ψ^m+ (∆ψ^m−∆ψ^m−1)u^m dx

= Z

Ω

u^m∆ψ^mdx− Z

Ω

u^m−1∆ψ^m−1dx+ Z

Ω

(u^m−u^m−1)(∆ψ^m−∆ψ^m−1) dx, and Young’s inequality in the following way

2 γ

Z

Ω

(u^m−u^m−1)(∆ψ^m−∆ψ^m−1)dx ≤ 1

γ²ku^m−u^m−1k²_L2(Ω)+k∆ψ^m−∆ψ^m−1k²_L2(Ω), 2

γ Z

Ω

u^m∆ψ^mdx ≤ 2

γ²ku^mk²_L2(Ω)+1

2k∆ψ^mk²_L2(Ω), and get

k∆ψ^kk²_L2(Ω)+ 4 γ

k

X

m=1

hk∇ψ^mk²_L2(Ω) =3k∆ψ⁰k²_L2(Ω)+ 6

γ²ku⁰k²_L2(Ω)+ 2

γ²ku^kk²_L2(Ω)

+ 1 γ³

k

X

m=1

hk∇u^mk²_L2(Ω)+ 2 γ²

k

X

m=1

hkψ^mk²_L2(Ω)

+ 2 γ²

k

X

m=1

hku^mk²_L2(Ω).

where we have summed both sides from m = 1 to m = k, (1 ≤ k ≤ M). The energy

estimate (3.23) and (3.25) give (3.28). 2

To indicate the dependence onM, we denote for anyM ∈Nthe solutions of (3.7)-(3.9) by(u^m_M, w^m_M, ψ_M^m). We define the piecewise linear

ˆ

u_M(x, t) =u^m+ t−mh

h (u^m−u^m−1) for t∈[(m−1)h, mh], (3.29) ψˆ_M(x, t) =ψ^m+ t−mh

h (ψ^m−ψ^m−1) for t∈[(m−1)h, mh], (3.30) as well as the constant interpolates

ˇ

u_M(x, t) =u^m for t∈[(m−1)h, mh], (3.31) ˇ

wM(x, t) = w^m+w^m−1

2 for t∈[(m−1)h, mh], (3.32) ψˇ_M(x, t) =ψ^m for t∈[(m−1)h, mh], (3.33)

for 1≤m≤M. With these notations we obtain

T

Z

0

Z

Ω

ˆ

u_M,tϕ dxdt+

+

T

Z

0

Z

Ω

(∇ˇu_M +µ_ε(ˇu_M)∇( ˇw_M + ˇψ_M))· ∇ϕ dxdt= 0, ∀ϕ∈L²(0, T;H¹(Ω))

γ

T

Z

0

Z

Ω

∇ψˆ_M,t· ∇φdx dt+

T

Z

0

Z

Ω

ψˇ_Mφ dxdt=

T

Z

0

Z

Ω

ˆ

u_M,tφdxdt, ∀φ ∈L²(0, T;H¹(Ω)).

(3.34)

We again get like in Remark 3 using ϕ= 1 and φ = 1 in (3.34) ˇ

u(t) =u₀,

T

Z

0

Z

Ω

ψˇ(x)dxdt= 0. (3.35)

By virtue of the energy estimate (3.23), γ

2

M

X

m=1

k∇(ψ^m−ψ^m−1)k²_L2(Ω)+ γ 2 sup

0≤t≤T

k∇ψˇ_M(t)k²_L2(Ω)+

T

Z

0

Z

Ω

|ψˇ_M|²dxdt

+

T

Z

0

Z

Ω

µ_ε(ˇu_M)|∇ˇv_M|²dxdt≤C₁,

(3.36)

where ˇv_M :=f_ε⁰(ˇu_M) + ˇw_M + ˇψ_M. Using (3.35) and the generalized Poincaré inequality we find from (3.25) that

kˇu_Mk_L²_(0,T;H¹_(Ω)) ≤C₄(√ T).

