Abstract
In this work, we study the nematic–isotropic phase transition based on the dynamics of the Landau–De Gennes theory of liquid crystals. At the critical temperature, 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 solution 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.
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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 \(3\times 3\) traceless and symmetric matrix-valued function Q(x) as order parameter to characterize the orientation of molecules near a material point x (cf. [8]). The matrix Q, also called Q-tensor, can be interpreted as the second moment of a number density function
where f(x, p) corresponds to the number density of liquid crystal molecules which orient along the direction \(p\in {{\mathbb {S}}^2}\) near the material point x (cf. [5]). The configuration space of the Q-tensor is the 5-dimensional linear space
By elementary linear algebra, each such Q can be written as
for some \(s,t\in {\mathbb {R}}\) and \(\mathrm {u},\mathrm {v}\in {{\mathbb {S}}^2}\) which 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 Q has repeated eigenvalues, it is called uniaxial. These Q’s form a 3-dimensional manifold in \({\mathcal {Q}}\), denoted by
with a conical singularity at \(s=0\). Here the parameter s is called the degree of orientation. To study static configurations of the liquid crystal material in a physical domain \(\Omega \), a natural approach is to consider the Ginzburg–Landau type energy
where \(\Omega \subset {\mathbb {R}}^d\) is a bounded domain with smooth boundary, \(|\nabla Q|{=}\sqrt{\sum _{ijk}|\partial _k Q_{ij}|^2}\), and F(Q) is the bulk energy density
Here \(a,b,c \in {\mathbb {R}}^+\) are material and temperature dependent constants, and \(\varepsilon \) denotes the relative intensity of elastic and bulk energy, which is usually quite small. It can be proved that all critical points of F(Q) are uniaxial (1.4), (cf. [23]), and thus
Moreover, F(Q) has two families of stable local minimizers corresponding to the following choices of \(s=s_\pm \):
In this work we shall consider the bistable case when
By rescaling, one can choose \(a=3,b=9,c=1\). 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 of F(Q):
where
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
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 \(Q\in {\mathcal {Q}}\) as order parameter, Ericksen’s model uses \((s,\mathrm {u})\in {\mathbb {R}}\times {{\mathbb {S}}^2}\) and 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 when \(s=0\). 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-\(\varepsilon \) limit 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
where \(\nabla _q F(Q)\) is the variation of F(Q) in space \({\mathcal {Q}}\):
The system (1.13a) is the \(L^2\)-gradient flow of energy (1.5) on the slow time scale \(\varepsilon \).
Our main result, Theorem 2.1, states that starting from initial conditions with a reasonable nematic–isotropic phase transition from a nematic region \(\Omega ^+(0)\) into an isotropic region \(\Omega ^-(0)\), before the occurrence of topological changes, the solution \(Q_\varepsilon \) of (1.13) converges to the isotropic phase \(Q\equiv 0\) in \(\Omega ^-(t)\) and to a field \(Q\in {\mathcal {N}}\) taking values in the nematic phase in \(\Omega ^+(t)\), where the interface between \(\Omega ^+(t)\) and \(\Omega ^-(t)\) moves by mean curvature flow. Furthermore, we show that the limit Q is a harmonic map heat flow from \(\Omega ^+(t)\) into the closed manifold \({\mathcal {N}}\). Finally, if the region \(\Omega ^+(t)\) is simply-connected, there exists a director field \(\mathrm {u}\) such that \(Q=s_+(\mathrm {u}\otimes \mathrm {u}-\frac{1}{3} I_3)\), \(\mathrm {u}\) is a harmonic map heat flow from \(\Omega ^+(t)\) into \({{\mathbb {S}}^2}\), and satisfies homogenous Neumann boundary conditions on the evolving boundary \(\partial \Omega ^+(t)\).
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 \(\partial \Omega ^+(t)\). (ii) A version of Chen–Shatah’s wedge-product trick in the sense that (1.13) implies
where \([\cdot ,\cdot ]\) denotes the commutator.
