On an Incompressible Navier-Stokes/Cahn-Hilliard System with Degenerate Mobility
aa r X i v : . [ m a t h . A P ] O c t On an Incompressible Navier-Stokes/Cahn-HilliardSystem with Degenerate Mobility
Helmut Abels ∗ , Daniel Depner † , and Harald Garcke ‡ Abstract
We prove existence of weak solutions for a diffuse interface model forthe flow of two viscous incompressible Newtonian fluids in a boundeddomain by allowing for a degenerate mobility. The model has beendeveloped by Abels, Garcke and Gr¨un for fluids with different den-sities and leads to a solenoidal velocity field. It is given by a non-homogeneous Navier-Stokes system with a modified convective termcoupled to a Cahn-Hilliard system, such that an energy estimate is ful-filled which follows from the fact that the model is thermodynamicallyconsistent.
Key words:
Two-phase flow, Navier-Stokes equations, diffuse interfacemodel, mixtures of viscous fluids, Cahn-Hilliard equation, degenerate mo-bility
AMS-Classification:
Primary: 76T99; Secondary: 35Q30, 35Q35, 76D03,76D05, 76D27, 76D45
Classically the interface between two immiscible, viscous fluids has beenmodelled in the context of sharp interface approaches, see e.g. [Mue85]. Butin the context of sharp interface models it is difficult to describe topologicalchanges, as e.g. pinch off and situations where different interfaces or differentparts of an interface connect. In the last 20 years phase field approaches havebeen a promising new approach to model interfacial evolution in situationswhere interfacial energy effects are important, see e.g. [Che02]. In phasefield approaches a phase field or order parameter is introduced which rapidly ∗ Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, Germany, e-mail: [email protected] † Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, Germany, e-mail: [email protected] ‡ Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, Germany, e-mail: [email protected] ∂ t ( ρ ( ϕ ) v ) + div( v ⊗ ( ρ ( ϕ ) v + e J )) − div(2 η ( ϕ ) D v ) + ∇ p = − div( a ( ϕ ) ∇ ϕ ⊗ ∇ ϕ ) in Q T , div v = 0 in Q T ,∂ t ϕ + v · ∇ ϕ = div ( m ( ϕ ) ∇ µ ) in Q T ,µ = Ψ ′ ( ϕ ) + a ′ ( ϕ ) |∇ ϕ | − div ( a ( ϕ ) ∇ ϕ ) in Q T , e J = − ˜ ρ − ˜ ρ m ( ϕ ) ∇ µ , Q T = Ω × (0 , T ) for 0 < T < ∞ , and Ω ⊂ R d , d = 2 ,
3, is a sufficiently smooth bounded domain. We close the system withthe boundary and initial conditions v | ∂ Ω = 0 on ∂ Ω × (0 , T ) ,∂ n ϕ | ∂ Ω = ∂ n µ | ∂ Ω = 0 on ∂ Ω × (0 , T ) , ( v , ϕ ) | t =0 = ( v , ϕ ) in Ω , where ∂ n ϕ = n · ∇ ϕ and n denotes the exterior normal at ∂ Ω. Here v is thevolume averaged velocity, ρ = ρ ( ϕ ) is the density of the mixture of the twofluids, ϕ is the difference of the volume fractions of the two fluids and weassume a constitutive relation between ρ and the order parameter ϕ givenby ρ ( ϕ ) = (˜ ρ + ˜ ρ ) + (˜ ρ − ˜ ρ ) ϕ , see [ADG12] for details. In addition, p is the pressure, µ is the chemical potential associated to ϕ and ˜ ρ , ˜ ρ are the specific constant mass densities of the unmixed fluids. Moreover, D v = ( ∇ v + ∇ v T ), η ( ϕ ) > m ( ϕ ) ≥ ϕ ) is the homogeneous freeenergy density for the mixture and the (total) free energy of the system isgiven by E free ( ϕ ) = Z Ω (cid:18) Ψ( ϕ ) + a ( ϕ ) |∇ ϕ | (cid:19) dx for some positive coefficient a ( ϕ ). The kinetic energy is given by E kin ( ϕ, v ) = R Ω ρ ( ϕ ) | v | dx and the total energy as the sum of the kinetic and free energyis E tot ( ϕ, v ) = E kin ( ϕ, v ) + E free ( ϕ )= Z Ω ρ ( ϕ ) | v | dx + Z Ω (cid:18) Ψ( ϕ ) + a ( ϕ ) |∇ ϕ | (cid:19) dx. (1.1)In addition there have been further modelling attempts for two phase flowwith different densities. We refer to Boyer [Boy02] and the recent work ofAki et al. [ADGK12]. We remark that for the model of Boyer no energyinequalities are known and the model of Aki et al. does not lead to velocityfields which are divergence free.In [ADG12] an existence result for the above Navier-Stokes/Cahn-Hil-liard model has been shown in the case of a non-degenerate mobility m ( ϕ ).As is discussed in [AGG12] the case with non-degenerate mobility can leadto Ostwald ripening effects, i.e., in particular larger drops can grow to theexpense of smaller ones. In many applications this is not reasonable and aspointed out in [AGG12] degenerate mobilities avoid Ostwald ripening andhence the case of degenerate mobilities is very important in applications. Inwhat follows we assume that m ( ϕ ) = 1 − ϕ for | ϕ | ≤ R . In this way we do not allow for diffusion through the bulk,3.e., the region where ϕ = 1 resp. ϕ = −
1, but only in the interfacial region,where | ϕ | <