Moreover we find from (3.34) and (3.36)

T

Z

0

Z

Ω

ˆ

u_M,tϕdxdt

≤

T

Z

0

Z

Ω

µ_ε(ˇu_M)∇ˇv_M · ∇ϕ dxdt

≤ 1 2





T

Z

0

Z

Ω

µ_ε(ˇu_M)|∇ˇv_M|²dxdt





1/2 



T

Z

0

Z

Ω

|∇ϕ|²dxdt





1/2

≤ C₁ 2





T

Z

0

Z

Ω

|∇ϕ|²dxdt





1/2

(3.37)

for all ϕ∈L²(0, T;H¹(Ω)). We get

kˆu_M,tk_L²_(0,T;H¹_(Ω)^∗₎ = sup

ϕ∈L²(0,T;H¹(Ω))

|RT 0

R

ΩuˆM,t(x, t)ϕdxdt|

kϕk_L²_(0,T;H¹_(Ω)) ≤C₅. Thus, we find

T

Z

0

Z

Ω

∇ψˆ_M,t· ∇φdxdt

≤

T

Z

0

Z

Ω

ˆ

u_M,tφdxdt

+





T

Z

0

Z

Ω

|ψˇ_M|²dxdt





1/2



T

Z

0

Z

Ω

|φ|²dxdt





1/2

≤C₆kφk²_L2(0,T;H¹(Ω)),

(3.38)

and we find

k∇ψˆ_M,tk_L²_(0,T;H¹_(Ω)^∗₎ ≤C₆, sup

0≤t≤T

k∆ ˇψ_M(t)k²_L2(Ω) ≤C₃(T).

In addition, (3.25), (3.29) and (3.31) imply that k∇ˇuM − ∇ˆuMk²_L2(0,T;L²(Ω)) = T

3M

M

X

m=1

k∇u^m_M − ∇u^m−1_M k²_L2(Ω) →0 asM → ∞.

We also get from (3.29), (3.31) and Remark 7 that kˇu_M −uˆ_Mk²_L2(0,T;L²(Ω)) = T

3M

M

X

m=1

ku^m_M −u^m−1_M k²_L2(Ω) →0 asM → ∞.

We obtain using the generalized Poincaré inequality the following convergence

kˇu_M −uˆ_Mk_L²_(0,T;H¹_(Ω)) →0as M → ∞. (3.39) Moreover we have with Remark 7

kψˇ_M −ψˆ_Mk²_L2(0,T;L²(Ω)) = T 3M

M

X

m=1

kψ_M^m −ψ_M^m−1k²_L2(Ω) = 0, and by (3.30), (3.33) and (3.23), as M → ∞

k∇ψˇ_M − ∇ψˆ_Mk_L^∞_(0,T;L²_(Ω))= max

0≤t≤Tk∇ψ_M^m − ∇ψ^m−1_M k_L²_(Ω) →0. (3.40)

In conclusion, there are functionsu,ˇ uˆ_t,ψ,ˇ ψˆ_t, such that forM → ∞, possibly after selecting subsequences,

ˇ

uM −→ uˇ weakly in L²(0, T;H¹(Ω)), ˆ

u_M,t −→ uˆ_t weakly in L²(0, T;H¹(Ω)^∗), ψˇ_M −→ ψˇ weakly in L²(0, T;L²(Ω)),

∇ψˇ_M −→ ∇ψˇ weakly-star in L^∞(0, T;H¹(Ω)),

∇ψˆ_M,t −→ ∇ψˆ_t weakly in L²(0, T;H¹(Ω)^∗).

(3.41)

Taking (3.39) and (3.40) into account, we see thatuˇ= ˆuand ψˇ= ˆψ. It follows from (3.41) that we may pass to the limit as M → ∞ in (3.34). The convergence of the linear terms in (3.34) are standard. We take a closer look on the convergence of the nonlinear term

T

Z

0

Z

Ω

(µ_ε(u)∇(w+ψ)−µ_ε(u_M)∇(w_M +ψ_M))· ∇ϕ dxdt

=

T

Z

0

Z

Ω

(µ_ε(u)−µ_ε(u_M))∇(w+ψ)· ∇ϕ dxdt

+

T

Z

0

Z

Ω

µ_ε(u_M)∇[(w−w_M) + (ψ−ψ_M)]· ∇ϕ dxdt.

