Digital Object Identifier (DOI) https://doi.org/10.1007/s00205-021-01681-0
Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective
Geometric Motions
Tim Laux & Yuning Liu
Communicated byF. Lin
Abstract
In this work, we study the nematic–isotropic phase transition based on the dy- namics of the Landau–De Gennes theory of liquid crystals. At the critical tempera- ture, the Landau–De Gennes bulk potential favors the isotropic phase and nematic phase equally. When the elastic coefficient is much smaller than that of the bulk potential, a scaling limit can be derived by formal asymptotic expansions: the so- lution gradient concentrates on a closed surface evolving by mean curvature flow.
Moreover, on one side of the surface the solution tends to the nematic phase which is governed by the harmonic map heat flow into the sphere while on the other side, it tends to the isotropic phase. To rigorously justify such a scaling limit, we prove a convergence result by combining weak convergence methods and the modulated energy method. Our proof applies as long as the limiting mean curvature flow remains smooth.
1. Introduction
Nematic liquid crystals react to shear stress like a conventional liquid while the molecules are oriented in a crystal-like way. One of the successful continuum theories modeling nematic liquid crystals is the Q-tensor theory, also referred to as Landau–De Gennes theory, which uses a 3×3 traceless and symmetric matrix- valued functionQ(x)as order parameter to characterize the orientation of molecules near a material pointx(cf. [8]). The matrixQ, also calledQ-tensor, can be inter- preted as the second moment of a number density function
Q(x)=
S2(p⊗p−13I3)f(x,p)dp, (1.1) where f(x,p)corresponds to the number density of liquid crystal molecules which orient along the directionp∈S2near the material pointx(cf. [5]). The configura-
tion space of theQ-tensor is the 5-dimensional linear space
Q= {Q∈R3×3|Q=QT, trQ=0}. (1.2) By elementary linear algebra, each suchQcan be written as
Q=s
u⊗u−1 3I3
+t
v⊗v−1 3I3
(1.3) for somes,t∈Rand u,v∈S2which are perpendicular. In the physics literature, for instance De Gennes–Prost [8], such a representation is called the biaxial nematic configurations, cf. [23]. In case Qhas repeated eigenvalues, it is called uniaxial.
TheseQ’s form a 3-dimensional manifold inQ, denoted by U:=
Q∈QQ=s
u⊗u−1 3I3
for somes∈Rand u∈S2
, (1.4) with a conical singularity ats =0. Here the parameters is called the degree of orientation. To study static configurations of the liquid crystal material in a physical domain, a natural approach is to consider the Ginzburg–Landau type energy
Eε(Q)=
ε
2|∇Q|2+1 εF(Q)
dx, (1.5)
where⊂Rdis a bounded domain with smooth boundary,|∇Q|= i j k|∂kQi j|2, andF(Q)is the bulk energy density
F(Q)= a
2tr(Q2)−b
3tr(Q3)+c 4
tr(Q2)2
. (1.6)
Herea,b,c∈R+are material and temperature dependent constants, andεdenotes the relative intensity of elastic and bulk energy, which is usually quite small. It can be proved that all critical points ofF(Q)are uniaxial (1.4), (cf. [23]), and thus
F(Q)= s2
27(9a−2bs+3cs2)=: f(s), ifQis uniaxial(1.4). (1.7) Moreover,F(Q)has two families of stable local minimizers corresponding to the following choices ofs=s±:
s−=0, s+= b+√
b2−24ac
4c . (1.8)
In this work we shall consider the bistable case when
b2=27ac, anda,c>0. (1.9)
By rescaling, one can choosea =3,b =9,c=1. From the physics view point, such choices of the coefficients correspond to the critical temperature at which the system favors the nematic phase and the isotropic phase equally [8, Section
2.3]. Analytically, it can be shown that, in this case, the two families of minimizers corresponding to (1.8) are the only global minimizers ofF(Q):
F(Q)0 and the equality holds if and only ifQ∈ {0} ∪N, (1.10) where
N:=
Q∈QQ=s+
u⊗u−1 3I3
for some u∈S2
,
withs+= 3a
c . (1.11)
At this point we digress to mention that the Landau–De Gennes model (1.5) is closely related to Ericksen’s model, where the energy is
eE(s,u):=
κ|∇s|2+s2|∇u|2+ψ(s)
dx. (1.12)
This model was introduced by Ericksen [9] for the purpose of studying line defects.