1. The degenerate mobility leads to the physically reasonablebound | ϕ | ≤ ϕ , which is the difference of volumefractions and therefore we can consider in this work a smooth homogeneousfree energy density Ψ in contrast to the previous work [ADG12].For the Cahn-Hilliard equations without the coupling to the Navier-Stokes equations Elliott and Garcke [EG96] considered the case of a de-generate mobility, see also Gr¨un [Gru95]. We will use a suitable testingprocedure from the work [EG96] to get a bound for the second derivativesof a function of ϕ in the energy estimates of Lemma 3.7. We point out thatour result is also new for the case of model H with degenerate mobility,i.e., ˜ ρ = ˜ ρ , which implies e J = 0 in the above Navier-Stokes/Cahn-Hilliardsystem.The structure of the article is as follows: In Section 2 we summarizesome notation and preliminary results. Then, in Section 3, we reformulatethe Navier-Stokes/Cahn-Hilliard system suitably, define weak solutions andstate our main result on existence of weak solutions. For the proof of theexistence theorem in Subsections 3.2 and 3.3 we approximate the equationsby a problem with positive mobility m ε and singular homogeneous free en-ergy density Ψ ε . For the solution ( v ε , ϕ ε , J ε ) of the approximation (with J ε = − m ε ( ϕ ε ) ∇ µ ε ) we derive suitable energy estimates to get weak limits.Then we extend the weak convergences to strong ones by using methodssimilar to the previous work of the authors [ADG12], careful estimates ofthe additional singular free energy density and by an additional subtle ar-gument with the help of time differences and a theorem of Simon [Sim87].We remark that this last point would be easier in the case of a constantcoefficient a ( ϕ ) in the free energy. Finally we can pass to the limit ε → v ε , ϕ ε , J ε ) and recover the identitiesfor the weak solution of the main problem. We denote a ⊗ b = ( a i b j ) di,j =1 for a, b ∈ R d and A sym = ( A + A T ) for amatrix A ∈ R d × d . If X is a Banach space and X ′ is its dual, then h f, g i ≡ h f, g i X ′ ,X = f ( g ) , f ∈ X ′ , g ∈ X, denotes the duality product. We write X ֒ → ֒ → Y if X is compactly em-bedded into Y . Moreover, if H is a Hilbert space, ( · , · ) H denotes its innerproduct. Moreover, we use the abbreviation ( . , . ) M = ( . , . ) L ( M ) . Function spaces: If M ⊆ R d is measurable, L q ( M ), 1 ≤ q ≤ ∞ , denotesthe usual Lebesgue-space and k . k q its norm. Moreover, L q ( M ; X ) denotesthe set of all strongly measurable q -integrable functions if q ∈ [1 , ∞ ) and4ssentially bounded strongly measurable functions, if q = ∞ , where X is aBanach space.Recall that, if X is a Banach space with the Radon-Nikodym property,then L q ( M ; X ) ′ = L q ′ ( M ; X ′ ) for every 1 ≤ q < ∞ by means of the duality product h f, g i = R M h f ( x ) , g ( x ) i X ′ ,X dx for f ∈ L q ′ ( M ; X ′ ), g ∈ L q ( M ; X ). If X is reflexive or X ′ is separable, then X hasthe Radon-Nikodym property, cf. Diestel and Uhl [DU77].Moreover, we recall the Lemma of Aubin-Lions: If X ֒ → ֒ → X ֒ → X are Banach spaces, 1 < p < ∞ , 1 ≤ q < ∞ , and I ⊂ R is a bounded interval,then (cid:26) v ∈ L p ( I ; X ) : dvdt ∈ L q ( I ; X ) (cid:27) ֒ → ֒ → L p ( I ; X ) . (2.1)See J.-L. Lions [Lio69] for the case q > q = 1.Let Ω ⊂ R d be a domain. Then W kq (Ω), k ∈ N , 1 ≤ q ≤ ∞ , de-notes the usual L q -Sobolev space, W kq, (Ω) the closure of C ∞ (Ω) in W kq (Ω), W − kq (Ω) = ( W kq ′ , (Ω)) ′ , and W − kq, (Ω) = ( W kq ′ (Ω)) ′ . We also use the abbrevi-ation H k (Ω) = W k (Ω).Given f ∈ L (Ω), we denote by f Ω = | Ω | R Ω f ( x ) dx its mean value.Moreover, for m ∈ R we set L q ( m ) (Ω) := { f ∈ L q (Ω) : f Ω = m } , ≤ q ≤ ∞ . Then for f ∈ L (Ω) we observe that P f := f − f Ω = f − | Ω | Z Ω f ( x ) dx is the orthogonal projection onto L (Ω). Furthermore, we define H ≡ H (Ω) = H (Ω) ∩ L (Ω) , ( c, d ) H (Ω) := ( ∇ c, ∇ d ) L (Ω) . Then H (Ω) is a Hilbert space due to Poincar´e’s inequality. Spaces of solenoidal vector-fields:
For a bounded domain Ω ⊂ R d wedenote by C ∞ ,σ (Ω) in the following the space of all divergence free vectorfields in C ∞ (Ω) d and L σ (Ω) is its closure in the L -norm. The correspondingHelmholtz projection is denoted by P σ , cf. e.g. Sohr [Soh01]. We note that P σ f = f − ∇ p , where p ∈ W (Ω) ∩ L (Ω) is the solution of the weakNeumann problem( ∇ p, ∇ ϕ ) Ω = ( f, ∇ ϕ ) for all ϕ ∈ C ∞ (Ω) . (2.2)5 paces of continuous vector-fields: In the following let I = [0 , T ]with 0 < T < ∞ or let I = [0 , ∞ ) if T = ∞ and let X be a Banachspace. Then BC ( I ; X ) is the Banach space of all bounded and continu-ous f : I → X equipped with the supremum norm and BU C ( I ; X ) is thesubspace of all bounded and uniformly continuous functions. Moreover, wedefine BC w ( I ; X ) as the topological vector space of all bounded and weaklycontinuous functions f : I → X . By C ∞ (0 , T ; X ) we denote the vector spaceof all smooth functions f : (0 , T ) → X with supp f ⊂⊂ (0 , T ). We say that f ∈ W p (0 , T ; X ) for 1 ≤ p < ∞ , if and only if f, dfdt ∈ L p (0 , T ; X ), where dfdt denotes the vector-valued distributional derivative of f . Finally, we note: Lemma 2.1.
Let
X, Y be two Banach spaces such that
Y ֒ → X and X ′ ֒ → Y ′ densely. Then L ∞ ( I ; Y ) ∩ BU C ( I ; X ) ֒ → BC w ( I ; Y ) . For a proof, see e.g. Abels [Abe09a].
