pdfs.semanticscholar.org › f602 › 695bf121721d377f4887e33… · ON MICRO-MACRO-MODELS FOR TWO-PHASE FLOW WITH DILUTE POLYMERIC SOLUTIONS MODELING AND ANALYSIS GÜNTHER GRÜN AND

ON MICRO-MACRO-MODELS FOR TWO-PHASE FLOW WITHDILUTE POLYMERIC SOLUTIONS MODELING AND ANALYSIS

GÜNTHER GRÜN AND STEFAN METZGER

Abstract. By methods from nonequilibrium thermodynamics, we derive a diuse-interface model for two-phase ow of incompressible uids with dissolved noninteractingpolymers. The polymers are modeled by dumbbells subjected to general elastic spring-force potentials including in particular Hookean and nitely extensible, nonlinear elastic(FENE) potentials. Their density and orientation are described by a Fokker-Planck-typeequation which is coupled to a Cahn-Hilliard and a momentum equation for phase-eldand gross velocity/pressure. Henry-type energy functionals are used to describe dierentsolubility properties of the polymers in the dierent phases or at the liquid-liquid inter-face. Taking advantage of the underlying energetic/entropic structure of the system, weprove existence of a weak solution globally in time for the case of FENE-potentials. Asa by-product in the case of Hookean spring potentials, we derive a macroscopic diuse-interface model for two-phase ow of Oldroyd-B-type liquids.

1. Introduction

In this paper, we are concerned with modeling and analysis of two-phase ow of dilutepolymeric solutions. We assume the solvents to be incompressible and immiscible, andwe formulate a diuse interface model that couples a momentum equation and a CahnHilliard type phase-eld equation to a FokkerPlanck equation which models orientationand elongation of the polymers. To the best of our knowledge this is the rst paperwhich derives multi-scale models for such ow phenomena based on approaches from non-equilibrium thermodynamics.In recent years, many authors have contributed to a mathematical theory for multi-scalemodeling of dilute polymeric ows (see [7, 10, 9, 13, 16, 25, 29, 32, 30, 34, 35, 37, 40, 44] andthe references therein). Neglecting for the sake of brevity models which couple Langevinequations to macroscopic equations (cf. [29]) and focussing instead on FokkerPlanckNavierStokes approaches, a generic single phase model reads as

∂tu + (u · ∇)u− divx 2ηDu+∇xp = divx

ˆD

ψU ′(12|q|2)q⊗ q−

ˆD

ψ1

, (1.1a)

divx u = 0 in Ω× (0, T ) , (1.1b)

∂tψ + u · ∇xψ + divq ψ∇xu · q = cq divq ψ∇qµψ+ cx divx ψ∇xµψ , (1.1c)

µψ = log(ψM

)in Ω×D × (0, T ) (1.1d)

with η, cq > 0 and cx ≥ 0.In this generic model, polymers are modeled for simplicity by dumbbells, and the

Date: September 15, 2015.2010 Mathematics Subject Classication. 35Q35,76A05,76D03, 76D05,76T99,82C31,82D60 .Key words and phrases. Two-phase ow, Cahn-Hilliard equation, Fokker-Planck equation, Navier-

Stokes equations, polymeric ow models, nitely extensible nonlineare elastic, existence of weak solutions.

1

2 G. GRÜN AND S. METZGER

underlying entropic forces are represented by an elastic spring potential U . The functionψ(x, t) : D → R+

0 describes a distribution on the conguration spaceD ⊂ Rd of admissibleelongations/orientations of the polymers. Associated with U , there comes the Maxwellian

M(q) :=exp

(−U(

12|q|2))

´D

exp(−U(

12|q|2))dq

.

The pair (u, p) stands for the velocity and pressure eld, respectively, and the tensoron the right-hand side of equation (1.1a) is the Kramers stress tensor (see [26, 12] andthe references therein) which models eects exerted by the polymers on the solvent ow.Equation (1.1c) is the usual evolution equation for the probability density on the con-guration space. Note that divx ψ∇xµψ describes the center-of-mass diusion of thedumbbells. Although the parameter cx is magnitudes smaller than one, it seems reason-able to keep the corresponding diusion term for mathematical reasons: It guaranteesparabolicity of (1.1c). For approaches setting cx ≡ 0, see [34, 35] and the referencestherein.In the present paper, we suggest a class of diuse interface models for incompressibletwo-phase ow of polymeric solutions which looks in its basic version as follows (case ofsolvents with identical mass densities, setting mass density and surface tension to one).

∂tu + (u · ∇x)u− divx 2η(φ)Du+∇xp

= µφ∇xφ+

ˆD

µψ∇xψ + divx

ˆD

ψU ′q⊗ q−ˆD

ψ1

, (1.2a)

divx u = 0 , (1.2b)

∂tφ+ u · ∇xφ− divx m(φ)∇xµφ = 0 , (1.2c)

µφ = −δ∆xφ+ 1δW ′(φ) + β′(φ)

ˆD

ψ in Ω× (0, T ) , (1.2d)

∂tψ + u · ∇xψ + divq ψ∇xu · q = cq divq ψ∇qµψ+ cx divx ψ∇xµψ , (1.2e)

µψ = log(ψM

)+ β(φ) in Ω×D × (0, T ) . (1.2f)

Energetically, the model is based on a physical energy given as the sum of the kinetic en-ergy 1

2

´Ω|u|2, a Cahn-Hilliard energy δ

2

´Ω|∇xφ|2 + 1

δ

´ΩW (φ), where W (φ) = 1

4(1− φ2)

2

is a standard double-well potential, an entropic component´

Ω×D ψ(log(ψM

)− 1), and the

so called Henry energy´

Ωβ(φ)

´Dψ dq dx. Note that choosing β(φ) appropriately allows

to model discrepancies in the solubility of the polymers (cf. [3]). In diuse interfacemodels, the interface separating the two uid phases is assumed to be of nite width. Aphase-eld parameter φ representing pure phases by values ±1 is supposed to modelthe evolution of the ow. Two advantages of diuse interface models compared to sharp-interface models are well known. First, no articial additional conditions are necessaryto model topology changes or to guarantee conservation of individual masses. Secondly,in many cases it is possible to prove global existence of solutions and to formulate con-vergent numerical schemes (see e.g. [2], [20] and the references therein). During thelast two decades, much eort has been invested to construct starting from the famousmodel H of Halperin and Hohenberg [23] diuse interface models for the case of liquid

TWO-PHASE FLOW WITH DILUTE POLYMERIC SOLUTIONS 3

constituents with dierent mass densities which are consistent with thermodynamics inthe sense that the physical energy is a Lyapunov functional for the evolution providedno external forces are applied (cf. [3, 4, 33]). Onsager's variational principle (cf. [39]and the references therein) turned out to be an ecient workhorse in the derivation ofsuch models (although not applied in each of the aforementioned papers). Similarly, theenergetic structure underlying the model (1.1) would allow a derivation along these lines(although Barrett and Süli followed a dierent pathway in [9]).In the present paper we use Onsager's approach to suggest a diuse interface model con-sistent with thermodynamics for general mass densities and dumbbell models with genericspring potentials.In the second part, we prove existence of weak solutions in two and in three spatial di-mensions. Here, we conne ourselves to the case of identical solvent mass densities andto FENE potentials U where FENE stands for nitely extensible, nonlinear elastic.That is, we take a ball D = BQmax ⊂ R

d for the conguration space and we considerlogarithmic potentials on D becoming singular on the boundary ∂D.The outline of the present paper is as follows. Section 2 is devoted to the derivation of themodel based on the diuse interface approach. The idea is to model the two-phase owby a coupled system of momentum equations, phase-eld equations and a FokkerPlanckequation for the congurational density ψ. The latter will be dened on Ω×D×R+

0 , whereD is the set of admissible elongations of the dumbbells (in the case of FENE-models, forinstance, a ball in Rd centered at the origin). In particular, the marginal

´Dψ(x,q, t) dq,

stands for the number density of polymer chains with barycenter in x at time t.To derive this system in a thermodynamically consistent way, we take advantage of On-sager's variational principle. For this, we formulate generic conservation laws for themomentum ρu, the phase-eld parameter φ and the congurational density ψ. Excludingpolymer-polymer interactions and assuming the contribution of the polymers to the totalmomentum to be negligible ("dilute solution"), we dene the momentum ρ(x, t)u(x, t) viathe sum of the mass densities of the solvent constituents ρ(x, t) := ρ1(x, t) + ρ2(x, t) andthe gross velocity u which is obtained by volume averaging from the individual solventvelocities, compare [15], [3]. As a consequence of volume averaging, the gross velocityeld will be solenoidal. For the phase-eld parameter φ, we take the dierence φ2 − φ1,where φi := ρi(x,t)

ρiare the volume fractions of solvent i (ρi denoting the mass densities in

the pure phases).To obtain an evolution equation for the phase-eld φ, we decompose the ux in the con-servation law for ρ into a term representing transport with the volume averaged velocityu and a correction term to be determined later on. The conservation law for the cong-urational density ψ being dened on Ω × D × R+

0 can be decomposed in a similarway. First, the divergence operator on Ω × D is split into two parts reecting spatialand congurational contributions, respectively. Decomposing the spatial part into a dragterm based on the gross velocity u and an initially unknown ux term, and mimicking thisprocedure for the congurational part, results in a conservation equation being analogousto that one derived in [9] which has been obtained there starting from Langevin's equationfrom statistical mechanics (cf. also [41]).Similarly, the conservation laws for momentum and phase-eld contain ux terms andforce terms which will be determined by Onsager's variational principle. For this, wex the total energy of the system to be the sum of the kinetic energy of the solvent, a


uid-uid interfacial energy (CahnHilliard energy), the Henry energy which describesthe solubility discrepancies of the polymer, and nally an entropic term which penalizesthe deviation of the congurational density ψ from its equilibrium density given by anappropriate Maxwellian. We obtain a micro-macro diuse-interface model for two-phasedilute polymeric ow with generic spring potentials and general mass densities.In some special cases, it is possible to deduce corresponding macroscopic models forviscoelastic two-phase ow. As an example, we consider the case of a Hookean springpotential associated with the conguration space D = Rd. Mimicking the strategies fromthe single phase case (see e.g. [9]), we easily formulate a two-phase Oldroyd-B model.

In Section 3, we prove existence of weak solutions for the aforementioned diuse-interfacemodel in dimensions d = 2, 3. For simplicity, we conne ourselves to the case of matcheddensities ρ1 = ρ2. It turns out that a formulation that mollies the drag term in theFokkerPlanck equation as well as the corresponding stress tensor and the solubility en-ergy is advantageous. First, a directional mollication in the drag term is well-justiedby the modelling approach in [9]. Secondly, mollifying the Henry energy refers to thefact that the polymers cannot be considered to be point objects. Finally, the nonlin-ear coupling of the Cahn-Hilliard equation with the remaining equations causes technicaldiculties: Mollication of the solubility energies, however, allows to work with the stan-dard Dirichlet functional in the CahnHilliard energy. Otherwise, a s-Dirichlet functional(s > 2) seems to be the only remedy cf. e.g. [1, 18].The general concept follows the strategies of Barrett and Süli [9], i.e. discretization w.r.t.time and passage to the limit with the time-increment. To establish the existence ofdiscrete solutions, it turns out that the phase-eld equations can be separated from theremaining equations by adapting ideas used for the numerical analysis of splitting schemesfor NavierStokesCahnHilliard systems ([5, 22] and the references therein). For the pas-sage to the limit, higher regularity of the marginals ω(x, t) =

´Dψ(x,q, t) dq is needed.

First establishing a discrete version of the energy estimate using Young's complemen-tary functions in a suitable way we show the number density ω to be contained inL∞(L2(Ω)) ∩ L2(H1(Ω)). This will be sucient to control the dierence quotients of thevelocity eld in negative Sobolev-spaces. Similarly, compactness in time is readily estab-lished for the phase-eld. Using the linearity of the FokkerPlanck equation, the resultfollows.

Notation: In this paper, Ω denotes the spatial domain of the ow, and D stands forthe conguration space, both sets being contained in Rd with d ∈ 2, 3. The positionvector in Ω ×D is split into a spatial component x and a congurational component q.Consequently, the gradient with respect to x (q) is denoted by ∇x (∇q). By "·", wedenote the Euclidean scalar product on Rd. Sometimes, we write ΩT for the space-timecylinder Ω × (0, T ). By W k,p(Ω), we denote the space of k-times weakly dierentiable

functions with weak derivatives in Lp(Ω). The symbol W k,p0 (Ω) stands for the closure

of C∞0 (Ω) in W k,p(Ω). Corresponding spaces of vector- and matrix-valued functions aredenoted in boldface. A similar notation is used for function spaces dened on D or Ω×D.

For the denition of additional function spaces which will be used in Section 3, we referto the Appendix.


For a Banach space X and a time interval I, the symbol Lp(I;X) stands for the par-abolic space of Lp-integrable functions on I with values in X. We use the notation∂−τ f(t) := 1

τ(f(t)− f(t− τ)) for backward dierence quotients in time of functions in

L1(I;X). By Sγ we denote the Helmholtz-Stokes operator mapping the dual space ofH1

σ,0(Ω) := W ∈H10 (Ω) : divx w = 0 onto H1

σ,0(Ω) (cf. Appendix).

