Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

Подождите немного. Документ загружается.

78 G. Bizhanova

We can represent the derivatives ∂

∂

W (x, t), i =1,...,n, μ=1,...,n−

1, j =0, 1, in the form

∂

W (x, t)=



dτ





g(y



,τ) − g(x



,τ)



erfc

√

aτ

×∂

∂



(x −y,t −τ) − Γ

(x −y

∗

,t− τ)



by an identity



n−1

∂



(x − y, t − τ) − Γ

(x −y

∗

,t− τ)





=0.

Applying the notations (A.9), (A.10) and the estimates (A.6) for Γ

(x, t),

(A.2) for erfc

√

,(A.8)forJ

|ξ|

−ξ

≤ c

−ξ

,β≥ 0, (A.12)

and integrating with respect to y



we obtain

|W (x, t)|≤c



√

t − τ

,t− τ, τ;8)dτ ≤ c

−

8at

; (A.13)

|∂

∂

W (x, t)|≤c



s−2−α

(t −τ)

1+j/2−α/2

,t− τ, τ;8)dτ

≤ c

s−1−j

−

8at

,j=0, 1,

andfromhereweshallhavetheestimates

|∂

W (x, t)|≤c

s−1

−

8at

, |∂

∂

W (x, t)|≤c

s−2

−

8at

, (A.14)

i =1,...,n, μ =1,...,n− 1,s∈ [2, 2+α].

We evaluate ∂

W (x, t), ∂

W (x, t). For that we write down the derivative

∂

W (x, t) as follows:

∂

W (x, t)=



dτ



n−1

g(y



,τ)Γ

n−1



− y



,t− τ)J

,t− τ, τ) dy



, (A.15)

(·)=−



∞

erfc

√

aτ

∂



−y

,t− τ)+Γ

+ y

,t− τ)



where Γ

n−1

,Γ

are deﬁned by (A.5). Integrating J

by parts and applying formula

(A.7) we obtain

,t− τ, τ)=2Γ

,t− τ) −

2aπ



(t − τ)τ

,t−τ,τ;4)



,t− τ) − Γ

,t)



On the Classical Solvability 79

and the derivative (A.15) takes the form

∂

W (x, t)=2





,t− τ) − Γ

,t)



dτ



n−1

g(y



,τ)Γ

n−1



− y



,t− τ) dy



From here with the help of the formula



n−1



−y



,t− τ) dy



= 1 (A.16)

we derive

∂

W (x, t)=2



∂



,t− τ) − Γ

,t)



dτ



n−1



g(y



,τ) − g(x



,t)



n−1



− y



,t− τ) dy



+2g(x



,t)



∂



,t− τ) − Γ

,t)



dτ

:= J

(x, t)+J

(x, t). (A.17)

Taking into account the notations (A.9)–(A.11), formula (A.16), the inequal-

ities (A.12) and

−

4a(t−τ)

≤ e

−

4at

,τ∈ (0,t),

then integrating with respect to τ we ﬁnd

|∂

W (x, t)|≤c

1/2

−

4at

(A.18)

and

(x, t)|≤c

+ M

)



s−2−α



(t −τ )

3−α

−

4a(t−τ)

(t − τ)

α/2

3/2

−

4at



dτ

≤ c

+ M

) t

s−2

−

4at

, (A.19)

(x, t)=

g(x



,t)



√

aπt

−

4at

− erfc

√



, (A.20)

(x, t) ≤ c

−

8at

. (A.21)

We make use of estimates (A.19), (A.21) in (A.17) and take into account that

+ M

= |g|

s−2,R

, e

−ξ

≤ e

−ξ

, then we obtain

|∂

W (x, t)|≤c



+ t

s−2

+ M

)



−

8at

≤ c

|g|

(α)

s−2,R

−

8at

(A.22)

(here s ∈ [2, 2+α]).

80 G. Bizhanova

From the equation in (A.1) with the help of formulas (A.17), (A.20) we ﬁnd

the time derivative ∂

W (x, t),

∂

W (x, t)=aΔ



W (x, t)+aJ

(x, t)+g(x



,t)

√

aπt

−

4at

and applying the estimates (A.14), (A.19), (A.12) to it we derive

|∂

W (x, t)|≤c



+ t

s−2

+ M

)



−

8at

≤ c

|g|

(α)

s−2,R

−

8at

. (A.23)

Gathering obtained estimates (A.13), (A.14), (A.18), (A.22), (A.23) we shall

have an estimate (A.3) of the norm |W |

(2)

s,δ

, δ

, deﬁned by (1.10). 