In this section we prove an existence result for the Navier-Stokes/Cahn-Hilliard system from the introduction for a situation with degenerate mo-bility. Since in this case we will not have a control of the gradient ofthe chemical potential, we reformulate the equations by introducing a flux J = − m ( ϕ ) ∇ µ consisting of the product of the mobility and the gradient ofthe chemical potential. In this way, the complete system is given by: ∂ t ( ρ v ) + div( ρ v ⊗ v ) − div(2 η ( ϕ ) D v ) + ∇ p + div( v ⊗ β J ) = − div( a ( ϕ ) ∇ ϕ ⊗ ∇ ϕ ) in Q T , (3.1a)div v = 0 in Q T , (3.1b) ∂ t ϕ + v · ∇ ϕ = − div J in Q T , (3.1c) J = − m ( ϕ ) ∇ (cid:18) Ψ ′ ( ϕ ) + a ′ ( ϕ ) |∇ ϕ | − div ( a ( ϕ ) ∇ ϕ ) (cid:19) in Q T , (3.1d) v | ∂ Ω = 0 on S T , (3.1e) ∂ n ϕ | ∂ Ω = ( J · n ) | ∂ Ω = 0 on S T , (3.1f)( v , ϕ ) | t =0 = ( v , ϕ ) in Ω , (3.1g)where we set β = ˜ ρ − ˜ ρ and J = − m ( ϕ ) ∇ µ as indicated above. Theconstitutive relation between density and phase field is given by ρ ( ϕ ) = (˜ ρ + ˜ ρ ) + (˜ ρ − ˜ ρ ) ϕ as derived in Abels, Garcke and Gr¨un [AGG12],where ˜ ρ i > ϕ is the difference of the volume fractions of the fluids. By introducing J , we omitted the chemical potential µ in our equations and we search from6ow on for unknowns ( v , ϕ, J ). In the above formulation and in the fol-lowing, we use the abbreviations for space-time cylinders Q ( s,t ) = Ω × ( s, t )and Q t = Q (0 ,t ) and analogously for the boundary S ( s,t ) = ∂ Ω × ( s, t ) and S t = S (0 ,t ) . Equation (3.1e) is the no-slip boundary condition for viscousfluids, ( J · n ) | ∂ Ω = 0 resulting from ∂ n µ | ∂ Ω = 0 means that there is no massflux of the components through the boundary, and ∂ n ϕ | ∂ Ω = 0 describesa contact angle of π/ In the following we summarize the assumptions needed to formulate thenotion of a weak solution of (3.1a)-(3.1g) and an existence result.
Assumption 3.1.
We assume that Ω ⊂ R d , d = 2 , , is a bounded domainwith smooth boundary and additionally we impose the following conditions. ( i ) We assume a, Ψ ∈ C ( R ) , η ∈ C ( R ) and < c ≤ a ( s ) , η ( s ) ≤ K forgiven constants c , K > . ( ii ) For the mobility m we assume that m ( s ) = (cid:26) − s , if | s | ≤ , , else . (3.2)We remark that other mobilities which degenerate linearly at s = ± a ( ϕ ) inthe free energy, so that we can replace the two terms with a ( ϕ ) in equation(3.1d) by a single one. To this end, we introduce the function A ( s ) := R s p a ( τ ) dτ . Then A ′ ( s ) = p a ( s ) and − p a ( ϕ ) ∆ A ( ϕ ) = a ′ ( ϕ ) |∇ ϕ | − div ( a ( ϕ ) ∇ ϕ )resulting from a straightforward calculation. By reparametrizing the poten-tial Ψ through e Ψ : R → R , e Ψ( r ) := Ψ( A − ( r )) we see Ψ ′ ( s ) = p a ( s ) e Ψ ′ ( A ( s ))and therefore we can replace line (3.1d) with the following one: J = − m ( ϕ ) ∇ (cid:16)p a ( ϕ ) (cid:16) e Ψ ′ ( A ( ϕ )) − ∆ A ( ϕ ) (cid:17)(cid:17) . (3.3)We also rewrite the free energy with the help of A to E free ( ϕ ) = Z Ω (cid:18) e Ψ( A ( ϕ )) + |∇ A ( ϕ ) | (cid:19) dx . emark 3.2. With the above notation and with the calculation − div( a ( ϕ ) ∇ ϕ ⊗ ∇ ϕ )= − div( a ( ϕ ) ∇ ϕ ) ∇ ϕ − a ( ϕ ) ∇ (cid:18) |∇ ϕ | (cid:19) = − div( a ( ϕ ) ∇ ϕ ) ∇ ϕ + ∇ ( a ( ϕ )) |∇ ϕ | − ∇ (cid:18) a ( ϕ ) |∇ ϕ | (cid:19) = (cid:18) − div( a ( ϕ ) ∇ ϕ ) + a ′ ( ϕ ) |∇ ϕ | (cid:19) ∇ ϕ − ∇ (cid:18) a ( ϕ ) |∇ ϕ | (cid:19) = − p a ( ϕ )∆ A ( ϕ ) ∇ ϕ − ∇ (cid:18) a ( ϕ ) |∇ ϕ | (cid:19) we rewrite line (3.1a) with a new pressure g = p + a ( ϕ ) |∇ ϕ | into: ∂ t ( ρ v ) + div( ρ v ⊗ v ) − div(2 η ( ϕ ) D v ) + ∇ g + div( v ⊗ β J )= − p a ( ϕ )∆ A ( ϕ ) ∇ ϕ . (3.4) We remark that in contrast to the formulation in [ADG12] we do not usethe equation for the chemical potential here.
Now we can define a weak solution of problem (3.1a)-(3.1g).
Definition 3.3.
Let T ∈ (0 , ∞ ) , v ∈ L σ (Ω) and ϕ ∈ H (Ω) with | ϕ | ≤ almost everywhere in Ω . If in addition Assumption 3.1 holds, we call thetriple ( v , ϕ, J ) with the properties v ∈ BC w ([0 , T ]; L σ (Ω)) ∩ L (0 , T ; H (Ω) d ) ,ϕ ∈ BC w ([0 , T ]; H (Ω)) ∩ L (0 , T ; H (Ω)) with | ϕ | ≤ a.e. in Q T , J ∈ L (0 , T ; L (Ω) d ) and ( v , ϕ ) | t =0 = ( v , ϕ ) a weak solution of (3.1a) - (3.1g) if the following conditions are satisfied: − ( ρ v , ∂ t ψ ) Q T + (div( ρ v ⊗ v ) , ψ ) Q T + (2 η ( ϕ ) D v , D ψ ) Q T − (( v ⊗ β J ) , ∇ ψ ) Q T = − (cid:16)p a ( ϕ )∆ A ( ϕ ) ∇ ϕ, ψ (cid:17) Q T (3.5) for all ψ ∈ [ C ∞ (Ω × (0 , T ))] d with div ψ = 0 , − Z Q T ϕ ∂ t ζ dx dt + Z Q T ( v · ∇ ϕ ) ζ dx dt = Z Q T J · ∇ ζ dx dt (3.6) for all ζ ∈ C ∞ ((0 , T ; C (Ω)) and Z Q T J · η dx dt = − Z Q T (cid:16)p a ( ϕ ) (cid:16) e Ψ ′ ( A ( ϕ )) − ∆ A ( ϕ ) (cid:17)(cid:17) div( m ( ϕ ) η ) dx dt (3.7) for all η ∈ L (0 , T ; H (Ω) d ) ∩ L ∞ ( Q T ) d which fulfill η · n = 0 on S T . emark 3.4. The identity (3.7) is a weak version of J = − m ( ϕ ) ∇ (cid:16)p a ( ϕ ) (cid:16) e Ψ ′ ( A ( ϕ )) − ∆ A ( ϕ ) (cid:17)(cid:17) . Our main result of this work is the following existence theorem for weaksolutions on an arbitrary time interval [0 , T ], where
T > Theorem 3.5.