2. Derivation of the model

2.1. FENE-Model. In this subsection, we will briey summarize some basic ideas formodelling polymer chains. Polymers can be seen as long chains consisting of multiple(partly up to several thousand) monomers, which themselves consist of multiple atoms.Tracking the position of all these atoms is often neither practical nor necessary. In fact,it is sucient to regard just N selected beads in the polymer chain. This approach leadsto the so called Rouse chain model. The distances between those beads are governedby an entropic potential which may be described by a spring potential. For additionalinformations, we refer to the work by Le Bris and Lelièvre ([28]).In this paper, for the ease of presentation, we will conne ourselves to the case N = 2,

q(t) = rH(t)− rT(t)

lbarycenter r(t)

H

T

Figure 2.1. Sketch of a dumbbell

the dumbbell model. As sketched in gure 2.1, the polymer chain is represented bytwo beads (T and H) which are connected by a massless, elastic spring. Therefore, theposition of the dumbbell can either be described by the position of the two beads (rH(t),rT (t)) or by the position of the barycenter r(t) := 1

2(rH(t) + rT (t)) and the elongation

vector q(t) := rH(t)− rT (t). Concerning the potential of the elastic spring, we choose theFENE-potential. In contrast to the Hookean potential, the FENE-potential

U(

12|q|2)

= −Q2max

2ln(

1− |q|2Q2max

)(2.1)

bounds the set of admissible elongations. More precisely, it requires

q ∈ D := B(0, Qmax) , (2.2)

where B(0, Qmax) is a bounded, open, origin centered ball with radius Qmax. For furtherinformation about the choice of the spring potential, we refer to [9, 28]. The energetically


most favourable probability density of the elongation of a polymer chain is given by thenormalized Maxwellian

M(q) :=exp

(−U(

12|q|2))

´D

exp(−U(

12|q|2))

dq. (2.3)

A straightforward computation yields

M∇qM−1 = −M−1∇qM = U ′q . (2.4)

For further reference, we collect some properties of the FENE-potential and the Maxwellian,which were proven in [9].

Lemma 2.1. The spring potential U dened in (2.1) and the associated Maxwellian Mhave the following properties on the corresponding set of admissible elongations D :=B(0, Qmax) ⊂ Rd.

(P1) q 7→ U(

12|q|2)∈ C∞(D), nonnegative,

(P2) q 7→ U ′(

12|q|2)is positive on D,

(P3) there exist constants ci > 0 (i = 1, ..., 5) that for κ = Q2max

2the following inequalities

hold true:

c1[dist (q, ∂D)]κ ≤M(q) ≤ c2[dist (q, ∂D)]κ ∀q ∈ D ,

c3 ≤ [dist (q, ∂D)]U ′(

12|q|2)≤c4 , [dist (q, ∂D)]2

∣∣U ′′(12|q|2)∣∣ ≤ c5 ∀q ∈ D ,

(P4)´D

[1 +

(1 + |q|2

)((U)2 + |q|2 (U ′)2)]M dq <∞.

Under the additional assumption Qmax > 2, the FENE-Potential satises the followingestimates:

(P5) There exist constants c6, c7 > 0, such that for B(

0, ( dc7

)1/2)⊂⊂ D

(U ′)2 − U ′′ ≥ c6 ∀q ∈ D and (U ′)

2 − U ′′ ≥ 2c7U′ ∀q ∈ D : |q|2 ≥ d

c7,

In a last step, we dene the congurational density function ψ : Ω×D×R+0 → R. Note

that the marginal

ω(x, t) :=

ˆD

ψ(x,q, t) dq (2.5)

gives the number density of polymer chains with barycenter in x at time t.

Remark 2.2. The denition of the set of admissible elongations D in (2.2) implies thatthere are no other conditions which prohibit a certain elongation. In fact, near the bound-ary of the spatial domain Ω the set of admissible elongations has to be smaller as thepolymer chains may not collide with the boundary ∂Ω. Since the length of the polymers istypically very small in comparison with the macroscopical length, this eect occurs onlyin a small part of Ω. Therefore, following the strategy in [9] (see also [11, 38]), we willneglect the x-dependence of D and retain to the denition in (2.2).


2.2. Conservation laws. We begin with balance equations for mass and momentum ofthe uids. To describe the evolution of the two uid phases, we apply the diuse-interfacemethod, which was used e.g. in [33], [15] and [3] and which allows for a smooth transitionbetween the phases. Starting from the conservation laws for the individual mass densities

∂tρi + divx viρi = 0 , (2.6)

we follow the approach of [15, 3], and we dene the volume fractions φi (i ∈ 1, 2) byφi := ρi

ρi, where ρi denotes the mass density of the ith uid in a pure phase and ρi denotes

the actual mass density in the two-phase ow. Assuming φ1 + φ2 ≡ 1 and dening thephase eld parameter φ as φ := φ2 − φ1, we obtain

ρ(φ) = ρ1+ρ22

+ ρ2−ρ12φ

for the total mass ρ(φ) := ρ1(φ) + ρ2(φ). Note that conservation of´

Ωφ(x, t) dx implies

the conservation of the individual masses (for details, we refer to [3]).Following [3], we get the basic conservation laws

∂tφ+ u · ∇xφ+ divx Jφ = 0 , (2.7)

ρ(φ)∂tu +((ρ(φ)u + ∂ρ

∂φJφ

)· ∇x

)u = divx S−∇xp+ k , (2.8)

where u := φ1v1 + φ2v2 is the volume averaged velocity eld, S is a symmetric stresstensor and k incorporates all additional forces. A straightforward computation showsthat u is solenoidal. In addition to (2.7) and (2.8), we impose the boundary conditions

u = 0 on ∂Ω , (2.9a)

Jφ · nx = 0 on ∂Ω , (2.9b)

which guarantee conservation of mass.To describe the evolution of the congurational density ψ, we follow the same recipe andwe propose the conservation law

∂tψ + divx vxψ+ divq vqψ = 0 (2.10)

on Ω×D. Following the approach in [3] for the mass densities, we rewrite ψvx = ψu+Jψ,xwith a correction ux Jψ,x to be determined later. Similarly, we split ψvq into a driftterm with velocity

uq(x,q, t) :=u(x + ε

2q, t)− u

(x− ε

2q, t)

ε, (2.11)

and some correction ux Jψ,q. Note that uq is intended to model the drift caused bydierent velocities at head and tail of the dumbbell. To reect the dierent length scalesin Ω and D, we introduced the parameter ε in (2.11) which is dened as

ε :=typical length of a polymer chain

typical macroscopic length. (2.12)

Altogether, we end up with

∂tψ + divx uψ + Jψ,x+ divq uqψ + Jψ,q = 0 . (2.13)


By using a directional mollier, one may rewrite (2.11) as

uq(x,q, t) = ∇x

1ε

ˆ ε2

− ε2

u(x + θq, t)dθ

· q . (2.14)

One may argue that ε is such small that we may approximate the integral by u(x, t).Nevertheless, we will follow the approach in [9] and keep the mollication, but use anisotropic mollier instead of the directional one.In the scalar case, we dene the isotropic mollier Jε as follows: Starting with a nonneg-ative, rotationally symmetric ζ ∈ W 1,∞(

Rd)satisfying supp ζ ⊂ B(0, 1) having mass one,

we dene ζε(x) := ε−dζ(ε−1x) and Jε : L1(Ω)→ W 1,∞(Ω) by

Jεf(x) :=

ˆΩ

ζε(x− y)f(y)dy ∀x ∈ Ω , ∀f ∈ L1(Ω) . (2.15)

For further reference, we list some the properties of the isotropic mollier Jε.

‖Jεf‖L2(Ω) ≤ ‖f‖L2(Ω) ∀f ∈ L2(Ω) , (2.16a)

‖(I − Jε)f‖L2(Ω) → 0 as ε 0 ∀f ∈ L2(Ω) , (2.16b)ˆΩ

Jεfg dx =

ˆΩ

fJεg dx ∀f, g ∈ L2(Ω) , (2.16c)

∂xiJεf = Jε∂xif ∀f ∈ H10 (Ω) , i = 1, ..., d , (2.16d)

‖Jεf‖H1(Ω) ≤ ‖f‖H1(Ω) ∀f ∈ H10 (Ω) , (2.16e)

‖Jεf‖W 1,∞(Ω) ≤ C(ε) ‖f‖L1(Ω) ∀f ∈ L1(Ω) . (2.16f)

In the same spirit, we dene the vector- and tensor-valued mollier J ε and Jε compo-nentwise. Obviously, they enjoy properties equivalent to (2.16a)-(2.16f).Replacing the directional mollier in (2.13) and (2.14) by the isotropic one, we get

∂tψ + u · ∇xψ + divq ∇xJ εu · qψ = − divx Jψ,x − divq Jψ,q . (2.17)

In addition, we impose the following boundary conditions to guarantee that the numberof polymers stays constant in the course of the process:

Jψ,x · nx = 0 on ∂Ω×D ×R+, (2.18a)

Jψ,q · nq = −ψ∇xJ ε(u) : nq ⊗ q on Ω× ∂D ×R+. (2.18b)

To determine the unknown quantities in our model, we will use Onsager's variationalprinciple of minimum energy dissipation (cf. [39]). It is supposed to model the mostprobable behaviour of an irreversible process and it postulates the relation

δ(Jφ,Jψ,x,Jψ,q,S)

(dEdt

+ Φ

)!

= 0 (2.19)

between thermodynamical forces (given by the rate of energy in the system) and anappropriate dissipation function Φ which we will dene in (2.26). In (2.19), δ(Jφ,Jψ,x,Jψ,q,S)denotes the rst variation with respect to the quantities Jφ, Jψ,x, Jψ,q, and S.We assume the energy of the system to be the sum of

• the kinetic energy´

Ω12ρ(φ) |u|2 of the macroscopic uid motion,


• the CahnHilliard free energy´

Ωδ2|∇xφ|2 + 1

δW (φ),1

• an entropic term´

Ω×DMψM

(log(ψM

)− 1)which measures the deviation of the

density ψ from the Maxwellian M ,• the Henry energy

´Ω×D Jεβ(φ)ψ which refers to dierent solubilities of the poly-

mer chains in the two dierent uids.

Altogether, we end up with

E(φ, ψ,u) :=

ˆΩ

12ρ(φ) |u|2 +

ˆΩ

δ2|∇xφ|2 + 1

δW (φ)

+

ˆΩ×D

ψ(log(ψM

)− 1)

+

ˆΩ×DJεβ(φ)ψ .

(2.20)

Remark 2.3. In contrast to previous publications (see [3] and the references therein), weconsider a mollication of the Henry energy. Physically, this refers to the nite size of thepolymers, mathematically it circumvents a lack of integrability which presently can onlybe overcome by considering s-Dirichlet integrals in the CahnHilliard energy density (see[1], [18]).

Taking the time-derivative in the physical energy, noting the identityˆΩ

u ·((ρu + ∂ρ

∂φJφ

)· ∇x

)u = −1

2

ˆΩ

∂ρ∂φ|u|2 (divx Jφ + u · ∇xφ) , (2.21)

and using the balance laws (2.7), (2.8), and (2.17), we obtain

d

dtE(φ, ψ,u) = −

ˆΩ

S : Du +

ˆΩ

∇xµφ · Jφ +

ˆΩ×D∇xµψ · Jψ,x +

ˆΩ×D∇qµψ · Jψ,q

+

ˆΩ

u ·(k− µψ∇xψ − divx

Jε

ˆD

∇qµψ ⊗ qψ

− µφ∇xφ

). (2.22)

Here, we introduced the chemical potential µφ of the phase eld as the rst variation ofthe free energy with respect to φ, i.e.

µφ := −δ∆xφ+ 1δW ′(φ) + β′(φ)Jε

ˆD

ψ

. (2.23)

Analogously, the rst variation of the free energy with respect to ψ gives the chemicalpotential of the congurational distribution, namely

µψ := log(ψM

)+ Jεβ(φ) . (2.24)

We note that entropy production is only due to gradients of velocities and of chemicalpotentials (cf. [14, 3]). As we do not consider external forces which may change themechanical work in the system, we follow the argumentation in [3] and in [24] to identify

k = µψ∇xψ + divx

Jε

ˆD

∇qµψ ⊗ qψ

+ µφ∇xφ . (2.25)

For simplicity, we assume a dissipation functional given by

Φ(S,Jφ,Jψ,x,Jψ,q) :=

ˆΩ

|S|24η(φ)

+

ˆΩ

|Jφ|22m(φ)

+

ˆΩ×D

|Jψ,x|22cxψ

+

ˆΩ×D

|Jψ,q|22cqψ

. (2.26)

1This is a diuse-interface approximation of a surface Dirac measure supported on the interface sepa-rating the two phases.


By (2.19), we infer

S = 2η(φ)Du , (2.27a)

Jφ = −m(φ)∇xµφ , (2.27b)

Jψ,x = −cxψ∇xµψ , (2.27c)

Jψ,q = −cqψ∇qµψ . (2.27d)

The second term on the right-hand side of (2.25) can be interpreted as an additional stresstensor. Noting (2.24), (2.4) gives

Jε

ˆD

∇qµψ ⊗ qψ

= Jε

ˆD

∇q log(ψM

)⊗ qψ

= Jε

ˆD

M∇qψM⊗ q

= Jε

ˆD

ψM∇qM−1 ⊗ q +

ˆD

∇qψ ⊗ q

. (2.28a)

Remark 2.4. In Lemma 3.9, we will show thatˆD

ψM∇qM−1 ⊗ q +

ˆD

∇qψ ⊗ q =

ˆD

ψU ′q⊗ q−ˆD

ψ1 (2.28b)

on Ω×R+ provided ψM

is contained in the weighted Sobolev space H1(D;M), for a de-nition see (3.42).Up to the mollication of −

´Dψ1, the expression on the right-hand side equals the

Kramers expression which has been used by Barrett and Süli in [9]. Due to the struc-ture of this additional stress tensor, this dierence only inuences the pressure.It should be emphasized that the Onsager approach pursued in this paper would yield sim-ilar results as in [9] if applied to the single-solvent setting.

Collecting the results of (2.7), (2.8), (2.9), (2.17), (2.18), (2.23), (2.24) and (2.27), wederive the following set of equations:

ρ(φ)∂tu +((ρ(φ)u−m(φ) ∂ρ

∂φ∇xµφ

)· ∇x

)u− divx 2η(φ)Du+∇xp

= µφ∇xφ+

ˆD

µψ∇xψ + divx

Jε

ˆD

M∇qψM⊗ q

, (2.29a)

divx u = 0 , (2.29b)


µφ = −δ∆xφ+ 1δW ′(φ) + β′(φ)Jε

ˆD

ψ

, (2.29d)

∂tψ + u · ∇xψ + divq ψ∇xJ εu · q = divq cqψ∇qµψ+ divx cxψ∇xµψ , (2.29e)

µψ = log(ψM

)+ Jεβ(φ) , (2.29f)


with the boundary conditions

u = 0 on ∂Ω×R+, (2.30a)

∇xφ · nx = 0 on ∂Ω×R+, (2.30b)

∇xµφ · nx = 0 on ∂Ω×R+, (2.30c)

ψ∇xµψ · nx = 0 on ∂Ω×D ×R+, (2.30d)

ψ (∇xJ ε(u)q− cq∇qµψ) · nq = 0 on Ω× ∂D ×R+. (2.30e)

Note that (2.30e) can equivalently be written in the form(cq∇qψ + ψ

(cqU

′(12|q|2)q−∇xJ εu · q

))· nq = 0 on Ω× ∂D ×R+.

Remark 2.5. Due to the derivation via the principle of least energy dissipation, the model(2.29) is formally consistent with thermodynamics, i.e. in absence of external forces, theenergy of the system is not increasing. This can be proven by testing (2.29a) by u, (2.29c)by µφ, (2.29d) by ∂tφ, (2.29e) by µψ and (2.29f) by ∂tψ.

2.3. A two-phase Oldroyd-B model. It is well known that micro-macro models maybe used to derive deterministic visco-elastic models by taking the expected value in theFokker-Planck equation. If the elastic potential is of Hookean type, i.e. U(s) = s and D =Rd, particular versions of the Oldroyd-B model are obtained (see [9] and the references

therein). In this subsection, we use model (2.29) applying identity (2.28b) for theKramers stress tensor to formulate a Oldroyd-B-type model for two-phase ow. Thesketch of derivation follows the standard methods to be found e.g. in [9]. Assuminga suciently fast decay of ψ and of ∇qψ at innity, multiplying (2.29e) by q ⊗ q andintegrating over the conguration space D = R

d, we formally obtain

∂tC[ψ] + (u · ∇x)C[ψ]−∇xJ εuC[ψ]−C[ψ](∇xJ εu)T

= 2cqω1− 2cqC[ψ] + cx∆xC[ψ] + cx divx (C[ψ]⊗∇xJεβ(φ)).(2.31)

Here, ω(x, t) :=´Dψ(x,q, t) dq denotes the number density of the polymers and

C[ψ](x, t) :=

ˆD

ψ(x,q, t)q⊗ q dq =

ˆD

ψ(x,q, t)U ′(12|q|2)q⊗ q dq

is the expected value for the Kramers stress tensor. Note that we used in particularthe identity ∇qµψ = 1

ψ∇qψ + q which follows in the Hookean case from (2.4). Deriving

analogously an evolution equation for the number density ω and redening the pressurep = p−

´Dψ log

(ψM

)−´Dψ, we end up with the following closed system

ρ(φ)∂tu +((ρ(φ)u−m(φ) ∂ρ

∂φ∇xµφ

)· ∇x

)u− divx 2η(φ)Du+∇xp

= µφ∇xφ+ Jεβ(φ)∇xω + divx JεC− ω1 , (2.32a)

divx u = 0 , (2.32b)


µφ = −δ∆xφ+ 1δW ′(φ) + β′(φ)Jεω , (2.32d)


∂tC + (u · ∇x)C−∇xJ εuC−C(∇xJ εu)T

= 2cqω1− 2cqC + cx∆xC + cx divx (C⊗∇xJεβ(φ)) .(2.32e)

∂tω + (u · ∇x)ω − cx∆xω = cx divx (ω∇xJεβ(φ)). (2.32f)

in Ω× (0, T ) subjected to

(∇xC + C⊗∇xJεβ(φ)) · nx = 0 ,

(∇xω + ω∇xJεβ(φ)) · nx = 0(2.33)

on ∂Ω× (0, T ), no-slip boundary conditions for u, no-ux boundary conditions for ω andφ as well as homogeneous Neumann conditions for φ. For dierent approaches to modelviscoelastic two-phase ow, we refer to [43, 45] and the references therein.

3. Existence of weak solutions

In this section, we will study model (2.29) for FENE-potentials in the case of identicalmass densities of the solvents and of a constant mobility in the Cahn-Hilliard equation.Without loss of generality, we assume ρ ≡ 1, m ≡ 1, and δ ≡ 1. In this case, (2.29a) issimplied to

∂tu + (u · ∇x)u− divx η(φ)Du+∇xp

= µφ∇xφ+

ˆD

µψ∇xψ + divx

Jε

ˆD

M∇qψM⊗ q

(3.1)

after an appropriate redenition of the pressure.We make the following general assumptions:

(A1) Ω ⊂ Rd (d ∈ 2, 3) is a bounded domain, of class C1,1 or alternatively convexwith Lipschitz boundary,

(A2) the set of admissible elongations is the ball dened in (2.2),(A3) the potential U satises (P1)-(P5) of Lemma 2.1,(A4) the molliers Jε, J ε, Jε have the properties (2.16a)-(2.16f),(A5) initial data (φ0,u0, ψ0) are contained in H2(Ω)×H1

σ,0(Ω)×L2(Ω×D;M−1) withψ0 being nonnegative almost everywhere.

(A6) the double-well potential W is of the form W (φ) = 14(1− φ2)

2,

(A7) β, η ∈ C∞(R), and there exist constants c1, c2 > 0 such that

c1 ≤ β(s) ≤ c2 , c1 ≤ η(s) ≤ c2 ∀s ∈ R .

Remark 3.1. As it was shown in [9], the usage of an appropriate mollier on the initialdatum u0 allows to choose u0 ∈ w ∈ L2(Ω) : divx w = 0.

For future reference we note the following estimate for all χ ∈ X (for a denition of X,cf. (4.9) in the Appendix):

ˆΩ

∣∣∣∣ˆD

M∇qχM⊗ q dq

∣∣∣∣2 dx ≤ C

ˆΩ×D

M∣∣∇q

χM

∣∣2 dq dx . (3.2)

holds true.


3.1. Existence of time-discrete solutions. In a rst step, we prove the existence of aquadruple (

φn, µnφ,un, ψn

)∈ H2(Ω)×H1(Ω)×H1

σ,0(Ω)× L2(Ω×D;M−1

),

given a triple (φn−1,un−1, ψn−1) with similar regularity properties, such that for a xed,small time increment τ the following time-discrete formulation of (3.1), (2.29b)-(2.29f),completed by the boundary conditions (2.30), is solved.ˆ

Ω

∂−τ φnθ −

ˆΩ

u∗ · φn−1∇xθ +

ˆΩ

∇xµnφ · ∇xθ = 0 ∀θ ∈ H1(Ω) , (3.3a)

with u∗ := un−1 − τφn−1∇xµnφ , (3.3b)

ˆΩ

µnφθ =

ˆΩ

∇xφn · ∇xθ+

ˆΩ

(W ′

+(φn) +W ′−(φn−1

))θ+

ˆΩ

β′(φn, φn−1

)Jεˆ

D

ψn−1

θ

∀θ ∈ H1(Ω) , (3.3c)

ˆΩ

∂−τ un ·w +

ˆΩ

(un−1 · ∇x

)un ·w +

ˆΩ

2η(φn−1

)Dun : Dw

=

ˆΩ

divx

Jε

ˆD

M∇qψn

M⊗ q

·w +

ˆΩ

µnφ∇xφn−1 ·w +

ˆΩ×DJεβ(φn)∇xψ

n ·w

∀w ∈H1σ,0(Ω) , (3.3d)

ˆΩ×D

∂−τ ψn θM

+

ˆΩ×D

θMun · ∇xψ

n −ˆ

Ω×Dψn(∇xJ εun · q) · ∇q

θM

+

ˆΩ×D

cqM∇qψn

M· ∇q

θM

+

ˆΩ×D

cxM∇xψn

M· ∇x

θM

+

ˆΩ×D

cxψn∇xJεβ(φn) · ∇x

θM

= −En,n−1

(12

ˆΩ×D

|q|2cq‖∇xJ εun‖2

L∞ ψn θM

+ 12cx

ˆΩ×D‖∇xJεβ(φn)‖2

L∞ ψn θM

)∀θ ∈ X , (3.3e)

with the backward time dierence quotient ∂−τ an := 1

τ(an − an−1), the dierence quotient

β′(an, an−1

):=

β(an)−β(an−1)

an−an−1 if an 6= an−1 ,β′(an−1) if an = an−1 ,

the shift terms En,n−1(an) := an − an−1, and a convex-concave splitting of W into W+

and W−. Note that due to solenoidality of w, the last term of (3.3d) is identical to´Ω×D µ

nψ∇xψ

n ·w.This discretization, which is inspired by the decoupling technique in [22, 5, 36], the time-discretization in [21] and the regularization in [9], decouples the phase-eld equation fromthe momentum and FokkerPlanck equations, and therefore allows to prove the existenceof a suitable φn and µnφ before dealing with the existence of un and ψn.

Lemma 3.2. Let the assumptions (A1)-(A4), (A6) and (A7) hold true. Then for givenun−1 ∈ H1

σ,0(Ω), ψn−1 ∈ L2(Ω×D;M−1) and φn−1 ∈ H2(Ω), there exists a tuple(φn, µnφ

)∈ H2(Ω)×H1(Ω) which solves (3.3a)-(3.3c).


Remark 3.3. From ψn−1 ∈ L2(Ω×D;M−1), one may easily deduce ωn−1 :=´Dψn−1 ∈

L2(Ω).

Proof of Lemma 3.2. Consider a complete orthonormal system in L2(Ω) given by eigen-

functions θk, k ∈ N, of the Laplace-operator subjected to homogeneous Neumann condi-

tions. On Hk := spanθ1, ..., θk

, we look for Galerkin solutions φnk , µ

nφ,k to the discrete

version of (3.3a) -(3.3c) given byˆΩ

∂−τ φnkθk −

ˆΩ

un−1φn−1k · ∇xθk +

ˆΩ

(1 + τ

∣∣φn−1k

∣∣2)∇µnφ,k · ∇θk = 0 ∀θk ∈ Hk(Ω) ,

(3.4a)

ˆΩ

µnφ,kθk =

ˆΩ

∇xφnk ·∇xθk+

ˆΩ

(W ′

+(φnk) +W ′−(φn−1k

))θk+

ˆΩ

β′(φnk , φ

n−1k

)Jεωn−1

θk

∀θk ∈ Hk(Ω) . (3.4b)

We denote elements of Hk(Ω) by an index k.Observe that φn−1

k is dened by the orthogonal L2-projection of φn−1 onto Hk. In partic-ular, ∥∥φn−1

k

∥∥H2(Ω)

≤ C∥∥φn−1

∥∥H2(Ω)

(3.5)

uniformly in k ∈ N due to the stability of the orthogonal L2-projection on H2∗ (Ω) := u ∈

H2(Ω)| ∂∂νu ≡ 0 on ∂Ω.

The existence of discrete solutions(φnk , µ

nφ,k

)∈ Hk×Hk can easily be obtained by a slight

variation of the proof of Lemma 3.2 in [19]. To pass to the limit k → ∞, we have toestablish k-independent bounds for φnk and µ

nφ,k. By testing (3.4a) by τµnφ,k and (3.4b) by

τ∂−τ φnk we obtain

12

ˆΩ

(|∇xφ

nk |

2 +∣∣∇xφ

nk −∇xφ

n−1k

∣∣2 − ∣∣∇xφn−1k

∣∣2)+

ˆΩ

W (φnk)−W(φn−1k

)+

ˆΩ

(1 + τ

∣∣φn−1k

∣∣2) |∇µφ,k|2 ≤ C , (3.6)

with a constant C which depends on un−1, ωn−1 and β, but not on k. Together with (3.5),one may extract subsequences (again denoted by φnk , µ

nφ,k, φ

n−1k ) with

φnk φn in H1(Ω) ,

µnφ,k µnφ in H1(Ω) ,

φn−1k →φn−1 in H2(Ω) ,

φn−1k →φn−1 in L∞(Ω) .