In (i) we basically follow [12] but need to carefully regularize the metric \(d^F\) on Q induced by the conformal structure F(Q) in order to exploit the fine properties of its derivative \(\nabla _q d^F_\varepsilon \). In particular, we will use the crucial commutator relation \(\left[ \nabla _q d^F_\varepsilon (Q_\varepsilon ), Q_\varepsilon \right] =0\) 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 \(\partial \Omega ^+(t)\) can be naturally encoded in the distributional formulation of (1.15) by enlarging the space of test functions. This however, requires uniform \(L^2\)-estimates on the commutators \([\partial _t Q_\varepsilon ,Q_\varepsilon ]\) and \([\nabla Q_\varepsilon ,Q_\varepsilon ]\), which are one order of \(\varepsilon \) better 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
starting from a closed smooth surface \(I_0\subset \Omega \). Let \(\Omega ^+(t)\) be the domain enclosed by \(I_t\), and \(d(x,I_t)\) be the signed-distance from x to \(I_t\) which takes positive values in \(\Omega ^+(t)\), and negative values in \(\Omega ^-(t)=\Omega \backslash \overline{\Omega ^+(t)}\), where
Moreover, for each \(T>0\) we shall denote the ‘distorted’ parabolic cylinder by
For \(\delta >0\), the \(\delta \)-neighborhood of \(I_t\) is denoted by
Thus there exists a sufficiently small number \(\delta _I\in (0,1)\) such that the nearest point projection \(P_{I}(\cdot ,t): I_t(\delta _I) \rightarrow I_t\) is smooth for any \(t\in [0,T]\), and the interface (2.1) stays at least \(\delta _I\) distance away from the boundary of the domain \(\partial \Omega \).
To introduce the modulated energy for (1.13), we extend the inner normal vector field \({\text {n}}_{I}\) of \(I_t\) to a neighborhood of it by
where \(\eta \) is a cutoff function satisfying
Following [12, 16], we define the modulated energy by
where
and the convolution is understood in the space \({\mathcal {Q}}\simeq {\mathbb {R}}^5\). Moreover, we set
a family of mollifiers in the 5-dimensional configuration space (1.2). Here \(\phi \) is smooth, non-negative, having support in \(B_1^{\mathcal {Q}}\) (the unit ball in \({\mathcal {Q}}\)), and isotropic, i.e. for any orthogonal matrix \(R\in O(3)\) and any \(q\in {\mathcal {Q}}\) it holds \(\phi (R^TqR)=\phi (q)\). The function \(d^F\) in (2.8b) is the quasi-distance function
which was introduced by Sternberg [26] and independently by Tartar-Fonseca [13] for the study of the singular perturbation problem. Some properties of \(d^F\) are stated in Lemma 3.1 below, and interested readers can find the proof in [19, 26]. One can refer to Section 3 for more details of these functions. Throughout, we will assume an \(L^\infty \)-bound of \(Q_\varepsilon \), i.e.
for some fixed constant \(c_0\). Such an estimate can be obtained by assuming an uniform \(L^\infty \)-bound of the initial data \(Q_\varepsilon ^{in}\) and then applying maximum principle to (1.13a), see Lemma 3.3 in the sequel. Note that the choice \(K=4\) in (2.8a) is due to a technical reason, and is used in the proof of Lemma 4.1.
The main result of this work is the following:
Theorem 2.1
Assume the surface \(I_t\) (2.1) evolves by mean curvature flow and encloses a simply-connected domain \(\Omega ^+(t)\). If the initial datum \(Q_\varepsilon ^{in}\) of (1.13) is well-prepared in the sense that
for some constant \(c_1\) that does not depend on \(\varepsilon \), then for some \(\varepsilon _k\downarrow 0\) as \(k\uparrow +\infty \),
where \(s_\pm \) are given by (1.8) and
Moreover, \(\mathrm {u}\) is a harmonic map heat flow into \({{\mathbb {S}}^2}\) with homogenous Neumann boundary conditions in the sense that
where \(\wedge \) is the wedge product in \({\mathbb {R}}^3\).