Let Assumption 3.1 hold, v ∈ L σ (Ω) and ϕ ∈ H (Ω) with | ϕ | ≤ almost everywhere in Ω . Then there exists a weak solution ( v , ϕ, J ) of (3.1a) - (3.1g) in the sense of Definition 3.3. Moreover for some b J ∈ L ( Q T ) it holds that J = p m ( ϕ ) b J and E tot ( ϕ ( t ) , v ( t )) + Z Q ( s,t ) η ( ϕ ) | D v | dx dτ + Z Q ( s,t ) | b J | dx dτ ≤ E tot ( ϕ ( s ) , v ( s )) (3.8) for all t ∈ [ s, T ) and almost all s ∈ [0 , T ) including s = 0 . The total energy E tot is the sum of the kinetic and the free energy, cf. (1.1) . In particular, J = 0 a.e. on the set {| ϕ | = 1 } . The proof of the theorem will be done in the next two subsections. Butfirst of all we consider a special case which can then be excluded in thefollowing proof. Due to | ϕ | ≤ R Ω − ϕ dx ∈ [ − , R Ω − ϕ dx = 1 we can then conclude that ϕ ≡ ϕ ≡ J ≡ v be the weak solution of the incompressible Navier-Stokesequations without coupling to the Cahn-Hilliard equation, where ρ and η areconstants. The situation where R Ω − ϕ dx = − Z Ω − ϕ dx ∈ ( − , , which will be needed for the reference to the previous existence result of theauthors [ADG12] and for the proof of Lemma 3.7, ( iii ). In the following we substitute problem (3.1a)-(3.1g) by an approximationwith positive mobility and a singular homogeneous free energy density, whichcan be solved with the result from the authors in [ADG12]. For the weaksolutions of the approximation we then derive energy estimates.First we approximate the degenerate mobility m by a strictly positive m ε as m ε ( s ) := m ( − ε ) for s ≤ − ε ,m ( s ) for | s | < − ε ,m (1 − ε ) for s ≥ − ε .
9n addition we use a singular homogeneous free energy density Ψ ε given byΨ ε ( s ) := Ψ( s ) + ε Ψ ln ( s ) , whereΨ ln ( s ) := (1 + s ) ln(1 + s ) + (1 − s ) ln(1 − s ) . Then Ψ ε ∈ C ([ − , ∩ C (( − , s →± Ψ ′ ε ( s ) = ±∞ , Ψ ′′ ε ( s ) ≥ κ for some κ ∈ R and lim s →± Ψ ′′ ε ( s )Ψ ′ ε ( s ) = + ∞ . To deal with the positive coefficient a ( ϕ ), we set similarly as above e Ψ ln ( r ) :=Ψ ln ( A − ( r )) and e Ψ ε ( r ) := Ψ ε ( A − ( r )) for r ∈ [ a, b ] := A ([ − , m by m ε and Ψ by Ψ ε and consider the following ap-proximate problem, this time for unknowns ( v , ϕ, µ ): ∂ t ( ρ v ) + div ( ρ v ⊗ v ) − div (2 η ( ϕ ) D v ) + ∇ g + div ( v ⊗ βm ε ( ϕ ) ∇ µ ) = − p a ( ϕ )∆ A ( ϕ ) ∇ ϕ in Q T , (3.9a)div v = 0 in Q T , (3.9b) ∂ t ϕ + v · ∇ ϕ = div( m ε ( ϕ ) ∇ µ ) in Q T , (3.9c) µ = p a ( ϕ ) (cid:16) e Ψ ′ ε ( A ( ϕ )) − ∆ A ( ϕ ) (cid:17) in Q T , (3.9d) v | ∂ Ω = 0 on S T , (3.9e) ∂ n ϕ | ∂ Ω = ∂ n µ | ∂ Ω = 0 on S T , (3.9f)( v , ϕ ) | t =0 = ( v , ϕ ) in Ω . (3.9g)From [ADG12] we get the existence of a weak solution ( v ε , ϕ ε , µ ε ) with theproperties v ε ∈ BC w ([0 , T ]; L σ (Ω)) ∩ L (0 , T ; H (Ω) d ) ,ϕ ε ∈ BC w ([0 , T ]; H (Ω)) ∩ L (0 , T ; H (Ω)) , Ψ ′ ε ( ϕ ε ) ∈ L (0 , T ; L (Ω)) ,µ ε ∈ L (0 , T ; H (Ω)) and( v ε , ϕ ε ) | t =0 = ( v , ϕ )in the following sense: − ( ρ ε v ε , ∂ t ψ ) Q T + (div( ρ ε v ε ⊗ v ε ) , ψ ) Q T + (2 η ( ϕ ε ) D v ε , D ψ ) Q T − (( v ε ⊗ βm ε ( ϕ ε ) ∇ µ ε ) , ∇ ψ ) Q T = ( µ ε ∇ ϕ ε , ψ ) Q T (3.10)for all ψ ∈ [ C ∞ (Ω × (0 , T ))] d with div ψ = 0, − ( ϕ ε , ∂ t ζ ) Q T + ( v ε · ∇ ϕ ε , ζ ) Q T = − ( m ε ( ϕ ε ) ∇ µ ε , ∇ ζ ) Q T (3.11)10or all ζ ∈ C ∞ ((0 , T ); C (Ω)) and µ ε = p a ( ϕ ε ) (cid:16) e Ψ ′ ε ( A ( ϕ ε )) − ∆ A ( ϕ ε ) (cid:17) almost everywhere in Q T . (3.12)Moreover, E tot ( ϕ ε ( t ) , v ε ( t )) + Z Q ( s,t ) η ( ϕ ε ) | D v ε | dx dτ + Z Q ( s,t ) m ε ( ϕ ε ) |∇ µ ε | dx dτ ≤ E tot ( ϕ ε ( s ) , v ε ( s )) (3.13)for all t ∈ [ s, T ) and almost all s ∈ [0 , T ) has to hold (including s = 0).Herein ρ ε is given as ρ ε = (˜ ρ + ˜ ρ ) + (˜ ρ − ˜ ρ ) ϕ ε . Note that due tothe singular homogeneous potential Ψ ε we have | ϕ ε | < Remark 3.6.
