On sharp interface limits for diffuse interface models for two-phase flows

DOI 10.4171/IFB/324

On sharp interface limits for diffuse interface models for two-phase flows

HELMUTABELS

Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, Germany

E-mail:helmut.abels@mathematik.uni-regensburg.de

DANIELLENGELER

Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg, Germany

E-mail:daniel.lengeler@mathematik.uni-regensburg.de

[Received 14 January 2013 and in revised form 6 March 2014]

We discuss the sharp interface limit of a diffuse interface model for a two-phase flow of two partly miscible viscous Newtonian fluids of different densities, when a certain parameter" > 0related to the interface thickness tends to zero. In the case that the mobility stays positive or tends to zero slower than linearly in"we will prove that weak solutions tend to varifold solutions of a corresponding sharp interface model. But, if the mobility tends to zero faster than"³we will show that certain radially symmetric solutions tend to functions, which will not satisfy the Young-Laplace law at the interface in the limit.

2010 Mathematics Subject Classification: Primary 76T99; Secondary 35Q30, 35Q35, 35R35, 76D05, 76D45.

Keywords:Two-phase flow, diffuse interface model, sharp interface limit, Navier–Stokes system, free boundary problems

1. Introduction

The present contribution is devoted to the study of the relations between so-called diffuse and sharp interface models for the flow of two viscous incompressible Newtonian fluids. Such two-phase flows play a fundamental role in many fluid dynamical applications in physics, chemistry, biology, and the engineering sciences. There are two basic types of models namely the (classical) sharp interface models, where the interface .t /between the fluids is modeled as a (sufficiently smooth) surface and so-called diffuse interface models, where the “sharp” interface .t /is replaced by an interfacial region, where a suitable order parameter (e.g., the difference of volume fractions) varies smoothly, but with a large gradient between two distinguished values (e.g.,˙1for the difference of volume fractions). Then the natural question arises how diffuse and sharp interface models are related if a suitable parameter" > 0, which is related to the width of the diffuse interface, tends to zero. There are several results on this question, which are based on formally matched asymptotics calculations.

But so far there are very few mathematically rigorous convergence results.

More precisely, we study throughout the paper the sharp interface limit of the following diffuse interface model:

@_tvC

vC @

@cJ

rvdiv..c/Dv/C rpD "div.a.c/rc˝ rc/inQ; (1.1)

divvD0 inQ; (1.2)

c European Mathematical Society 2014

@_tcCv rcDdiv.m_".c/r/ inQ; (1.3) D"¹f⁰.c/"c inQ; (1.4)

vj_@˝ D0 onS; (1.5)

n_@˝ rcj_@˝ Dn_@˝ rj_@˝ D0 onS; (1.6)

.v; c/j_tD0D.v₀; c₀/ in˝; (1.7) whereQD ˝.0;1/; S D@˝.0;1/,˝ Rⁿis a suitable domain, andJ D m_".c/r: Herec D c₂c₁ is the volume fraction difference of the fluids, D .c/is the density of the fluid mixture, depending explicitly oncthrough.c/D ^{Q2 Q1}

2 c^{Q1C Q2}

2 , whereQ_j is the specific density of fluidj D 1; 2, andf is a suitable “double-well potential”, e.g.,f .c/ D ¹

8.1c²/². Precise assumptions will be made below. Moreover," > 0 is a small parameter related to the interface thickness,is the so-called chemical potential,m_".c/ > 0a mobility coefficient related to the strength of diffusion in the mixture anda.c/is a coefficient in front of thejrcj²-term in the free energy of the system. Finally,n_@˝ denotes the exterior normal of@˝. The model was derived by A., Garcke, and Gr¨un [5]. In the case.c/ const:it coincides with the so-called

“Model H” in Hohenberg and Halperin [14], cf. also Gurtin et al. [13]. Existence of weak solutions for this system in the case of a bounded, sufficiently smooth domain˝and for a suitable class of singular free energy densitiesf was proved by A., Depner, and Garcke [4]. We refer to the latter article for further references concerning analytic results for this diffuse interface model in the case .c/const:and related models.

In [5] the sharp interface limit " ! 0 was discussed with the method of formally matched asymptotics. It was shown that for the scalingm_".c/ Qm"^˛with˛D0; 1,m > 0, solutions of theQ system (1.1)–(1.5) converges to solutions of

^˙@_tvC.^˙vC^{Q2 Q}₂¹J/ rvdivT^˙.v; p/D0 in˝^˙.t /; t > 0; (1.8) divvD0 in˝^˙.t /; t > 0; (1.9) m₀D0 in˝^˙.t /; t > 0; (1.10) nŒT.v; p/DHn on .t /; t > 0; (1.11) V nvj _.t/D Œ^m₂⁰n ron .t /; t > 0; (1.12) j _.t/DH on .t /; t > 0; (1.13) withJD m₀r. Herendenotes the unit normal of .t /that points inside˝^C.t /andV andH the normal velocity and scalar mean curvature of .t /with respect ton. Moreover, byŒwe denote the jump of a quantity across the interface in direction ofn, i.e.,Œf .x/ D lim_h!0.f .xChn/

f .xhn//forx 2 .t /. Furthermore,is a surface tension coefficient determined uniquely byf andm₀D Qmif˛D0andm₀D0if˛D1is a mobility constant. Implicitly it is assumed thatv;

do not jump across .t /, i.e.,

ŒvDŒD0 on .t /; t > 0:

In the following we close the system with the boundary and initial conditions

vj_@˝D0 on@˝; t > 0; (1.14)

n_@˝m_".c/rj_@˝ D0 on@˝; t > 0; (1.15)

˝^C.0/D˝₀^C; (1.16)

vj_tD0Dv₀ in˝; (1.17)

wherev₀; ˝₀^Care given initial data satisfying@˝₀^C\@˝ D ;. Equations (1.8)–(1.9) describe the conservation of linear momentum and mass in both fluids, and (1.11) is the balance of forces at the boundary. The equations forvare complemented by the non-slip condition (1.14) at the boundary of

˝. The conditions (1.10), (1.15) describe together with (1.12) a continuity equation for the mass of the phases, and (1.13) relates the chemical potentialto theL²-gradient of the surface area, which is given by the mean curvature of the interface.

We note that in the case˛ D 1, i.e.,m₀ D 0, (1.12) describes the usual kinematic condition that the interface is transported by the flow of the surrounding fluids and (1.8)-(1.17) reduces to the classical model of a two-phase Navier–Stokes flow. Existence of strong solutions locally in time was first proved by Denisova and Solonnikov [11]. We refer to Pr¨uss and Simonett [19] and K¨ohne et al. [15] for more recent results and further references. Existence of generalized solutions globally in times was shown by Plotnikov [18] and A. [1,2]. On the other hand, if˛D0,m₀> 0, respectively, the equations (1.10), (1.13), (1.15) are a variant of the Mullins–Sekerka flow of a family of interfaces with an additional convection termnvj _.t/. In the caseQ₁ D Q₂existence of weak solutions for large times and general initial data was proved by A. and R¨oger [6] and existence of strong solutions locally in time and stability of spherical droplets was proved by A. and Wilke [8].

In the following we address the following question: Under which assumptions on the behavior of

m_".c/as"!0do weak solutions of (1.1)–(1.7) converge to weak/generalized solutions of (1.8)–

(1.17)? In this paper we provide a partial answer to that question. If one assumes e.g.m_".c/D Qm"^˛, the results in the following will show that convergence holds true in the case˛ 2 Œ0; 1/. More precisely, we will show that weak solutions of (1.1)–(1.7) converge to so-called varifold solutions of (1.8)–(1.17), which are defined in the spirit of Chen [10]. But in the case˛ 2 .3;1/we will construct radially symmetric solutions of (1.1)-(1.4) in the domain˝ D fx 2RW1 <jxj < Mg with suitable inflow and outflow boundary conditions, which do not converge to a solution of (1.8)–

(1.13). In particular, the pressurepin the limit"!0satisfies ŒpD .t /H on .t /D@B_R.t/.0/;

whereR.t /; .t /!_t!11andvis independent oftand smooth in˝. This shows that the Young- Laplace law (1.11) is not satisfied. We note that these results are consistent with the numerical studies of Jacqmin, where a scaling of the mobility asm_".c/D Qm"^˛with˛2Œ1; 2/was proposed and considered.

The structure of the article is as follows: First we introduce some notation and preliminary results in Section2. Then we prove our main result on convergence of weak solutions of (1.1)–(1.7) to varifold solutions of (1.8)–(1.17) in the case that the mobilitym_".c/tends to zero as"!0slower than linearly in Section3. Finally, in Section4, we consider certain radially symmetric solutions of (1.1)–(1.7) and show that these do not converge to a solution of (1.8)–(1.13) if the mobility tends to zero too fast as"!0.

2. Notation and preliminaries

LetXbe a locally compact separable metric space and letC₀.XIR^N/by the closure of compactly supported continuous functionsfWX !R^N,N 2N, in the supremum norm. Moreover, denote by

M.XIR^N/the space of all finiteR^N-valued Radon measures,M.X /WDM.XIR/. Then by Riesz representation theoremM.XIR^N/ D C₀.XIR^N/⁰, cf., e.g., Ambrosio et al. [9, Theorem 1.54].

Given2M.XIR^N/we denote byjjthe total variation measure defined by jj.A/Dsup

(₁ X

kD0

j.A_k/j WA_k 2B.X /pairwise disjoint; AD [1 kD0

A_k )

for everyA2B.X /, whereB.X /denotes the-algebra of Borel sets ofX. Moreover,_jjWX !R^N denotes the Radon-Nikodym derivative of with respect tojj. The restriction of a measure to a-measurable set Ais denoted by.bA/.B/ D .A\B/. Furthermore, thes-dimensional Hausdorff measure onR^d,06s6d, is denoted byH^s. Recall that

BV .U /D˚

f 2L¹.U /W rf 2M.UIR^d/

kfk_{BV .U /} D kfk_L1.U /C krfk_M.UIRd/;

where rf denotes the distributional derivative and U Rⁿ is an open set. Moreover, BV .UI f0; 1g/denotes the set of allX2BV .U /such thatX.x/2 f0; 1gfor almost allx 2U.