Remark 2.2
Note that (2.15) encodes both the harmonic map heat flow into \({{\mathbb {S}}^2}\) and the boundary conditions. Indeed, if \((\partial _t\mathrm {u},\nabla ^2 \mathrm {u})\) is continuous up to the boundary of \(\Omega ^+(t)\), then the weak formulation (2.15) implies that \(\mathrm {u}\) is a harmonic heat flow into \({{\mathbb {S}}^2}\) with Neumann boundary conditions on \(I_t\):
If \(\Omega ^+(t)\) is multi-connected, for instance when \(\Omega ^+(t)\) is the region outside \(I_t\), 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.
Theorem 2.1 solves 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 2.3 below. 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 data \(Q_\varepsilon ^{in}\) satisfying (2.12). Let \(I_0\subset \Omega \) be a smooth closed surface and let \(I_0(\delta _0)\) be a neighborhood in which the signed distance function \(d(x,I_0)\) is smooth. Let \(\zeta (z)\) be a cut-off function such that
Then we define
where S(z) is given by the optimal profile
Note that (2.19) is the solution of the ODE
Proposition 2.3
For every \(\mathrm {u}^{in}\in H^1(\Omega ;{{\mathbb {S}}^2})\), the initial datum defined by
satisfies \(Q_\varepsilon ^{in}\in H^1(\Omega ;{\mathcal {Q}})\cap L^\infty (\Omega ;{\mathcal {Q}})\) and
Moreover, there exists a constant \(c_1>0\) which only depends on \(I_0\) and \(\Vert \mathrm {u}^{in}\Vert _{H^1(\Omega )}\) such that \(Q_\varepsilon ^{in}\) is well-prepared in the sense of (2.12).
The rest of this work will be organized as follows: in Section 3 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 Proposition 2.3. In Section 4 we establish a relative-entropy type inequality for the parabolic system (1.13). Based on the various estimates given by such an inequality, in Section 5 we study the limit \(\varepsilon \downarrow 0\) of (1.13) and give the proof of Theorem 2.1.
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 \(d^F(q)\) is locally Lipschitz in \({\mathcal {Q}}\) with point-wise derivative satisfying
Moreover,
where \(c^F\) is the 1-d energy of the minimal connection between \({\mathcal {N}}\) and 0:
By elementary linear algebra, any \(Q\in {\mathcal {Q}}\) can be expressed by \(Q=\sum _{i=1}^3 \lambda _i(Q)P_i(Q)\) with \(\sum _{i=1}^3 \lambda _i(Q)=0\), where \(P_i(Q)=\mathrm {n}_i\otimes \mathrm {n}_i\) denotes the projection onto the i-th eigenspace, and \(\lambda _1(Q)\leqq \lambda _2(Q)\leqq \lambda _3(Q)\) are the eigenvalues ordered increasingly. Furthermore using the identities \(\sum _{j=1}^3 \lambda _j(Q)=0\) and \(I_3=\sum _{j=1}^3 P_j(Q)\), we may write
The next lemma gives a precise form of \(d^F\) for uniaxial Q-tensors.
Lemma 3.2
If \(Q=s_0\left( \mathrm {u} \otimes \mathrm {u}-\frac{1}{3} I_3\right) \) for some \(s_0\in [0,s_+]\) and \(\mathrm {u} \in {{\mathbb {S}}^2}\), then
where f(s) is given by (1.7).
