Existence and weak–strong uniqueness for Maxwell–Stefan–Cahn–Hilliard systems

Existence and weak–strong uniqueness for Maxwell–Stefan–Cahn–Hilliard systems

Item Type Article

Authors Huo, Xiaokai; Jüngel, Ansgar; Tzavaras, Athanasios

Citation Huo, X., Jüngel, A., & Tzavaras, A. E. (2023). Existence and weak–

strong uniqueness for Maxwell–Stefan–Cahn–Hilliard systems.

Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire.

https://doi.org/10.4171/aihpc/89 Eprint version Post-print

DOI 10.4171/aihpc/89

Publisher European Mathematical Society - EMS - Publishing House GmbH Journal Annales de l'Institut Henri Poincaré C, Analyse non linéaire Rights Archived with thanks to Annales de l'Institut Henri Poincaré C,

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Link to Item http://hdl.handle.net/10754/677939

MAXWELL–STEFAN–CAHN–HILLIARD SYSTEMS

XIAOKAI HUO, ANSGAR J ¨UNGEL, AND ATHANASIOS E. TZAVARAS

Abstract. A Maxwell–Stefan system for fluid mixtures with driving forces depending on Cahn–Hilliard-type chemical potentials is analyzed. The corresponding parabolic cross- diffusion equations contain fourth-order derivatives and are considered in a bounded do- main with no-flux boundary conditions. The nonconvex part of the energy is assumed to have a bounded Hessian. The main difficulty of the analysis is the degeneracy of the diffusion matrix, which is overcome by proving the positive definiteness of the matrix on a subspace and using the Bott–Duffin matrix inverse. The global existence of weak solutions and a weak-strong uniqueness property are shown by a careful combination of (relative) energy and entropy estimates, yieldingH²(Ω) bounds for the densities, which cannot be obtained from the energy or entropy inequalities alone.

1. Introduction

The evolution of fluid mixtures is important in many scientific fields like biology and nanotechnology to understand the diffusion-driven transport of the species. The transport can be modeled by the Maxwell–Stefan equations [33, 35], which consist of the mass bal- ance equations and the relations between the driving forces and the fluxes. The driving forces involve the chemical potentials of the species, which in turn are determined by the (free) energy. When the fluid is immiscible, the energy can be assumed to consist of the thermodynamic entropy and the phase separation energy, given by a density gradient [6].

The gradient energetically penalizes the formation of an interface and restrains the segre- gation. This leads to a system of cross-diffusion equations with fourth-order derivatives.

The aim of this paper is to provide a global existence and weak-strong uniqueness analysis for multicomponent systems of Maxwell–Stefan–Cahn–Hilliard-type.

Date: February 26, 2023.

2000Mathematics Subject Classification. 35A02, 35G20, 35G31, 35K51, 35K55, 35Q35.

Key words and phrases. Cross-diffusion systems, global existence, weak-strong uniqueness, relative en- tropy, relative free energy, parabolic fourth-order equations, Maxwell–Stefan equations, Cahn–Hilliard equations.

XH and AJ acknowledge partial support from the Austrian Science Fund (FWF), grants P33010, W1245, and F65. AET acknowledges support from baseline funds of the King Abdullah University of Science and Technology (KAUST). This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, ERC Advanced Grant no. 101018153.

1

1.1. Model equations and state of the art. The equations for the partial densities c_i and partial velocitiesu_i are given by

∂_tc_i+ div(c_iu_i) = 0, i= 1, . . . , n, (1)

c_i∇µ_i− c_i Pn

k=1c_k

n

X

j=1

c_j∇µ_j =−

n

X

j=1

K_ij(c)c_ju_j, (2)

n

X

j=1

c_ju_j = 0, (3)

supplemented by the initial and boundary conditions

(4) c(·,0) = c⁰ in Ω, c_iu_i·ν =∇c_i·ν= 0 on ∂Ω, t >0, i= 1, . . . , n,

where Ω⊂R^d(d= 1,2,3) is a bounded domain,ν is the exterior unit normal vector on the boundary∂Ω,c= (c₁, . . . , c_n) is the density vector, andK_ij(c) are the friction coefficients.

The left-hand side of (2) can be interpreted as the driving forces of the thermodynamic system, and the right-hand side is the sum of the friction forces. The chemical potentials

(5) µi = δE

δc_i = logci−∆ci, i= 1, . . . , n, are the variational derivatives of the (free) energy

(6) E(c) =H(c) + 1 2

n

X

i=1

Z

Ω

|∇c_i|²dx, H(c) =

n

X

i=1

Z

Ω

c_i(logc_i−1) + 1 dx,

and H(c) is the thermodynamic entropy. Note that this energy is convex; we show in Section 5 that our results still hold if the energy contains a nonconvex part with bounded Hessian (like the potential in [13]). We assume that Pn

i=1K_ij(c) = 0 for j = 1, . . . , n, meaning that the linear system in∇µ_j is invertible only on a subspace, and thatPn

i=1c⁰_i = 1 in Ω, which implies that Pn

i=1c_i(t) = 1 in Ω for all time t > 0. This means that the mixture is saturated andc_i can be interpreted as volume fraction. For simplicity, we have normalized all physical constants.