A setEU is said to have finite perimeter inU ifXE2BV .U /. By the structure theorem of sets of finite perimeterjrXEj D H^d1b@E, where@Eis the so-called reduced boundary ofE and for all'2C₀.U;R^d/

hrX_E;'i D Z

@E

'n_EdH^d1; wheren_E.x/D _jr^rX_X^E

Ej, cf., e.g., [9]. Note that, ifEis a domain withC¹-boundary, then@ED@E andn_E coincides with the interior unit normal.

As usual the space of smooth and compactly supported functions in an open setU is denoted byC₀¹.U /. Moreover,C¹.U /denotes the set of all smooth functionsfWU ! Csuch that all derivatives have continuous extensions onU. For0 < T 6 1, we denote by L^p_loc.Œ0; T /IX /, 16 p6 1, the space of all strongly measurablefW.0; T /!X such thatf 2 L^p.0; T⁰IX /for all0 < T⁰ < T. HereL^p.M /andL^p.MIX /denote the standard Lebesgue spaces for scalar and X-valued functions, respectively. Furthermore,C_0;¹.˝/D f'2C₀¹.˝/^d Wdiv'D0gand

L².˝/ WD C_0;¹.˝/^L2.˝/:

IfY DX⁰is a dual space andQR^N is open, thenL¹_!.QIY /denotes the space of all functions WQ!Y that are weakly-measurable and essentially bounded, i.e.,

x7! h_x; F .x; :/i_X0;X

is measurable for eachF 2L¹.QIX /and kk_L1

!.QIY /WDess sup_x2Qk_xk_Y <1:

Moreover, we note that there is a separable Banach spaceX such thatX⁰ DBV .˝/, cf. [9]. As a consequence [12] we obtain thatL¹_!.0; TIBV .˝//D

L¹.0; TIX /

and that uniformly bounded sets inL¹_!.0; TIBV .˝//are weakly*-precompact.

3. Sharp interface limit

In this section we discuss the relation between (1.8)–(1.17) and its diffuse interface analogue (1.1)–

(1.7).

ASSUMPTION 3.1 We assume that the domain ˝ R^d, d D 2; 3 is bounded and smooth.

Furthermore, we assume that there exist constantsc₀; C₀> 0such that

f 2C³.R/,f .c/>0,f .c/D0if and only ifcD 1; 1, andf⁰⁰.c/>c₀jcj^p2ifjcj>1c₀ for some constantp>3

; a; 2C¹.R/withc₀6; a; 6C₀and .c/D Q₂C Q₁

2 C Q₂ Q₁

2 c

forc2Œ1; 1

m_"; m₀2C¹.R/,06m_"; m₀6C₀,m_" !_"!0m₀inC¹.R/, and eitherm₀>c₀orm₀0. If m₀0, thenm_">m_"for constantsm_"> 0withm_"!_"!00.

The stronger assumptionp > 3(compared top > 2in [10]) is needed here for the uniform estimate ofv_" rc_"Ddiv.v_"c_"/inL².0; TIH¹.˝//. A possible choice for the homogeneous free energy density isf .s/D.s²1/². Moreover, let DR₁

1

pf .s/=2 dsandA.s/DR_s

0

pa. / d . Now, let us consider the energy identities corresponding to our two systems. We recall that every sufficiently smooth solution of the Navier–Stokes/Mullins–Sekerka system (1.8)–(1.17) satisfies

d dt

1 2

Z

˝

.c/jvj²dxC d

dtH^d1. /D Z

˝

.c/jDvj²dx Z

˝

m₀.c/jrj²dx; (3.1) wherec.t; x/D 1C2_˝C.t/.x/. On the other hand, every sufficiently smooth solution of (1.1)–

(1.7) satisfies d dt

1 2

Z

˝

.c/jvj²dxC d

dtE".c/D Z

˝

.c/jDvj²dx Z

˝

m_".c/jrj²dx; (3.2)

where

E".c/D Z

˝

"jrA.c/j²

2 Cf .c/

"

dx

is the free energy. Moreover, by Modica and Mortola [17] or Modica [16], forA.c/Dc, we have

E"!_"!0P w.r.t.L¹--convergence;

where

P.u/D

(H^d1.@E/ ifuD 1C2_EandEhas finite perimeter;

C1 else:

Here,@Edenotes the reduced boundary. Note that@ED@EifEis a sufficiently regular domain.

Therefore, we see that the energy identity (3.1) is formally identical to the sharp interface limit of the energy identity (3.2) of the diffuse interface model (1.1)–(1.7).

We will now adapt the arguments of Chen [10], see also A. and R¨oger [6], to show that, as"!0, solutions of the diffuse interface model (1.1)–(1.7) converge to varifold solutionsof the system (1.8)–(1.17). LetQ D˝.0;1/andG_d1 WD S^d1= where0 1for0;1 2S^d1iff ₀D ˙₁andS^d1is the unit sphere inR^d.

DEFINITION3.2 Letv₀ 2 L².˝/andE₀ ˝ be a set of finite perimeter. Then.v; E; ; V /if m₀> 0and.v; E; V /else is called avarifold solutionof (1.8)–(1.17) with initial values.v₀; E₀/if the following conditions are satisfied:

1. v 2 L²..0;1/IH¹.˝/^d/ \ L¹..0;1/IL².˝//; 2 L²_loc.Œ0;1/IH¹.˝//;r 2 L²..0;1/IL².˝/^d/ifm₀> 0.