Proof
The argument here is similar to that in [24]. Let \(\gamma \) be any curve connecting \({\mathcal {N}}\) to Q. When expressed in the form of eigenframe \(\gamma (t)=\sum _{i=1}^3 \lambda _i(t)\mathrm {n}_i(t)\otimes \mathrm {n}_i(t),\) we claim that \(\mathrm {n}_i\) are constant in t. Actually using the identity
we calculate
This implies that the global minimum is achieved by a path \(\gamma (t)\) with constant eigenframe. So by (3.4) we may write
then by (1.6)
Writing \(\sqrt{3} s+i r =:\rho e^{i\theta }\) with \(i=\sqrt{-1}\), we have \(3\sqrt{3}(s^3-sr^2)=\rho ^3\cos 3\theta \), and thus
It is clear that this energy is minimized when \(\theta \equiv 0\), which corresponds to the uniaxial solution \(r(t)\equiv 0\). In view of (1.7)
One can verify that the minimum of the right hand side can be achieved by a monotone function s(t), and thus (3.6) follows from a change of variable. \(\square \)
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 of \(d^F\) like (3.6) (cf. [26, Part 2, Lemmas 5 and 7]).
Before giving the proof of Proposition 2.3, we digress here and discuss the convolution in (2.8b). The space \({\mathcal {Q}}\) (1.2) is equipped with the inner product \(A:B= {\text {tr}}A^TB\), and one can easily verify that \(\{E_i\}_{i=1}^5\) defined below form an orthonormal basis:
This establishes an isometry \({\mathcal {Q}}\simeq {\mathbb {R}}^5\) and thus the convolution operation in (2.8b) can be interpreted as an integration in \({\mathbb {R}}^5\). Concerning the choice of \(\phi \) in (2.9), one can simply choose \(g\in C_c^\infty ({\mathbb {R}})\) and set \(\phi (q){:=} g({\text {tr}}(q^2))\), which is obviously isotropic in q.
Proof of Proposition 2.3
As a consequence of the choice of the cutoff function \(\zeta \) satisfying (2.17), we deduce that (2.22) is fulfilled and \({\tilde{S}}_\varepsilon \) is smooth. To compute the modulated energy (2.7) of the initial data \(Q^{in}_\varepsilon \), we write (2.18) by
where
It follows from the exponential decay of (2.19) that
for some constant \(C>0\) that only depends on \(I_0\). So we can write
Recalling the form of the bulk energy (1.7) for uniaxial Q-tensors, in view of (1.9), we have
and \(F(Q^{in}_\varepsilon )=f(S +{\hat{S}}_\varepsilon )\). Thus the integrand of \(E_\varepsilon [Q_\varepsilon | I](0)\) can be written as
The first line on the right hand side vanishes due to the identity \(S'(z)=\sqrt{3f(S(z))}\) as a consequence of (2.19) or equivalently the ODE (2.20):
On the other hand, since \(0\leqq {\tilde{S}}_\varepsilon \leqq s_+\), by Lemma 3.2,
This together with (2.6) and (3.15) implies
Adding up (3.19) and (3.21) yields
By the exponential decay of (2.19) and (3.15),
To treat the last term, we first deduce from the exponential decay of (2.19) that
for some C that only depends on \(I_0\). This together with (2.6) implies
Substituting the above estimate into (3.23) and use (2.8a), we arrive at
On the other hand, by (2.9) and (3.1), we have
where \(L=L(M,\phi , F)\). This pointwise estimate implies
which together with (3.26) implies (2.12).
\(\square \)
The next result is concerned with a maximum modulus estimate of (1.13a).