Model (1)–(5) has been derived rigorously in [24] in the high-friction limit from a multi- component Euler–Korteweg system for a general convex energy functional depending onc and∇c. A thermodynamics-based derivation can be found in [34]. When the energy equals E(c) = H(c), the model reduces to the classical Maxwell–Stefan equations, analyzed first in [4, 21, 22] for local-in-time smooth solutions and later in [30] for global-in-time weak solu- tions. In the single-species case, model (1)–(5) becomes a fourth-order Cahn–Hilliard-type equation with convex potentialφ(c) =c(logc−1). Such a model, additionally including a nonconvex potential, was analyzed in, e.g., [13, 27]. Convergence from the Euler-Korteweg in the high-friction limit to the Cahn-Hilliard equation with nonconvex potential is pro- vided in [20]. Only few works are concerned with the multi-species situation, and all of them require additional conditions. The mobility matrix in [5, 32] is assumed to be diago- nal and that one in [31] has constant entries, while the works [11, 14] suppose a particular

(but nondiagonal) structure of the mobility matrix. We also mention the works [2, 3] on related models with free energies of the typeH.

The proof of the uniqueness of solutions to cross-diffusion or fourth-order systems is quite delicate due to the lack of a maximum principle and regularity of the solutions.

The uniqueness of strong solutions to Maxwell–Stefan systems has been shown in [22, 26], and uniqueness results for weak solutions in a very special case can be found in [8]. A weak-strong uniqueness result was proved for reaction-diffusion systems in [18] and for Maxwell–Stefan systems in [25]. Concerning uniqueness results for fourth-order equations, we refer to [9] for single-species Cahn–Hilliard equations, [28] for single-species thin-film equations, and [17] for the quantum drift-diffusion equations. Up to our knowledge, there are no uniqueness results for multicomponent Cahn–Hilliard-type systems. In this paper, we analyze these equations in a general setting for the first time.

1.2. Key ideas of the analysis. Before stating the main results, we explain the math- ematical ideas needed to analyze model (1)–(5). First, we rewrite (2) by introducing the matrix D(c)∈R^n×n with entries

D_ij(c) = 1

√ci

K_ij(c)√ c_j

in the unknowns (√

c₁u₁, . . . ,√ c_nu_n):

(7)

√c_i∇µ_i−

√c_i Pn

k=1c_k

n

X

j=1

c_j∇µ_j =−

n

X

j=1

D_ij(c)√ c_ju_j,

n

X

i=1

√c_i √ c_iu_i

= 0.

We show in Lemma 3 that this linear system has a unique solution in the space L(c) :=

{z ∈Rⁿ:Pn i=1

√c_iz_i = 0}, and the solution reads as

√c_iu_i =−

n

X

j=1

D_ij^BD(c)√

c_j∇µ_j,

where D^BD(c) is the so-called Bott–Duffin matrix inverse; see Lemmas 3 and 4 for the definition and some properties. Then, defining the matrix B(c)∈R^n×n with elements

(8) B_ij(c) =√

c_iD_ij^BD(c)√

c_j, i, j = 1, . . . , n, system (1)–(2) can be formulated as (see Section 2.1 for details)

∂_tc_i = div

n

X

j=1

B_ij(c)∇µ_j, i= 1, . . . , n.

The matrix B(c) is often called Onsager or mobility matrix in the literature. The major difficulty of the analysis consists in the fact that the matrixB(c) is singular and degenerates

when c_i →0 for some i∈ {1, . . . , n}. Computing formally the energy identity dE

dt(c) +

n

X

i,j=1

Z

Ω

B_ij(c)∇µ_i· ∇µ_jdx = 0,

the degeneracy atc_i = 0 prevents uniform estimates for∇µ_i inL²(Ω). In some works, this issue has been compensated. For instance, there exists an entropy equality for the model of [14] yielding an L²(Ω) bound for ∆c_i, and the decoupled mobilities in [7, 32] allow for decoupled entropy estimates. In our model, the energy identity does not provide a gradient estimate for the full vector (∇µ₁, . . . ,∇µ_n) but only for a projection:

dE

dt(c) +C₁

n

X

i=1

Z

Ω

n

X

j=1

(δ_ij −√ c_ic_j)√

c_j∇µ_j

2

dx≤0,

where δij is the Kronecker delta; see Lemma 5. (The constant C1 > 0 and all constants that follow do not depend on c.) To address the degeneracy issue, we compute the time derivative of the entropy:

dH dt (c) +

n

X

i,j=1

Z

Ω

B_ij(c)∇logc_i· ∇µ_jdx= 0.

This does not provide a uniform estimate for ∆c_i, but we show (see Lemma 5) that dH

dt (c) +C₂

n

X

i=1

Z

Ω

(∆c_i)²dx≤C₃

n

X

i=1

Z

Ω

n

X

j=1

(δ_ij −√ c_ic_j)√

c_j∇µ_j

2

dx.

Combining the energy and entropy inequalities in a suitable way, the last integral cancels:

(9) d

dt

H(c) + C₃ C₁E(c)

+C₂

n

X

i=1

Z

Ω

(∆c_i)²dx≤0.

This provides the desired H²(Ω) bound for ci. Note that the energy or entropy inequality alone does not give estimates for ci. The combined energy-entropy inequality is the key idea of the paper for both the existence and weak-strong uniqueness analysis. Observe that the termH(c) + (C₃/C₁)E(c) can also be written as (1 +C₃/C₁)H(c) +¹₂Pn

i=1

R

Ω|∇c_i|²dx.

1.3. Main results. We make the following assumptions:

(A1) Domain: Ω ⊂ R^d with d ≤ 3 is a bounded domain. We set Q_T = Ω×(0, T) for T >0.

(A2) Initial data: c⁰_i ∈H¹(Ω) satisfies c⁰_i ≥0 in Ω,i= 1, . . . , n, and Pn

i=1c⁰_i = 1 in Ω.