Therefore, we are able to pass to the limit in (3.4) to get the existence of(φn, µnφ

)∈

H1(Ω)×H1(Ω). Since φn is a weak solution of the Poisson problemˆΩ

∇xξ · ∇xθ =

ˆΩ

µnφθ −ˆ

Ω

(W ′

+(φn) +W ′−(φn−1

))θ −

ˆΩ

β′(φn, φn−1

)Jεωn−1

θ


with homogeneous Neumann boundary conditions, a xed mean value and a right-handside in L2(Ω), standard results of elliptic regularity theory give

‖φn‖H2(Ω) ≤∥∥µnφ∥∥L2(Ω)

+∥∥W ′

+(φn) +W ′−(φn−1

)∥∥L2(Ω)

+∥∥∥β(φn, φn−1

)Jεωn−1

∥∥∥L2(Ω)

+

∣∣∣∣ Ω

φ dx

∣∣∣∣ (3.7)

The next step is to prove the existence of ψn ∈ X and un ∈ H1σ,0(Ω) if

(φn, µnφ

)∈

H2(Ω)×H1(Ω) are given. We rewrite (3.3d) as

b[un−1](un,w) = l[ψn](w) ∀w ∈H1σ,0(Ω) , (3.8)

with

l[χ](w) :=

ˆΩ

un−1 ·w + τ

ˆΩ

µnφ∇xφn−1 ·w + τ

ˆΩ×DJεβ(φn)∇xχ ·w

+ τ

ˆΩ

divx

Jε

ˆD

M∇qχM⊗ q

·w ∀χ ∈ X and w ∈H1

σ,0(Ω) , (3.9a)

and

b[v](w1,w2) :=

ˆΩ

w1 ·w2 + τ

ˆΩ

(v · ∇x)w1 ·w2 + τ

ˆΩ

2η(φn−1

)Dw1 : Dw2

∀v,w1,w2 ∈H1σ,0(Ω) . (3.9b)

Analogously, we can rewrite (3.3e) as

a[un](ψn, θ) = k(θ) ∀θ ∈ X , (3.10)

with

k(θ) :=

ˆΩ×D

ψn−1θM

+ 12τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞ψn−1 θ

M

+ 12τ

ˆΩ×D

cx∥∥∇xJε

β(φn−1

)∥∥2

L∞ψn−1 θ

M∀θ ∈ X , (3.11a)

and

a[v](χ, θ) :=

ˆΩ×D

χθM

+ τ

ˆΩ×D

v · ∇xχθM− τ

ˆΩ×D∇xJεvqχ · ∇q

θM

+ τ

ˆΩ×D

cqM∇qχM· ∇q

θM

+ τ

ˆΩ×D

cxM∇xχM· ∇x

θM

+ τ

ˆΩ×D

cxχ∇xJεβ(φn) · ∇xθM

+ 12τ

ˆΩ×D‖∇xJ εv‖2

L∞|q|2cqχ θM

+ 12τ


L∞ cxχθM

∀v ∈ Y r, χ, θ ∈ X . (3.11b)

Lemma 3.4. Let the assumptions (A1)-(A4), (A6), (A7) hold true and suppose thatφn, µnφ, φ

n−1 ∈ H1(Ω), un−1 ∈ H1σ,0(Ω), ψn−1 ∈ L2(Ω×D;M−1) and that ψn−1 is non-

negative a.e.. Then


(1) for given u ∈ Y r, there exists a unique ψ ∈ X+ solving

a[u](ψ, θ) = k(θ) ∀θ ∈ X ,

(2) for given ψ ∈ X, there exists a unique u ∈H1σ,0(Ω) solving

b[un−1](u,w) = l[ψ](w) ∀w ∈H1σ,0(Ω) .

Proof. This result is an adaptation of the corresponding result in [9], see the argumenton page 527. Using Young's inequality on the third and sixth term in (3.11b) yields

a[u](χ, χ) ≥ c ‖χ‖2X , (3.12)

independently of u. A Hölder argument shows

a[u](χ, θ) ≤ C ‖χ‖X ‖θ‖X + τ

ˆΩ×D

u · ∇xχθM

+ τ

ˆΩ×D

cxχ∇xJεβ(φn) · ∇xθM.

Noting Jεβ(φn) ∈ W 1,∞(Ω), we combine Hölder's inequality with the estimateˆΩ×D

u · ∇xχθM≤c

ˆD

1M‖u‖L3(Ω) ‖∇xχ‖L2(Ω) ‖θ‖H1(Ω)

≤c ‖u‖L3(Ω)

[ˆΩ×D

M∣∣∇x

χM

∣∣2]1/2[ˆΩ×D

|θ|2M

+

ˆΩ×D

M∣∣∇x

θM

∣∣2]1/2

to conclude that a[u](χ, θ) ≤ C ‖χ‖X ‖θ‖X . Therefore, the Lax-Milgram theorem gives the

existence of a unique ψ ∈ X. By testing (3.13b) by ψ− := min(

0, ψ), a straightforward

computation shows ψ ∈ X+ (cf. [9]).The proof of the existence of u ∈H1

σ,0(Ω) is analogous.

To prove the existence of solutions to the coupled system, we follow the ideas in [9] anduse a xed-point argument exploiting the unique solvability of the decoupled equations.Due to Lemma 3.4 the mapping G : Y r → Y r, u 7→ u := G(u) which is given by

a[u](ψ, θ) = k(θ) ∀θ ∈ X , (3.13a)

b[un−1](u,w) = l[ψ](w) ∀w ∈H1σ,0(Ω) (3.13b)

is well-dened.

Lemma 3.5. Under the assumptions of Lemma 3.4 the mapping G dened by (3.13) hasa xed point.

Proof. The proof is a slight adaption of that one of Lemma 3.2 in [9]. To apply Schaefer'sxed-point theorem (see e.g. [17]), it is sucient to show the following conditions to besatised:

i) G : Y r → Y r is continuous,ii) G is compact,iii) there exists C∗ > 0 such that

‖w‖Lr(Ω) ≤ C∗

for all w ∈ Y r satisfying w = γG(w) with γ ∈ (0, 1].


To prove i), we have to show that for a sequenceu(i)i≥0

in Y r strongly converging to

u ∈ Y r with respect to the Lr-norm, the sequenceu

(i)G

i≥0

:=G(u(i))

i≥0converges

strongly towards G(u) in Lr(Ω) as i → ∞. The occurring intermediate values (i.e. the

solutions of (3.13a)) will be denoted by ψ(i)G . Combining (3.12), (3.13a) and (3.11a) yields

c∥∥∥ψ(i)

G

∥∥∥2

X≤ˆ

Ω×D

ψn−1ψ(i)G

M+ 1

2τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞ψn−1 ψ

(i)G

M

+ 12τ

ˆΩ×D

|q|2cq

∥∥∇xJεβ(φn−1

)∥∥2

L∞ψn−1 ψ

(i)G

M.

By Hölder's inequality,ˆΩ×D

∣∣∣ψ(i)G

∣∣∣2M

+

ˆΩ×D

M

∣∣∣∣∇xψ(i)G

M

∣∣∣∣2 +

ˆΩ×D

M

∣∣∣∣∇qψ(i)G

M

∣∣∣∣2 ≤ C(

1τ,un−1, φn−1, ψn−1

), (3.14)

independently of u(i). Analogously (3.9b), (3.13b), (3.9a) and (2.28) yield

c(τ)∥∥∥u(i)

G

∥∥∥2

H1(Ω)≤ˆ

Ω

µnφ∇xφn−1 · u(i)

G

+


(i)G · u

(i)G +

ˆΩ

divx

Jε

ˆD

M∇qψ(i)G

M⊗ q

· u(i)

G

=: I + II + III .

Using

|I| ≤C∥∥µnφ∥∥H1(Ω)

∥∥∇xφn−1∥∥L2(Ω)

∥∥∥u(i)G

∥∥∥H1(Ω)

,

|III| ≤∣∣∣∣ˆ

Ω×DM∇q

ψ(i)G

M⊗ q : Jε

∇xu

(i)G

∣∣∣∣ ,together with Hölder's inequality,(3.2), (2.16a) and (3.14), we obtain∥∥∥u(i)

G

∥∥∥H1(Ω)

≤ C(

1τ,un−1, φn−1, ψn−1, µnφ

)(3.15)

Therefore, we may extract a subsequence (again denoted by (ψ(i)G , u

(i)G )i≥0) with the

following properties:

ψ(i)G

M1/2 ψGM1/2 in L2(Ω×D) ,

M1/2∇xψ(i)G

MM1/2∇x

ψGM

in L2(Ω×D) ,

M1/2∇qψ(i)G

MM1/2∇q

ψGM

in L2(Ω×D) ,

u(i)G uG in H1(Ω) ,

u(i)G → uG in Lr(Ω) ,

J ε

u

(i)G

→ J εuG in W 1,∞(Ω) .

Hence, we have not only the strong convergence ofu

(i)G

i∈N

in Lr(Ω), but we may

also pass to the limit in (3.13) which permits to identify uG := limi→∞G(u(i))with


G(u) ∈ H1σ,0(Ω). Due to the uniqueness of the solutions of the bilinear forms, we may

extend this result to the original sequence, which proves i).The second claim is a direct result of the compact embedding of H1

σ,0(Ω) into Y r.As regards iii), we note (3.15) and compute

‖w‖Lr(Ω) ≤ ‖G(w)‖Lr(Ω) ≤ C(

1τ,un−1, φn−1, ψn−1, µnφ

)=: C∗ .

3.2. Compactness in space and time. In this subsection, we will derive some τ -independent bounds for the time-discrete solution

(φn, µnφ,u

n, ψn). In contrast to [9],

we are not able to prove the regularity of un independently of ψn since the coupling be-tween the equations is stronger: Due to the additional β-term, it is not possible to writeµψ∇xψ as a gradient. Hence, the third term on the right-hand side of (3.9a) remains. Toovercome this obstacle, we use the thermodynamical consistency of the model.Although ψ is conserved in the original model, in the time-discrete setting we only getthe pseudo-conservation relation

ˆΩ×D

[1 + 1

2τ |q|

2

cq‖∇xJ εun‖2

L∞ + 12τcx ‖∇xJεβ(φn)‖2

L∞

]ψn

=

ˆΩ×D

[1 + 1

2τ |q|

2

cq

∥∥∇xJ ε

u0∥∥2

L∞+ 1

2τcx

∥∥∇xJεβ(φ0)∥∥2

L∞

]ψ0 (3.16)

due to the regularization terms. To prove (3.16), it is sucient to test (3.3e) by τM andto sum over all the time-steps.

Remark 3.6. In the next step, we would like to test (3.3e) by τMµnψ := τM log(ψn

M

)+

τMJεβ(φn). As Mµnψ is not admissible, we instead use the regularized version Mµnψ :=

M(log(ψn

M+ ν)

+ Jεβ(φn)). Noting Jεβ(φn) ∈ L∞(Ω) and log (ν) ≤ log

(ψn

M+ ν)≤

ψn

M+ ν, a straightforward computation yields

ˆΩ×D

M |µnψ|2 ≤ C(Ω, β(φn), ν, ψn) .

Noting 0 ≤(

Mψn+Mν

)2

≤ 1ν2, we also infer

ˆΩ×D

M∣∣∇qµ

nψ

∣∣2 ≤ ˆΩ×D

M(

Mψn+Mν

)2 ∣∣∇qψn

M

∣∣2 ≤ C(ν, ψn) ,

andˆΩ×D

M |∇xµψ|2 ≤ 2

ˆΩ×D

M(

Mψn+Mν

)2 ∣∣∇xψn

M

∣∣2 + 2

ˆΩ×D

M |∇xJεβ(φn)|2

≤ C(Ω, β(φn), ν, ψn) .

Therefore, we have Mµnψ ∈ X.