Note that equation (3.10) can be rewritten with the help ofthe identity ( µ ε ∇ ϕ ε , ψ ) Q T = − (cid:16)p a ( ϕ ε )∆ A ( ϕ ε ) ∇ ϕ ε , ψ (cid:17) Q T . This can be seen by testing (3.12) with ∇ ϕ ε · ψ and noting that ψ is diver-gence free. For the weak solution ( v ε , ϕ ε , µ ε ) we get the following energy estimates: Lemma 3.7.
For a weak solution ( v ε , ϕ ε , µ ε ) of problem (3.9a) - (3.9g) wehave the following energy estimates: ( i ) sup ≤ t ≤ T Z Ω (cid:18) ρ ε ( t ) | v ε ( t ) | |∇ ϕ ε ( t ) | + Ψ ε ( ϕ ε ( t )) (cid:19) dx + Z Q T η ( ϕ ε ) | D v ε | dx dt + Z Q T m ε ( ϕ ε ) |∇ µ ε | dx dt ≤ C , ( ii ) sup ≤ t ≤ T Z Ω G ε ( ϕ ε ( t )) dx + Z Q T | ∆ A ( ϕ ε ) | dx dt ≤ C , ( iii ) ε Z Q T | Ψ ′ ln ( ϕ ε ) | dx dt ≤ C , ( iv ) Z Q T | b J ε | dx dt ≤ C , where b J ε = − p m ε ( ϕ ε ) ∇ µ ε . Here G ε is a non-negative function defined by G ε (0) = G ′ ε (0) = 0 and G ′′ ε ( s ) = m ε ( s ) p a ( s ) for s ∈ [ − , .Proof. ad ( i ): This follows directly from the estimate (3.13) derived in thework of Abels, Depner and Garcke [ADG12]. We just note that for the esti-mate of ∇ ϕ ε we use ∇ A ( ϕ ε ) = p a ( ϕ ε ) ∇ ϕ ε and the fact that a is boundedfrom below by a positive constant due to Assumption 3.1.11d ( ii ): From line (3.11) we get that ∂ t ϕ ε ∈ L (0 , T ; (cid:0) H (Ω) (cid:1) ′ ), since ∇ µ ε ∈ L ( Q T ) and v · ∇ ϕ = div( v ϕ ) with v ϕ ∈ L ( Q T ). Then we derivefor a function ζ ∈ L (0 , T ; H (Ω)) the weak formulation Z t h ∂ t ϕ ε , ζ i dτ + Z Q t v ε · ∇ ϕ ε ζ dx dτ = − Z Q t m ε ( ϕ ε ) ∇ µ ε · ∇ ζ dx dτ = Z Q t p a ( ϕ ε ) (cid:16) e Ψ ′ ε ( A ( ϕ ε )) − ∆ A ( ϕ ε ) (cid:17) div( m ε ( ϕ ε ) ∇ ζ ) dx dτ, (3.14)where we additionally used (3.12) to express µ ε . Now we set as test function ζ = G ′ ε ( ϕ ε ), where G ε is defined by G ε (0) = G ′ ε (0) = 0 and G ′′ ε ( s ) = m ε ( s ) A ′ ( s ) for s ∈ [ − , G ε is a non-negative function, whichcan be seen from the representation G ε ( s ) = R s (cid:16)R r m ε ( τ ) A ′ ( τ ) dτ (cid:17) dr . With ζ = G ′ ε ( ϕ ε ) it holds that ∇ ζ = G ′′ ε ( ϕ ε ) ∇ ϕ ε = 1 m ε ( ϕ ε ) ∇ ( A ( ϕ ε )) and thereforediv ( m ε ( ϕ ε ) ∇ ζ ) = ∆ ( A ( ϕ ε )) . Hence we derive Z t h ∂ t ϕ ε , G ′ ε ( ϕ ε ) i dτ + Z Q t v ε · ∇ ϕ ε G ′ ε ( ϕ ε ) dx dτ = Z Q t p a ( ϕ ε ) (cid:16) e Ψ ′ ε ( A ( ϕ ε )) − ∆ A ( ϕ ε ) (cid:17) ∆ A ( ϕ ε ) dx dτ = Z Q t Ψ ′ ε ( ϕ ε )∆ A ( ϕ ε ) dx dτ − Z Q t p a ( ϕ ε ) | ∆ A ( ϕ ε ) | dx dτ . (3.15)With this notation we deduce Z t h ∂ t ϕ ε , G ′ ε ( ϕ ε ) i dt = Z Ω G ε ( ϕ ( t )) dx − Z Ω G ε ( ϕ ) dx and Z Q t v ε · ∇ ϕ ε G ′ ε ( ϕ ε ) dx dt = Z Q t v ε · ∇ ( G ε ( ϕ ε )) dx dt = − Z Q t div v ε G ε ( ϕ ε ) dx dt = 0 . For the first term on the right side of (3.15) we observe Z Q t Ψ ′ ε ( ϕ ε )∆ A ( ϕ ε ) dx dτ = Z Q t Ψ ′ ( ϕ ε )∆ A ( ϕ ε ) dx dτ + ε Z Q t Ψ ′ ln ( ϕ ε )∆ A ( ϕ ε ) dx dτ − Z Q t Ψ ′′ ( ϕ ε ) ∇ ϕ ε · ∇ A ( ϕ ε ) dx dt = − Z Q t Ψ ′′ ( ϕ ε ) p a ( ϕ ε ) |∇ ϕ ε | dx dt. Herein the estimate Z Q t Ψ ′ ln ( ϕ ε )∆ A ( ϕ ε ) dx dτ ≤ ϕ ε by ϕ αε = αϕ ε for 0 < α < | ϕ αε | < α < Z Q t Ψ ′ ln ( ϕ αε )∆ A ( ϕ αε ) dx dτ = − Z Q t Ψ ′′ ln ( ϕ αε ) ∇ ϕ αε · ∇ A ( ϕ αε ) dx dτ ≤ , where we used integration by parts. To pass to the limit for α ր ϕ αε → ϕ ε in L (0 , T ; H (Ω)). Hence together with thebound | Ψ ′ ln ( ϕ αε ) | ≤ | Ψ ′ ln ( ϕ ε ) | we can use Lebesgue’s dominated convergencetheorem to conclude Z Q t Ψ ′ ln ( ϕ αε )∆ A ( ϕ αε ) dx dτ −→ Z Q t Ψ ′ ln ( ϕ ε )∆ A ( ϕ ε ) dx dτ for α ր . With the bound from below a ( s ) ≥ c > Z Ω G ε ( ϕ ( t )) dx + Z Q t | ∆ A ( ϕ ε ) | dx dτ ≤ C (cid:18)Z Ω G ε ( ϕ ) dx + Z Q t Ψ ′′ ( ϕ ε ) p a ( ϕ ε ) |∇ ϕ ε | dx dτ (cid:19) . Now we use m ε ( τ ) ≥ m ( τ ) to observe the inequality G ε ( s ) = Z s (cid:18) Z r m ε ( τ ) A ′ ( τ ) | {z } = √ a ( τ ) dτ (cid:19) dr ≤ Z s (cid:18)Z r m ( τ ) p a ( τ ) dτ (cid:19) dr =: G ( s ) for s ∈ ( − , . Due to the special choice of the degenerate mobility m in (3.2) we concludethat G can be extended continuously to the closed interval [ − ,