The assumptiond≤3 is made for convenience, it can be relaxed for higher space dimen- sion, by choosing another regularization in the existence proof; see (88). The constraint Pn

i=1c⁰_i = 1 expresses the saturation of the mixture and it propagates to the solution. We introduce the matrix D_ij(c) = (1/√

c_i)K_ij(c)√

c_j for i, j = 1, . . . , n and set (10) L(c) = {x∈Rⁿ :√

c·x= 0}, L^⊥(c) = span{√ c},

where √

c = (√

c₁, . . . ,√

c_n). The projections P_L(c), P_L^⊥(c) ∈ R^n×n on L(c), L(c)^⊥, respectively, are given by

(11) P_L(c)_ij =δ_ij −√

c_ic_j, P_L^⊥(c)_ij =√

c_ic_j for i, j = 1, . . . , n.

We impose for any given c∈[0,1]ⁿ the following assumptions onD(c) = (D_ij(c))∈R^n×n: (B1) D(c) is symmetric and ranD(c) = L(c), ker(D(c)P_L(c)) =L^⊥(c).

(B2) For all i, j = 1, . . . , n, D_ij ∈C¹([0,1]ⁿ) is bounded.

(B3) The matrix D(c) is positive semidefinite, and there exists ρ > 0 such that all eigenvaluesλ 6= 0 of D(c) satisfy λ≥ρ.

(B4) For all i, j = 1, . . . , n, K_ij(c) = √

c_iD_ij(c)/√

c_j is bounded in [0,1]ⁿ.

Examples of matricesD(c) satisfying these assumptions are presented in Section 6. Our first main result is the global existence of weak solutions.

Theorem 1 (Global existence). Let Assumptions (A1)–(A2) and (B1)–(B4) hold. Then there exists a weak solution c to (1)–(5) satisfying 0≤c_i ≤1, Pn

i=1c_i = 1 in Ω×(0,∞), c_i ∈L^∞_loc(0,∞;H¹(Ω))∩L²_loc(0,∞;H²(Ω)), ∂_tc_i ∈L²_loc(0,∞;H¹(Ω)⁰),

the initial condition in (4) is satisfied in the sense of H¹(Ω)⁰, and for all φ_i ∈ C₀^∞(Ω× (0,∞)),

0 =− Z ∞

0

Z

Ω

c_i∂_tφ_idxdt+

n

X

j=1

Z ∞ 0

Z

Ω

B_ij(c)∇logc_i· ∇φ_idxdt (12)

+

n

X

j=1

Z ∞ 0

Z

Ω

div(B_ij(c)∇φ_i)∆c_jdxdt, where B_ij(c) is defined in (8). Furthermore,

H(c(·, T)) +C1E(c(·, T)) +C2

Z T 0

Z

Ω

(|∇√

c|²+|∆c|²)dxdt (13)

+C₂ Z T

0

Z

Ω

|ζ|²dxdt≤ H(c⁰) +C₁E(c⁰),

where C₁ >0 depends on ρ, n, kD(c)k_F and C₂ >0 depends on n, kD(c)k_F (k · k_F is the Frobenius matrix norm and ρ is introduced in Assumption (B3)). Moreover, ζ is the weak L²(Ω) limit of an approximating sequence of Pn

j=1P_L(c)_ij√

c_j∇µ_j.

Some comments are in order. First, by Assumption (B2), the elements of the matrixD(c) are bounded for anyc∈[0,1]ⁿ and therefore, the quantitykD(c)k_F is bounded uniformly inc. Second, the weak formulation (12) makes sence sinceB_ij(c)∇logc_i ∈L²(Q_T). Indeed, by the definition of B(c), we have

B_ij(c)∇logc_j =√

c_iD^BD_ij (c) 1

√c_j∇c_j,

and the matrix √

c_iD_ij^BD(c)/√

c_j is bounded for all c ∈ [0,1]ⁿ; see Lemma 4 (iii) below.

However, note that the expressionPn

j=1B_ij(c)∇µ_j is generally not an element ofL²(Q_T).

In particular, we cannot expect that ∇∆c_i ∈ L²(Q_T). Third, we have not been able to identify the weak limit ζ because of low regularity. However, if Pn

j=1P_L(c)_ij√

c_j∇µ_j ∈ L²_loc(0,∞;L²(Ω)) holds for alli= 1, . . . , n, then we can identifyζ_i =Pn

j=1P_L(c)_ij√

c_j∇µ_j; see Lemma 9.

To prove Theorem 1, we first introduce a truncation with parameterδ∈(0,1) as in [14]

to avoid the degeneracy. Then we reduce the cross-diffusion system ton−1 equations by replacing c_n by 1−Pn−1

i=1 c_i. The advantage is that the diffusion matrix of the reduced system is positive definite (with a lower bound depending onδ). The existence of solutions c^δ_i to the truncated, reduced system is proved by an approximation as in [29] and the Leray–Schauder fixed-point theorem; see Section 3.1. An approximate version of the free energy estimate (13) (proved in Lemma 8 in Section 3.2) provides suitable uniform bounds that allow us to perform the limit δ → 0. The approximate densities c^δ_i may be negative but, by exploiting the entropy bound for c^δ_i, its limitc_i turns out to be nonnegative. The limitδ →0 is then performed in Section 3.3, using the uniform estimates and compactness arguments.

Our second main result is concerned with the weak-strong uniqueness. For this, we define the relative entropy and free energy in the spirit of [19] by, respectively,

H(c|¯c) :=H(c)− H(¯c)− ∂H

∂c (¯c)·(c−c) =¯

n

X

i=1

Z

Ω

c_ilogc_i

¯

c_i −(c_i−¯c_i)

dx, (14)

E(c|¯c) :=E(c)− E(¯c)− ∂E

∂c(¯c)·(c−c) =¯ H(c|¯c) + 1 2

n

X

i=1

Z

Ω

|∇(c_i−¯c_i)|²dx.