It can be formally obtained by plugging the uniaxial Ansatz (1.4) into (1.5). In contrast to (1.5) which uses Q ∈ Qas order parameter, Ericksen’s model uses (s,u)∈R×S2and is very useful to describe liquid crystal defects. The analysis of this model is very challenging, mainly due to the reason that the geometry of the uniaxial configuration (1.4) corresponds to a double-cone, and the energy (1.12) is highly degenerate whens =0. The analytical aspects of such a model have been investigated by many authors, for instance, by Lin [18], Hardt–Lin–Poon [20], Bedford [7], Alper–Hardt–Lin [2], and Alper [1].
To model nematic–isotropic phase transitions in the framework of Landau–De Gennes theory, we shall investigate the small-εlimit of the natural gradient flow dynamics of (1.5) with initial data undergoing a sharp transition near a smooth interface. To be more precise, we consider the system
∂tQε =Qε− 1
ε2∇qF(Qε), in×(0,T), (1.13a)
Qε(x,0)=Qi nε (x), in, (1.13b)
Qε(x,t)=0, on∂×(0,T), (1.13c) where∇qF(Q)is the variation ofF(Q)in spaceQ:
(∇qF(Q))i j =a Qi j−b 3 k=1
Qi kQk j+c|Q|2Qi j +b
3|Q|2δi j. (1.14) The system (1.13a) is the L2-gradient flow of energy (1.5) on the slow time scale ε.
Our main result, Theorem2.1, states that starting from initial conditions with a reasonable nematic–isotropic phase transition from a nematic region+(0)into an isotropic region−(0), before the occurrence of topological changes, the solution Qεof (1.13) converges to the isotropic phaseQ≡0 in−(t)and to a fieldQ∈N
taking values in the nematic phase in+(t), where the interface between+(t) and−(t)moves by mean curvature flow. Furthermore, we show that the limit Qis a harmonic map heat flow from+(t)into the closed manifoldN. Finally, if the region+(t)is simply-connected, there exists a director field u such that Q =s+(u⊗u−13I3), u is a harmonic map heat flow from+(t)intoS2, and satisfies homogenous Neumann boundary conditions on the evolving boundary
∂+(t).
The proof consists of two key steps: (i) an adaptation of the modulated energy inequality in [12] to the vector-valued case to control the leading-order energy contribution, which is of order O(1)and comes from the phase transition across
∂+(t). (ii) A version of Chen–Shatah’s wedge-product trick in the sense that (1.13) implies
[∂tQε,Qε] = ∇ · [∇Qε,Qε], (1.15) where[·,·]denotes the commutator.
In (i) we basically follow [12] but need to carefully regularize the metricdF on Qinduced by the conformal structureF(Q)in order to exploit the fine prop- erties of its derivative ∇qdεF. In particular, we will use the crucial commutator relation
∇qdεF(Qε),Qε
=0 for a.e.(x,t). This seems to lie beyond the realm of generalized chain rules as in [3], which was employed in the work of Simon and one of the authors in [17]. Regarding (ii), we emphasize that the Neumann boundary condition along the free boundary∂+(t)can be naturally encoded in the distributional formulation of (1.15) by enlarging the space of test functions.
This however, requires uniformL2-estimates on the commutators[∂tQε,Qε]and [∇Qε,Qε], which are one order ofεbetter than the a priori estimates suggest. We show that these estimates are guaranteed by our bounds on the modulated energy.
2. Main Results To state the main result of this work, we assume that
I =
t∈[0,T]
(It× {t}) is a smoothly evolving closed surface in, (2.1)
starting from a closed smooth surfaceI0⊂. Let+(t)be the domain enclosed byIt, andd(x,It)be the signed-distance fromxtoIt which takes positive values in+(t), and negative values in−(t)=\+(t), where
±(t):={x∈|d(x,It)≷0}. (2.2) Moreover, for eachT >0 we shall denote the ‘distorted’ parabolic cylinder by
±T:=
t∈(0,T)
±(t)× {t}
. (2.3)
Forδ >0, theδ-neighborhood ofItis denoted by
It(δ):={x∈: |d(x,It)|< δ}. (2.4)
Thus there exists a sufficiently small numberδI ∈(0,1)such that the nearest point projection PI(·,t): It(δI)→ It is smooth for anyt ∈ [0,T], and the interface (2.1) stays at leastδI distance away from the boundary of the domain∂.