In the next step, we test (3.3a) by τµnφ, (3.3c) by (φn − φn−1), (3.3d) by τun and (3.3e) by

τMµnψ = τM(log(ψn

M+ ν)

+ Jεβ(φn))which is a regularized version of the chemical


potential µnψ. Using the convex-concave decomposition of W entails

ˆΩ

(φn − φn−1

)µnφ − τ

ˆΩ

φn−1un−1 · ∇xµnφ + τ 2

ˆΩ

∣∣φn−1∣∣2 ∣∣∇xµ

nφ

∣∣2 + τ

ˆΩ

∣∣∇xµnφ

∣∣2 = 0 ,

(3.17a)ˆΩ

µnφ(φn − φn−1

)≥ˆ

Ω

12

(|∇xφ

n|2 +∣∣∇xφ

n −∇xφn−1∣∣2 − ∣∣∇xφ

n−1∣∣2)

+

ˆΩ

(W (φn)−W

(φn−1

))+

ˆΩ

(β(φn)− β

(φn−1

))Jεˆ

D

ψn−1

,

(3.17b)

0 =

ˆΩ

12

(|un|2 +

∣∣un − un−1∣∣2 − ∣∣un−1

∣∣2)+ τ

ˆΩ

2η(φn−1

)|Dun|2 + τ

ˆΩ

φn−1∇xµnφ · un

−ˆ

Ω×DJεβ(φn)∇xψ

n · un − τˆ

Ω

divx

Jε

ˆD

M∇qψn

M⊗ q

· un , (3.17c)

0 =

ˆΩ×D

(ψn − ψn−1

)[log(ψn

M+ ν)

+ Jεβ(φn)]

+ τ

ˆΩ×D

un · ∇xψnJεβ(φn)

− τˆ

Ω×D(∇xJ εun · qψn) · ∇q log

(ψn

M+ ν)

+ τcq

ˆΩ×D

M∇qψn

M· ∇q log

(ψn

M+ ν)

+ τcx

ˆΩ×D

M∇xψn

M· ∇x log

(ψn

M+ ν)

+ τcx

ˆΩ×D

M∇xψn

M· ∇xJεβ(φn)

+ τcx

ˆΩ×D

ψn∇xJεβ(φn) · ∇x log(ψn

M+ ν)

+ τcx

ˆΩ×D

ψn |∇xJεβ(φn)|2

+ 12τ

ˆΩ×D


L∞ ψn[log(ψn

M+ ν)

+ Jεβ(φn)]

− 12τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞ψn−1

[log(ψn

M+ ν)

+ Jεβ(φn)]

+ 12τ

ˆΩ×D

cx ‖∇xJεβ(φn)‖2L∞ ψ

n[log(ψn

M+ ν)

+ Jεβ(φn)]

− 12τ

ˆΩ×D

cx∥∥∇xJε

β(φn−1

)∥∥2

L∞ψn−1

[log(ψn

M+ ν)

+ Jεβ(φn)]

=: R1 +R2 +R3 +R4 +R5 +R6 +R7 +R8 +R9 +R10 +R11 +R12 . (3.17d)

As we are interested in the limit ν 0, we can assume 0 < ν < 1. With C we will denotea generic constant which may depend on Ω, D and initial data, but not on τ , ν or thesolution itself. By convexity of g(s) := s log s− s,

ˆΩ×D

(ψn − ψn−1

)log(ψn

M+ ν)≥ˆ

Ω×DMg(ψn

M+ ν)−Mg

(ψn−1

M+ ν). (3.18)


We rewrite the fourth term on the right-hand side of (3.17d) as

τcq

ˆΩ×D

M∇qψn

M· ∇q log

(ψn

M+ ν)

= τcq

ˆΩ×D

M2

ψn+νM

∣∣∇qψn

M

∣∣2 (3.19)

and the fth, sixth, seventh and eighth term in the form

ˆΩ×D

M∇xψn

M· ∇x log

(ψn

M+ ν)

+

ˆΩ×D

M∇xψn

M· ∇xJεβ(φn)

+

ˆΩ×D

ψn∇xJεβ(φn) · ∇x log(ψn

M+ ν)

+

ˆΩ×D

ψn |∇xJεβ(φn)|2

=

ˆΩ×D

(ψn + νM)∣∣∇x

[log(ψn

M+ ν)

+ Jεβ(φn)]∣∣2

−ˆ

Ω×DνM∇xJεβ(φn) · ∇x log

(ψn

M+ ν)−ˆ

Ω×DνM |∇xJεβ(φn)|2 . (3.20)

Dening χn := ψn

M+ ν, we rewrite the logarithmic parts of R9 and R10 as

12τ

ˆΩ×D

|q|2cq

[‖∇xJ εun‖2

L∞ ψn −

∥∥∇xJ ε

un−1

∥∥2

L∞ψn−1

]log(ψn

M+ ν)

=12τ

ˆΩ×D


L∞Mχn logχn

− 12τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞Mχn−1 logχn

− 12τ

ˆΩ×D


L∞Mν logχn

+ 12τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞Mν logχn = I + II + III + IV .

(3.21)

To deal with those terms, we will use the following version of Young's inequality forcomplementary functions which can be found e.g. in [27]:

a b ≤a log+ a+ b if b ∈ [0, 1] ,a log+ a+ eb−1 if b> 1 ,

∀a, b ≥ 0 . (3.22)

Here, log+ a := max0, log a. Dening A := (x,q) ∈ Ω×D : χn(x,q) ≥ 1, B :=(Ω×D)\A, noting χn ≥ 0 and using (2.16f), we may estimate

I =12τ

Â


L∞Mχn logχn + 12τ

ˆB


L∞Mχn logχn

≥12τ

Â


L∞Mχn logχn − 12τ

∣∣∣∣∣mins∈R+

0

s log s

∣∣∣∣∣ ‖∇xJ εun‖2L∞

ˆΩ×D

|q|2Mcq

≥12τ

ˆΩ×D


L∞Mχn log+ χn − τC

ˆΩ

|un|2 .

(3.23)


By using (3.22) and (3.16) we infer

II =12τ

ˆB

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞Mχn−1 |logχn|

− 12τ

Â

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞Mχn−1 log+ χ

n

≥− 12τ

Â

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞M(χn−1 log+ χ

n−1 + max(1, elog+ χ

n−1))

≥− 12τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

un−1

∥∥2


n−1 − τCˆ

Ω

∣∣un−1∣∣2 .

(3.24)

With a similar approach, using (3.16) and noting that χn ≥ logχn, we may estimate theremaining terms via

III + IV ≥12τ

ˆB


L∞Mν |logχn| − 12τ

Â


L∞Mνχn

+ 12τ

Â

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞Mν logχn

+ 12τ

ˆB

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞Mν logχn

≥− 12τ

Â


L∞Mνχn + 12τ

ˆB

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞Mν log ν

≥− τνCˆ

Ω

|un|2 − τ |ν log ν|Cˆ

Ω

∣∣un−1∣∣2 .

(3.25)

Analogously, we obtain for the logarithmic parts of R11 and R12 by using (2.16f)

12τ

ˆΩ×D

cx

[‖∇xJεβ(φn)‖2

L∞ ψn −

∥∥∇xJεβ(φn−1

)∥∥2

L∞ψn−1

]log(ψn

M+ ν)

≥12τ

ˆΩ×D

cx ‖∇xJεβ(φn)‖2L∞Mχn log+ χ

n − τCˆ

Ω

|β(φn)|2

− 12τ

ˆΩ×D

cx∥∥∇xJε

β(φn−1

)∥∥2


n−1

− τCˆ

Ω

∣∣β(φn−1)∣∣2 − 1

2τνC

ˆΩ

|β(φn)|2 − τ |ν log ν|Cˆ

Ω

∣∣β(φn−1)∣∣2 .

(3.26)


Using (3.18), (3.19), (3.20), (3.21), (3.23), (3.24), (3.25), (3.26), (2.16f) and´

Ω|β(φn)| ≤

C, we obtain from (3.17d)

ˆΩ×D

Mg(ψn

M+ ν)

+

ˆΩ×D

ψnJεβ(φn)+ τ

ˆΩ×D


+ τ

ˆΩ×D

cqM2

ψn+νM

∣∣∇qψn

M

∣∣2 + τ

ˆΩ×D

cx(ψn + νM)∣∣∇x

[log(ψn

M+ ν)

+ Jεβ(φn)]∣∣2

+ 12τ

ˆΩ×D


L∞Mχn log+ χn+ 1

2τ

ˆΩ×D


n

≤ˆ

Ω×DMg(ψn−1

M+ ν)

+

ˆΩ×D

ψn−1Jεβ(φn)+ τ

ˆΩ×D

(∇xJ εun · q ψnM

ψn+Mν

)· ∇q

ψn

M

+ τ

ˆΩ×D

cxνM∇xJεβ(φn) · ∇x log(ψn

M+ ν)

+ τ

ˆΩ×D

cxνM |∇xJεβ(φn)|2

+ τC

ˆΩ

|un|2 + 12τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

un−1

∥∥2


n−1

+ τC

ˆΩ

∣∣un−1∣∣2 + 1

2τ

ˆΩ×D

cx∥∥∇xJε

β(φn−1

)∥∥2


n−1 + τC . (3.27)

Using (3.16), one may compute for the third term on the right-hand side

τ

∣∣∣∣ˆΩ×D

(∇xJ εun · q ψnM

ψn+νM

)· ∇q

ψn

M

∣∣∣∣ ≤ατ ˆΩ×D

M2

ψn+νM

∣∣∇qψn

M

∣∣2+ Cατ

ˆΩ×D

|ψn|2ψn+νM

|∇xJ εun|2 |q|2 .

(3.28)

Similarly, the fth term on the left-hand side may be rewritten as

ˆΩ×D

(ψn + νM)∣∣∇x

[log(ψn

M+ ν)

+ Jεβ(φn)]∣∣2 =

ˆΩ×D

M2 1ψn+νM

∣∣∇xψn

M

∣∣2+ 2

ˆΩ×D

M∇xψn

M· ∇xJεβ(φn)+

ˆΩ×D

(ψn + νM) |∇xJεβ(φn)|2 . (3.29)

The fourth term on the right-hand side of (3.27) can be written as

τ

ˆΩ×D

cxνM∇xJεβ(φn) · ∇x log(ψn

M+ ν)

= −12τ

ˆΩ×D

cxνM |∇xJεβ(φn)|2

+ 12τ

ˆΩ×D

cxνM[∇xJεβ(φn)+∇x log

(ψn

M+ ν)]2− 1

2τ

ˆΩ×D

cxνM∣∣∇x log

(ψn

M+ ν)∣∣2(3.30)

The second term on the right-hand side can be absorbed in the fth term on the left-handside of (3.27), the remaining terms have a good sign but will vanish in the limit ν 0.


Summing up, we get

ˆΩ×D

Mg(ψn

M+ ν)

+

ˆΩ×D

ψnJεβ(φn)+ τ

ˆΩ×D


τ(1− α)

ˆΩ×D

cqM2

ψn+νM

∣∣∇qψn

M

∣∣2 + τ

ˆΩ×D

cxψn∣∣∇x

[log(ψn

M+ ν)

+ Jεβ(φn)]∣∣2

+ 12τ

ˆΩ×D

cxνM∣∣∇x

[log(ψn

M+ ν)

+ Jεβ(φn)]∣∣2 + 1

2τ

ˆΩ×D

cxνM∣∣∇x log

(ψn

M+ ν)∣∣2

12τ

ˆΩ×D


L∞Mχn log+ χn + 1

2τ

ˆΩ×D


n

≤ Cατ

ˆΩ×D

|ψn|2ψn+νM

|∇xJ εun|2 |q|2 +

ˆΩ×D

Mg(ψn−1

M+ ν)

+

ˆΩ×D

ψn−1Jεβ(φn)

+ 12τ

ˆΩ×D

cxνM |∇xJεβ(φn)|2 + τC

ˆΩ

∣∣un−1∣∣2 + τC

ˆΩ

|un|2

+ 12τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

un−1

∥∥2


n−1

+ 12τ

ˆΩ×D

cx∥∥∇xJε

β(φn−1

)∥∥2


n−1 + τC . (3.31)

By (2.16f) the fourth term on the right-hand side is of order O(ν).Let us now pass to the limit ν 0 with the terms on the left-hand side. The rst andfourth term may be treated by monotone convergence.Using (3.29), we observe that the remaining terms on the left-hand side may be controlledby Lebesgue's theorem using in particular (2.16f) to control the terms involving convo-lutions. Similarly, convergence of the terms on the right-hand side can be established.Using these results, we may pass to the limit in (3.28) by using monotone convergence.Altogether, we end up with

ˆΩ×D

Mg(ψn

M

)+

ˆΩ×D

(ψn − ψn−1

)Jεβ(φn)+ τ

ˆΩ×D


− τˆ

Ω×D(∇xJ εun · qM) · ∇q

ψn

M+ (1− 2α)τ

ˆΩ×D

cqψn∣∣∇q log

(ψn

M

)∣∣2+ τ

ˆΩ×D

cxψn∣∣∇x

[log(ψn

M

)+ Jεβ(φn)

]∣∣2 + 12τ

ˆΩ×D


L∞ ψn log+

(ψn

M

)+ 1

2τ


L∞ ψn log+

(ψn

M

)≤ˆ

Ω×DMg(ψn−1

M

)+ 1

2τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞ψn−1 log+

(ψn−1

M

)+ 1

2τ

ˆΩ

cx∥∥∇xJε

β(φn−1

)∥∥2

L∞ψn−1 log+

(ψn−1

M

)+ τC

ˆΩ

|un|2 + τC

ˆΩ

∣∣un−1∣∣2 + τC .

(3.32)

At this point we are able to state our rst regularity result.


Lemma 3.7. Let T>0 be arbitrary, but xed. Suppose the time-increment τ is sucientlysmall and dene N :=

[Tτ

]+ 1. Let (φ0,u0, ψ0) satisfy assumptions (A1)-(A7) and as-

sume(φn, µnφ,u

n, ψn)n=1,...,N

to solve (3.3a)-(3.3e) for all n ∈ 1, ..., N. Then a positive

constant C depending on T and initial data exists such thatˆΩ×D

ψn(log(ψn

M

)− 1)

+

ˆΩ×DJεβ(φn)ψn +

ˆΩ

12|∇xφ

n|2 +

ˆΩ

W (φn) +

ˆΩ

12|un|2

+ 12

n∑k=1

ˆΩ

∣∣∇xφk −∇xφ

k−1∣∣2 +

n∑k=1

ˆΩ

14

∣∣uk − uk−1∣∣2

+ 12τ

ˆΩ×D


L∞ ψn log+

(ψn

M

)+ 1

2τ

ˆΩ×D


n log+

(ψn

M

)+(1− 2α)τ

n∑k=1

ˆΩ×D

cqψk∣∣∣∇q log

(ψk

M

)∣∣∣2+τn∑k=1

ˆΩ×D

cxψk∣∣∣∇x

[log(ψk

M

)+ Jε

β(φk)]∣∣∣2

+ τ

n∑k=1

ˆΩ

∣∣∇xµkφ

∣∣2 + τn∑k=1

ˆΩ

2η(φk−1

) ∣∣Duk∣∣2 ≤ C

(φ0,u0, ψ0, T

), (3.33)

for all n ∈ 1, ..., N.