2. EDS

t>0E_t ftgis a measurable subset of˝Œ0;1/such that_E 2C.Œ0;1/IL¹.˝//\

L¹_!..0;1/IBV .˝//andjE_tj D jE₀jfor allt >0.

3. V is a Radon measure on˝G_d1.0;1/such thatV DV^tdtwhereV^tis a Radon measure on˝G_d1for almost allt 2 .0;1/, i.e., a general varifold in˝. Moreover, for almost all t 2.0;1/ V^thas the representation

Z

˝G_d1

.x;p/ d V^t.x;p/D Xd iD1

Z

˝

b_i^t.x/

x;p^t_i.x/

d ^t.x/ (3.3) for all 2C.˝G_d1/. Here, for almost allt 2.0;1/,^tis a Radon measure on˝, and the ^t-measurable functionsb_i^t;p^t_iareR- andG_d1-valued, respectively, such that

06b^t_i 61;

Xd iD1

b_i^t>1;

Xd iD1

p^t_i˝p^t_iDI; ^t-a.e.

and jr_Etj

^t 6 1

2:

4. Forc WD 1C2_E,J WD m₀.c/rifm₀ > 0andJ WD 0else as well asJQ WD ^@

@c.c/Jwe

have Z

Q

.c/v@_t'v˝

.c/vC QJ

W r'C.c/DvWD' d.x; t /

Z

˝

.cj_tD0/v₀'j_tD0dxD Z ₁

0

hıV^t;'idt (3.4) for all'2C₀¹.Œ0;1/IC_0;¹.˝//and

2 Z

E

@_t Cdiv. v/ d.x; t /C Z

Q

J r d.x; t /C Z

E0

j_tD0dxD0 (3.5) for all 2C₀¹.Œ0;1/˝/. Here

hıV^t;'i WD Z

˝G_d1

.Ip˝p/W r'd.x;p/ for all'2C¹.˝IR^d/:

Furthermore, ifm₀> 0we have 2

Z

Et

div./ dxD˝ ıV^t;˛

(3.6) for all2C₀¹.˝IR^d/and almost allt 2.0;1/.

5. Finally, for almost all0 < s < t <1 1

2 Z

˝

.c.t //jv.t /j²dxC^t.˝/C Z _t

s

Z

˝

.c/jDvj²J r d.x; /

6 1 2 Z

˝

.c.s//jv.s/j²dxC^s.˝/: (3.7) We define the free energy density by

e_".c/D"jrA.c/j²

2 C f .c/

" :

In [3] the existence of global weak solutions is shown for a class of singular free energies. We note that this proof can be easily carried over to the present situation with only minor modifications and even some simplifications sincef is non-singular. Throughout this paper we will use the definition of weak solutions in [3]. By this definition we have

v_"2BC_!.Œ0;1/IL².˝//\L²..0;1/IH¹.˝/^d/;

c_"2BC_!.Œ0;1/IH¹.˝//\L²_loc.Œ0;1/IH².˝//; f .c_"/2L²_loc.Œ0;1/IL².˝//;

2L²_loc.Œ0;1/IL².˝//;r2L².Œ0;1/IL².˝/^d/;

and Z

Q

.c_"/v_"@_t'v_"˝

.c_"/v_"C QJ_"

W r'C.c_"/Dv_"WD'd.x; t /

Z

˝

.c_0;"/v_0;"'j_tD0dxD Z

Q

" a.c_"/rc_"˝ rc_"W r'd.x; t / (3.8) forJ_"WD m_".c_"/r_",JQ_"WD ^@

@c.c_"/J_", and all'2C₀¹.Œ0;1/IC_0;¹.˝//, as well as Z

Q

c_".@_t Cdiv. v_"// d.x; t /C Z

˝

c_0;" j_tD0dxD Z

Q

m_".c_"/r_" r d.x; t / (3.9) for all 2C₀¹.Œ0;1/˝/, and

_"D f⁰.c_"/

" C" a⁰.c_"/jrc_"j²

2 "div.a.c_"/rc_"/ a.e. inQ; (3.10)

n_@˝ rc_"D0 a.e. on.0;1/@˝: (3.11)

Moreover, we have Z

˝

.c_".t //jv_".t /j²

2 dxCE".c_".t //C

Z _t

s

Z

˝

.c_"/jDv_"j²dx d

C Z _t

s

Z

˝

m_".c_"/jr_"j²d.x; t /6 Z

˝

.c_".s//jv_".s/j²

2 dxCE_".c_".s// (3.12)

for allt >sand almost everys>0includingsD0.

THEOREM3.3 For all"2.0; 1, let initial data.v_0;"; c_0;"/2L².˝/H¹.˝/be given such that

j˝1j

R

˝c_0;"dxD Nc2.1; 1/and

Z

˝

jv_0;"j²

2 dxCE".c_0;"/6R (3.13)

for someR > 0. Furthermore, let.v_"; c_"; _"/be weak solutions of (1.1)–(1.7) in the intervalŒ0;1/.

Then there exists a sequence."_k/_k2N, converging to0ask! 1, such that the following assertions are true.