To introduce the modulated energy for (1.13), we extend the inner normal vector field nI ofIt to a neighborhood of it by
ξ(x,t):=η (d(x,It))nI(PI(x,t),t) , (2.5) whereηis a cutoff function satisfying
ηis even inRand decreases in[0,∞);
η(z)=1−z2, for|z|δI/2; η(z)=0 for|z|δI. (2.6) Following [12,16], we define the modulated energy by
Eε[Qε|I](t):=
ε
2|∇Qε(·,t)|2+1
εFε(Qε(·,t))−ξ· ∇ψε(·,t)
dx, (2.7) where
Fε(q):=F(q)+εK−1withK =4, (2.8a) ψε(x,t):=dεF◦Qε(x,t), anddεF(q):=(φε∗dF)(q), ∀q ∈Q, (2.8b) and the convolution is understood in the spaceQR5. Moreover, we set
φε(q):=ε−5Kφ ε−Kq
, (2.9)
a family of mollifiers in the 5-dimensional configuration space (1.2). Hereφ is smooth, non-negative, having support inB1Q(the unit ball inQ), and isotropic, i.e.
for any orthogonal matrix R ∈ O(3)and anyq ∈ Qit holdsφ(RTq R)=φ(q).
The functiondF in (2.8b) is the quasi-distance function dF(q):=inf
1
0
2F(γ (t))|γ(t)|dtγ ∈C0,1([0,1];Q), γ (0)∈N, γ (1)=q
, (2.10) which was introduced by Sternberg [26] and independently by Tartar-Fonseca [13]
for the study of the singular perturbation problem. Some properties ofdFare stated in Lemma3.1below, and interested readers can find the proof in [19,26]. One can refer to Section3for more details of these functions. Throughout, we will assume anL∞-bound ofQε, i.e.
QεL∞(×(0,T))c0 (2.11) for some fixed constant c0. Such an estimate can be obtained by assuming an uniform L∞-bound of the initial dataQi nε and then applying maximum principle to (1.13a), see Lemma3.3in the sequel. Note that the choice K =4 in (2.8a) is due to a technical reason, and is used in the proof of Lemma4.1.
The main result of this work is the following:
Theorem 2.1.Assume the surface It (2.1) evolves by mean curvature flow and encloses a simply-connected domain+(t). If the initial datum Qi nε of (1.13) is well-prepared in the sense that
Eε[Qε|I](0)c1ε, (2.12)
for some constant c1that does not depend onε, then for someεk↓0as k↑ +∞, Qεk −−−→k→∞ Q=s±
u(x,t)⊗u(x,t)−13I3
,
strongly in C([0,T];Lloc2 (±(t))), (2.13) where s±are given by(1.8)and
u∈H1(+T;S2). (2.14)
Moreover, u is a harmonic map heat flow into S2 with homogenous Neumann boundary conditions in the sense that
+T∂tu∧u·ϕdxdt= − d
j=1
+T ∂ju∧u·∂jϕdxdt
∀ϕ ∈C1(× [0,T];R3), (2.15)
where∧is the wedge product inR3.
Remark 2.2.Note that (2.15) encodes both the harmonic map heat flow intoS2 and the boundary conditions. Indeed, if(∂tu,∇2u)is continuous up to the boundary of+(t), then the weak formulation (2.15) implies that u is a harmonic heat flow intoS2with Neumann boundary conditions onIt:
∂tu=u+ |∇u|2u in+T, ∂nIu=0 on
t∈(0,T)
(It × {t}) . (2.16)
If+(t)is multi-connected, for instance when+(t)is the region outsideIt, then a well-known orientability issue arises and the conclusion (2.14) usually only holds away from defects. See the work of Bedford [7] for more discussions of such issues.
Theorem2.1solves a special case of the Keller–Rubinstein–Sternberg problem [25] using the energy method. A similar result has been established previously by Fei et al. [10,11] using matched asymptotic expansions and spectral gap estimates. Our approach has the superiority that it allows more flexible initial data, as indicated by Proposition 2.3below. The general case of the Keller–Rubinstein–Sternberg problem is fairly sophisticated and remains open. We refer the interested readers to a recent work of Lin–Wang [21] for the well-posedness of the limiting system.