Proof. First note the estimate

τ

ˆΩ

φn−1∇xµnφ ·(un − un−1

)≥ −τ 2

ˆΩ

∣∣φn−1∣∣2 ∣∣∇xµ

nφ

∣∣2 − 14

ˆΩ

∣∣un − un−1∣∣2 , (3.34)

which holds for the dierence of the second term in (3.17a) and the third term in (3.17c),and which is a direct consequence of Young's inequality. One may easily add up (3.17a),(3.17b), (3.17c) and (3.32) to obtainˆ

Ω×Dψn(log(ψn

M

)− 1)

+

ˆΩ×DJεβ(φn)ψn +

ˆΩ

12|∇xφ

n|2 +

ˆΩ

W (φn) +

ˆΩ

12|un|2

+ 12

ˆΩ

∣∣∇xφn −∇xφ

n−1∣∣2 +

ˆΩ

14

∣∣un − un−1∣∣2

+ 12τ

ˆΩ×D


L∞ ψn log+

(ψn

M

)+ 1

2τ

ˆΩ×D


n log+

(ψn

M

)+ τ(1− 2α)

ˆΩ×D

cqψn∣∣∇q log

(ψn

M

)∣∣2 + τ

ˆΩ×D

cxψn∣∣∇x

[log(ψn

M

)+ Jεβ(φn)

]∣∣2+ τ

ˆΩ

∣∣∇xµnφ

∣∣2 + τ

ˆΩ

2η(φn−1

)|Dun|2

≤ˆ

Ω×Dψn−1

(log(ψn−1

M

)− 1)

+

ˆΩ×DJεβ(φn−1

)ψn−1 +

ˆΩ

12

∣∣∇xφn−1∣∣2 +

ˆΩ

W(φn−1

)+

ˆΩ

12

∣∣un−1∣∣2 + 1

2τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞ψn−1 log+

(ψn−1

M

)+ 1

2τ

ˆΩ×D

cx∥∥∇xJε

β(φn−1

)∥∥2

L∞ψn−1 log+

(ψn−1

M

)+τC

ˆΩ

|un|2+τC

ˆΩ

∣∣un−1∣∣2+τC .

(3.35)


Using the telescope structure of (3.35), recalling (2.28) and performing a discrete integra-tion in time, we may bound the left-hand side of 3.33 by

l.h.s. ≤ˆ

Ω×DMg(ψ0

M

)+

ˆΩ×DJεβ(φ0)ψ0 +

ˆΩ

12

∣∣∇xφ0∣∣2 +

ˆΩ

W(φ0)

+

ˆΩ

∣∣u0∣∣2

+12τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

u0∥∥2

L∞ψ0 log+

(ψ0

M

)+1

2τ

ˆΩ×D

cx∥∥∇xJε

β(φ0)∥∥2

L∞ψ0 log+

(ψ0

M

)+ τC0

ˆΩ

|un|2 + τCn−1∑k=0

ˆΩ

∣∣uk∣∣2 + TC . (3.36)

As C0 does not depend on τ or the solution itself, we may safely assume τC0 ≤ 14.

Absorbing τC0

´Ω|un|2 on the left-hand side allows us to apply the discrete version of

Gronwall's lemma (cf. e.g. [6, Lemma 4.2.3.]) to conclude

ˆΩ

∣∣uk∣∣2 ≤ C(φ0,u0, ψ0, T

)∀k ≤ n . (3.37)

Therefore, Lemma 3.7 is proven.

The τ -independent bounds for un may be used to obtain higher regularity for ψ.

Lemma 3.8. Under the assumptions of Lemma 3.7, there is a positive constant C =C(φ0,u0, ψ0, T ) such that

12

ˆΩ×D

|ψn|2M

+ 12

n∑k=1

ˆΩ×D

|ψk−ψk−1|2M

+ 12τcq

n∑k=1

ˆΩ×D

M∣∣∣∇q

ψk

M

∣∣∣2+ 1

2τcx

n∑k=1

ˆΩ×D

M∣∣∣∇x

ψk

M

∣∣∣2 ≤ C(φ0,u0, ψ0, T

),

for all n ∈ 1, ..., N.

Proof. We test (3.3e) by τψn and use Young's inequality to obtain

12

ˆΩ×D

|ψn|2+|ψn−ψn−1|2−|ψn−1|2M

+ 12τ

ˆΩ×D

cqM∣∣∇q

ψn

M

∣∣2 + 12τ

ˆΩ×D

cxM∣∣∇x

ψn

M

∣∣2≤ 1

2τ

ˆΩ×D

|q|2cq

∥∥∇xJ ε

un−1

∥∥2

L∞ψn−1 ψn

M+ 1

2τ

ˆΩ×D

cx∥∥∇xJε

β(φn−1

)∥∥2

L∞ψn−1 ψn

M.

(3.38)

Combining (A7), (2.16f), and Lemma 3.7 and noting´

Ωφn =

´Ωφ0 for all n ≤ N yields

‖∇xJεβ(φn−1)‖2L∞ ≤ C(β, ε, φ0,u0, ψ0, T ). Recalling the inequality∥∥∇xJ ε

un−1

∥∥2

L∞≤ C

(φ0,u0, ψ0, T

)for all n ≤ N,


we again use Young's inequality and obtain by summation

12

ˆΩ×D

|ψn|2M

+ 12

n∑k=1

ˆΩ×D

|ψk−ψk−1|2M

+ 12τcq

n∑k=1

ˆΩ×D

M∣∣∣∇q

ψk

M

∣∣∣2+ 1

2τcx

n∑k=1

ˆΩ×D

M∣∣∣∇x

ψk

M

∣∣∣2 ≤ 12

ˆΩ×D

|ψ0|2M

+ 12τ

ˆΩ×D

|ψn|2M

+ τC

n−1∑k=0

ˆΩ×D

|ψk|2M

. (3.39)

Analogously to the proof of the last lemma, we can absorb 12τ´

Ω×D|ψn|2M

on the left-handside and apply Gronwall's lemma to complete the proof.

In the next step, we will use this gain in regularity of the congurational densities toderive the classical formula for the Kramers stress tensor and to improve our estimatesfor it. More precisely, the following lemma holds true.

Lemma 3.9. Under the assumptions of Lemma 3.7, we have the identityˆD

ψnM∇qM−1 ⊗ q +

ˆD

∇qψn ⊗ q =

ˆD

ψnU ′q⊗ q−ˆD

ψn1 , (3.40)

for all n ∈ 1, ..., N. Moreover, the Kramers tensor satises

ˆΩ

∣∣∣∣ˆD

M∇qψn

M⊗ q

∣∣∣∣2 =

ˆΩ

∣∣∣∣ˆD

ψnU ′q⊗ q

∣∣∣∣2 ≤ C

ˆΩ

|ψn|2M

(3.41)

for all n ∈ 1, ..., N.

We will sketch the proof following the lines of [8] which is the long version of [10].

Proof. It takes advantage of the dense embedding of C∞(D)in

H1(D;M) :=

u ∈ H1

loc(D) :

ˆD

Mu2 +

ˆD

M |∇qu|2 <∞

(3.42)

which allows integration by parts with vanishing boundary terms at an approximationlevel. The rst conclusion then follows by density. Following the lines of [9], one maycompute

ˆΩ

∣∣∣∣ˆD

ψnU ′q⊗ q

∣∣∣∣2 ≤ C

ˆΩ×D

|ψn|2M

, (3.43)

with a constant C depending on the spring potential and on D.

Lemma 3.10. Under the assumptions of Lemma 3.7, there is a positive constant C de-pending only on initial data and on time T such that we have for ωn :=

´Dψn and for all

n ∈ 1, ..., N

12

ˆΩ

|ωn|2 + 12

n∑k=1

ˆΩ

∣∣ωk − ωk−1∣∣2 + 1

2τcx

n∑k=1

ˆΩ

∣∣∇xωk∣∣2 ≤ C .


Proof. Let us use the functionMωn = M´Dψn as a testing function in (3.3e). By ψn ∈ X,

we infer its admissibility. One easily computes

12

ˆΩ

(|ωn|2 +

∣∣ωn − ωn−1∣∣2 − ∣∣ωn−1

∣∣2)+ τ

ˆΩ

cx |∇xωn|2

+ 12τ

ˆΩ

1cq‖∇xJ εun‖2

L∞

(ˆD

ψn |q|2)ωn + 1

2τ

ˆΩ

cx ‖∇xJεβ(φn)‖2L∞ |ω

n|2

= −τˆ

Ω

cxωn∇xJεβ(φn) · ∇xω

n + 12τ

ˆΩ

1

cq

∥∥∇xJ ε

un−1

∥∥2

L∞

(ˆD

ψn−1 |q|2)ωn

+ 12τ

ˆΩ

cx∥∥∇xJε

β(φn−1

)∥∥2

L∞ωn−1ωn (3.44)

Noting |q| ≤ Qmax, using Young's inequality and summing up with respect to time yields

12

ˆΩ

|ωn|2 + 12

n∑k=1

ˆΩ

∣∣ωk − ωk−1∣∣2 + 1

2τcx

n∑k=1

ˆΩ

∣∣∇xωk∣∣2

≤ 12

ˆΩ

∣∣ω0∣∣2 + Cτ

ˆΩ

|ωn|2 + Cτn−1∑k=0

ˆΩ

∣∣ωk∣∣2 . (3.45)

Again we assume Cτ ≤ 14and use Gronwall's lemma to complete the proof.

Recalling the last step of the proof of the Lemma 3.2 and noting (A5), (A7), Lemma 3.7,and Lemma 3.10, we immediately obtain the following result:

Lemma 3.11. Under the assumptions of Lemma 3.7 and (A7), there exists a constantC > 0 depending only on initial data and on time T such that the estimate

τn∑k=1

∥∥φk∥∥2

H2(Ω)< C

holds true for all n ∈ 1, ..., N.

Now, we are in the position to prove compactness in time for un and φn.

Lemma 3.12. Let Sγ be the HelmholtzStokes operator which is dened in (4.2) withsome xed γ > 0. Under the assumptions of Lemma 3.7, there is a positive constant Cdepending only on initial data and on time T such that

n∑k=1

τ

[ˆΩ

∣∣∇xSγ∂−τ u

k∣∣2]2/d

+n∑k=1

τ

[ˆΩ

∣∣Sγ∂−τ uk∣∣2]2/d

≤ C (3.46)

for all n ∈ 1, ..., N.


Proof. Testing (3.3d) by Sγ∂−τ un and using (4.3) yields

ˆΩ

[γ∣∣∇xSγ

∂−τ u

n∣∣2 +

∣∣Sγ∂−τ un∣∣2] = −ˆ

Ω

2η(φn−1

)Dun : ∇xSγ

∂−τ u

n

−ˆ

Ω

φn−1∇xµnφ · Sγ

∂−τ u

n

+


n · Sγ∂−τ u

n

−ˆ

Ω

(un−1 · ∇x

)un · Sγ

∂−τ u

n−ˆ

Ω×D

(M∇q

ψn

M⊗ q

): Jε

∇xSγ

∂−τ u

n

=: I + II + III + IV + V . (3.47)

By using Young's inequality, Lemma 3.7, and the solenoidality of Sγ∂−τ u several timestogether with (2.16f), (A7), Lemma 3.10 and the GagliardoNirenberg inequality, one mayestimate

|I| ≤αγˆ

Ω


n∣∣2 + Cα,γ

ˆΩ

2η(φn−1

)|Dun|2 , (3.48)

|II| ≤∥∥Sγ∂−τ un∥∥L4(Ω)

∥∥φn−1∥∥L4(Ω)

∥∥∇xµnφ

∥∥L2(Ω)

≤C∥∥∇xSγ

∂−τ u

n∥∥

L2(Ω)

∥∥∇xµnφ

∥∥L2(Ω)

≤αγ∥∥∇xSγ

∂−τ u

n∥∥2

L2(Ω)+ Cα,γ

∥∥∇xµnφ

∥∥2

L2(Ω),

(3.49)

|III| =∣∣∣∣ˆ

Ω

∇xJεβ(φn)Sγ∂−τ u

nωn∣∣∣∣

≤α∥∥Sγ∂−τ un∥∥2

L2(Ω)+ Cα ,

(3.50)

|IV | =∣∣∣∣ˆ

Ω

(∇xSγ

∂−τ u

n· un−1

)· un

∣∣∣∣≤αγ

ˆΩ


n∣∣2 + Cα,γ

ˆΩ

∣∣un−1∣∣2 |un|2

≤αγˆ

Ω


n∣∣2 + Cα,γ

[(ˆΩ

|∇xun|2)d/2

+

(ˆΩ

∣∣∇xun−1∣∣2)d/2] .

(3.51)

For the last term on the right-hand side, we estimate using (2.28), (2.16c), Young's in-equality (P4) and Lemma 3.8

|V | =∣∣∣∣ˆ

Ω×D(ψnU ′q⊗ q) : Jε

∇xSγ

∂−τ u

n∣∣∣∣

≤αγˆ

Ω


n∣∣2 + Cα,γ .

(3.52)

By discrete integration with respect to time, using in particular Lemma 3.7, the claimfollows.

Later on, this result will permit to apply the compactness lemma of Aubin and Lions toidentify strongly converging subsequences of the velocity eld. To prove compactness intime for the phase-eld, we derive a Nikolsk'ii estimate.