1. There arev2L²..0;1/IH¹.˝/^d/\L¹..0;1/IL².˝/^d/,v₀2L².˝/such that, ask! 1,

v_"k *v inL²..0;1/IH¹.˝/^d/; (3.14)

v_"k !v inL²_loc.Œ0;1/IL².˝//; (3.15)

v_0;"k *v₀ inL².˝/: (3.16)

Ifm₀> 0, there exists a2L²_loc.Œ0;1/IH¹.˝//withr2L²..0;1/IL².˝/^d/and such that

_"k * inL²_loc.Œ0;1/IH¹.˝//: (3.17)

2. There are measurable setsE˝Œ0;1/andE₀˝such that, ask! 1,

c_"k ! 1C2_E a.e. in˝.0;1/and inC_loc¹⁹ .Œ0;1/IL².˝// (3.18)

c_0;"k ! 1C2_E0 a.e. in˝: (3.19)

In particular, we have_Ej_tD0D_E0inL².˝/.

3. There exist Radon measuresand_ij,i; j D1; : : : ; don˝Œ0;1/such that for everyT > 0, i; j D1; : : : ; d, ask! 1,

e_"k.c_"k/ dx dt * inM.˝Œ0; T /; (3.20)

"_ka.c_"k/ @_xic_"k@_xjc_"kdx dt *_ij inM.˝Œ0; T /: (3.21) 4. There exists a Radon measureV DV^tdton˝G_d1.0;1/such that.v; E; ; V /ifm₀> 0 and.v; E; V /else is a varifold solution of (1.8)–(1.17) in the sense of Definition3.2with initial values.v₀; E₀/and DR₁

1

pf .s/=2 ds. Furthermore, Z _T

0

hıV^t;idtD Z _T

0

Z

˝

rW

d I .d _ij/^d_i;jD1

dt (3.22)

for all2C₀¹.˝Œ0; T IR^d/.

5. Ifv_0;"k !v₀inL².˝/andE_".c_0;"/!2jr_E0j.˝/ask! 1, then (3.7) holds for almost allt 2.0;1/,sD0, and⁰.˝/replaced by2jr_E0j.˝/.

By (3.12) and the assumptions on the initial data we obtain Z

˝

.c_".t //jv_".t /j²

2 dxCE".c_".t //C

Z _t

0

Z

˝

.c_"/jDv_"j²Cm_".c_"/jr_"j²dx dt 6R (3.23)

for allt >0.

From this estimate, Korn’s inequality, and (3.13) we deduce that there exists a sequence"_k&0 ask! 1such that (3.14), (3.16), (3.17), (3.20), and (3.21) hold. Using the assumptions onf, we further deduce that

Z

˝

jc_".t /j^pdx6C.1CR/; (3.24)

Z

˝

.jc_".t /j 1/²dx6C "R (3.25)

for allt > 0. In particular, for (3.25) we used thatf .c/ > C.jcj 1/²for allc 2 Rand some constantC > 0which follows from the positivity off⁰⁰.˙1/and thep-growth off for largec.

With the definitions (cf. [10]) W .c/D

Z _c

1

q

2f .s/ ds;Q wheref .s/Q Dmin

f .s/; 1C jsj²

;

and

w_".x; t /DW

c_".x; t /

;

the functionsw_"are uniformly bounded inL¹

.0;1/IBV .˝/

since Z

˝

jrw_".x; t /jdxD Z

˝

q 2fQ

c_".x; t /

jrc_".x; t /jdx6 Z

˝

e_"

c_".x; t /

dx6R: (3.26) Moreover, note that by the assumptions onf, there exist constantsC₀; C₁ > 0such that for all c₀; c₁2R

C₀jc₀c₁j²6jW .c₀/W .c₁/j6C₁jc₀c₁j.1C jc₀j C jc₁j/: (3.27) Here, for the first inequality we used again thatf .s/>C.jsj 1/²for alls2R.

LEMMA3.4 There exists a constantC > 0such that kw_"k

C¹⁸.Œ0;1/IL¹.˝//C kc_"k

C¹⁸.Œ0;1/IL².˝// 6C:

Proof. The proof is a modification of [10, Proof of Lemma 3.2]. Therefore, we only give a brief presentation. For sufficiently small > 0,x2˝, andt>0let

c_".x; t /D Z

B1

!.y/ c_".xy; t / dy;

where!is a standard mollifying kernel andc_"is extended to a small neighborhood of˝as in [10, Proof of Lemma 3.2]. Then, there exist constantsC; C⁰> 0such that

krc_".t /k_L2.˝/6C ¹kc_".t /k_L2.˝/6C⁰¹ (3.28)

kc_".t /c_".t /k²

L².˝/6C krw_".t /k_L1.˝/6C⁰ (3.29)

for all sufficiently small > 0, cf. [10, Proof of Lemma 3.2]. From (3.28) and (3.9) we deduce that for all06 < t <1such thatjtj61

Z

˝

c_".x; t /c_".x; /

c_".x; t /c_".x; / dx

D Z _t

Z

˝

m_".c_".x; s//r_".x; s/v_".x; s/c_".x; s/

rc_".x; t / rc_".x; / d.x; s/

6C.R/.t /¹² sup

s2.;t/

krc_".s/k_L2.˝/6C.R/¹.t /¹²: (3.30)

Here, we used the fact that for all; tas above we have m_".c_"/r_"v_"c_"

L².˝.;t//6C.R/;

since the sequences.v_"/L²..0;1/IL⁶.˝//and.c_"/L¹..0;1/IL³.˝//are bounded due to the assumptionsd 63andp >3. Now, combining (3.30), (3.29) and using H¨older’s and Young’s inequality we conclude that for; ;andt as above we have

kc_".t /c_". /k_L²_2.˝/6C.C¹jtj¹²/:

Choosing D .t /¹⁴ for sufficiently smallt we conclude the claim concerningc_". Using (3.27), one derives the claim concerningw_"as in [10].