On the other hand, the static problem has been investigated by Lin et al. [19]. It is worthy to mention that recently Golovaty et al. [14,15] studied a model problem based on highly disparate elastic constants. Most recently, Lin–Wang [22] studied isotropic-nematic transitions based on an anisotropic Ericksen’s model.
Now we turn to the construction of initial dataQi nε satisfying (2.12). LetI0⊂ be a smooth closed surface and let I0(δ0)be a neighborhood in which the signed distance functiond(x,I0)is smooth. Letζ(z)be a cut-off function such that
ζ(z)=0 for|z|1, andζ(z)=1 for|z|1/2. (2.17) Then we define
S˜ε(x):=ζ
d(x,I0) δ0
S
d(x,I0) ε
+
1−ζ
d(x,I0) δ0
s+1+(0), (2.18) whereS(z)is given by the optimal profile
S(z):=s+ 2
1+tanh √
a 2 z
, z∈R. (2.19)
Note that (2.19) is the solution of the ODE
−S(z)+a S(z)−b
3S2(z)+2
3cS3(z)=0, S(−∞)=0,S(+∞)=s+. (2.20)
Proposition 2.3.For everyui n∈ H1(;S2), the initial datum defined by Qi nε (x):= ˜Sε(x)
ui n(x)⊗ui n(x)−1 3I3
(2.21)
satisfies Qi nε ∈ H1(;Q)∩L∞(;Q)and
Qi nε (x)=
⎧⎪
⎨
⎪⎩
s+(ui n⊗ui n−13I3) if x∈+(0)\I0(δ0), S
d(x,I0) ε
(ui n⊗ui n−13I3) if x∈ I0(δ0/2), 0 if x∈−(0)\I0(δ0).
(2.22)
Moreover, there exists a constant c1>0which only depends on I0andui nH1()
such that Qi nε is well-prepared in the sense of (2.12).
The rest of this work will be organized as follows: in Section3 we discuss some properties of the quasi-distance function (2.10) and use them to construct the well-prepared initial data (2.21) and thus prove Proposition2.3. In Section4we establish a relative-entropy type inequality for the parabolic system (1.13). Based on the various estimates given by such an inequality, in Section5we study the limit ε↓0 of (1.13) and give the proof of Theorem2.1.
3. Preliminaries
We start with a lemma about the quasi-distance function (2.10), which was originally due to [13,26].
Lemma 3.1.The function dF(q)is locally Lipschitz inQwith point-wise derivative satisfying
|∇qdF(q)| =
2F(q) for a.e.q ∈Q. (3.1) Moreover,
dF(q)=
0 if q ∈N,
cF if q =0, (3.2)
where cFis the 1-d energy of the minimal connection betweenN and0:
cF:=inf 1
0
|γ(t)|2
2 +F(γ (t)
dtγ ∈C0,1([0,1];Q), γ (0)∈N, γ (1)=0
. (3.3) By elementary linear algebra, any Q ∈ Q can be expressed by Q
= 3i=1λi(Q)Pi(Q)with 3i=1λi(Q)=0, wherePi(Q)=ni⊗ni denotes the projection onto thei-th eigenspace, andλ1(Q)λ2(Q)λ3(Q)are the eigen- values ordered increasingly. Furthermore using the identities 3j=1λj(Q)=0 and I3= 3j=1Pj(Q), we may write
Q= s+r
3 P3(Q)−1 3I3
+2r
3
P2(Q)−1 3I3
, (3.4)
where s(Q)=3
2λ3(Q),r(Q)= 3
2(λ2(Q)−λ1(Q)) . (3.5) The next lemma gives a precise form ofdF for uniaxialQ-tensors.
Lemma 3.2.If Q=s0
u⊗u−13I3
for some s0∈ [0,s+]andu∈S2, then
dF(Q)= 2
√3 s+
s0
f(τ)dτ =:g(s0), (3.6)
where f(s)is given by(1.7).