(15)

Theorem 2 (Weak-strong uniqueness). Let Assumptions (A1)–(A2), (B1)–(B4) hold, let c be a weak solution to (1)–(5) with initial datum c⁰, and let c¯ be a strong solution to (1)–(5) with initial datum c¯⁰. We assume that the weak solution c satisfies

(16)

n

X

j=1

P_L(c)_ij√

c_j∇µ_j ∈L²_loc(0,∞;L²(Ω)) for i, j = 1, . . . , n

(see (11) for the definition of P_L(c)) and for all T >0 the energy and entropy inequalities E(c(T)) +

n

X

i,j=1

Z T 0

Z

Ω

B_ij(c)∇µ_i· ∇µ_jdxdt≤ E(c⁰), (17)

H(c(T)) +

n

X

i,j=1

Z T 0

Z

Ω

B_ij(c)∇logc_i· ∇µ_jdxdt≤ H(c⁰).

(18)

The strong solution c¯ is supposed to be strictly positive, i.e., there exists m >0 such that

¯

c_i ≥m in Ω, t >0, and satisfies the regularity

¯

c_i ∈L^∞_loc(0,∞;W^3,∞(Ω)), ∇div 1

¯

c_iB_ij(¯c)∇¯µ_j

∈L^∞_loc(0,∞;L^∞(Ω))

for i= 1, . . . , n, as well as for any T > 0 the energy and entropy conservation identities E(¯c(T)) +

n

X

i,j=1

Z T 0

Z

Ω

B_ij(¯c)∇¯µ_i· ∇¯µ_jdxdt=E(¯c⁰), (19)

H(¯c(T)) +

n

X

i,j=1

Z T 0

Z

Ω

B_ij(¯c)∇log ¯c_i· ∇¯µ_jdxdt=H(¯c⁰), (20)

where µi = logci−∆ci and µ¯i = log ¯ci−∆¯ci. Then, for any T >0, there exist constants C₁, only depending on kD(c)k_F, n, ρ, and C₂(T) >0, only depending on T, meas(Ω), n, ρ, such that

(21) H(c(T)|¯c(T)) +C₁E(c(T)|¯c(T))≤C₂(T) H(c⁰|¯c⁰) +C₁E(c⁰|c¯⁰) . In particular, if c⁰ = ¯c⁰ then the weak and strong solutions coincide.

Observe that we need stronger assumptions on the weak solutions than those obtained in Theorem 1. Assumption (16) guarantees that the flux Pn

j=1B_ij(c)∇µ_j lies in L²(Q_T).

Indeed, we prove in Lemma 4 (i) in Section 2 that D^BD_ij (c) is bounded for c ∈ [0,1]ⁿ. Therefore, sinceD^BD(c) = D^BD(c)P_L(c), assumption (16) and c_i ∈L^∞(Q_T) imply that (22)

n

X

j=1

Bij(c)∇µj =√ ci

n

X

j,k=1

D^BD_ik (c)PL(c)kj

√cj∇µj ∈L²(QT).

By the way, it follows fromPn

j=1P_L(c)_ij√

c_j∇logc_j = 2∇√

c_i ∈L²(Q_T) that (23)

n

X

j=1

P_L(c)_ij√

c_j∇∆c_j =

n

X

j=1

P_L(c)_ij√

c_j∇(logc_j −µ_j)∈L²(Q_T).

Since ∇∆c_i may be not in L²(Q_T), we interpret (23) in the sense of distributions, i.e. for all Φ∈C₀^∞(Ω;R^d),

n

X

j=1

P_L(c)_ij√

c_j∇∆c_j,Φ

=−

n

X

j=1

Z

Ω

∇(P_L(c)_ij√

c_j)·Φ +P_L(c)_ij√

c_jdiv Φ

∆c_jdx.

For the proof of Theorem 2, we estimate first the time derivative of the relative entropy (14):

dH

dt (c|¯c) +C₁

n

X

i=1

Z

Ω

n

X

j=1

P_L(c)_ij√

c_j∇logc_j

¯ c_j

2

dx+C₁

n

X

i=1

Z

Ω

(∆(c_i−¯c_i))²dx

≤C₂

n

X

i=1

Z

Ω

n

X

j=1

P_L(c)_ij√

c_j∇(µ_j −µ¯_j)

2

dx+C₃ Z

Ω

E(c|¯c)dx,

where C_i > 0 are some constants depending only on the data. The first term on the right-hand side can be handled by estimating the time derivative of the relative energy (15):

dE

dt(c|c) +¯ C₄

n

X

i=1

Z

Ω

n

X

j=1

P_L(c)_ij√

c_j∇(µ_j −µ¯_j)

2

dx

≤θ

n

X

i=1

Z

Ω

n

X

j=1

P_L(c)_ij√

c_j∇logc_j

¯ c_j

2

dx+θ

n

X

i=1

Z

Ω

(∆(c_i −c¯_i))²dx +C₅(θ)

Z

Ω

E(c|¯c)dx,

where θ > 0 can be arbitrarily small. Choosing θ = C₁C₄/C₂, we can combine both estimates leading to

d dt

H(c|¯c) + C2

C₄E(c|¯c)

≤

C3+C2C5

C₄

E(c|¯c),

and the theorem follows after applying Gronwall’s lemma. As the computations are quite involved, we compute first in Section 4.1 the time derivative of the relative entropy and energy for smooth solutions. The rigorous proof of the combined relative entropy-energy inequality for weak solutions c and strong solutions ¯c is then performed in Section 4.2.