REMARK 3.5 It is possible to understand the proof of Lemma3.4 from a more general point of view. From (3.9) and (3.23) we easily deduce that the distributional time-derivative of

.c_"/ is uniformly bounded in L²..0;1/; H¹.˝//. In particular, .c_"/ is uniformly bounded

in C¹⁼².Œ0;1/; H¹.˝//. On the other hand, the computations leading to (3.29) show

that .c_"/ is uniformly bounded in L¹..0;1/IB₂¹⁼³₁.˝//. This follows from B₂¹⁼³₁.˝/ D

.L².˝/; H¹.˝//_1=3;1 and the definition of the real interpolation spaces with the aid of the K- method. By interpolation, we obtain uniform boundedness inC¹⁼⁸.Œ0;1/; L².˝//.

The proof of the following lemma is literally the same as the proof of [10, Lemma 3.3].

LEMMA 3.6 There exists a sub-sequence (again denoted by"_k) and a measurable setE ˝ Œ0;1/such that, ask! 1,

w_"k !2_E a.e. in˝.0;1/and inC_loc¹⁹

Œ0;1/IL¹.˝/

c_"k ! 1C2_E a.e. in˝.0;1/and inC_loc¹⁹

Œ0;1/IL².˝/

Moreover,_E 2 L¹_!..0;1/IBV .˝//\C¹⁴.Œ0;1/IL¹.˝//and for allt > 0we havejE_tj D jE₀j D ^{1C Nc}

2 j˝j.

LEMMA3.7 There exist constantsC; "₀> 0such that k_".t /k_H1.˝/6C

E".c_".t //C kr_".t /k_L2.˝/

(3.31)

for almost allt > 0and0 < "6"₀. Usingm_">m_"we deduce from (3.31) and (3.23) that m_"

Z _T

0

k_".t /k²

L².˝/dt6C.R; T / for all0 < T <1: (3.32)

Proof. Let us suppress the time variable. Due to Poincar´e’s inequality it suffices to control the average of_". Equation (3.10) can be written in the form

_" D f⁰.c_"/

" "p

a.c_"/ A.c_"/: (3.33)

Multiplying by rc_"for2C¹.˝IR^d/, integrating over˝, and integrating by parts yields Z

˝

rc_""dxD Z

˝

rW

e_".c_"/ I"rA.c_"/˝ rA.c_"/ dxC

Z

@˝

e_".c_"/n_@˝dH^d1: (3.34) Now we can proceed exactly as in the proof of [10, Lemma 3.4].

LEMMA3.8 There exists a sub-sequence (again denoted by"_k) such that, ask! 1,

v_"k!v inL²_loc

Œ0;1/IL².˝/

v_"k.t /!v.t / inL².˝/for almost everyt > 0:

Furthermore, there exists a measurable, non-increasing functionE.t /,t > 0, such that for almost all t > 0

E_"k c_"k.t /

!E.t / and jr_Etj.˝/6 1

2E.t /6 1

2R: (3.35)

Proof. Let us fix someT > 0and letP WL².˝/^d !L².˝/denote the Helmholtz projection. In order to prove the claim concerningv_"kit suffices to show that for a sub-sequence we have

P

.c_"k/v_"k

!_k!1P .c/v

in L²

.0; T /I

L².˝/\H¹.˝/^d0

(3.36) since then

Z _T

0

Z

˝

.c_"k/jv_"kj²d dt D Z _T

0

Z

˝

P..c_"k/v_"k/v_"kdx dt

!_k!1 Z _T

0

Z

˝

P..c/v/vdx dtD Z _T

0

Z

˝

.c/jvj²dx dt;

and from this convergence, the strong convergence of.c_"k/, and the strict positivity ofwe easily deduce the claim, cf. [3]. But (3.36) follows from the Aubin-Lions lemma by noting that, firstly,

L².˝/ ,!,!.L².˝/\H¹.˝/^d/⁰,!.L².˝/\W^1;1.˝//⁰

and that, secondly, the distributional time-derivative of.P..c_"k/v_"k//is uniformly bounded in L⁸⁼⁷..0; T /I.L².˝/\W^1;1.˝//⁰/. This last bound follows by estimating each term in (3.8). We have (abbreviatingL^p..0; T /IL^q.˝//byL^pL^q)

k.c_"/v_"˝v_"k_L2L³⁼² 6k.c_"/v_"k_L1L²kv_"k_L2L⁶; kv_"˝ QJ_"k_L8=7L⁴⁼³ 6kv_"˝ QJ_"k³⁼⁴

L¹L³⁼²kv_"˝ QJ_"k¹⁼⁴

L²L¹

6Ckv_"k³⁼⁴

L²L⁶kv_"k¹⁼⁴

L¹L²km.c_"/jr_"j²k_L1L¹; k.c_"/Dv_"k_L2L² 6CkDv_"k_L2L²;

k" a.c_"/rc_"˝ rc_"k_L1L¹ 6Ck"jrA.c_"/j²k_L1L¹:

Concerning the remaining claims we note that the total energies E^tot_" .t /WD 1

2kv_".t /k²

L².˝/CE_".c_".t //; t >0;

form a sequence of bounded, non-increasing functions and thatv_"k.t /!_k!1 v.t /for almost all t > 0inL².˝/. Now, we can proceed exactly as in the proof of [10, Lemma 3.3].