Proof. The argument here is similar to that in [24]. Letγ be any curve connecting NtoQ. When expressed in the form of eigenframeγ (t)= 3i=1λi(t)ni(t)⊗ni(t), we claim that ni are constant int. Actually using the identity
λ2i ni2=λi2
3 j=1
nj ·ni2
=λ2i
j=i
nj ·ni2
, (3.7)
we calculate
|γ|2=(λ1)2+(λ2)2+(λ3)2+2 3 i=1
λ2i ni2 +
3 k=1
1i<j3
4λiλj
ni·nj nj ·ni
=(λ1)2+(λ2)2+(λ3)2+ 3 k=1
1i<j3
2 λi
nj ·ni +λj
ni·nj
2
(λ1)2+(λ2)2+(λ3)2. (3.8)
This implies that the global minimum is achieved by a pathγ (t)with constant eigenframe. So by (3.4) we may write
γ (t)=diag
−s(t)+r(t)
3 ,−s(t)−r(t) 3 ,2s(t)
3
,
with(s,r)|t=0=(s+,0), (s,r)|t=1=(s0,0). (3.9) then by (1.6)
F(γ (t))=a
9(3s2+r2)+ c
81(3s2+r2)2−2b
27(s3−sr2)=: ˜F(r,s). (3.10) Writing√
3s+ir =:ρeiθ withi =√
−1, we have 3√
3(s3−sr2)=ρ3cos 3θ, and thus
1
0
|γ(t)|
2F(γ (t))dt
=2 3
1 0
3s2+r2
F(s,˜ r)dt
=2 3
1
0
ρ2+ρ2θ2
aρ2 9 +cρ4
81 − 2bρ3 81√
3cos 3θdt.
It is clear that this energy is minimized when θ ≡ 0, which corresponds to the uniaxial solutionr(t)≡0. In view of (1.7)
1 0
|γ(t)|
2F(γ (t))dt 2
√3 1
0
|s(t)|
f(s(t))dt. (3.11) One can verify that the minimum of the right hand side can be achieved by a monotone functions(t), and thus (3.6) follows from a change of variable.
At this point we would like to remark that for the general Keller–Rubinstein–
Sternberg problem it is very hard to obtain a precise form ofdFlike (3.6) (cf. [26, Part 2, Lemmas 5 and 7]).
Before giving the proof of Proposition2.3, we digress here and discuss the convolution in (2.8b). The spaceQ(1.2) is equipped with the inner productA:B= trATB, and one can easily verify that{Ei}5i=1defined below form an orthonormal basis:
E1=
⎡
⎢⎣
√3−3
6 0 0
0
√3+3
6 0
0 0 −√33
⎤
⎥⎦, E2=
⎡
⎢⎣
√3+3
6 0 0
0
√3−3
6 0
0 0 −√33
⎤
⎥⎦,
E3=
⎡
⎢⎣
0 √22 0
√2 2 0 0 0 0 0
⎤
⎥⎦, E4=
⎡
⎢⎣
0 0 √22 0 0 0
√2 2 0 0
⎤
⎥⎦, E5=
⎡
⎢⎣ 0 0 0 0 0
√2 2
0 √22 0
⎤
⎥⎦. (3.12)
This establishes an isometryQR5and thus the convolution operation in (2.8b) can be interpreted as an integration in R5. Concerning the choice ofφin (2.9), one can simply chooseg ∈ Cc∞(R)and setφ(q):=g(tr(q2)), which is obviously isotropic inq.