The paper is organized as follows. The Bott–Duffin matrix inverse is introduced in Section 2, some properties of the mobility matrix B(c) are proved, and the combined energy-entropy inequality (9) is derived for smooth solutions. The global existence of solutions (Theorem 1) is shown in Section 3, while Section 4 is concerned with the proof of the weak-strong uniqueness property (Theorem 2). The case of nonconvex energies is investigated in Section 5, showing that Theorems 1 and 2 still hold if the convex part of the energy equals the Boltzmann entropy (see (6)) and the Hessian of the nonconvex part is bounded. Finally, we present some examples verifying Assumptions (B1)–(B4) in Section 6.

Notation. Elements of the matrix A∈ R^n×n are denoted byAij, i, j = 1, . . . , n, and the elements of a vectorc∈Rⁿarec₁, . . . , c_n. We use the notationf(c) = (f(c₁), . . . , f(c_n)) for c ∈ Rⁿ and a function f : R → R. The expression |∇f(c)|² is defined by Pn

i=1|∇f(c_i)|² and | · | is the usual Euclidean norm. The matrix R(c) ∈ R^n×n is the diagonal matrix with elements √

c₁, . . . ,√

c_n, i.e. R_ij(c) = √

c_iδ_ij for i, j = 1, . . . , n, where δ_ij denotes the Kronecker delta. We understand by ∇µ the matrix with entries ∂_xiµ_j. Furthermore, C >0, C_i >0 are generic constants with values changing from line to line.

2. Properties of the mobility matrix and a priori estimates

We wish to express the fluxesc_iu_ias a linear combination of the gradients of the chemical potentials. SinceK(c) has a nontrivial kernel, we need to use a generalized matrix inverse, the Bott–Duffin inverse. This inverse and its properties are studied in Section 2.1. The properties allow us to derive in Section 2.2 some a priori estimates for the Maxwell–Stefan–

Cahn–Hilliard system.

2.1. The Bott–Duffin inverse. We wish to invert (2) or, equivalently, (7). We recall definition (11) of the projection matrices P_L(c) ∈ R^n×n on L(c) and P_L^⊥(c) ∈ R^n×n on L^⊥(c), where L(c) and L^⊥(c) are defined in (10). Then (7) is equivalent to the problem:

(24) Solve D(c)z =−P_L(c)R(c)∇µ in the spacez ∈L(c), where z_i =√

c_iu_i, recalling that R(c) = diag(√ c).

Lemma 3 (Solution of (24)). Suppose that D(c) satisfies Assumption (B1). The Bott–

Duffin inverse

D^BD(c) =P_L(c) D(c)P_L(c) +P_L^⊥(c)−1

is well-defined, symmetric, and satisfies kerD^BD(c) = L^⊥(c). Furthermore, for any y ∈ L(c), the linear problem D(c)z = y for z ∈ L(c) has a unique solution given by z = D^BD(c)y.

We refer to [25, Lemma 17] for the proof. The property for the kernel follows from kerD^BD(c) = kerP_L(c) = L^⊥(c). Since P_L(c)R(c)∇µ ∈ L(c) (this follows from the definition of P_L(c) and Pn

i=1c_i = 1), we infer from Lemma 3 that (24) has the unique solutionz =−D^BD(c)P_L(c)R(c)∇µ∈L(c) or, componentwise,

c_iu_i =√

c_iz_i =−

n

X

j=1

√c_i D^BD(c)P_L(c)

ij

√c_j∇µ_j =−

n

X

j=1

√c_iD^BD(c)_ij√ c_j∇µ_j

for i = 1, . . . , n, where the last equality follows from D^BD(c)P_L(c) = D^BD(c); see [25, (81)]. Then we can formulate equation (1) as

(25) ∂_tc_i = div

n

X

j=1

B_ij(c)∇µ_j, where B_ij(c) = √

c_iD_ij^BD(c)√

c_j, i, j = 1, . . . , n.

The boundary conditions c_iu_i·ν = 0 on∂Ω yield (26)

n

X

j=1

B_ij(c)∇µ_j ·ν= 0 on ∂Ω, t >0, i= 1, . . . , n.

We recall some properties of the Bott–Duffin inverse.

Lemma 4(Properties ofD^BD(c)). Suppose thatD(c)∈R^n×n satisfies Assumptions (B1)–

(B4). Then:

(i) The coefficients D^BD_ij ∈C¹([0,1]ⁿ) are bounded for i, j = 1, . . . , n.

(ii) Let λ(c) be an eigenvalue of (D(c)P_L(c) +P_L^⊥(c))⁻¹. Then λ_m ≤ λ(c) ≤ λ_M, where

λ_m = (1 +nkD(c)k_F)⁻¹, λ_M = max{1, ρ⁻¹},

k · k_F is the Frobenius matrix norm, and ρ >0 is a lower bound for the eigenvalues of D(c); see Assumption (B3).

(iii) The functions c7→√

c_iD^BD_ij (c)/√

c_j are bounded in [0,1]ⁿ for i, j = 1, . . . , n.

A consequence of (ii) are the inequalities

(27) λ_m|P_L(c)z|² ≤z^TD^BD(c)z ≤λ_M|P_L(c)z|² for z ∈Rⁿ.

Note that the Frobenius norm of D(c) is bounded uniformly in c ∈ [0,1]ⁿ, since D_ij is bounded by Assumption (B1).

Proof. The points (i) and (ii) are proved in [25, Lemma 11] in an interval [m,1]ⁿ for some m >0. In fact, we can conclude (i)–(ii) in the full interval [0,1]ⁿ, since our Assumptions (B2)–(B3) are stronger than those in [25].