Acknowledgment

The consideration of these problems was initiated by Professor Herbert Amann.

The author is grateful to him for this and for his very valuable remarks and advices.

References

[1] V.S. Belonosov, Estimates of the solutions of the parabolic systems in the weighted

H¨older classes and some of their applications. Mat. Sb. 110 (1979), N 2, 163–188 (in

Russian).

[2] V.S. Belonosov and T.I. Zelenyak, Nonlocal Problems in the Theory of Quasilinear

Parabolic Equations. Novosibirsk Gos. Univ., Novosibirsk, 1975 (in Russian).

[3] G.I. Bizhanova, Solution in H¨older spaces of boundary-value problems for parabolic

equations with nonconjugate initial and boundary data. Sovremennaya Matematika.

Fundamental’nye Napravleniya. 36 (2010), 12–23 (in Russian) (English transl. Jour-

nal of Math. Sciences, 171 (2010), N. 1, 9–21).

[4] G.I. Bizhanova, Classical solution of a nonregular onjunction problem for the heat

equations.Matem.Jurnal10 (2010), N. 3, 37–48.

[5] Handbook of mathematical functions. Edited by M. Abramowitz, I. Stegun, Moscow,

Nauka, 1979.

[6] O.A. Ladyˇzenskaja, V.A. Solonnikov and N.N. Ural’¸ceva, Linear and quasilinear

equations of parabolic type. “Nauka”, Moscow, 1967; translation by S. Smith, Trans-

lations of Mathematical Monographs, American Mathematical Society, Providence,

R.I. 23, 1967.

[7] Y. Martel and Ph. Souplet, Small time boundary behavior of solutions of parabolic

equations with noncompatible data. J. Math. Pures Appl. 79 (2000), 603–632.

[8] V.A. Solonnikov, On an estimate of the maximum of a derivative modulus for a

solution of a uniform parabolic initial-boundary value problem.LOMIPreprintN

P-2-77, 1977 (in Russian).

Galina Bizhanova

Institute of Mathematics

Pushkin str., 125

Almaty 050010, Kazakhstan

e-mail: galya@math.kz, galina math@mail.ru

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 81–93

 2011 Springer Basel AG

On the Maxwell-Stefan Approach

to Multicomponent Diﬀusion

Dieter Bothe

Dedicated to Herbert Amann on the occasion of his 70

anniversary

Abstract. We consider the system of Maxwell-Stefan equations which describe

multicomponent diﬀusive ﬂuxes in non-dilute solutions or gas mixtures. We

apply the Perron-Frobenius theorem to the irreducible and quasi-positive ma-

trix which governs the ﬂux-force relations and are able to show normal el-

lipticity of the associated multicomponent diﬀusion operator. This provides

local-in-time wellposedness of the Maxwell-Stefan multicomponent diﬀusion

system in the isobaric, isothermal case.

Mathematics Subject Classiﬁcation (2000). Primary 35K59; Secondary 35Q35,

76R50, 76T30, 92E20.

Keywords. Multicomponent diﬀusion, cross-diﬀusion, quasilinear parabolic

systems.

1. Introduction

On the macroscopic level of continuum mechanical modeling, ﬂuxes of chemical

components (species) are due to convection and molecular ﬂuxes, where the latter

essentially refers to diﬀusive transport. The almost exclusively employed constitu-

tive “law” to model diﬀusive ﬂuxes within continuum mechanical models is Fick’s

law, stating that the ﬂux of a chemical component is proportional to the gradient

of the concentration of this species, directed against the gradient. There is no in-

ﬂuence of the other components, i.e., cross-eﬀects are ignored although well known

to appear in reality. Actually, such cross-eﬀects can completely divert the diﬀusive

ﬂuxes, leading to so-called reverse diﬀusion (up-hill diﬀusion in direction of the

gradient) or osmotic diﬀusion (diﬀusion without a gradient). This has been proven

in several experiments, e.g., in a classical setting by Duncan and Toor; see [7].