Proof of Proposition2.3. As a consequence of the choice of the cutoff functionζ satisfying (2.17), we deduce that (2.22) is fulfilled andS˜ε is smooth. To compute the modulated energy (2.7) of the initial dataQi nε , we write (2.18) by
S˜ε(x)=S
d(x,I0) ε
+ ˆSε(x), (3.13)
where
Sˆε(x):=
1−ζ
d(x,I0) δ0
s+1+(0)−S
d(x,I0) ε
. (3.14) It follows from the exponential decay of (2.19) that
ˆSεL∞()+ ∇ ˆSεL∞() Ce−Cε, (3.15) for some constantC >0 that only depends on I0. So we can write
|∇Qi nε |2= 2
3ε2S2+2S2|∇ui n|2+O(e−C/ε)(|∇ui n|2+1) (3.16) Recalling the form of the bulk energy (1.7) for uniaxialQ-tensors, in view of (1.9), we have
f(s)=c
9s2(s−s+)2, f(s)=
√c
3 s|s−s+|, for alls∈ [0,s+], (3.17) andF(Qi nε )= f(S+ ˆSε). Thus the integrand ofEε[Qε|I](0)can be written as
ε 2
∇Qi nε 2+1
εF(Qi nε )−2S ε
f(S) 3
=S2
3ε + f(S) ε −2S
ε
f(S) 3
+εS2|∇ui n|2+O(e−C/ε)(|∇ui n|2+1)+ f(S+ ˆSε)− f(S)
ε . (3.18)
The first line on the right hand side vanishes due to the identityS(z)=√
3f(S(z)) as a consequence of (2.19) or equivalently the ODE (2.20):
ε 2
∇Qi nε 2+1
εF(Qi nε )−2S ε
f(S) 3
=εS2|∇ui n|2+O(e−C/ε)(|∇ui n|2+1)+ f(S+ ˆSε)− f(S)
ε . (3.19)
On the other hand, since 0S˜ε s+, by Lemma3.2, dF(Qi nε (x))= 2
√3 s+
˜ Sε(x)
f(τ)dτ. (3.20)
This together with (2.6) and (3.15) implies
−(ξ· ∇)dF(Qi nε )
= −η(d(x,I0))2S
√3ε
f(S)−ξ ·nI
2S
√3ε
f(S)−
f(S+ ˆSε)
+O(e−C/ε).
(3.21) Adding up (3.19) and (3.21) yields
ε 2
∇Qi nε 2+1
εF(Qi nε )−(ξ· ∇)dF(Qi nε )
=εS2|∇ui n|2+O(e−C/ε)(|∇ui n|2+1)+ f(S+ ˆSε)− f(S) ε
+(1−η(d(x,I0)))2S ε
f(S)
3 −ξ·nI
2S ε
⎛
⎝ f(S)
3 −
f(S+ ˆSε) 3
⎞
⎠. (3.22) By the exponential decay of (2.19) and (3.15),
ε∇Qi nε 2
2
+F(Qi nε )
ε −(ξ· ∇)dF(Qi nε )
εS2|∇ui n|2+O(e−C/ε)(|∇ui n|2+1)+(1−η(d(x,I0)))2S ε
f(S) 3 .
(3.23) To treat the last term, we first deduce from the exponential decay of (2.19) that
####
#
d(x,I0) ε
2
S
d(x,I0) ε
#####
L∞(I0(δ0))
C (3.24)
for someCthat only depends onI0. This together with (2.6) implies (1−η(d(x,I0)))2S
ε
f(S) 3
=ε2d2(x,I0) ε2
2S ε
f(S)
3 −η(d(x,I0))1{d(x,I0)>δ0/2}2S ε
f(S) 3
εC(I0)+Ce−Cε. (3.25)
Substituting the above estimate into (3.23) and use (2.8a), we arrive at
ε 2
∇Qi nε 2+1
εFε(Qi nε )−(ξ· ∇)dF(Qi nε )
dx
ε+O(e−C/ε)
|∇ui n|2dx+ε2||. (3.26) On the other hand, by (2.9) and (3.1), we have
|dεF(q)−dF(q)| = |(φε∗dF)(q)−dF(q)|εL, if|q|M, (3.27) whereL =L(M, φ,F). This pointwise estimate implies
(ξ· ∇)dF(Qi nε )−(ξ· ∇)dεF(Qi nε ) dx
=
(∇ ·ξ)
dF(Qi nε )−dεF(Qi nε ) dx
Lε, (3.28)
which together with (3.26) implies (2.12).
The next result is concerned with a maximum modulus estimate of (1.13a).
Lemma 3.3.Assume Qε is the solution of (1.13) satisfying Qi nε L∞() C0
for some fixed constant C0. Then there exists an ε-independent constant c0 = c0(a,b,c,C0) >0such that
QεL∞(×(0,T))c0. (3.29) Proof. On the one hand, by (1.13a),|Qε|2fulfills the following identity
∂t|Qε|2−|Qε|2+ |∇Qε|2= − 2 ε2
a|Qε|2−btrQ3ε+c|Qε|4
. (3.30) On the other hand, there existsμ >0 (sufficiently large) such that|Q|μimplies
a|Q|2−btrQ3+c|Q|4>0.
Assume|Qε|(x,t)achieves its maximum at(xε,tε)∈×(0,T). If|Qε(xε,tε)|
μ, then we obtain the desired estimate. Otherwise there holds∂t|Qε|2−|Qε|20, and the weak maximum principle implies the maximum must be achieved on the parabolic boundary(∂×(0,T))∪(× {0}), on which|Qε|is bounded by our
assumptions.