Lemma 3.3
Assume \(Q_\varepsilon \) is the solution of (1.13) satisfying \(\Vert Q_\varepsilon ^{in}\Vert _{L^\infty (\Omega )}\leqq C_0\) for some fixed constant \(C_0\). Then there exists an \(\varepsilon \)-independent constant \(c_0=c_0(a,b,c,C_0)>0\) such that
Proof
On the one hand, by (1.13a), \(|Q_\varepsilon |^2\) fulfills the following identity
On the other hand, there exists \(\mu >0\) (sufficiently large) such that \(|Q|\geqq \mu \) implies
Assume \(|Q_\varepsilon |(x,t)\) achieves its maximum at \((x_\varepsilon ,t_\varepsilon )\in \overline{\Omega \times (0,T)}\). If \(|Q_\varepsilon (x_\varepsilon ,t_\varepsilon )|\leqq \mu \), then we obtain the desired estimate. Otherwise there holds \(\partial _t |Q_\varepsilon |^2-\Delta |Q_\varepsilon |^2\leqq 0\), and the weak maximum principle implies the maximum must be achieved on the parabolic boundary \(\left( \partial \Omega \times (0,T)\right) \cup \left( \Omega \times \{0\}\right) \), on which \(|Q_\varepsilon |\) is bounded by our assumptions. \(\square \)
4 The Modulated Energy Inequality
As the gradient flow of (1.5), the system (1.13a) has the following energy dissipation law
Due to the concentration of \(\nabla Q_\varepsilon \) near the interface \(I_t\), this estimate is not sufficient to derive the convergence of \(Q_\varepsilon \). 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 \({\text {n}}_{I}\) of the interface \(I_t\) to a neighborhood of it. We also extend the mean curvature vector \(\mathrm {H}_I\) of (2.1) to a neighborhood by
where \({{\tilde{\eta }}}\in C^\infty _c((-\delta _I,\delta _I))\) is a cut-off which is identically equal to 1 for \(s\in (-\delta _I/2,\delta _I/2)\), and \(P_I(x,t)=x-\nabla d(x,I_t) d(x,I_t)\) is the projection onto \(I_t\). The definitions (2.5) and (4.2) of \(\xi \) and \(\mathrm {H}_I\), respectively, imply the following relations:
where \(\nabla \mathrm {H}_I{:=}\{\partial _j (\mathrm {H}_I)_i\}_{1\leqq i,j\leqq d}\) is a matrix with i being the row index. Actually in \(I_t(\delta _I/2)\) there holds \(\partial _t d(x,I_t)=-\mathrm {n}_I\cdot \mathrm {H}_I(P_I(x,t))\) and \(\nabla d(x,I_t)=\mathrm {n}_I(P_I(x,t))\). So we obtain (4.3) by chain rule. Moreover,
and since \(\mathrm {H}_I\) is extended constantly in normal direction, we have
Moreover, by the choice of \(\delta _I\) at the beginning of Section 2, we have
Finally, we have the following regularity
We denote the phase-field analogs of the mean curvature and normal vectors by
respectively, where \(\psi _\varepsilon \) is defined by (2.8b). Here and throughout we use the convention that : denotes the contraction in the indices i, j in three-tensors like \(\partial _k Q_{i,j}\), i.e., the scalar product in the state space \({\mathcal {Q}}\).
By chain rule and (2.8b)
This motivates the definition of the following projection of \(\partial _i Q_\varepsilon \) onto the span of \(\nabla _q d^F_\varepsilon (Q_\varepsilon )\)
Hence, (4.9) implies
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 convolution \(d_\varepsilon ^F\) is as good as if \(d^F\) was \(C^{1,1/2}\) and it simply follows from the fact that the modulus \(|\nabla d^F|\) is \(C^{1/2}\).
Lemma 4.1
For each \(c_0>0\) there exists \(\varepsilon _0\in {\mathbb {R}}^+\) such that
Proof
Recall (2.8a), i.e. \(F_\varepsilon (q)=F(q)+\varepsilon ^{K-1}\) with \(K=4\). It follows from (2.9), (3.1) and \(\int _{{\mathbb {R}}^5} \phi _\varepsilon (p) \mathrm{d}p=1\) that
where in the last step \(C_0\) is a local Lipschitz constant of F(q) for \(|q|\leqq c_0\). By (2.9) and the assumption that \(\phi \) is supported in the unit ball of \({\mathcal {Q}}\), the integral in the last step can be treated as follows
Finally choosing \(\varepsilon _0\) sufficiently small leads to (4.12).\(\square \)
We shall apply the above lemma with \(c_0\) being the constant in (3.29).