Lemma 3.13. Under the assumptions of Lemma 3.7, there is a positive constant C de-pending only on initial data and on time T such that

τ

N−l∑k=1

ˆΩ

∣∣φk+l − φk∣∣2 ≤ Clτ (3.53)

for all l ∈ 1, ..., N.

Proof. Testing (3.3a) by τ(φk+l − φk

)with 0 ≤ k ≤ N − l and summing from n = k + 1

to k + l yields

ˆΩ

(φk+l − φk

)2 − τk+l∑

n=k+1

ˆΩ

φn−1un−1 · ∇x

(φk+l − φk

)+ τ 2

k+l∑n=k+1

ˆΩ

∣∣φn−1∣∣2∇xµ

nφ · ∇x

(φk+l − φk

)+ τ

k+l∑n=k+1

ˆΩ

∇xµnφ · ∇x

(φk+l − φk

)= 0 .

(3.54)

Computing

∣∣∣∣∣τk+l∑

n=k+1

ˆΩ

φn−1un−1 · ∇x

(φk+l − φk

)∣∣∣∣∣≤ τ

k+l∑n=k+1

∥∥un−1∥∥L6(Ω)

∥∥φn−1∥∥L3(Ω)

∥∥∇xφk+l −∇xφ

k∥∥L2(Ω)

≤ Cl−1∑m=0

τ 1/2∥∥∇xu

k+m∥∥L2(Ω)

∥∥∇xφk+m

∥∥L2(Ω)

τ 1/2∥∥∇xφ

k+m −∇xφk∥∥L2(Ω)

, (3.55)

∣∣∣∣∣τ 2

k+l∑n=k+1

ˆΩ

∣∣φn−1∣∣2∇xµ

nφ · ∇x

(φk+l − φk

)∣∣∣∣∣≤ τ 2

k+l∑n=k+1

∥∥φn−1∥∥2

L∞(Ω)

∥∥∇xµnφ

∥∥L2(Ω)

∥∥∇xφk+l −∇xφ

k∥∥L2(Ω)

≤ C

l−1∑m=0

τ∥∥φk+m

∥∥2

H2(Ω)τ 1/2

∥∥∇xµk+m+1φ

∥∥L2(Ω)

τ 1/2∥∥∇xφ

k+l −∇xφk∥∥L2(Ω)

, (3.56)


we again multiply by τ and sum from k = 1 to N − l to obtain

τN−l∑k=1

ˆΩ

(φk+l − φk

)2 ≤Cτl−1∑m=0

(N−l∑k=1

τ∥∥∇xu

k+m∥∥2

L2(Ω)

)1/2(sup

k=1,...,N−l

∥∥∇xφk+m

∥∥2

L2(Ω)

)1/2

×

(N−l∑k=1

τ∥∥∇xφ

k+l −∇xφk∥∥2

L2(Ω)

)1/2

+Cτl−1∑m=0

(N−l∑k=1

τ∥∥φk+m

∥∥2

H2(Ω)

)(N−l∑k=1

τ∥∥∇xµ

k+m+1φ

∥∥2

L2(Ω)

)1/2

×

(N−l∑k=1

τ∥∥∇xφ


L2(Ω)

)1/2

+Cτl−1∑m=0

(N−l∑k=1

τ∥∥∇xµ

k+m+1φ

∥∥2

L2(Ω)

)1/2

×

(N−l∑k=1

τ∥∥∇xφ


L2(Ω)

)1/2

≤Clτ ,(3.57)

due to Lemma 3.7 and Lemma 3.11.

In order to pass to the limit τ 0, we dene time-interpolants of the time-discretefunctions un, n = 0, ..., N, and introduce some related index-free notations as follows.

uτ (., t) := t−tn−1

τun(.) + tn−t

τun−1(.) t ∈

[tn−1, tn

], n ≥ 1 , (3.58a)

uτ,+(., t) := un(.), uτ,−(., t) := un−1(.) t ∈(tn−1, tn

], n ≥ 1 , (3.58b)

Furthermore, we will use the abbreviation aτ,± if an estimate is valid for aτ , aτ,+, and aτ,−.

For φ, µφ, ψ, and ω, analogous notations will be used. Let us x T > 0 and let us chooseN, τ such that T = Nτ. Using these notations and summing (3.3a)-(3.3e) from n = 1 toN , we may restate our set of equations asˆ

ΩT

∂

∂tφτθ −

ˆΩT

(uτ,− − τφτ,−∇xµ

τ,+φ

)· φτ,−∇xθ +

ˆΩT

∇xµτ,+φ · ∇xθ = 0

∀θ ∈ L2(0, T ;H1(Ω)

), (3.59a)

ˆΩT

µτ,+φ θ =

ˆΩT

∇xφτ,+ · ∇xθ +

ˆΩT

(W ′

+

(φτ,+

)+W ′

−(φτ,−

))θ

+

ˆΩT

β′(φτ,+, φτ,−

)Jεˆ

D

ψτ,−θ

∀θ ∈ L2(0, T ;H1(Ω)

), (3.59b)


ˆΩT

∂

∂tuτ ·w +

ˆΩT

((uτ,− · ∇x

)uτ,+

)·w +

ˆΩT

2η(φτ,−

)Duτ,+ : Dw

=

ˆΩT

divx

Jε

ˆD

ψτ,+U ′q⊗ q

·w +

ˆΩT

µτ,+φ ∇xφτ,− ·w

+

ˆ T

0

ˆΩ×DJεβ(φτ,+

)∇xψ

τ,+ ·w ∀w ∈ L2(0, T ;H1

σ,0(Ω)), (3.59c)

ˆ T

0

ˆΩ×D

∂

∂tψτ θ

M+

ˆ T

0

ˆΩ×D

θMuτ,+ · ∇xψ

τ,+−ˆ T

0

ˆΩ×D

ψτ,+(∇xJ ε

uτ,+

· q)· ∇q

θM

+

ˆ T

0

ˆΩ×D

cqM∇qψτ,+

M· ∇q

θM

+

ˆ T

0

ˆΩ×D

cxM∇xψτ,+

M· ∇x

θM

+

ˆ T

0

ˆΩ×D

cxψτ,+∇xJε

β(φτ,+

)· ∇x

θM

= 12

ˆ T

0

ˆΩ×D

|q|2cq

θM

(∥∥∇xJ ε

uτ,+

∥∥2

L∞ψτ,+ −

∥∥∇xJ ε

uτ,−

∥∥2

L∞ψτ,−

)+ 1

2cx

ˆ T

0

ˆΩ×D

θM

(∥∥∇xJεβ(φτ,+

)∥∥2

L∞ψτ,+ −

∥∥∇xJεβ(φτ,−

)∥∥2

L∞ψτ,−

)∀θ ∈ X . (3.59d)

From (A5), (3.43), Lemma 3.7, Lemma 3.8, Lemma 3.10, Lemma 3.11, Lemma 3.12 and(A7) we obtain

supt∈[0,T ]

ˆΩ

∣∣∇xφτ,(±)

∣∣2 +

ˆ T

0

∥∥φτ,(±)∥∥2

H2(Ω)+

ˆ T

0

ˆΩ

∣∣∇xµτ,+φ

∣∣2+ sup

t∈[0,T ]

ˆΩ

∣∣ωτ,(±)∣∣2 +

ˆ T

0

ˆΩ

∣∣∇xωτ,+∣∣2

+ supt∈[0,T ]

ˆΩ

∣∣uτ,(±)∣∣2 +

ˆ T

0

ˆΩ

∣∣Duτ,(±)∣∣+

ˆ T

0

∥∥Sγ∂uτ∂t ∥∥4/d

H1(Ω)

+ supt∈[0,T ]

ˆΩ×D

|ψτ,(±)|2M

+ supt∈[0,T ]

ˆΩ

∣∣∣∣Jε

ˆD

ψτ,(±)U ′q⊗ q

∣∣∣∣2+

ˆ T

0

ˆΩ×D

M∣∣∣∇x

ψτ,+

M

∣∣∣2 +

ˆ T

0

ˆΩ×D

M∣∣∣∇q

ψτ,+

M

∣∣∣2≤ C

(T, φ0,u0, ψ0

). (3.60)

3.3. Passage to the limit τ → 0. In this section, we prove the existence of weaksolutions. Let us begin with a result on convergence properties.


Lemma 3.14. Under the assumptions (A1)-(A7), there exists a subsequence (again de-

noted byφτ,(±), µ

τ,(±)φ ,uτ,(±), ψτ,(±)

τ) and functions (φ, µφ,u, ψ) satisfying

φ ∈ L∞(0, T ;H1(Ω)

)∩ L2

(0, T ;H2(Ω)

),

µφ ∈ L2(0, T ;H1(Ω)

),

u ∈ L∞(0, T ;L2(Ω)

)∩ L2

(0, T ;H1

σ,0(Ω))∩W 1,4/d

(0, T ;

(H1

σ,0(Ω))′)

,

ψ ∈ L2(0, T ;X+) ∩ L∞(0, T ;L2

(Ω×D;M−1

)),

ω :=

ˆD

ψ ∈ L∞(0, T ;L2(Ω)

)∩ L2

(0, T ;H1(Ω)

),

ω ≥ 0 f.a.e. t ∈ [0, T ] , x ∈ Ω ,

such that, as τ 0,

uτ,(±) ∗ u in L∞(0, T ;L2(Ω)

), (3.61a)

uτ,(±) u in L2(0, T ;H1

σ,0(Ω)), (3.61b)

Sγ∂uτ

∂t

Sγ

∂u∂t

in L4/d

(0, T ;H1

σ,0(Ω)), (3.61c)

uτ,(±) → u in L2(0, T ;Y s) , s ∈ [1, 2dd−2

) (3.61d)

φτ,(±) ∗ φ in L∞(0, T ;H1(Ω)

), (3.61e)

φτ,(±) φ in L2(0, T ;H2(Ω)

), (3.61f)

φτ,(±) → φ in Lp(0, T ;Ls(Ω)) ∀p <∞ , s ∈ [1, 2dd−2

) , (3.61g)

µτ,+φ µφ in L2(0, T ;H1(Ω)

), (3.61h)

ψτ,(±)

M1/2

∗ ψ

M1/2 in L∞(0, T ;L2(Ω×D)

), (3.61i)

ψτ,(±) ∗ ψ in L∞(0, T ;L2(Ω×D)

), (3.61j)

M1/2∇xψτ,+

M M1/2∇x

ψM

in L2(0, T ;L2(Ω×D)

), (3.61k)

M1/2∇qψτ,+

M M1/2∇q

ψM

in L2(0, T ;L2(Ω×D)

), (3.61l)

ωτ,(±) ∗ ω in L∞(0, T ;L2(Ω)

), (3.61m)

ωτ,+ ω in L2(0, T ;H1(Ω)

), (3.61n)

J ε

uτ,(±)

→ J εu in L2

(0, T ;W 1,∞(Ω)

), (3.61o)

and

Jε

ˆD

ψτ,(±)U ′q⊗ q

∗ Jε

ˆD

ψU ′q⊗ q

in L∞

(0, T ;L2(Ω)

), (3.61p)

β′(φτ,+, φτ,−

)Jεˆ

D

ψτ,+∗ β′(φ)Jε

ˆD

ψ

in L∞(ΩT ) . (3.61q)

Proof. The convergence expressed in (3.61a) and (3.61b) is a direct consequence of thebound in (3.60). To prove (3.61c), we note that Sγ constitutes the inverse Rieszisomorphism on H1

σ,0. Hence, weak∗-convergence of ∂uτ

∂ttowards ∂u

∂tis implied by (3.46).

Observe that we have the injections H1σ,0 ⊂⊂ Y s ⊂

(H1

σ,0

)′. Hence, uτ → u strongly in


L2(0, T ;Y s) for any s ∈ [1, 2dd−2

) by the Lemma of Aubin-Lions [31]. Noting Lemma 3.7,one may conclude ∥∥uτ − uτ,±

∥∥2

L2(0,T ;L2(Ω))≤ Cτ ,

which implies the result for uτ,±. The assertions (3.61e) to (3.61i) and (3.61m) to (3.61n)are entailed by the corresponding bounds in (3.60), appealing for the strong convergencein (3.61g) to (3.53) and Simon's compactness result [42]. Obviously, (3.61i) implies (3.61j)due to the boundedness ofM . AsM does not depend on x, the weak convergence (3.61k)is immediate. To prove (3.61l), we argue along the lines of (4.32) in [7]. Finally, (3.61o)follows combining (3.61d) and (2.16f). Similarly, (3.61p) is a consequence of (3.61j) and(2.16c) and (P4) together with (3.60). (3.61q) follows using (3.61m) and the regularityproperties of Jε. The nonnegativity of ψ and ω is inherited from ψτ,(±).

Our existence result reads as follows.

Theorem 3.15. Let d ∈ 2, 3. Under the assumptions (A1)-(A7), there is a quadru-ple (φ, µ,u, ψ) which solves the equal-density version of system (2.29) combined with theboundary conditions (2.30) in the following weak sense.