4. The Modulated Energy Inequality
As the gradient flow of (1.5), the system (1.13a) has the following energy dissipation law
Eε(Qε(·,T))+ T
0
ε|∂tQε|2dxdt =Eε(Qi nε (·)), for allT 0. (4.1) Due to the concentration of∇Qεnear the interfaceIt, this estimate is not sufficient to derive the convergence ofQε. Following a recent work of Fisher et al. [12] we shall develop in this section a calibrated inequality, which modulates the surface energy.
Recall in (2.5) that we extend the normal vector field nI of the interface It to a neighborhood of it. We also extend the mean curvature vector HI of (2.1) to a neighborhood by
HI(x,t)= ˜η(d(x,It))HI(PI(x,t),t)
= ˜η(d(x,It))(∇ ·nI)(PI(x,t),t)nI(PI(x,t),t), (4.2) where η˜ ∈ Cc∞((−δI, δI)) is a cut-off which is identically equal to 1 fors ∈ (−δI/2, δI/2), andPI(x,t)=x− ∇d(x,It)d(x,It)is the projection ontoIt. The definitions (2.5) and (4.2) ofξ and HI, respectively, imply the following relations:
∂tξ = −(HI· ∇) ξ−(∇HI)Tξ+O(d(x,It)), (4.3a)
∂t|ξ|2= −(HI· ∇)|ξ|2+O
d2(x,It)
, (4.3b)
where∇HI:={∂j(HI)i}1i,jdis a matrix withibeing the row index. Actually in It(δI/2)there holds∂td(x,It)= −nI·HI(PI(x,t))and∇d(x,It)=nI(PI(x,t)). So we obtain (4.3) by chain rule. Moreover,
−∇ ·ξ =HI·ξ+O(d(x,It)), (4.4) and since HIis extended constantly in normal direction, we have
(ξ· ∇)HI =0 for all(x,t)such that|d(x,It)|< δI/2. (4.5) Moreover, by the choice ofδIat the beginning of Section2, we have
ξ =0 on∂and HI =0 on∂. (4.6)
Finally, we have the following regularity
|∇ξ| + |HI| + |∇HI|C(I0). (4.7) We denote the phase-field analogs of the mean curvature and normal vectors by
Hε(x,t):= −
εQε−∇qF(Qε) ε
: ∇Qε
|∇Qε|, (4.8a) nε(x,t):= ∇ψε(x,t)
|∇ψε(x,t)|, (4.8b)
respectively, whereψεis defined by (2.8b). Here and throughout we use the con- vention that:denotes the contraction in the indicesi,jin three-tensors like∂kQi,j, i.e., the scalar product in the state spaceQ.
By chain rule and (2.8b)
∇ψε(x,t)= ∇qdεF(Qε): ∇Qε(x,t) for a.e.(x,t)∈×(0,T). (4.9) This motivates the definition of the following projection of∂iQεonto the span of
∇qdεF(Qε) Qε∂iQε =
$ ∂iQε: |∇∇qdεF(Qε)
qdεF(Qε)|
∇
qdεF(Qε)
|∇qdεF(Qε)|, if∇qdεF(Qε)=0,
0, otherwise. (4.10)
Hence, (4.9) implies
|∇ψε| = |Qε∇Qε||∇qdεF(Qε)| for a.e.(x,t)∈×(0,T), (4.11a) Qε∇Qε= |∇ψε|
|∇qdεF(Qε)|2∇qdεF(Qε)⊗nε for a.e.(x,t)∈×(0,T), (4.11b) The next inequality will be crucial to show the non-negativity of the modulated energy (2.7) and various lower bounds of it. It states that the upper bound for the gradient of the convolutiondεFis as good as ifdFwasC1,1/2and it simply follows from the fact that the modulus|∇dF|isC1/2.