For the proof of (iii), dropping the argument c and observing that RDR⁻¹ = K, we obtain

RD^BDR⁻¹ =RP_L(DP_L+P_L^⊥)⁻¹R⁻¹ =RP_L(R⁻¹R)(DP_L+P_L^⊥)⁻¹R⁻¹

=RP_LR⁻¹ R(DP_L+P_L^⊥)R⁻¹−1

=RP_LR⁻¹ RDR⁻¹RP_LR⁻¹+RP_L^⊥R⁻¹−1

=RP_LR⁻¹ KRP_LR⁻¹+RP_L^⊥R⁻¹−1

. The determinant of the expression in the brackets equals

det R(DP_L+P_L^⊥)R⁻¹

= det(DP_L+P_L^⊥).

Therefore, denoting by “adj” the adjugate matrix, it follows that (28) RD^BDR⁻¹ = RP_LR⁻¹adj(KRP_LR⁻¹+RP_L^⊥R⁻¹)

det(DP_L+P_L^⊥) .

By Assumption (B3), the eigenvalues of D are not smaller than ρ > 0. The proof of [25, Lemma 11] shows that the eigenvalues ofDP_L+P_L^⊥ are not smaller thanρ >0, too. This implies that det(DP_L+P_L^⊥)≥ρⁿ⁻¹ >0. The coefficients

(RP_LR⁻¹)_ij =δ_ij −c_i, (RP_L^⊥R⁻¹)_ij =c_i

are bounded forc∈[0,1]ⁿand, by Assumption (B4), the coefficients ofK are also bounded.

Therefore, all elements of adj(KRP_LR⁻¹+RP_L^⊥R⁻¹) are bounded. We conclude from (28) that the entries ofRD^BDR⁻¹ are bounded in [0,1]ⁿ, i.e., point (iii) holds.

The most important property is the positive definiteness of D^BD(c) on L(c); see (27).

This property implies the a priori estimates proved in the following subsection.

2.2. A priori estimates. We show an energy inequality for smooth solutions.

Lemma 5 (Free energy inequality). Let c ∈ C^∞(Ω×(0,∞);Rⁿ) be a positive, bounded, smooth solution to (1)–(5). Then, for any 0< λ < λ_m,

d dt

H(c) + (λ_M −λ)² λ_mλ E(c)

+ 2λ

Z

Ω

|∇√

c|²dx+λ Z

Ω

|∆c|²dx

+ (λ_M −λ)² 2λ

Z

Ω

|P_L(c)R(c)∇µ|²dx≤0.

where the entropy H(c) and the free energy E(c) are given by (6) and λm, λM are defined in Lemma 4.

Proof. We derive first the energy inequality. To this end, we multiply equation (25) for ci

by µi = (∂E/∂ci)(c), integrate over Ω, integrate by parts (using the boundary conditions (26)), and take into account the lower bound (27) for D^BD(c):

dE dt(c) =

n

X

i=1

Z

Ω

∂E

∂c_i(c)∂_tc_idx=−

n

X

i,j=1

Z

Ω

B_ij(c)∇µ_i· ∇µ_jdx (29)

=−

n

X

i,j=1

D_ij^BD(c)(√

c_i∇µ_i)·(√

c_j∇µ_j)dx≤ −λ_m Z

Ω

|P_L(c)R(c)∇µ|²dx.

The entropy inequality is derived by multiplying (25) by logc_i, integrating over Ω, and integrating by parts (using the boundary conditions (26)):

dH dt (c) =

n

X

i=1

Z

Ω

(logci)∂tcidx=−

n

X

i,j=1

Z

Ω

Bij(c)∇logci· ∇µjdx.

To estimate the right-hand side, we set G=RPLR (omitting the argument c) and M :=

B−λG for λ∈(0, λ_m). Then (30) dH

dt (c) =−

n

X

i,j=1

Z

Ω

M_ij∇logc_i· ∇µ_jdx−λ

n

X

i,j=1

Z

Ω

G_ij∇logc_i· ∇µ_jdx=:I₁+I₂. Before estimating the integrals I₁ and I₂, we start with some preparations. We use Lemma 4 (ii) and P_L^TP_L=P_L to obtain

z^TBz = (Rz)^TD^BDRz ≥λ_m|P_LRz|² =λ_m(P_LRz)^T(P_LRz) =λ_mz^TGz for z ∈Rⁿ. The matrix M is positive semidefinite since for any z ∈Rⁿ,

(31) z^TMz=z^TBz−λz^TGz ≥(λ_m−λ)z^TGz = (λ_m−λ)|P_LRz|². Furthermore, by Lemma 4 (ii) again, we have the upper bound

(32) z^TMz =z^T(B−λG)z ≤(λ_M −λ)z^TGz = (λ_M −λ)|P_LRz|².

We are now in the position to estimate the integral I₁, using Young’s inequality for any θ >0:

I1 ≤ θ 2

n

X

i,j=1

Z

Ω

Mij∇logci· ∇logcjdx+ 1 2θ

n

X

i,j=1

Z

Ω

Mij∇µi· ∇µjdx (33)

≤ θ

2(λ_M −λ) Z

Ω

|P_LR∇logc|²dx+λ_M −λ 2θ

Z

Ω

|P_LR∇µ|²dx

= 2θ(λ_M −λ) Z

Ω

|∇√

c|²dx+λ_M −λ 2θ

Z

Ω

|P_LR∇µ|²dx,

where the last step follows from Pn

j=1(P_L)_ijR_j∇logc_j = 2∇√

c_i, which is a consequence of Pn

j=1∇c_j = 0. For the integral I₂, we use the definitions G_ij = c_iδ_ij −c_ic_j and µ_j = logc_j −∆c_j:

I₂ =−λ

n

X

i,j=1

Z

Ω

(c_iδ_ij −c_ic_j)∇c_i

c_i · ∇(logc_j−∆c_j)dx

=−λ

n

X

i=1

Z

Ω

∇c_i· ∇(logc_i −∆c_i)dx+λ Z

Ω n

X

i=1

∇c_i·

n

X

j=1

c_j∇(logc_j−∆c_j)dx

=−λ

n

X

i=1

Z

Ω

∇c_i· ∇(logc_i −∆c_i)dx=−λ Z

Ω

4|∇√

c|²+|∆c|² dx,

where we integrated by parts in the last step.