1] and thattherefore the integral R Ω G ( ϕ ) dx and in particular the integral R Ω G ε ( ϕ ) dx is bounded.Moreover, since Ψ ′′ ( s ) is bounded in | s | ≤ R Ω |∇ ϕ ε ( t ) | dx in ( i ), we proved ( ii ).13d ( iii ): To show this estimate we will argue similarly as in the time-discrete situation of Lemma 4.2 in Abels, Depner and Garcke [ADG12].We multiply equation (3.12) with P ϕ ε , integrate over Ω and get almosteverywhere in t the identity Z Ω µ ε P ϕ ε dx = Z Ω Ψ ′ ( ϕ ε ) P ϕ ε dx + ε Z Ω Ψ ′ ln ( ϕ ε ) P ϕ ε dx − Z Ω p a ( ϕ ε )∆ A ( ϕ ε ) P ϕ ε dx. (3.16)By using in identity (3.11) a test function which depends only on time t andnot on x ∈ Ω, we derive the fact that ( ϕ ε ) Ω = ( ϕ ) Ω and by assumption thisnumber lies in ( − α, − α ) for a small α >
0. In addition with the propertylim s →± Ψ ′ ln ( s ) = ±∞ we can show the inequality Ψ ′ ln ( s )( s − ( ϕ ) Ω ) ≥ C α | Ψ ′ ln ( s ) | − c α in three steps in the intervals [ − , − α ], [ − α , − α ]and [1 − α ,
1] successively. Altogether this leads to the following estimate: ε Z Ω | Ψ ′ ln ( ϕ ε ) | dx ≤ C (cid:18) ε Z Ω Ψ ′ ln ( ϕ ε ) P ϕ ε dx + 1 (cid:19) . (3.17)We observe the fact that R Ω µ ε P ϕ ε dx = R Ω ( P µ ε ) ϕ ε dx and due to integra-tion by parts − Z Ω p a ( ϕ ε )∆ A ( ϕ ε ) P ϕ ε dx = Z Ω p a ( ϕ ε ) ∇ A ( ϕ ε ) · ∇ ϕ ε dx + Z Ω a ( ϕ ε ) − ∇ ϕ ε · ∇ A ( ϕ ε ) P ϕ ε dx = Z Ω a ( ϕ ε ) |∇ ϕ ε | dx + Z Ω P ϕ ε |∇ ϕ ε | dx . Combining estimate (3.17) with identity (3.16) we are led to ε Z Ω | Ψ ′ ln ( ϕ ε ) | dx ≤ C (cid:18)Z Ω | ( P µ ε ) ϕ ε | dx + Z Ω | Ψ ′ ( ϕ ε ) P ϕ ε | dx + Z Ω | p a ( ϕ ε )∆ A ( ϕ ε ) P ϕ ε | dx + 1 (cid:19) ≤ C (cid:0) k P µ ε k L (Ω) + k∇ ϕ ε k L (Ω) + 1 (cid:1) ≤ C (cid:0) k∇ µ ε k L (Ω) + 1 (cid:1) . In the last two lines we have used in particular the facts that ϕ ε is boundedbetween − ′ is continuous, the energy estimate from ( ii ) forsup ≤ t ≤ T k∇ ϕ ε k L (Ω) and the Poincar´e inequality for functions with meanvalue zero. 14ith the last inequality we can estimate the integral of µ ε by simplyintegrating identity (3.12) over Ω: (cid:12)(cid:12)(cid:12)(cid:12)Z Ω µ ε dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Ω | Ψ ′ ( ϕ ε ) | dx + ε Z Ω | Ψ ′ ln ( ϕ ε ) | dx + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω p a ( ϕ ε )∆ A ( ϕ ε ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:0) k∇ µ ε k L (Ω) + 1 (cid:1) , where we used similarly as above integration by parts for the integral over p a ( ϕ ε )∆ A ( ϕ ε ). By the splitting of µ ε into µ ε = P µ ε + ( µ ε ) Ω we arrive at k µ ε k L (Ω) ≤ C (cid:16) k∇ µ ε k L (Ω) + 1 (cid:17) . Then, again from identity (3.12), we derive ε | Ψ ′ ln ( ϕ ε ) | ≤ C (cid:0) | µ ε | + | ∆ A ( ϕ ε ) | + 1 (cid:1) and together with the last estimates and an additional integration over time t this leads to ε k Ψ ′ ln ( ϕ ε ) k L ( Q T ) ≤ C (cid:16) k∇ µ ε k L ( Q T ) + 1 (cid:17) . Note that we used the bound k ∆ A ( ϕ ε ) k L ( Q T ) ≤ C from ( ii ). Furthermore,due to the bounds in ( i ), we see ε k∇ µ ε k L ( Q T ) ≤ C since m ε ( s ) ≥ ε for | s | ≤ ε k Ψ ′ ln ( ϕ ε ) k L ( Q T ) ≤ C . ad ( iv ): This follows directly from ( i ). In this subsection we use the energy estimates to get weak limits for the se-quences ( v ε , ϕ ε , J ε ), where J ε = p m ε ( ϕ ε ) b J ε (= − m ε ( ϕ ε ) ∇ µ ε ). With somesubtle arguments we extend the weak convergences to strong ones, so thatwe are able to pass to the limit for ε → v ε ⇀ v in L (0 , T ; H (Ω) d ) ,ϕ ε ⇀ ϕ in L (0 , T ; H (Ω)) , b J ε ⇀ b J in L (0 , T ; L (Ω) d ) and J ε ⇀ J in L (0 , T ; L (Ω) d )15or v ∈ L (0 , T ; H (Ω) d ) ∩ L ∞ (0 , T ; L σ (Ω)), ϕ ∈ L ∞ (0 , T ; H (Ω)) and b J , J ∈ L (0 , T ; L (Ω) d ). Here and in the following all limits are meant to be forsuitable subsequences ε k → k → ∞ .With the notation J ε = − m ε ( ϕ ε ) ∇ µ ε the weak solution of problem(3.9a)-(3.9g) fulfills the following equations: − (cid:0) ρ ε v ε ,∂ t ψ (cid:1) Q T + (div( ρ ε v ε ⊗ v ε ) , ψ ) Q T + (2 η ( ϕ ε ) D v ε , D ψ ) Q T − (( v ε ⊗ β J ε ) , ∇ ψ ) Q T = − (cid:16)p a ( ϕ ε )∆ A ( ϕ ε ) ∇ ϕ ε , ψ (cid:17) Q T (3.18)for all ψ ∈ [ C ∞ (Ω × (0 , T ))] d with div ψ = 0, − Z Q T ϕ ε ∂ t ζ dx dt + Z Q T ( v ε · ∇ ϕ ε ) ζ dx dt = Z Q T J ε · ∇ ζ dx dt (3.19)for all ζ ∈ C ∞ ((0 , T ; C (Ω)) and Z Q T J ε · η dx dt = − Z Q T (cid:16) Ψ ′ ε ( ϕ ε ) − p a ( ϕ ε )∆ A ( ϕ ε ) (cid:17) div( m ε ( ϕ ε ) η ) dx dt (3.20)for all η ∈ L (0 , T ; H (Ω) d ) ∩ L ∞ ( Q T ) d with η · n = 0 on S T . For the lastline we used that for functions η with η · n = 0 on S T it holds Z Q T J ε · η dx dt = Z Q T ∇ µ ε · m ε ( ϕ ε ) η dx dt = − Z Q T µ ε div( m ε ( ϕ ε ) η ) dx dt = − Z Q T (cid:16) Ψ ′ ε ( ϕ ε ) − p a ( ϕ ε )∆ A ( ϕ ε ) (cid:17) div( m ε ( ϕ ε ) η ) dx dt . Now we want to pass to the limit ε → ∂ t ϕ ε is bounded in L (0 , T ; (cid:0) H (Ω) (cid:1) ′ ) and ϕ ε is bounded in L ∞ (0 , T ; H (Ω)) . Therefore we can deduce from the Lemma of Aubins-Lions (2.1) the strongconvergence ϕ ε → ϕ in L (0 , T ; L (Ω))and ϕ ε → ϕ pointwise almost everywhere in Q T .From the bound of ∆ A ( ϕ ε ) in L ( Q T ) and from ∇ A ( ϕ ε ) · n = p a ( ϕ ε ) ∇ ϕ ε · n = 0 on S T ,
16e get from elliptic regularity theory the bound k A ( ϕ ε ) k L (0 ,T ; H (Ω)) ≤ C .
This yields A ( ϕ ε ) ⇀ g in L (0 , T ; H (Ω))at first for some g ∈ L (0 , T ; H (Ω)), but then, due to the weak convergence ∇ ϕ ε ⇀ ∇ ϕ in L (0 , T ; L (Ω)) and due to the pointwise almost everywhereconvergence a ( ϕ ε ) → a ( ϕ ) in Q T we can identify g with A ( ϕ ) to get A ( ϕ ε ) ⇀ A ( ϕ ) in L (0 , T ; H (Ω)) . The next step is to strengthen the convergence of ∇ ϕ ε in L ( Q T ). To thisend, we remark that by definition A is Lipschitz-continuous with | A ( r ) − A ( s ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z rs p a ( τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | r − s | . Furthermore from the bound of ∂ t ϕ ε in L (0 , T ; (cid:0) H (Ω) (cid:1) ′ ) we get with thenotation ϕ ε ( . + h ) for a shift in time k ϕ ε ( . + h ) − ϕ ε k L (0 ,T − h ;( H (Ω)) ′ ) ≤ Ch , which leads to the estimate k A ( ϕ ε ( . + h )) − A ( ϕ ε ) k L (0 ,T − h ;( H (Ω)) ′ ) ≤ C k ϕ ε ( . + h ) − ϕ ε k L (0 ,T − h ;( H (Ω)) ′ ) ≤ Ch −→ h → . Together with the bound of A ( ϕ ε ) in L (0 , T ; H (Ω)) we can use a theoremof Simon [Sim87, Th. 5] to conclude the strong convergence A ( ϕ ε ) → A ( ϕ ) in L (0 , T ; H (Ω)) . From ∇ A ( ϕ ε ) = p a ( ϕ ε ) ∇ ϕ ε we get then in particular the strong conver-gence ∇ ϕ ε → ∇ ϕ in L (0 , T ; L (Ω)) . In addition we want to use an argument of Abels, Depner and Garcke from[ADG12, Sec. 5.1] which shows that due to the a priori estimate in Lemma3.7 and the structure of equation (3.18) we can deduce the strong conver-gence v ε → v in L (0 , T ; L (Ω) d ). In few words we show with the helpof some interpolation inequalities the bound of ∂ t ( P σ ( ρ ε v ε )) in the space L ( W ∞ (Ω) ′ ) and together with the bound of P σ ( ρ ε v ε ) in L (0 , T ; H (Ω) d )17his is enough to conclude with the Lemma of Aubin-Lions the strong con-vergence P σ ( ρ ε v ε ) → P σ ( ρ v ) in L (0 , T ; L (Ω) d ) . From this we can derive v ε → v in L (0 , T ; L (Ω) d ). For the details we referto [ADG12, Sec. 5.1 and Appendix].With the last convergences and the weak convergence J ε ⇀ J in L ( Q T )we can pass to the limit ε → ϕ ε and J ε in L ( Q T ) and the strong ones of v ε and ∇ ϕ ε in L ( Q T ).Finally, the convergence in line (3.20) can be seen as follows: The leftside converges due to the weak convergence of J ε and for the right side wecalculate Z Q T (cid:16) Ψ ′ ε ( ϕ ε ) − p a ( ϕ ε )∆ A ( ϕ ε ) (cid:17) div( m ε ( ϕ ε ) η ) dx dt = Z Q T Ψ ′ ( ϕ ε ) div( m ε ( ϕ ε ) η ) dx dt + ε Z Q T Ψ ′ ln ( ϕ ε ) div( m ε ( ϕ ε ) η ) dx dt − Z Q T p a ( ϕ ε )∆ A ( ϕ ε ) div( m ε ( ϕ ε ) η ) dx dt . (3.21)The first and the third term can be treated similarly as in Elliott and Garcke[EG96]. For the