To account for such important phenomena, a multicomponent diﬀusion ap-

proach is required for realistic models. The standard approach within the theory

82 D. Bothe

of Irreversible Thermodynamics replaces Fickian ﬂuxes by linear combinations of

the gradients of all involved concentrations, respectively chemical potentials. This

requires the knowledge of a full matrix of binary diﬀusion coeﬃcients and this

diﬀusivity matrix has to fulﬁll certain requirements like positive semi-deﬁniteness

in order to be consistent with the fundamental laws from thermodynamics. The

Maxwell-Stefan approach to multicomponent diﬀusion leads to a concrete form

of the diﬀusivity matrix and is based on molecular force balances to relate all

individual species velocities. While the Maxwell-Stefan equations are successfully

used in engineering applications, they seem much less known in the mathemati-

cal literature. In fact we are not aware of a rigorous mathematical analysis of the

Maxwell-Stefan approach to multicomponent diﬀusion, except for [8] which mainly

addresses questions of modeling and numerical computations, but also contains

some analytical results which are closely related to the present considerations.

2. Continuum mechanical modeling of multicomponent ﬂuids

We consider a multicomponent ﬂuid composed of n chemical components A

.Start-

ing point of the Maxwell-Stefan equations are the individual mass balances, i.e.,

∂

+div(ρ

)=R

tot

, (1)

where ρ

= ρ

(t, y) denotes the mass density and u

= u

(t, y) the individual

velocity of species A

. Note that the spatial variable is denoted as y, while the usual

symbol x will refer to the composition of the mixture. The right-hand side is the

total rate of change of species mass due to all chemical transformations. We assume

conservation of the total mass, i.e., the production terms satisfy

i=1

tot

=0.

Let ρ denote the total mass density and u be the barycentric (i.e., mass averaged)

velocity, determined by

ρ :=

i=1

,ρu :=

i=1

Summation of the individual mass balances (1) then yields

∂

ρ +div(ρu)=0, (2)

i.e., the usual continuity equation.

In principle, a full set of n individual momentum balances should now be

added to the model; cf. [11]. But in almost all engineering models, a single set

of Navier-Stokes equations is used to describe the evolution of the velocity ﬁeld,

usually without accounting for individual contributions to the stress tensor. One

main reason is a lack of information about appropriate constitutive equations for

the stress in multicomponent mixtures; but cf. [16]. For the multicomponent, single

momentum model the barycentric velocity u is assumed to be determined by the

Navier-Stokes equations. Introducing the mass diﬀusion ﬂuxes

:= ρ

− u)(3)

On Maxwell-Stefan Multicomponent Diﬀusion 83

and the mass fractions Y

:= ρ

/ρ, the mass balances (1) can be rewritten as

ρ∂

+ ρu ·∇Y

+divj

= R

tot

. (4)

In the present paper, main emphasis is on the aspect of multicomponent diﬀu-

sion, including the cross-diﬀusion eﬀects. Therefore, we focus on the special case

of isobaric, isothermal diﬀusion. The (thermodynamic) pressure p is the sum of

partial pressures p

and the latter correspond to c

RT in the general case with c

denoting the molar concentration, R the universal gas constant and T the absolute

temperature; here c

= ρ

with M

the molar mass of species A

. Hence iso-

baric conditions correspond to the case of constant total molar concentration c

tot

where c

tot

i=1

. Still, species diﬀusion can lead to transport of momentum

because the M

are diﬀerent. Instead of u we therefore employ the molar averaged

velocity deﬁned by

tot

v :=

i=1

. (5)

Note that other velocities are used as well; only the diﬀusive ﬂuxes have to be

adapted; see, e.g., [20]. With the molar averaged velocity, the species equations

(1) become

∂

+div(c

v + J

)=r

tot

(6)

with r

tot

:= R

tot

and the diﬀusive molar ﬂuxes

:= c

− v). (7)

Below we exploit the important fact that

i=1

=0. (8)

As explained above we may now assume v = 0 in the isobaric case. In this case the

species equations (6) simplify to a system of reaction-diﬀusion equations given by

∂

+divJ

= r

tot

, (9)

where the individual ﬂuxes J

need to be modeled by appropriate constitutive

equations. The most common constitutive equation is Fick’s law which states that

= −D

grad c

(10)

with diﬀusivities D

> 0. The diﬀusivities are usually assumed to be constant,

while they indeed depend in particular on the composition of the system, i.e.,

= D

(c)withc := (c

,...,c

). Even if the dependence of the D

is taken

into account, the above deﬁnition of the ﬂuxes misses the cross-eﬀects between

the diﬀusing species. In case of concentrated systems more realistic constitutive

equations are hence required which especially account for such mutual inﬂuences.