Finally, we define the discrepancy function by ^".c_"/WD "

2jrA.c_"/j²1

"f .c_"/:

THEOREM 3.9 For all sufficiently small > 0 there exists a constant C./ such that for all sufficiently small" > 0(the maximal"may depend on) we have

Z _T

0

Z

˝

^".c_"/C

d.x; t /6 Z _T

0

Z

˝

e_".c_"/ d.x; t /C" C./

Z _T

0

Z

˝

j_"j²d.x; t /:

Combining this estimate with the assumption"=m_"!_"!00and (3.32) we deduce that

"lim!0

Z _T

0

Z

˝

^".c_"/C

d.x; t /D0 for all0 < T <1:

Proof. The proof is based on the elliptic equation (3.33) which can be written in the form _"a

A.c_"/1=2D .f ıA¹/⁰ A.c_"/

" " A.c_"/:

Letc˙WDA.˙1/,B.c/WDc^cC₂^cC^cCC₂^c, andf .c/Q WDf .A¹.B.c///=.B⁰/²forc2R. Then fQfulfills Assumption3.1, and forcQ_" WDB¹.A.c_"//we have

_"a

A.c_"/1=2

.B⁰/¹D fQ⁰.Qc_"/

" " cQ_":

Since the functiona.A.c_"//¹⁼².B⁰/¹is uniformly bounded, [10, Theorem 3.6] yields Z _T

0

Z

˝

.Q^".c_"//^Cd.x; t /6 Z _T

0

Z

˝

Q

e_".c_"/ d.x; t /C" C./

Z _T

0

Z

˝

j_"j²d.x; t / (3.37)

where

Q^".cQ_"/WD "

2jr Qc_"j²1

"

f .Q cQ_"/D^".c_"/=.B⁰/²

eQ^".cQ_"/WD "

2jr Qc_"j²C 1

"

f .QcQ_"/De^".c_"/=.B⁰/²:

This proves the claim.

Using the previous statements, we can now easily finish the proof of Theorem 3.3 by the arguments of [10, Section 3.5]. To be more precise, item 1 follows from (3.13) and Lemmas3.7 and3.8. Item 2 follows from Lemma3.6and the energy inequality (3.23). Item 3 follows from (3.23) as well. Furthermore, we note thatD^tdtfor Radon measures^ton˝since

.AI /6.˝I /D lim

k!1

Z

I

E_"k. / d 6jIjR

for any measurableA ˝; I Œ0;1/. Similarly, we also get ^t.˝/ D E.t / for almost all t 2.0;1/due to (3.35). From (3.12) we deduce that

^t.˝/D lim

k!1E_"k c_"k.t / 6lim inf

k!1

Z _t

s

Z

˝

.c_"k/jDv_"kj²Cm_"k.c_"k/jr_"kj² d.x; /

C lim

k!1

E_"k

c_"k.s/

C 1 2 Z

˝

c_"k.s/

jv_"k.s/j²dx1 2

Z

˝

c_"k.t /

jv_"k.t /j²dx

6 Z _t

s

Z

˝

.c/jDvj²dxJ r

d.x; /C^s.˝/

C 1 2 Z

˝

c.s/

jv.s/j²dx1 2 Z

˝

c.t /

jv.t /j²dx

for almost all0 < s < t <1wherec WD 1C2_E. This is (3.7). Item 5 follows similarly. We can proceed as in [10, Section 3.5] to construct the varifoldV. Therefore, we only give a sketch. We deduce from Theorem3.9that for all₀;₁2C¹.˝IR^d/and all0 < T <1

Z _T

0

Z

˝₀˝₁W.d _ij/6

Z _T

0

Z

˝

j₀jj₁jd :

This proves the existence of-measurableR-valued, non-negative functions_i and-measurable unit vector fieldsi,iD1; : : : ; d, such that

._ij/DX^d

iD1

_ii˝_i and Xd iD1

_i61;

Xd iD1

_i˝_iDI -a.e.

We denote the equivalence class ofi.x; t /inG_d1byp^t_i.x/, define the functionsb^t_iby b_i^t.x/WD_i.x; t /C 1

d1

1X^d

iD1

_i.x; t /

and define the varifoldV as in (3.3). Then item 3 in Definition 3.2follows taking into account (3.35). Furthermore, in the casem₀> 0we infer from (3.34) that

Z

˝

2_Etdiv./ dxD Z

˝

rW

d I .d _ij/^d_i;jD1 D

Z

˝

rWX^d

iD1

b_i^t

I p^t_i˝p^t_i d

D˝ ıV^t;˛

for all 2 C₀¹.˝IR^d/and almost all t 2 .0;1/. This is (3.6). Furthermore, these calculations prove (3.22). Similarly, (3.4) and (3.5) follow from (3.8) and (3.9), respectively, where one uses that

Z

Q

"a.c_"/rc_"˝ rc_"W r'd.x; t /D Z

Q

' rc_""d.x; t /!_"!0˝ ıV^t;'˛

for all ' 2 C¹.Œ0;1/IC₀¹.˝//. This proves item 4 in Definition 3.2. Finally, Item 2 in Definition3.2follows from Lemma3.6. This concludes the proof of Theorem3.3.