ˆΩT

(φ0 − φ)∂

∂tθ −

ˆΩT

φu · ∇xθ +

ˆΩT

∇xµφ · ∇xθ = 0

∀θ ∈ C1([0, T ];H1(Ω)

)with θ(., T ) ≡ 0, (3.62a)

ˆΩT

µφθ =

ˆΩT

∇xφ · ∇xθ +

ˆΩT

W ′(φ)θ +

ˆΩT

β′(φ)Jεˆ

D

ψ

θ

∀θ ∈ L2(0, T ;H1(Ω)

), (3.62b)

ˆΩT

〈 ∂∂t

u,w〉+

ˆΩT

(u · ∇x)u ·w +

ˆΩT

2η(φ)Du : Dw

=

ˆΩT

divx

Jε

ˆD

ψU ′q⊗ q

·w +

ˆΩT

µφ∇xφ ·w +

ˆ T

0

ˆΩ×DJεβ(φ)∇xψ ·w

∀w ∈ L4/(4−d)(0, T ;H1

σ,0(Ω)), (3.62c)

ˆ T

0

ˆΩ×D

(ψ0 − ψ)∂

∂tθM

+

ˆ T

0

ˆΩ×D

M θMu · ∇x

ψM−ˆ T

0

ˆΩ×D

ψ(∇xJ εu · q) · ∇qθM

+ cq

ˆ T

0

ˆΩ×D

M∇qψM· ∇q

θM

+ cx

ˆ T

0

ˆΩ×D

M∇xψM· ∇x

θM

+ cx

ˆ T

0

ˆΩ×D

ψ∇xJεβ(φ) · ∇xθM

= 0

(3.62d)


for all θ such that θM∈ C1

([0, T ];C∞

(Ω×D

))and θ(·, T ) ≡ 0. Moreover, the solution

has the following regularity properties.

u ∈ L∞(0, T ;L2(Ω)

)∩ L2

(0, T ;H1

σ,0(Ω))∩W 1,4/d

(0, T ;

(H1

σ,0(Ω))′)

, (3.63)

φ ∈ L∞(0, T ;H1(Ω)

)∩ L2

(0, T ;H2(Ω)

), (3.64)

µφ ∈ L2(0, T ;H1(Ω)

), (3.65)

ψ ∈ L2(0, T ;X+) ∩ L∞(0, T ;L2(Ω×D;M−1)

), (3.66)

ω :=

ˆD

ψ(·,q) dq ∈ L∞(0, T ;L2(Ω)

)∩ L2

(0, T ;H1(Ω)

), (3.67)

with ω ≥ 0 a.e. in ΩT .

Proof. This is a straightforward consequence of applying the convergence results collectedin (3.61) to the weak formulation (3.59).

4. Appendix

For the velocity eld u, we will use the well-known space of solenoidal functions withhomogeneous Dirichlet boundary data

H1σ,0(Ω) :=

w ∈H1

0 (Ω) : divx w = 0. (4.1)

By(H1

σ,0(Ω))′

we denote the dual space of H1σ,0(Ω). For given γ > 0 (γ = 1 would

be sucient), we dene the HelmholtzStokes operator Sγ :(H1

σ,0(Ω))′ → H1

σ,0(Ω),v 7→ Sγv, viaˆ

Ω

Sγv ·w dx + γ

ˆΩ

∇xSγv : ∇xw dx = 〈v,w〉 ∀w ∈H1σ,0(Ω) , (4.2)

where 〈., .〉 denotes the duality pairing between(H1

σ,0(Ω))′and H1

σ,0(Ω). For the readersconvenience, we collect some useful results to be found in [9]. First, we note

〈v,Sγv〉 =

ˆΩ

[γ |∇xSγv|2 + |Sγv|2

]dx ∀v ∈

(H1

σ,0(Ω))′, (4.3)

and ‖Sγ.‖H1(Ω) is a norm on(H1

σ,0(Ω))′. Besides, we note

‖Sγv‖2L2(Ω) + γ ‖∇xSγv‖2

L2(Ω) ≤ ‖v‖2L2(Ω) ∀v ∈ L2(Ω) , (4.4)

‖(I − Sγ)v‖2L2(Ω) + γ ‖∇x(I − Sγ)v‖2

L2(Ω)

= γ

ˆΩ

∇xv : ∇x(I − Sγ)v dx ≤ γ ‖∇xv‖2L2(Ω) ∀v ∈H1

σ,0(Ω) (4.5)

We need Lebesgue spaces of weakly solenoidal vector elds which are dened by

Y r :=

v ∈ Lr(Ω) :

ˆΩ

v · ∇xw = 0 ∀w ∈ W 1,rr−1

. (4.6)


Throughout this paper we will assume r ∈ (d, 6). This choice guarantees a compactembedding of H1

σ,0(Ω) into Y r and the validity ofˆΩ×D

u · ∇xχχM

= 12

ˆΩ×D

u · ∇x|χ|2M

= 0

for u ∈ Y r and χ ∈ X which is needed in the proof of Lemma 3.4.Adapted to the regularity properties of the congurational density ψ, we introduce theweighted Sobolev space

H1(Ω×D;M) :=θ ∈ L1

loc(Ω×D) : ‖θ‖H1(Ω×D;M) <∞

, (4.7)

with the associated norm

‖θ‖H1(Ω×D;M) :=

ˆΩ×D

M[|θ|2 + |∇xθ|2 + |∇qθ|2

]dq dx

1/2

. (4.8)

Moreover, we dene

X := M H1(Ω×D;M) , (4.9a)

X+ := θ ∈ X : θ(x,q) ≥ 0 for a.e. (x,q) ∈ Ω×D . (4.9b)

Obviously, X ⊂ L2(Ω×D;M−1) :=θ ∈ L1

loc(Ω×D) :

´Ω×D

|θ|2M

<∞. In addition, we

denote the intersection of the spaces Lq(Ω) for 1 ≤ q < p with Lp−(Ω). The notation‖w‖Lp− stands for ‖w‖Lp−ε for arbitrary, but xed ε > 0.

Acknowledgment: The authors gratefully acknowledge support by DFG (German Sci-ence Foundation) through the grant GR 1693/5-2 within the Priority Programme 1506Transport processes at uidic interfaces.

References

1. H. Abels, Existence of weak solutions for a diuse interface model for viscous incompressible uidswith general densities., Comm. Math. Phys. 289 (2009), no. 1, 4573.

2. H. Abels, D. Depner, and H. Garcke, On an incompressible Navier-Stokes/Cahn-Hilliard system withdegenerate mobility, Ann. Inst. H. Poincaré, Analyse non-linéaire 30 (2013), no. 6, 11751190.

3. H. Abels, H. Garcke, and G. Grün, Thermodynamically consistent, frame indierent diuse interfacemodels for incompressible two-phase ows with dierent densities, Math. Models Methods Appl. Sci.22 (2012), no. 3, 1150013.

4. G.L. Aki, W. Dreyer, J. Giesselmann, and C. Kraus, A quasi-incompressible diuse interface modelwith phase transition, Math. Models Methods Appl. Sci. 24 (2014), no. 5, 827861.

5. F. Armero and J.C. Simo, Formulation of a new class of fractional-step methods for the incompressiblemhd equations that retains the long-term dissipativity of the continuum dynamical system, Fields Inst.Commun. 10 (1996), 123.

6. W. Arndt, Gewöhnliche Dierentialgleichungen, Springer, 1991.7. J. W. Barrett, C. Schwab, and E. Süli, Existence of global weak solutions for some polymeric ow

models, Math. Models Methods Appl. Sci. 15 (2005), 939983.8. J. W. Barrett and E. Süli, Existence and equilibration of global weak solutions to nitely extensible

nonlinear bead-spring chain models for dilute polymers, arXiv:1004.1432.9. , Existence of global weak solutions to some regularized kinetic models for dilute polymers,

Multiscale Model. Simul. 6 (2007), 506546.


10. , Existence and equilibration of global weak solutions to kinetic models for dilute polymersi: Finitely extensible nonlinear bead-spring chains, Math. Models Methods Appl. Sci. 21 (2011),12111289.

11. P. Biller and F. Petruccione, The ow of dilute polymer solutions in conned geometries: a consistentnumerical approach, J. Non-Newtonian Fluid Mech. 25 (1987), no. 3, 347 364.

12. R.B. Bird and C.F. Curtiss: Polymer Symposia, Molecular theory expressions for the stress tensor inowing polymeric liquids, J. Polymer Sci. 73 (1985), 187199.

13. P. Constantin, Nonlinear FokkerPlanck NavierStokes systems, Commun. Math. Sci. 3 (2005), no. 4,531544.

14. S. R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North-Holland, 1969.15. H. Ding, P. D. M. Spelt, and C. Shu, Diuse interface model for incompressible two-phase ows with

large density ratios, J. Comput. Phys. 226 (2007), 20782095.16. W. E, T. Li, and P.-W. Zhang, Well-posedness for the dumbbell model of polymeric uids, Comm.

Math. Phys. 248 (2004), no. 2, 409427. MR 2073140 (2005d:35210)17. L. C. Evans, Partial dierential equations, Graduate studies in mathematics, vol. 19, AMS, 1998.18. M. A. Fontelos, G. Grün, and S. Jörres, On a phase-eld model for electrowetting and other electroki-

netic phenomena, SIAM J. Math. Anal. 43 (2011), no. 1, 527563.19. G. Grün, On the convergence of entropy consistent schemes for lubricationtype equations in multiple

space dimensions, Math.Comp. 72 (2003), no. 243, 12511279. MR 1972735 (2004c:65109)20. , On convergent schemes for diuse interface models for two-phase ow of incompressible

uids with general mass densities, SIAM J.Num.Anal. 51 (2013), no. 6, 30363061.21. G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin lm equation,

Num. Math. 87 (2000), 113152. MR 1800156 (2002h:76108)22. F. Guillén-González and G. Tierra, Splitting schemes for a NavierStokesCahnHilliard model for

two uids with dierent densities, Journal of Computational Mathematics 32 (2014), no. 6, 643664.23. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49

(1977), no. 3, 435479.24. J. Honerkamp and H. Römer, Klassische theoretische Physik, Springer-Verlag, 1993.25. B. Jourdain, T. Leliévre, and C. Le Bris, Existence of solution for a micromacro model of polymeric

uid: the FENE model, J. Funct. Anal. 209 (2004), no. 1, 162 193.26. H.A. Kramers, Het gedrag van macromoleculen in een stroomende vloeistof, Physica 11 (1944), 119.27. A. Kufner, J. Old°ich, and S. Fu£ík, Function spaces, Noordho, 1977.28. C. Le Bris and T. Lelièvre, Multiscale modelling of complex uids: A mathematical initiation, Mul-

tiscale Modeling and Simulation in Science (Björn Engquist, Per Lötstedt, and Olof Runborg, eds.),Lecture Notes in Computational Science and Engineering, vol. 66, Springer, Berlin, Heidelberg, DE,2009, pp. 49137.

29. , Micro-macro models for viscoelastic uids: modelling, mathematics and numerics, Sci. ChinaMath. 55 (2012), no. 2, 353384.

30. T. Li, H. Zhang, and P.-W. Zhang, Local existence for the dumbbell model of polymeric uids, Comm.Part. Di. Equations 29 (2004), no. 5-6, 903923.

31. J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod etGauthier-Villars, Paris, FR, 1969.

32. P. L. Lions and N. Masmoudi, Global existence of weak solutions to some micro-macro models, C. R.Math. Acad. Sci. Paris 345 (2007), no. 1, 1520.

33. J. Lowengrub and L. Truskinovsky, Quasi-incompressible CahnHilliard uids and topological transi-tions, R. Soc. Lond. Proc. Ser. A Math. Phys. Engrg. Sci 454 (1998), no. 1978, 26172654.

34. N. Masmoudi, Well-posedness for the fene dumbbell model of polymeric ows, Comm. Pure Appl.Math. 61 (2008), no. 12, 16851714.

35. , Global existence of weak solutions to the FENE dumbbell model of polymeric ows, Invent.Math. 191 (2013), no. 2, 427500.

36. S. Minjeaud, An unconditionally stable uncoupled scheme for a triphasic CahnHilliard/NavierfconstaStokes model., Num. Methods for PDE 29 (2013), no. 2, 584618.

37. F. Otto and A. Tzavaras, Continuity of velocity gradients in suspensions of rodlike molecules, Comm.Math. Phys. 277 (2008), no. 3, 729758 (English).


38. F. Petruccione and P. Biller, A consistent numerical-analysis of the tube ow of dilute polymer-solutions, J. Rheol. 32 (1988), no. 1, 121.

39. T. Qian, X.-P. Wang, and P. Sheng, A variational approach to the moving contact line hydrodynamics,J. Fluid Mech. 564 (2006), 333360.

40. M. Renardy, An existence theorem for model equations resulting from kinetic theories of polymersolutions, SIAM J.Math. Anal. 22 (1991), no. 2, 313327.

41. J.D. Schieber and H.C. Öttinger, The eects of bead inertia on the Rouse model, J. Chem. Phys. 89(1988), 6972 6981.

42. J. Simon, Compact sets in the space Lp(0, T ;B), Annali di Matematica Pura ed Applicata (IV) 146(1987), 6596. MR 916688 (89c:46055)

43. P. Yue, J.J. Feng, C. Liu, and J. Shen, A diuse-interface method for simulating two-phase ows ofcomplex uids, J. Fluid Mech. 515 (2004), 293317.

44. H. Zhang and P.-W. Zhang, Local existence for the fene-dumbbell model of polymeric uids, Arch.Rational Mech. Anal. 181 (2006), no. 2, 373400.

45. Y. Zhang, H. Wang, and T. Tang, Simulating two-phase viscoelastic ows using moving nite elementmethods, Commun. Comput. Phys. 7 (2010), no. 2, 333349.

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gen Germany

E-mail address: [email protected], [email protected]

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