Inserting the estimates for I₁ and I₂ into (30) yields dH

dt (c) + 4λ Z

Ω

|∇√

c|²dx+λ Z

Ω

|∆c|²dx

≤2θ(λM −λ) Z

Ω

|∇√

c|²dx+λ_M −λ 2θ

Z

Ω

|PLR∇µ|²dx.

We set θ=λ/(λ_M −λ) to conclude that

(34) dH

dt (c) + 2λ Z

Ω

|∇√

c|²dx+λ Z

Ω

|∆c|²dx≤ (λ_M −λ)² 2λ

Z

Ω

|P_LR∇µ|²dx.

The right-hand side can be absorbed by the corresponding term in (29). Indeed, adding the previous inequality to (29) times (λ_M −λ)²/(λ_mλ) finishes the proof.

Note that the energy inequality (29) or the entropy inequality (34) alone are not sufficient to control the derivatives of c but only a suitable linear combination. We will prove these inequalities rigorously in the following section for weak solutions; see Lemma 8.

3. Proof of Theorem 1

We prove the existence of global weak solutions to (1)–(4). For this, we construct an approximate system depending on a parameter δ > 0, similarly as in [14], and then pass to the limit δ→0.

3.1. An approximate system. In order to deal with the degeneracy of the matrixB(c) when a component of c vanishes, we introduce the cutoff function χ_δ :Rⁿ→Rⁿ by

(χ_δc)_i :=







δ for c_i < δ,

c_i for δ≤c_i ≤1−δ, 1−δ for c_i >1−δ, and define the approximate matrix

(35) B^δ(c) :=R(χ_δc)D^BD(χ_δc)R(χ_δc), recalling that R(χ_δc) = diag(√

χ_δc). We wish to solve the approximate problem

∂tc^δ_i = div

n

X

j=1

B_ij^δ(c^δ)∇µ^δ_j, µ^δ_j = ∂E^δ

∂c_j(c^δ) in Ω, t >0, (36)

c^δ_i(·,0) =c⁰_i in Ω,

n

X

j=1

B_ij^δ(c^δ)∇µ^δ_j ·ν = 0, ∇c^δ_i ·ν = 0 on ∂Ω, (37)

where i= 1, . . . , n, Pn

i=1c⁰_i = 1 and the approximate energy is defined by E^δ(c) :=H^δ(c) + 1

2

n

X

i=1

Z

Ω

|∇c_i|²dx, H^δ(c) :=

n

X

i=1

Z

Ω

h^δ_i(c_i)dx,

h^δ_i(r) =







rlogδ−δ/2 +r²/(2δ) for r < δ,

rlogr for δ≤r≤1−δ,

rlog(1−δ)−(1−δ)/2 +r²/(2(1−δ)) for r >1−δ.

(38)

Observe that the solutionsc^δ_i may be negative. We will show below that c^δ_i converges to a nonnegative function as δ →0. The approximate entropy density is chosen in such a way that h^δ_i ∈C²(R). Indeed, we obtain

(h^δ_i)⁰(c_i) =







logδ+ci/δ for ci < δ,

logci+ 1 for δ < ci <1−δ, log(1−δ) +c_i/(1−δ) for c_i >1−δ,

(h^δ_i)⁰⁰(c_i) = 1 (χ_δc)_i. With these definitions, we obtain µ^δ_i = (h^δ_i)⁰(c^δ_i)−∆c^δ_i for i= 1, . . . , n.

Theorem 6 (Existence for the approximate system). Let Assumptions (A1)–(A2) and (B1)–(B4) hold and let δ > 0. Then there exists a weak solution (c^δ,µ^δ) to (36)–(37) satisfying Pn

i=1c^δ_i(t) = 1 in Ω, t >0,

c^δ_i ∈L^∞_loc(0,∞;H¹(Ω))∩L²_loc(0,∞;H²(Ω)),

∂_tc_i ∈L²_loc(0,∞;H²(Ω)⁰), µ^δ_i ∈L²_loc(0,∞;H¹(Ω)), i= 1, . . . , n,

and the first equation in (36) as well as the initial condition in (37) are satisfied in the sense of L²_loc(0,∞;H²(Ω)⁰).

The proof of this theorem is deferred to Appendix A, since it is technical and involves well-established techniques. We show some properties of the matrix B^δ(c). We introduce the matrices P_L(χ_δc), P_L^⊥(χ_δc)∈R^n×n with entries

P_L(χ_δc)_ij =δ_ij −

p(χ_δc)_i(χ_δc)_j Pn

k=1(χ_δc)_k , P_L^⊥(χ_δc)_ij =

p(χ_δc)_i(χ_δc)_j Pn

k=1(χ_δc)_k , i, j = 1, . . . , n.

Lemma 7 (Properties of B^δ(c)). Suppose that D(c) satisfies Assumptions (B1)–(B4).