In the radially symmetric case we can prove a stronger statement concerning the discrepancy measure.

THEOREM3.10 Let˝DB₁.0/, and assume that the solutions.v_"; c_"; _"/are radially symmetric.

Assume, furthermore, thatA.c/Dcfor allc2R, and that the constantsm_"in the Assumptions3.1 satisfy

"^d11 =m_" !_"!00: (3.38) Then, for allT > 0, we have

lim

"&0

Z _T

0

Z

˝

j^".c_"/jd.x; t /D0:

For the proof we need the following result from [10, Lemma 4.4].

LEMMA3.11 There exist positive constantsC₀ and₀such that for every2 Œ0; ₀," 2 .0; 1, and every.u^"; v^"/2H².˝/L².˝/such that

v^"D "u^"C"¹f⁰.u^"/; n_@˝ ru^"j_@˝ D0 we have

Z

fx2˝Wu^"j>1g

e^".u^"/C"¹

f⁰.u^"/₂ 6C₀

Z

fx2˝Wju^"j61g

"jru^"j²dxCC₀"

Z

˝

jv^"j²dx (3.39) Proof of Theorem3.10:We can show exactly like in [10, Proof of Theorem 5.1] that there exists a constantC > 0such that for almost allt > 0we have

Z

Bı

e_"

c_".t /

dx6C ıM^".t / for allı2.0; 1/; (3.40)

j^"

c_".r; t /

C_".r; t /c_".r; t /j6C r^1dM^".t / for allr 2.0; 1/: (3.41)

Here, we use the notationrD jxjand

M^".t /WD1C k_".t /k_H1.˝/C"k_".t /k²

H¹.˝/: From (3.41) we deduce that for smallı; > 0

Z

˝

j^".c_".t //jdx6 Z

Bı[fjc".t/j>1ge_".c_".t // dxC Z

˝\fr>ı;jc".t/j<1g

j_".t /j.1/ dx

CCM^".t / Z

˝\fr>ı;jc".t/j<1gr^1ddx:

Using (3.40) and (3.39), the first integral on the right hand side may be estimated by C ıM^".t /CC⁰E_".t /CC⁰"k_".t /k²

L².˝/: By (3.25), the second integral is dominated by

k_".t /k_L2.˝/ˇˇ˚jc_".t /j>1ˇˇ¹⁼²6C⁰⁰./M^".t /"¹⁼²:

Finally, using (3.39) again, the third integral is smaller thanC⁰./M^".t /ı^1d". Summing up, we haveZ

˝

ˇˇ^"

c_".t /ˇˇdx6C⁰E".t /CC⁰"k_".t /k²

L².˝/CC⁰⁰./M^".t /."¹⁼²Cı^1d"Cı/:

Integrating this estimate from0toT and choosingsmall, the first term on the right hand side gets arbitrarily small. Choosing thenıD"^1=.2d2/and"small the other two terms get arbitrarily small, too. While this is obvious for the second term, concerning the third term we remark that it takes the form

C⁰⁰./

Z _T

0

M^".t / dt ."¹⁼²C"^1=.2d2//Do.1/ as"!0

due to (3.38).

4. Nonconvergence

In this section we show that solutions of (1.1)–(1.4) do not converge in general to solutions of (1.8)–

(1.12) ifm_".c/D Qm"^˛for some˛ > 3orm_".c/ 0, which corresponds to the case “˛ D 1”.

More precisely, we will determine radially symmetric solutions which converge as " ! 0 to a solution, which does not satisfy (1.11). Moreover, for these solutions the discrepancy measure_".c_"/ does not vanish in the limit"!0.

For simplicity of the following presentation we assume that .c/ ,.c/ . We will construct radially symmetric solutions of the form

v.x; t /Du.r; t /e_r; p.x; t /D Qp_".r; t /; c.x; t /D Qc_".r; t /; .x; t /D Q_".r; t /; (4.1) wherer D jxj;e_r D _j^x_xj. If.v; p; c; /are of this form, (1.1)–(1.4) reduce to

@_tuCu@_ru ¹

rⁿ¹@_r

rⁿ¹@_ru C@_rpQ_"D "ⁿ¹

r j@_rcQ_"j²"@_rj@_rcQ_"j² (4.2)

@_r.rⁿ¹u/D0 (4.3)

@_tcQ_"Cu@_rcQ_"Dm₀"^{˛ 1}

rⁿ¹@_r

rⁿ¹@_rQ_"

(4.4) Q

_"D " ¹

rⁿ¹@_r

rⁿ¹@_rcQ_"

C"¹f⁰.cQ_"/: (4.5) Here we have used

"div.rc˝ rc/D "div

j@_rcQ_"j²e_r˝e_r D ".n1/¹

rj@_rcQ_"j²e_r"@_rj@_rcQ_"j²e_r