As we shall not integrate the time variable t throughout this section, we shall abbreviate the spatial integration \(\int _\Omega \) by \(\int \) and sometimes we omit the \(\mathrm{d}x\). The following lemma shows that the energy \(E_\varepsilon [Q_\varepsilon | I]\) defined by (2.7) controls various quantities:
Lemma 4.2
There exists a universal constant \(C<\infty \) which is independent of \(t\in (0,T)\) and \(\varepsilon \) such that the following estimates hold for every \(t\in (0,T)\):
Proof of Lemma 4.2
Since \(|\xi \cdot \nabla \psi _\varepsilon |\leqq |\nabla \psi _\varepsilon |\), 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 \(|\nabla \psi _\varepsilon |\), the Lipschitz estimate (4.12) and then completing the square. Similarly, using the Lipschitz estimate (4.12) to the term \(\frac{1}{\varepsilon }F_\varepsilon (Q_\varepsilon )\) instead yields (4.13c)
Let us now turn to the estimate (4.13d). Completing the square and using (4.12) yield
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
Hence
This combined with (4.14), (4.13c) and the trivial estimate \(1-\xi \cdot \mathrm {n}_{\varepsilon }\leqq 2\) leads to (4.13d). Finally, by (2.6) and \(\delta _I\in (0,1)\) we have
Applying this to the second right-hand side integral of (4.13d) and then using (4.13a) yield (4.13e). \(\square \)
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 \(C=C(I_t)\) depending on the interface \(I_t\) such that
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 energy \(E_\varepsilon [Q_\varepsilon |I ]\):
Lemma 4.4
Under the assumptions of Theorem 2.1, the following identity holds:
where we use the notation
In order to prove the proposition, we only need to estimate the terms on the RHS of (4.19).
Proof of Proposition 4.3
According to Lemma 4.4, we only need to estimate the RHS of (4.19) by \(E_\varepsilon [Q_\varepsilon | I]\) up to a constant that only depends on \(I_t\). We start with (4.19a): it follows from the triangle inequality that
The first integral is controlled by (4.13c). Using (4.4), the second integral is controlled by (4.13d). The third integral can be treated using the relation \(\mathrm {H}_I=(\mathrm {H}_I\cdot \xi ) \xi +O(d(x,I_t))\) and (4.4), so it can be controlled by (4.13e).
The integrals in (4.19b) can be controlled using (4.13b) and (4.13d), recalling that
The first term in (4.19c) can be controlled using (4.13a), and the second term can be estimated by (4.13d). It remains to estimate (4.20) and (4.21). The last two terms on the RHS of \(J_\varepsilon ^1\) can be bounded using (4.13b), and the first integral can be rewritten using \(\mathrm {n}_\varepsilon =\mathrm {n}_\varepsilon -\xi +\xi \):
Note that in the last step we employed
and the fact that \((\xi \cdot \nabla ) \mathrm {H}_I\) vanishes in the neighborhood \(I_t(\frac{\delta _I}{2})\) by definition (4.2). Thus we employ (4.13d) and (4.13b), and (4.11a) to get
Finally applying the Cauchy-Schwarz inequality and then (4.13d) and (4.13b), we obtain
As for \(J_\varepsilon ^2\) (4.21), we employ (4.3a) and (4.3b) and yield
after applying (4.13d) and (4.13e). Therefore we have proved that the RHS of (4.19) is bounded by \(E_\varepsilon [Q_\varepsilon | I]\) up to a multiplicative constant which only depends on \(I_t\).