Here a common approach is the general constitutive law

= −

j=1

grad c

(11)

84 D. Bothe

with binary diﬀusivities D

= D

(c). Due to the structure of the driving forces,

as discussed below, the matrix D =[D

]isoftheformD(c)=L(c) G



(c)witha

positive deﬁnite matrix G



(c), the Hessian of the Gibbs free energy. Then, from

general principles of the theory of Irreversible Thermodynamics, it is assumed that

the matrix of transport coeﬃcients L =[L

]satisﬁes

• L is symmetric (the Onsager reciprocal relations)

• L is positive semideﬁnite (the second law of thermodynamics).

Under this assumption the quasilinear reaction-diﬀusion system

∂

c +div(−D(c) ∇c)=r(c), (12)

satisﬁes – probably after a reduction to n−1 species – parabolicity conditions suf-

ﬁcient for local-in-time wellposedness. Here r(c)isshortfor(r

tot

(c),...,r

tot

(c)).

A main problem now is how realistic diﬀusivity matrices together with their

dependence on the composition vector c can be obtained.

Let us note in passing that Herbert Amann has often been advocating that

general ﬂux vectors should be considered, accounting both for concentration depen-

dent diﬀusivities and for cross-diﬀusion eﬀects. For a sample of his contributions

to the theory of reaction-diﬀusion systems with general ﬂux vectors see [1], [2] and

the references given there.

3. The Maxwell-Stefan equations

The Maxwell-Stefan equations rely on inter-species force balances. More precisely,

it is assumed that the thermodynamical driving force d

of species A

is in local

equilibrium with the total friction force. Here and below it is often convenient to

work with the molar fractions x

:= c

tot

instead of the chemical concentrations.

From chemical thermodynamics it follows that for multicomponent systems which

are locally close to thermodynamical equilibrium (see, e.g., [20]) the driving forces

under isothermal conditions are given as

grad μ

(13)

with μ

the chemical potential of species A

. Equation (13) requires some more

explanation. Recall ﬁrst that the chemical potential μ

for species A

is deﬁned as

∂G

∂c

, (14)

where G denotes the (volume-speciﬁc) density of the Gibbs free energy. The chem-

ical potential depends on c

, but also on the other c

as well as on pressure and

temperature. In the engineering literature, from the chemical potential a part μ

depending on pressure and temperature is often separated and, depending on the

context, a gradient may be applied only to the remainder. To avoid confusion, the

common notation in use therefore is

∇μ

= ∇

T,p

∂μ

∂p

∇p +

∂μ

∂T

∇T.

On Maxwell-Stefan Multicomponent Diﬀusion 85

Here ∇

T,p

means the gradient taken under constant pressure and temperature.

In the isobaric, isothermal case this evidently makes no diﬀerence. Let us also note

that G is assumed to be a convex function of the c

for single phase systems, since

this guarantees thermodynamic stability, i.e., no spontaneous phase separations.

For concrete mixtures, the chemical potential is often assumed to be given by

= μ

+ RT ln a

(15)

with a

the so-called activity of the ith species; equation (15) actually implicitly

deﬁnes a

. In (15), the term μ

depends on pressure and temperature. For a mixture

of ideal gases, the activity a

equals the molar fraction x

. The same holds for

solutions in the limit of an ideally dilute component, i.e., for x

→ 0+. This is no

longer true for non-ideal systems in which case the activity is written as

= γ

(16)

with an activity coeﬃcient γ

which itself depends in particular on the full com-

position vector x.

The mutual friction force between species i and j is assumed to be propor-

tional to the relative velocity as well as to the amount of molar mass. Together

with the assumption of balance of forces this leads to the relation

= −

j=i

− u

) (17)

with certain drag coeﬃcients f

> 0; here f

= f

is a natural mechanical

assumption. Insertion of (13) and introduction of the so-called Maxwell-Stefan

(MS) diﬀusivities

=1/f

yields the system

grad μ

= −

j=i

− x

tot

for i =1,...,n. (18)

The set of equations (18) together with (8) forms the Maxwell-Stefan equations

of multicomponent diﬀusion. The matrix [

] of MS-diﬀusivities is assumed to be

symmetric in accordance with the symmetry of [f

]. Let us note that for ideal

gases the symmetry can be obtained from the kinetic theory of gases; cf. [9] and

[14]. The MS-diﬀusivities

will in general depend on the composition of the

system.