Then Lemmas 3 and 4 hold withP_L(c), P_L^⊥(c), andD^BD(c)replaced byP_L(χ_δc), P_L^⊥(χ_δc), and D^BD(χ_δc). As a consequence, the matrix B^δ(c), defined in (35), satisfies

(39) z^TB^δ(c)z≥λ_m|P_L(χ_δc)R(χ_δc)z|² for any z,c∈Rⁿ,

and the first (n−1)×(n−1) submatrix Be^δ(c) of B^δ(c) is positive definite and satisfies for η(δ) =λ_mδ²/n,

(40) ze^TBe^δ(c)ze≥η(δ)|z|e² for any ze∈Rⁿ⁻¹.

Proof. It can be verified that Assumptions (B1)–(B2) hold for D(χ_δc), so Lemmas 3 and 4 still hold for the matrix D(χ_δc). Inequality (39) is a direct consequence of Lemma 4 (ii).

It remains to prove (40). We define for given ze∈Rⁿ⁻¹ the vector z ∈Rⁿ with z_i =ze_i for i= 1, . . . , n−1 and z_n= 0. Then (39) becomes

(41) ze^TBe^δ(c)ze≥λ_m

eP_L(χ_δc)R(χe _δc)ze

2 =λ_m R(χe _δc)zeT

Pe_L(χ_δc) R(χe _δc)ze ,

where Ae denotes the first (n−1)×(n −1) submatrix of a given matrix A ∈ R^n×n. It follows from the Cauchy–Schwarz inequality that for anyζ ∈Rⁿ⁻¹,

ζ^TPe_L(χ_δc)ζ =

n−1

X

i=1

ζ_i²−

n−1

X

j=1

s

(χδc)j

Pn

k=1(χ_δc)_kζ_j

!²

≥ |ζ|²−

n−1

X

j=1

(χδc)j

Pn

k=1(χ_δc)_k|ζ|²

= (χ_δc)_n Pn

k=1(χ_δc)_k|ζ|² ≥ δ n|ζ|². Therefore, (41) becomes

ze^TBe^δ(c)ze≥ λ_mδ n

n−1

X

i=1

p(χ_δc)_ize_i

2 = λ_mδ n

n−1

X

i=1

(χ_δc)_i ez_i

2 ≥ λ_mδ² n |z|e²,

which proves (40).

3.2. Uniform estimates. We derive energy and entropy estimates for the solutions to (36), being uniform in δ.

Lemma 8 (Energy and entropy inequalities). Let c^δ be a weak solution to (36)–(37), constructed in Theorem 6. Then the following inequalities hold for any T >0,

E^δ(c^δ(·, T)) +

n

X

i,j=1

Z T 0

Z

Ω

B^δ_ij(c^δ)∇µ^δ_i · ∇µ^δ_jdxdt≤ E^δ(c⁰), (42)

H^δ(c^δ(·, T)) +

n

X

i,j=1

Z T 0

Z

Ω

B_ij^δ(c^δ)∇(h^δ_i)⁰(c^δ_i)· ∇µ^δ_jdxdt≤ H^δ(c⁰), (43)

H^δ(c^δ(·, T)) + (λ_M −λ)²

2λ_mλ E^δ(c^δ(·, T)) +λ

n

X

i=1

Z T 0

Z

Ω

|∇c^δ_i|² (χ_δc^δ)_idxdt (44)

+λ

n

X

i=1

Z T 0

Z

Ω

(∆c^δ_i)²dxdt+(λ_M −λ)² 2λ

Z T 0

Z

Ω

P_L(χ_δc^δ)R(χ_δc^δ)∇µ^δ

2dxdt

≤ H^δ(c⁰) + (λ_M −λ)²

2λ_mλ E^δ(c⁰),

where 0< λ < λ_m, λ_m, λ_M are introduced in Lemma 4, and R(χ_δc^δ) = diag(p χ_δc^δ).

Proof. Summing (94) with σ = 1 overk = 1, . . . , N, we find that Ee^δ(ce^(τ)(·, T)) +

n−1

X

i,j=1

Z T 0

Z

Ω

Be^δ_ij(ce^(τ))∇w_i^(τ)· ∇w^(τ)_j dxdt +ε

n

X

i=1

Z T 0

Z

Ω

(∆w^(τ)_i )²+ (w^(τ)_i )²

dxdt≤Ee^δ(ce⁰).

We know from (97) and the construction of χ_δ that (w^(τ)) is bounded in L²(0, T;H¹(Ω)) and (Be_ij^δ(ec)) is bounded in L^∞(Q_T) with respect to (ε, τ). Therefore, we can pass to the limit (ε, τ) → 0 in the previous inequality, and weak lower semicontinuity of the integral functionals leads to (42).

To show (43), we use (h^δ_i)⁰(c^δ_i)−(h^δ_i)⁰(c^δ_n) as a test function in the weak formulation of (87) and sum over i= 1, . . . , n−1:

H^δ(c(·, T)) +

n−1

X

i,j=1

Z T 0

Z

Ω

Be_ij^δ(ec^δ)∇ (h^δ_i)⁰(c^δ_i)−(h^δ_i)⁰(c^δ_n)

· ∇w^δ_jdxdt≤ H^δ(c⁰).

This inequality can be rewritten as (43) using w^δ_i = µ^δ_i −µ^δ_n. Finally, we derive (44) by combining (43) and (42) and proceeding as in the proof of Lemma 5.

3.3. Proof of Theorem 1. We perform the limit δ → 0 to finish the proof of Theorem 1. It follows from [15, Lemma 2.1] that for sufficiently small δ > 0, there exists C > 0 (independent of δ) such that for allr₁, . . . , r_n∈R satisfying Pn

i=1r_i = 1, (45)

n

X

i=1

h^δ_i(ri)≥ −C.