\(\square \)
The following lemma will be used in the proof of Lemma 4.4:
Lemma 4.5
Under the assumptions of Theorem 2.1,
Proof
The LHS of (4.26) can be written as
To treat the second term on the RHS of (4.26), we introduce the energy stress tensor \(T_\varepsilon \)
In view of (4.8a), we have the identity
Testing this identity by \(\mathrm {H}_I\), integrating by parts and using (4.6), we obtain
Adding zero leads to
Substituting this identity into (4.27) leads to (4.26). \(\square \)
Proof of Lemma 4.4
Using the energy dissipation law (4.1) and adding zero, we compute the time derivative of the energy (2.7) by
Due to the symmetry of the Hessian of \(\psi _\varepsilon \) and the boundary conditions (4.6), we have
Hence, the first integral on the RHS above can be rewritten as
Therefore
Using (4.26) in Lemma 4.5 to replace the third and fourth integrals on the RHS of the above identity and rewriting the last integral, we arrive at
First, note that the third to last and second to last integrals combine to \(J_\varepsilon ^2\). Next, by the property (4.11b) of the orthogonal projection (4.10), we can also find \(J_\varepsilon ^1\) on the right-hand side. Indeed,
Using the definition (4.8b) of \(\mathrm {n}_\varepsilon \), we may merge the second, third, and the last integral on the RHS of (4.32) to obtain
Now we complete squares for the first four terms on the RHS of (4.33): Reordering terms, we have
Using the definition (4.8b) of the normal \(\mathrm {n}_\varepsilon \) and the chain rule in form of (4.11a), the terms in (4.33) form the last missing square. Integrating over the domain \(\Omega \) and substituting into (4.33) we arrive at (4.19). \(\square \)
5 Convergence to the Harmonic Map Heat Flow
This section is devoted to the proof of Theorem 2.1. We start with a lemma about uniform estimates of \(Q_\varepsilon \).
Lemma 5.1
There exists a universal constant \(C=C(I_0)\) such that
Moreover, for any fixed \(\delta \in (0, \delta _I)\), if holds that
Proof
We first establish a priori estimates of the solutions \(Q_\varepsilon \) which are independent of \(\varepsilon \). It follows from (4.18) and the assumption (2.12) that
On the other hand, using the orthogonal projection (4.10), we obtain
This, together with (5.4), yields
The above two estimates together with (4.13b) implies (5.1). Moreover, (5.2) follows from (5.4) and (4.13e). Now we turn to the time derivative. It follows from (5.4) that
By (1.13a) and (4.8a) we have \(\mathrm {H}_\varepsilon =-\varepsilon \partial _t Q_\varepsilon :\frac{\nabla Q_\varepsilon }{|\nabla Q_\varepsilon |}\). Using this, we can expand the integrand in the above estimate and apply the Cauchy-Schwarz inequality to obtain
This implies
so combining (5.2) with (5.7) leads us to (5.3). \(\square \)
With the above uniform estimates, we can prove the following convergence result:
Proposition 5.2
There exists a subsequence of \(\varepsilon _k>0\) such that
for \(1\leqq i\leqq d\). Moreover,
where \(Q=Q(x,t)\) is represented as
for some unit vector field
Proof
We first deduce from (1.6) and (2.10) that \(d^F(Q)\) is an isotropic function, which only depends on the eigenvalue of \(Q\in {\mathcal {Q}}\). So by (2.8b), the mollified distance function \(d^F_\varepsilon (Q)\) is isotropic and smooth in Q. By [4] there exists a smooth symmetric function \(g(\lambda _1,\lambda _2,\lambda _3)\) such that \(d^F_\varepsilon (Q)=g(\lambda _1( Q),\lambda _2(Q),\lambda _3(Q))\). Let \(Q_0\in {\mathcal {Q}}\) be a matrix having distinct eigenvalues, then \(\lambda _i(Q)\) as well as the eigenvectors \(\mathrm {n}_i(Q)\) are real-analytic functions of Q near \(Q_0\), and then by chain rule
In a neighborhood of \(Q_0\), we also have \(Q=\sum _{k=1}^3\lambda _k(Q)\mathrm {n}_k(Q)\otimes \mathrm {n}_k(Q)\). So we have
holds in a neighborhood of \(Q_0\) having distinct eigenvalues, and thus for every \(Q\in {\mathcal {Q}}\) by continuity. Now in view of (4.10), we have
for \(1\leqq i\leqq d\). This together with (3.29) and (5.1) implies
for some C independent of \(\varepsilon \). Combining this estimate with weak compactness implies (5.2).