Due to the symmetry of [

], the model is in fact consistent with the Onsager

reciprocal relations (cf. [18] as well as below), but notice that the

are not to be

inserted into (11), i.e., they do not directly correspond to the D

there. Instead,

the MS equations have to be inverted in order to provide the ﬂuxes J

Note also that the Ansatz (17) implies

= 0 because of the symmetry of

], resp. of [

]. Hence

= 0 is necessary in order for (17) to be consistent.

It in fact holds because of (and is nothing but) the Gibbs-Duhem relation, see,

e.g., [12]. The relation

= 0 will be important below.

86 D. Bothe

Example (Binary systems). For a system with two components we have

(= −d

)=−

tot



− x



. (19)

Using x

+ x

=1andJ

+ J

= 0 one obtains

(= −J

)=−

grad μ

. (20)

Writing c and J instead of c

and J

, respectively, and assuming that the chemical

potential is of the form μ = μ

+ RT ln(γc) with the activity coeﬃcient γ = γ(c)

this ﬁnally yields

J = −



cγ



(c)

γ(c)



grad c. (21)

Inserting this into the species equation leads to a nonlinear diﬀusion equation,

namely

∂

c −Δφ(c)=r(c), (22)

where the function φ : R → R satisﬁes φ



(s)=

(1 + sγ



(s)/γ(s)) and, say,

φ(0) = 0. Equation (22) is also known as the ﬁltration equation (or, the generalized

porous medium equation) in other applications. Note that well-known pde-theory

applies to (22) and especially provides well-posedness as soon as φ is continuous

and nondecreasing; cf., e.g., [21]. The latter holds if s → sγ(s) is increasing which

is nothing but the fact that the chemical potential μ of a component should be an

increasing function of its concentration. This is physically reasonable in systems

without phase separation.

4. Inversion of the ﬂux-force relations

In order to get constitutive equations for the ﬂuxes J

from the Maxwell-Stefan

equations, which need to be inserted into (9), we have to invert (18). Now (18)

alone is not invertible for the ﬂuxes, since these are linearly dependent. Elimination

of J

by means of (8) leads to the reduced system

tot

⎡

⎣

n−1

⎤

⎦

= −B

⎡

⎣

n−1

⎤

⎦

, (23)

where the (n − 1) ×(n −1)-matrix B is given by

⎧

⎪

⎨

⎪

⎩



−



for i = j,

k=i

for i = j (with x

=1−

m<n

(24)

On Maxwell-Stefan Multicomponent Diﬀusion 87

Assuming for the moment the invertibility of B and letting μ

be functions of the

composition expressed by the molar fractions x =(x

,...,x

), the ﬂuxes are given

⎡

⎢

⎣

n−1

⎤

⎥

⎦

= −c

tot

−1

⎡

⎢

⎣

∇x

n−1

⎤

⎥

⎦

, (25)

where

Γ =[Γ

]withΓ

= δ

+ x

∂ ln γ

∂x

(26)

captures the thermodynamical deviations from the ideally diluted situation; here

denotes the Kronecker symbol.

Example (Ternary systems). We have

B =

⎡

⎢

⎣

+ x



−



−x



−



−x



−



+ x



−



⎤

⎥

⎦

(27)

and det(B − tI)=t

−tr B t +detB with

det B =

≥ min





(28)

and

tr B =

+ x

≥ 2min





. (29)

It is easy to check that (tr B)

≥ 3detB for this particular matrix and therefore

the spectrum of B

−1

is in the right complex half-plane within a sector of angle less

than π/6. This implies normal ellipticity of the diﬀerential operator B

−1

(x)(−Δx).

Recall that a second-order diﬀerential operator with matrix-valued coeﬃcients is

said to be normally elliptic if the symbol of the principal part has it’s spectrum

inside the open right half-plane of the complex plane; see section 4 in [2] for

more details. This notion has been introduced by Herbert Amann in [1] as the

appropriate concept for generalizations to more general situations with operator-

valued coeﬃcients.

Consequently, the Maxwell-Stefan equations for a ternary system are locally-

in-time wellposed if Γ = I, i.e., in the special case of ideal solutions. The latter

refers to the case when the chemical potentials are of the form (15) with γ

≡ 1

for all i. Of course this extends to any Γ which is a small perturbations of I, i.e.,

to slightly non-ideal solutions.

Let us note that Theorem 1 below yields the local-in-time wellposedness also

for general non-ideal solutions provided the Gibbs energy is strongly convex. Note

also that the reduction to n −1 species is the common approach in the engineering