convenience of the reader we give the details.First we observe the fact that m ε → m uniformly since for all s ∈ R itholds: | m ε ( s ) − m ( s ) | ≤ m (1 − ε ) → ε → . Hence we conclude with the pointwise convergence ϕ ε → ϕ a.e. in Q T that m ε ( ϕ ε ) → m ( ϕ ) a.e. in Q T . In addition with the convergences Ψ ′ ( ϕ ε ) → Ψ ′ ( ϕ ), a ( ϕ ε ) → a ( ϕ ) a.e. in Q T and with the weak convergence ∆ A ( ϕ ε ) → ∆ A ( ϕ ) in L ( Q T ) we are ledto Z Q T Ψ ′ ( ϕ ε ) m ε ( ϕ ε ) div η dx dt −→ Z Q T Ψ ′ ( ϕ ) m ( ϕ ) div η dx dt and Z Q T p a ( ϕ ε )∆ A ( ϕ ε ) m ε ( ϕ ε ) div η dxdt −→ Z Q T p a ( ϕ )∆ A ( ϕ ) m ( ϕ ) div η dxdt. The next step is to show that m ′ ε ( ϕ ε ) ∇ ϕ ε → m ′ ( ϕ ) ∇ ϕ in L ( Q T ). To this18nd we split the integral in the following way: Z Q T | m ′ ε ( ϕ ε ) ∇ ϕ ε − m ′ ( ϕ ) ∇ ϕ | dx dt = Z Q T ∩{| ϕ | < } | m ′ ε ( ϕ ε ) ∇ ϕ ε − m ′ ( ϕ ) ∇ ϕ | dx dt + Z Q T ∩{| ϕ | =1 } | m ′ ε ( ϕ ε ) ∇ ϕ ε − m ′ ( ϕ ) ∇ ϕ | dx dt . Since ∇ ϕ = 0 a.e. on the set {| ϕ | = 1 } , see for example Gilbarg andTrudinger [GT01, Lem. 7.7], we obtain Z Q T ∩{| ϕ | =1 } | m ′ ε ( ϕ ε ) ∇ ϕ ε − m ′ ( ϕ ) ∇ ϕ | dx dt = Z Q T ∩{| ϕ | =1 } | m ′ ε ( ϕ ε ) ∇ ϕ ε | dx dt ≤ C Z Q T ∩{| ϕ | =1 } |∇ ϕ ε | dx dt −→ C Z Q T ∩{| ϕ | =1 } |∇ ϕ | dx dt = 0 . Although m ′ ε is not continuous, we can conclude on the set {| ϕ ε | < } theconvergence m ′ ε ( ϕ ε ) → m ′ ( ϕ ) a.e. in Q T . Indeed, for a point ( x, t ) ∈ Q T with | ϕ ( x, t ) | < ϕ ε ( x, t ) → ϕ ( x, t ), it holds that | ϕ ε ( x, t ) | < − δ forsome δ > ε small enough and in that region m ′ ε and m ′ are continuous.Hence we have m ′ ε ( ϕ ε ) ∇ ϕ ε −→ m ′ ( ϕ ) ∇ ϕ a.e. in Q T (3.22)and the generalized Lebesgue convergence theorem now gives Z Q T ∩{| ϕ | < } | m ′ ε ( ϕ ε ) ∇ ϕ ε − m ′ ( ϕ ) ∇ ϕ | dx dt −→ , which proves finally m ′ ε ( ϕ ε ) ∇ ϕ ε → m ′ ( ϕ ) ∇ ϕ in L ( Q T ). Similarly as above,together with the convergences Ψ ′ ( ϕ ε ) → Ψ ′ ( ϕ ), a ( ϕ ε ) → a ( ϕ ) a.e. in Q T and with the weak convergence ∆ A ( ϕ ε ) → ∆ A ( ϕ ) in L ( Q T ) we are led to Z Q T Ψ ′ ( ϕ ε ) m ′ ε ( ϕ ε ) ∇ ϕ ε · η dx dt −→ Z Q T Ψ ′ ( ϕ ) m ′ ( ϕ ) ∇ ϕ · η dx dt and Z Q T p a ( ϕ ε )∆ A ( ϕ ε ) m ′ ε ( ϕ ε ) ∇ ϕ ε · η dx dt −→ Z Q T p a ( ϕ )∆ A ( ϕ ) m ′ ( ϕ ) ∇ ϕ · η dx dt. ε Z Ω T Ψ ′ ln ( ϕ ε ) div( m ε ( ϕ ε ) η ) dx dt = ε Z {| ϕ ε |≤ − ε } Ψ ′ ln ( ϕ ε ) div( m ε ( ϕ ε ) η ) dx dt + ε Z {| ϕ ε | > − ε } Ψ ′ ln ( ϕ ε ) div( m ε ( ϕ ε ) η ) dx dt =: ( I ) ε + ( II ) ε . On the set {| ϕ ε | ≤ − ε } we use that Ψ ′ ln ( ϕ ε ) = ln(1 + ϕ ε ) − ln(1 − ϕ ε ) + 2and therefore (cid:12)(cid:12) Ψ ′ ln ( ϕ ε ) (cid:12)(cid:12) ≤ | ln ε | + C to deduce that | ( I ) ε | ≤ ε ( | ln ε | + C ) Z Q T | div( m ε ( ϕ ε ) η ) | dx dt −→ . On the set {| ϕ ε | > − ε } , we use that m ε ( ϕ ε ) = ε (2 − ε ) to deduce( II ) ε = ε (2 − ε ) Z {| ϕ ε | > − ε } Ψ ′ ln ( ϕ ε ) div η dx dt ≤ Cε k Ψ ′ ln ( ϕ ε ) k L ( Q T ) = C √ ε (cid:16) ε k Ψ ′ ln ( ϕ ε ) k L ( Q T ) (cid:17) −→ , since the last term in brackets is bounded by the energy estimate formLemma 3.7.For the relation of b J and J we note that due to b J ε ⇀ b J , J ε ⇀ J in L ( Q T ), J ε = p m ε ( ϕ ε ) b J ε and p m ε ( ϕ ε ) → p m ( ϕ ) a.e. in Q T from (3.22)we can conclude J = p m ( ϕ ) b J . From the weak convergence b J ε ⇀ b J in L ( Q T ) we can conclude that Z Q ( s,t ) | b J | dx dτ ≤ lim inf ε → Z Q ( s,t ) m ε ( ϕ ε ) |∇ µ ε | dx dτ for all 0 ≤ s ≤ t ≤ T and this is enough to proceed as in Abels, Depner andGarcke [ADG12] to show the energy estimate.Finally we just remark that the continuity properties and the initialconditions can be derived with the same arguments as in [ADG12, Sec. 5.2,5.3], so that altogether we proved Theorem 3.5.20 cknowledgement This work was supported by the SPP 1506 ”Transport Processes at FluidicInterfaces” of the German Science Foundation (DFG) through the grant GA695/6-1. The support is gratefully acknowledged.
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