It follows from (5.2), (5.3), (3.29), and the Aubin-Lions lemma that, for any \(\delta >0\), there exists a subsequence \(\varepsilon _k=\varepsilon _k(\delta )>0\) such that
By a diagonal argument, we infer there exists
such that the convergence (5.2) as well as
hold for every \(\delta >0\) and every \(t\in [0,T]\). Moreover, by (5.17), the interpolation theory and (5.16c), we have
To prove (5.10), we first deduce that F(Q) has the same regularity as Q in (5.17), and thus by interpolation theory we obtain
We use (5.9c), (5.2), and Fatou’s lemma to deduce that
This, together with (1.10), implies
By taking the \(L^2\)-norm, we obtain two continuous functions:
On the other hand, by the choice of the initial condition (2.21) and the convergence (5.16d), we deduce that
for any \(\delta >0\) and thus for \(\delta =0\). This implies \(f^+(0)=s^+\sqrt{\tfrac{2}{3}|\Omega ^+(0)|},f^-(0)=0\) and thus
This, together with (5.19), implies
and thus (5.10) is proved.
By (5.10), (5.17), and the orientability theorem by Ball–Zarnescu [6, Section 3.2] implies that Q is uniaxial (5.10) for some
It remains to improve the integrability of \(\nabla _{x,t} \mathrm {u}\). To this end, we choose a sequence
Since \(|\mathrm {u}|=1\) a.e., by (5.8a), (5.8b) and (5.2), we deduce that for almost every \((x,t)\in \Omega ^+_T\), it holds that
where \(\partial _0{:=}\partial _t\). Note that for each fixed \(i\in \{0,\ldots ,3\}\),
where \((\partial _i \mathrm {u}\wedge \mathrm {u})_k\) denotes the k-th component of the 3-vector \(\partial _i \mathrm {u}\wedge \mathrm {u}\). Since \({\bar{S}}_i\) are \(L^2\) integrable in \(\Omega _T\), sending \(\ell \rightarrow \infty \) and applying the dominated convergence theorem to the above identity lead us to
Retaining that \(\mathrm {u}\) maps into \({{\mathbb {S}}^2}\), we deduce
so we improve (5.22) to (5.11). \(\square \)
Proof of Theorem 2.1
In the course of the proof, we shall adopt the notation \(A:B={\text {tr}}A^T B\) for any \(A,B\in {\mathbb {R}}^{3\times 3}\). We associate each testing vector field \(\varphi (x,t)=(\varphi _1,\varphi _2,\varphi _3)\in C^1(\overline{\Omega _T},{\mathbb {R}}^3)\) a matrix-valued function
Since \([\nabla _q F(Q_{\varepsilon _k}), Q_{\varepsilon _k}]=0\), applying the anti-symmetric product \([\cdot , Q_{\varepsilon _k}] \) to (1.13a) and integration by parts over \(\Omega _T\) yields
Note that no boundary integral will occur due to (1.13c). Recall that we denote \(I_t(\delta )\) the \(\delta -\) neighborhood of \(I_t\). Equivalently, we can write the above equation by
Using (5.2), (5.2) and (5.10), we can pass \(k\rightarrow \infty \) and yield
By \(|\mathrm {u}|=1\) a.e., (5.10), (5.27) and (5.25), we obtain the following identities :
Thus we obtain
Due to (5.11) we have the absolute continuity of \(\partial _t \mathrm {u}\wedge \mathrm {u}\) and \(\nabla \mathrm {u}\wedge \mathrm {u}\) in \(\Omega ^+_T\). Moreover, (5.2) implies the absolute continuity of \(\{{\bar{S}}_i\}_{0\leqq i\leqq d}\) in \(\Omega _T\). So we can pass to the limit \(\delta \rightarrow 0\) in the above identity, which yields
This concludes the proof of Theorem 2.1. \(\square \)
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Acknowledgements
T. Laux is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813. Y. Liu is partially supported by NSF of China under Grant 11971314.
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Laux, T., Liu, Y. Nematic–Isotropic Phase Transition in Liquid Crystals: A Variational Derivation of Effective Geometric Motions. Arch Rational Mech Anal 241, 1785–1814 (2021). https://doi.org/10.1007/s00205-021-01681-0
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DOI: https://doi.org/10.1007/s00205-021-01681-0