(3.42)

Because of the Lipschitz continuity of µ_ε and the compactness results in (3.41) the first term on the right hand side converges to zero. The second term converges to zero again by taking into account (3.41). Now we have proved the existence of solutions to theregularized problems (3.3)-(3.5).

3.1.2 Existence result for the original problem

Our aim is now to show the existence for the original problem (2.6)-(2.8) by showing the convergence ofε→0. To do this we need a priori estimates uniformly in the regularization parameter ε. Our starting point will again be the energy estimate.

We denote the solutions of the regularized problem by (u_ε, w_ε, ψ_ε).

Lemma 4. (energy estimate) There exists anε0, see Remark 6, such that for all0< ε≤ε0

the following estimate holds with constants C₇, C₈ independent of ε:

γ 2 max

0≤t≤T

Z

Ω

|∇ψ_ε(t)|²dx+

T

Z

0

Z

Ω

|ψ_ε|²dxdt+

T

Z

0

Z

Ω

µ_ε(u_ε)|∇v_ε|²dxdt≤C₇, (3.43)

T

Z

0

Z

Ω

|∇u_ε|²dxdt≤C₈. (3.44)

Proof. 1. The function v_ε =f_ε⁰(u_ε) +w_ε+ψ_ε ∈ L²(0, T;H¹(Ω)) is a valid testfunction in (3.3). Therefore we obtain

t

Z

0

h∂_tu_ε, f_ε⁰(u_ε) +w_ε+ψ_εidt =−

t

Z

0

Z

Ω

µ_ε(u_ε)|∇v_ε|²dxdt (3.45) for almost all t∈[0, T]. To prove this we define steklov averagedfunctions

u_εh(t, x) := 1 h

t

Z

t−h

u_ε(τ, x)dτ, (3.46)

where we set u_ε(t, x) = u₀(x) when t ≤0. From [18] it follows that u_εh converge strongly tou_ε inL²(0, T;H¹(Ω)). Because of (A2), (B3) and the continuity off_ε⁰ it is easily proven that

wεh −→ wε strongly in L²(0, T;H¹(Ω)),

f_εh⁰ (u_εh) −→ f_ε⁰(u_ε) strongly in L²(0, T;H¹(Ω)). (3.47) We define g_εh :=f_εh⁰ (u_εh) +w_εh, and v_εh :=g_εh+ψ_εh. By Lemma 5 we have

∇ψ_εh −→ ∇ψ_ε strongly in L²(0, T;L²(Ω)). (3.48) Furthermore, we can show ∂_tu_εh −→ ∂_tu_ε strongly in L²(0, T;H¹(Ω)^∗). For any ϕ ∈ L²(0, T;H¹(Ω)) we have

|h∂tuεh−∂tuε, ϕi|= 1 h

T

Z

0

* ^t Z

t−h

(∂tuε(τ)−∂tuε(t))dτ, ϕ +

dt

= 1 h

T

Z

0

* ⁰ Z

−h

(∂_tu_ε(t+s)−∂_tu_ε(t))ds, ϕ +

dt

≤ 1 h

0

Z

−h

T

Z

0

Z

Ω

(µ_ε(u_ε(t+s))∇v_ε−µ_ε(u_ε(t))∇v_ε)∇ϕdxdt

ds

≤ max

−h≤s≤0k(µ_ε(u_ε(t+s))∇v_ε(t+s)−µ_ε(u_ε(t))∇v_ε(t)k_L²_(QT)k∇ϕk_L²_(QT). We have

−h≤s≤0max kµ_ε(u_ε(t+s))∇v_ε(t+s)−µ_ε(u_ε(t))∇v_ε(t)k_L²_(QT)

≤ max

−h≤s≤0k[µ_ε(u_ε(t+s))−µ_ε(u_ε(t))]∇v_ε(t+s)k_L²_(QT) +C max

−h≤s≤0kgε(t+s)−gε(t)k_L²_(0,T;H¹_(Ω)) +C max

−h≤s≤0k∇ψ_ε(t+s)− ∇ψ_ε(t)k_L²_(0,T;L²_(Ω)).