Lemma 4.1.For each c0>0there existsε0∈R+such that
|∇qdεF(q)|
2Fε(q), ∀q ∈Q,|q|c0,∀ε∈(0, ε0). (4.12) Proof. Recall (2.8a), i.e.Fε(q)=F(q)+εK−1withK =4. It follows from (2.9), (3.1) and%
R5φε(p)dp=1 that
|∇qdεF(q)| =
R5φε(p)∇qdF(q−p)dp
R5
φε(p) φε(p)
2F(q−p)dp
R5φε(p)2F(q−p)dp
R5φε(p) (2F(q)+C0|p|)dp
where in the last stepC0is a local Lipschitz constant ofF(q)for|q|c0. By (2.9) and the assumption thatφis supported in the unit ball ofQ, the integral in the last step can be treated as follows
|∇qdεF(q)|
2F(q)+C0εK
R5φε(p)|p|ε−Kdp
2F(q)+C0εK. Finally choosingε0sufficiently small leads to (4.12).
We shall apply the above lemma withc0being the constant in (3.29).
As we shall not integrate the time variablet throughout this section, we shall abbreviate the spatial integration %
by %
and sometimes we omit the dx. The following lemma shows that the energyEε[Qε|I]defined by (2.7) controls various quantities:
Lemma 4.2.There exists a universal constant C < ∞which is independent of t ∈(0,T)andεsuch that the following estimates hold for every t ∈(0,T):
ε
2|∇Qε|2+1
εFε(Qε)− |∇ψε|
dx Eε[Qε|I](t), (4.13a) 1
2
√εQε∇Qε− 1
√ε
2Fε(Qε) 2
dx +ε
2 ∇Qε−Qε∇Qε2
dx Eε[Qε|I](t), (4.13b) 1
2
√εQε∇Qε− 1
√ε|∇qdεF(Qε)|
2
dx Eε[Qε|I](t), (4.13c)
√ε|∇Qε| − 1
√ε|∇qdεF(Qε)|
2
dx +
(1−ξ ·nε)ε
2Qε∇Qε2+ |∇ψε|
dx C Eε[Qε|I](t), (4.13d) ε
2|∇Qε|2+1
εFε(Qε)+ |∇ψε|
min
d2(x,It),1
dx C Eε[Qε|I](t).
(4.13e) Proof of Lemma4.2. Since|ξ·∇ψε||∇ψε|, we obtain the first estimate (4.13a).
The second estimate (4.13b) follows from the first one by using the chain rule in form of (4.11a) for the term|∇ψε|, the Lipschitz estimate (4.12) and then completing the square. Similarly, using the Lipschitz estimate (4.12) to the term 1εFε(Qε)instead yields (4.13c)
Let us now turn to the estimate (4.13d). Completing the square and using (4.12) yield
Eε[Qε|I] 1 2
√ε|∇Qε| − 1
√ε|∇qdεF(Qε)|
2
dx + |∇qdεF(Qε)||∇Qε| − |∇ψε|
dx +
(1−ξ·nε)|∇ψε|dx. (4.14)
By the chain rule in form of (4.11a), the second right-hand side integral is non- negative. Using (4.11b) and Young’s inequality, it holds that
εQε∇Qε2= |∇ψε| +√
εQε∇Qε&
√εQε∇Qε−|∇qdεF(Qε)|
√ε '
|∇ψε| +ε
2Qε∇Qε2+1 2
&
√εQε∇Qε−|∇qdεF(Qε)|
√ε '2
. (4.15) Hence
ε
2Qε∇Qε2|∇ψε| + 1 2
&
√εQε∇Qε−|∇qdεF(Qε)|
√ε '2
. (4.16)
This combined with (4.14), (4.13c) and the trivial estimate 1−ξ·nε 2 leads to (4.13d). Finally, by (2.6) andδI ∈(0,1)we have
1−ξ ·nε1−ηmin
&
d2(x,It),δ2I 4
' δ2I
4 min(d2(x,It),1). (4.17) Applying this to the second right-hand side integral of (4.13d) and then using
(4.13a) yield (4.13e).
The following result was first proved in [12] in the case of the Allen-Cahn equation, and can be generalized to the vectorial case:
Proposition 4.3.There exists a constant C =C(It)depending on the interface It
such that
d
dtEε[Qε|I] + 1
2ε ε2|∂tQε|2− |Hε|2 dx + 1
2ε ε∂tQε−(∇ ·ξ)∇qdεF(Qε)2dx + 1
2ε Hε−ε|∇Qε|HI2dxC Eε[Qε|I]. (4.18) The following lemma, the proof of which will be given at the end of this section, provides the exact computation of the time derivative of the energyEε[Qε|I]:
