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

$$\begin{aligned} Q(x)=\int _{{\mathbb {S}}^2} (p\otimes p-\tfrac{1}{3}I_3)f(x,p) \mathrm{d}p, \end{aligned}$$
(1.1)

where f(xp) 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

$$\begin{aligned} {\mathcal {Q}}=\{Q\in {\mathbb {R}}^{3\times 3}\mid Q =Q^T,~{\text {tr}}Q =0\}. \end{aligned}$$
(1.2)

By elementary linear algebra, each such Q can be written as

$$\begin{aligned} Q=s\left( \mathrm {u} \otimes \mathrm {u}-\frac{1}{3} I_3\right) +t\left( \mathrm {v} \otimes \mathrm {v}-\frac{1}{3} I_3\right) \end{aligned}$$
(1.3)

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

$$\begin{aligned} {\mathcal {U}}{:=}\left\{ Q\in {\mathcal {Q}}~\Big |~ Q=s\left( \mathrm {u} \otimes \mathrm {u}-\frac{1}{3} I_3\right) \quad \text {for some}~ s \in {\mathbb {R}}~\text {and}~ \mathrm {u} \in {{\mathbb {S}}^2}\right\} , \end{aligned}$$
(1.4)

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

$$\begin{aligned} E_\varepsilon (Q)=\int _{\Omega } \left( \frac{\varepsilon }{2} |\nabla Q|^2+\frac{1}{\varepsilon }F(Q) \right) \mathrm{d}x, \end{aligned}$$
(1.5)

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

$$\begin{aligned} F(Q)= \frac{a}{2}\mathrm {tr}(Q^2)-\frac{b}{3}{\text {tr}}(Q^3)+\frac{c}{4}\left( {\text {tr}}(Q^2)\right) ^2. \end{aligned}$$
(1.6)

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

$$\begin{aligned} F (Q)=\frac{s^2}{27}(9a-2bs+3cs^2)=:f(s),~\text {if}\, Q \, \text {is uniaxial} (1.4). \end{aligned}$$
(1.7)

Moreover, F(Q) has two families of stable local minimizers corresponding to the following choices of \(s=s_\pm \):

$$\begin{aligned} s_-=0,\qquad s_+=\frac{b+\sqrt{b^2-24ac}}{4c}. \end{aligned}$$
(1.8)

In this work we shall consider the bistable case when

$$\begin{aligned} b^2=27ac,\quad \text {and}~ a,c>0. \end{aligned}$$
(1.9)

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):

$$\begin{aligned} F(Q)\geqq 0~ \text {and the equality holds if and only if}~ Q\in \{0\}\cup {\mathcal {N}}, \end{aligned}$$
(1.10)

where

$$\begin{aligned}&{\mathcal {N}}{:=}\left\{ Q\in {\mathcal {Q}}\Big | Q=s_+\left( \mathrm {u}\otimes \mathrm {u}-\frac{1}{3}I_3\right) ~\text {for some}~\mathrm {u}\in {{\mathbb {S}}^2}\right\} ,\qquad \nonumber \\&\quad \text {with}~s_+=\sqrt{\frac{3a}{c}}. \end{aligned}$$
(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

$$\begin{aligned} e_{E}(s,\mathrm {u}){:=} \int _{\Omega }\left( \kappa |\nabla s|^{2}+s^{2}|\nabla \mathrm {u}|^{2}+\psi (s)\right) \, \mathrm{d} x. \end{aligned}$$
(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 \(Q\in {\mathcal {Q}}\) 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

$$\begin{aligned} \partial _{t} Q_\varepsilon&=\Delta Q_\varepsilon - \frac{1}{\varepsilon ^2}\nabla _q F(Q_\varepsilon ),\,\,\,\text {in}~ \Omega \times (0,T), \end{aligned}$$
(1.13a)
$$\begin{aligned} Q_\varepsilon (x,0)&=Q_\varepsilon ^{in}(x),\,\,\,\,\,\,\,\qquad \qquad \qquad ~\text {in}~\Omega , \end{aligned}$$
(1.13b)
$$\begin{aligned} Q_\varepsilon (x,t)&=0,\,\,\,\,\,\,\,\,\,\,\qquad \qquad \qquad \qquad ~\, ~ \text {on}~\partial \Omega \times (0,T), \end{aligned}$$
(1.13c)

where \(\nabla _q F(Q)\) is the variation of F(Q) in space \({\mathcal {Q}}\):

$$\begin{aligned} (\nabla _q F(Q))_{ij}=a Q_{i j}-b \sum _{k=1}^3Q_{i k} Q_{k j}+c|Q|^{2} Q_{i j}+\frac{b}{3} |Q|^{2} \delta _{i j}. \end{aligned}$$
(1.14)

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

$$\begin{aligned}{}[\partial _t Q_\varepsilon ,Q_\varepsilon ]=\nabla \cdot [\nabla Q_\varepsilon ,Q_\varepsilon ], \end{aligned}$$
(1.15)

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. (xt). 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.

Main Results

To state the main result of this work, we assume that

$$\begin{aligned} I=\bigcup _{t\in [0,T]}\left( I_t \times \{t\}\right) ~\text {is a smoothly evolving closed surface in}~\Omega , \end{aligned}$$
(2.1)

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

$$\begin{aligned} \Omega ^{\pm }(t){:=} \{x\in \Omega \mid d (x,I_t)\gtrless 0\}. \end{aligned}$$
(2.2)

Moreover, for each \(T>0\) we shall denote the ‘distorted’ parabolic cylinder by

$$\begin{aligned} \Omega ^\pm _T{:=}\bigcup _{t\in (0,T)}\left( \Omega ^\pm (t)\times \{t\}\right) . \end{aligned}$$
(2.3)

For \(\delta >0\), the \(\delta \)-neighborhood of \(I_t\) is denoted by

$$\begin{aligned} I_t(\delta ){:=} \{x\in \Omega : | d (x,I_t)|<\delta \}. \end{aligned}$$
(2.4)

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

$$\begin{aligned} \xi (x,t){:=}\eta \left( d(x, I_t)\right) {\text {n}}_{I}\left( P_{I}(x,t),t\right) , \end{aligned}$$
(2.5)

where \(\eta \) is a cutoff function satisfying

$$\begin{aligned}&\eta ~\text {is even in}~{\mathbb {R}}~\text {and decreases in}~[0,\infty );\nonumber \\&\eta (z) = 1- z^{2}, \text{ for } |z| \leqq \delta _I/2;\quad \eta (z) =0 \text{ for } |z| \geqq \delta _I. \end{aligned}$$
(2.6)

Following [12, 16], we define the modulated energy by

$$\begin{aligned} E_\varepsilon [Q_\varepsilon | I](t){:=}&\int _\Omega \left( \frac{\varepsilon }{2}\left| \nabla Q_\varepsilon (\cdot ,t)\right| ^2+\frac{1}{\varepsilon } {F_\varepsilon (Q_\varepsilon (\cdot ,t))}- \xi \cdot \nabla \psi _\varepsilon (\cdot ,t) \right) \mathrm{d}x, \end{aligned}$$
(2.7)

where

$$\begin{aligned} F_\varepsilon (q)&{{:=}F(q)+\varepsilon ^{K-1}~\text {with}~K=4,} \end{aligned}$$
(2.8a)
$$\begin{aligned} \psi _\varepsilon (x,t)&{:=} d^F_\varepsilon \circ Q_\varepsilon (x,t),\quad \text {and}~ d^F_\varepsilon (q){:=}(\phi _{\varepsilon } *d^F)(q),~\forall q\in {\mathcal {Q}}, \end{aligned}$$
(2.8b)

and the convolution is understood in the space \({\mathcal {Q}}\simeq {\mathbb {R}}^5\). Moreover, we set

$$\begin{aligned} \phi _{\varepsilon }(q){:=}\varepsilon ^{-5K}\phi \left( \varepsilon ^{-K}q\right) , \end{aligned}$$
(2.9)

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

(2.10)

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.

$$\begin{aligned} \Vert Q_\varepsilon \Vert _{L^\infty (\Omega \times (0,T))}\leqq c_0 \end{aligned}$$
(2.11)

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

$$\begin{aligned} {E_\varepsilon [Q_\varepsilon | I](0)\leqq c_1\varepsilon ,} \end{aligned}$$
(2.12)

for some constant \(c_1\) that does not depend on \(\varepsilon \), then for some \(\varepsilon _k\downarrow 0\) as \(k\uparrow +\infty \),

$$\begin{aligned}&Q_{\varepsilon _k}\xrightarrow {k\rightarrow \infty } Q=s_\pm \left( \mathrm {u}(x,t) \otimes \mathrm {u}(x,t)-\tfrac{1}{3} I_3\right) ,\nonumber \\&\quad ~\text {strongly in}~ C([0,T];L^2_{loc}(\Omega ^\pm (t))), \end{aligned}$$
(2.13)

where \(s_\pm \) are given by (1.8) and

$$\begin{aligned} \mathrm {u}\in H^1( \Omega ^+_T;{{\mathbb {S}}^2}). \end{aligned}$$
(2.14)

Moreover, \(\mathrm {u}\) is a harmonic map heat flow into \({{\mathbb {S}}^2}\) with homogenous Neumann boundary conditions in the sense that

$$\begin{aligned}&\int _{\Omega ^+_T} \partial _t \mathrm {u}\wedge \mathrm {u}\cdot \varphi \, \mathrm{d}x\mathrm{d}t=-\sum _{j=1}^d\int _{\Omega ^+_T} \partial _j \mathrm {u} \wedge \mathrm {u} \cdot \partial _j \varphi \, \mathrm{d}x\mathrm{d}t\nonumber \\&\quad \qquad \forall \varphi \in C^1({\overline{\Omega }}\times [0,T];{\mathbb {R}}^3), \end{aligned}$$
(2.15)

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 \(I_t\):

$$\begin{aligned} \partial _t \mathrm {u}=\Delta \mathrm {u}+ |\nabla \mathrm {u}|^2\mathrm {u}~\text {in}~\Omega _T^+,\qquad \partial _{\mathrm {n}_I} \mathrm {u}=0~\text {on}~\bigcup _{t\in (0,T)}\left( I_t\times \{t\}\right) . \end{aligned}$$
(2.16)

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

$$\begin{aligned} \zeta (z)=0~\text {for}~|z|\geqq 1,~\text {and}~\zeta (z)=1~\text {for}~|z|\leqq 1/2. \end{aligned}$$
(2.17)

Then we define

$$\begin{aligned} {\tilde{S}}_\varepsilon (x){:=} \zeta \left( \frac{d(x,I_0)}{\delta _0}\right) S\left( \frac{d(x,I_0)}{\varepsilon }\right) +\left( 1-\zeta \left( \frac{d(x,I_0)}{\delta _0}\right) \right) s_+{\mathbf {1}}_{\Omega ^+(0)},\qquad \end{aligned}$$
(2.18)

where S(z) is given by the optimal profile

$$\begin{aligned} S(z){:=}\frac{s_+}{2}\left( 1+\tanh \left( \frac{\sqrt{a}}{2} z\right) \right) ,\qquad z\in {\mathbb {R}}. \end{aligned}$$
(2.19)

Note that (2.19) is the solution of the ODE

$$\begin{aligned} -S''(z)+a S(z)-\frac{b}{3}S^2(z)+\frac{2}{3} c S^3(z)=0,~S(-\infty )=0,S(+\infty )=s_+. \end{aligned}$$
(2.20)

Proposition 2.3

For every \(\mathrm {u}^{in}\in H^1(\Omega ;{{\mathbb {S}}^2})\), the initial datum defined by

$$\begin{aligned} Q^{in}_\varepsilon (x){:=}{\tilde{S}}_\varepsilon (x)\left( \mathrm {u}^{in}(x)\otimes \mathrm {u}^{in}(x)-\frac{1}{3} I_3\right) \end{aligned}$$
(2.21)

satisfies \(Q_\varepsilon ^{in}\in H^1(\Omega ;{\mathcal {Q}})\cap L^\infty (\Omega ;{\mathcal {Q}})\) and

$$\begin{aligned} Q_\varepsilon ^{in}(x)=\left\{ \begin{array}{rl} s_+(\mathrm {u}^{in}\otimes \mathrm {u}^{in}-\frac{1}{3} I_3)&{} \quad \text {if}~ x\in \Omega ^+(0)\backslash I_0(\delta _0),\\ S\left( \frac{d(x,I_0)}{\varepsilon }\right) (\mathrm {u}^{in}\otimes \mathrm {u}^{in}-\frac{1}{3} I_3)&{} \quad \text {if}~ x\in I_0(\delta _0/2),\\ 0&{}\quad \text {if}~ x\in \Omega ^-(0)\backslash I_0(\delta _0). \end{array} \right. \end{aligned}$$
(2.22)

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.

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

$$\begin{aligned} |\nabla _q d^F(q)|= \sqrt{2 F(q)}~~\text {for}~~a.e.~q\in {\mathcal {Q}}. \end{aligned}$$
(3.1)

Moreover,

$$\begin{aligned} d^F(q)=\left\{ \begin{array}{ll} 0&{}\quad \text {if}q\in {\mathcal {N}},\\ c^F&{}\quad \text {if}q=0, \end{array} \right. \end{aligned}$$
(3.2)

where \(c^F\) is the 1-d energy of the minimal connection between \({\mathcal {N}}\) and 0:

$$\begin{aligned}&c^F{:=}\inf \left\{ \int _0^1\left( \frac{|\gamma '(t)|^2}{2}+ F(\gamma (t)\right) \, \mathrm{d}t \Big | \gamma \in C^{0,1}([0,1];{\mathcal {Q}}), \right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left. \gamma (0)\in {\mathcal {N}},\gamma (1)=0 \right\} . \end{aligned}$$
(3.3)

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

$$\begin{aligned}&Q=\left( s+\frac{r}{3}\right) \left( P_3(Q)-\frac{1}{3} I_3\right) +\frac{2r}{3}\left( P_2(Q)-\frac{1}{3} I_3\right) , \end{aligned}$$
(3.4)
$$\begin{aligned}&~\text {where}~\quad s(Q)= \frac{3}{2} \lambda _3(Q), ~r(Q) =\frac{3}{2} \left( \lambda _2(Q)-\lambda _1(Q)\right) . \end{aligned}$$
(3.5)

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

$$\begin{aligned} d^F(Q)= \frac{2}{\sqrt{3}} \int ^{s_+}_{s_0} \sqrt{ f(\tau ) } \, \mathrm{d}\tau =:g(s_0), \end{aligned}$$
(3.6)

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

$$\begin{aligned} \lambda _{i}^{2}\left| \mathrm {n}'_{i}\right| ^{2}=\lambda _{i}^{2} \sum _{j=1}^{3}\left( \mathrm {n}_{j} \cdot \mathrm {n}'_{i}\right) ^{2}=\lambda _{i}^{2} \sum _{j \ne i}\left( \mathrm {n}_{j} \cdot \mathrm {n}'_{i}\right) ^{2}, \end{aligned}$$
(3.7)

we calculate

$$\begin{aligned}&| \gamma '|^{2}=( \lambda _{1}')^{2}+(\lambda '_{2})^{2}+(\lambda '_{3})^{2}+2 \sum _{i=1}^3 \lambda _{i}^{2}\left| \mathrm {n}_{i}'\right| ^{2}\nonumber \\&\quad +\sum _{k=1}^{3} \sum _{1 \leqq i<j \leqq 3} 4 \lambda _{i} \lambda _{j}\left( \mathrm {n}_{i} \cdot \mathrm {n}'_{j}\right) \left( \mathrm {n}_{j} \cdot \mathrm {n}'_{i}\right) \nonumber \\ {}&=( \lambda _{1}')^{2}+(\lambda '_{2})^{2}+(\lambda '_{3})^{2}+\sum _{k=1}^{3} \sum _{1 \leqq i<j \leqq 3} 2\left( \lambda _{i}\left( \mathrm {n}_{j} \cdot \mathrm {n}'_{i}\right) +\lambda _{j}\left( \mathrm {n}_{i} \cdot \mathrm {n}'_{j}\right) \right) ^{2}\nonumber \\ {}&\geqq ( \lambda _{1}')^{2}+(\lambda '_{2})^{2}+(\lambda '_{3})^{2}.\end{aligned}$$
(3.8)

This implies that the global minimum is achieved by a path \(\gamma (t)\) with constant eigenframe. So by (3.4) we may write

$$\begin{aligned} \gamma (t)={\text {diag}}\left\{ -\frac{ s(t)+r(t)}{3},-\frac{s(t)-r(t)}{3}, \frac{2s(t)}{3}\right\} ,\nonumber \\ \text {with}~(s,r)|_{t=0}=(s_+,0),\qquad (s,r)|_{t=1}=(s_0,0). \end{aligned}$$
(3.9)

then by (1.6)

$$\begin{aligned} F(\gamma (t))=&\frac{a}{9} (3s^2+r^2)+\frac{c}{81}(3s^2+r^2)^2-\frac{2b}{27}(s^3-sr^2)=:{\tilde{F}}(r,s). \end{aligned}$$
(3.10)

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

$$\begin{aligned}&\int _0^1 |\gamma '(t)| \sqrt{2F(\gamma (t))}\, \mathrm{d}t \nonumber \\ =&\frac{2}{3}\int _0^1 \sqrt{3s'^2 + r'^2}\sqrt{{\tilde{F}}(s,r)}\, \mathrm{d}t \nonumber \\ =&\frac{2}{3}\int _0^1 \sqrt{\rho '^2+\rho ^2 \theta '^2}\sqrt{\frac{a\rho ^2 }{9}+\frac{c \rho ^4}{81}-\frac{2b \rho ^3}{81\sqrt{3}}\cos 3\theta }\, \mathrm{d}t. \end{aligned}$$

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)

$$\begin{aligned} \int _0^1 |\gamma '(t)| \sqrt{2 F(\gamma (t))} \, \mathrm{d}t\geqq \frac{2}{\sqrt{3}}\int _0^1 |s'(t)|\sqrt{f(s(t))}\, \mathrm{d}t. \end{aligned}$$
(3.11)

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:

$$\begin{aligned}&E_{1}=\left[ \begin{array}{ccc} \frac{\sqrt{3}-3}{6} &{} 0 &{} 0 \\ 0 &{} \frac{\sqrt{3}+3}{6} &{} 0 \\ 0 &{} 0 &{} -\frac{\sqrt{3}}{3} \end{array}\right] , \quad E_{2}=\left[ \begin{array}{ccc} \frac{\sqrt{3}+3}{6} &{} 0 &{} 0 \\ 0 &{} \frac{\sqrt{3}-3}{6} &{} 0 \\ 0 &{} 0 &{} -\frac{\sqrt{3}}{3} \end{array}\right] ,\nonumber \\&E_{3}=\left[ \begin{array}{ccc} 0 &{} \frac{\sqrt{2}}{2} &{} 0 \\ \frac{\sqrt{2}}{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{array}\right] , \quad E_{4}=\left[ \begin{array}{ccc} 0 &{} 0 &{} \frac{\sqrt{2}}{2} \\ 0 &{} 0 &{} 0 \\ \frac{\sqrt{2}}{2} &{} 0 &{} 0 \end{array}\right] , \quad E_{5}=\left[ \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{\sqrt{2}}{2} \\ 0 &{} \frac{\sqrt{2}}{2} &{} 0 \end{array}\right] . \end{aligned}$$
(3.12)

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

$$\begin{aligned} {\tilde{S}}_\varepsilon (x)= S\left( \frac{d(x,I_0)}{\varepsilon }\right) +{\hat{S}}_\varepsilon (x), \end{aligned}$$
(3.13)

where

$$\begin{aligned} {\hat{S}}_\varepsilon (x){:=}\left( 1-\zeta \left( \frac{d(x,I_0)}{\delta _0}\right) \right) \left( s_+{\mathbf {1}}_{\Omega ^+(0)} -S\left( \frac{d(x,I_0)}{\varepsilon }\right) \right) . \end{aligned}$$
(3.14)

It follows from the exponential decay of (2.19) that

$$\begin{aligned} \Vert {\hat{S}}_\varepsilon \Vert _{L^\infty (\Omega )}+\Vert \nabla {\hat{S}}_\varepsilon \Vert _{L^\infty (\Omega )}\leqq Ce^{-\frac{C}{\varepsilon }}, \end{aligned}$$
(3.15)

for some constant \(C>0\) that only depends on \(I_0\). So we can write

$$\begin{aligned} |\nabla Q^{in}_\varepsilon |^2=\frac{2}{3\varepsilon ^2}S'^2 +2S^2 |\nabla \mathrm {u}^{in}|^2+O(e^{-C/\varepsilon })(|\nabla \mathrm {u}^{in}|^2+1) \end{aligned}$$
(3.16)

Recalling the form of the bulk energy (1.7) for uniaxial Q-tensors, in view of (1.9), we have

$$\begin{aligned} f(s)=\frac{c}{9}s^2 (s-s_+)^2,\quad \sqrt{f(s)}=\frac{\sqrt{c}}{3} s|s-s_+|,~\text {for all}~ s\in [0,s_+], \end{aligned}$$
(3.17)

and \(F(Q^{in}_\varepsilon )=f(S +{\hat{S}}_\varepsilon )\). Thus the integrand of \(E_\varepsilon [Q_\varepsilon | I](0)\) can be written as

$$\begin{aligned}&\frac{\varepsilon }{2} \left| \nabla Q_\varepsilon ^{in}\right| ^2+\frac{1}{\varepsilon } F(Q_\varepsilon ^{in})-\frac{2S'}{\varepsilon }\sqrt{\frac{f(S)}{3}}\nonumber \\ =&\frac{S'^2}{3\varepsilon } +\frac{f(S)}{\varepsilon }-\frac{2S'}{\varepsilon }\sqrt{\frac{f(S)}{3}}\nonumber \\&+\varepsilon S^2 |\nabla \mathrm {u}^{in}|^2+O(e^{-C/\varepsilon })(|\nabla \mathrm {u}^{in}|^2+1)+\frac{f(S +{\hat{S}}_\varepsilon )-f(S)}{\varepsilon }. \end{aligned}$$
(3.18)

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):

$$\begin{aligned}&\frac{\varepsilon }{2} \left| \nabla Q_\varepsilon ^{in}\right| ^2+\frac{1}{\varepsilon } F(Q_\varepsilon ^{in})-\frac{2S'}{\varepsilon }\sqrt{\frac{f(S)}{3}}\nonumber \\ =&\varepsilon S^2 |\nabla \mathrm {u}^{in}|^2+O(e^{-C/\varepsilon })(|\nabla \mathrm {u}^{in}|^2+1)+\frac{f(S +{\hat{S}}_\varepsilon )-f(S)}{\varepsilon }. \end{aligned}$$
(3.19)

On the other hand, since \(0\leqq {\tilde{S}}_\varepsilon \leqq s_+\), by Lemma 3.2,

$$\begin{aligned} d^F(Q^{in}_\varepsilon (x))=\frac{2}{\sqrt{3}} \int ^{s_+}_{ {\tilde{S}}_\varepsilon (x)} \sqrt{ f(\tau ) } \, \mathrm{d}\tau . \end{aligned}$$
(3.20)

This together with (2.6) and (3.15) implies

$$\begin{aligned}&-(\xi \cdot \nabla ) d^F(Q_\varepsilon ^{in})\nonumber \\ =&-\eta (d(x,I_0))\frac{2S'}{\sqrt{3}\varepsilon }\sqrt{ f(S)}-\xi \cdot \mathrm {n}_I\frac{2S'}{\sqrt{3}\varepsilon }\left( \sqrt{ f(S)}-\sqrt{ f(S+{\hat{S}}_\varepsilon )}\right) +O(e^{-C/\varepsilon }). \end{aligned}$$
(3.21)

Adding up (3.19) and (3.21) yields

$$\begin{aligned}&\frac{\varepsilon }{2} \left| \nabla Q_\varepsilon ^{in}\right| ^2+\frac{1}{\varepsilon } F(Q_\varepsilon ^{in})-(\xi \cdot \nabla ) d^F(Q_\varepsilon ^{in})\nonumber \\ =&\varepsilon S^2 |\nabla \mathrm {u}^{in}|^2+O(e^{-C/\varepsilon })(|\nabla \mathrm {u}^{in}|^2+1)+\frac{f(S +{\hat{S}}_\varepsilon )-f(S)}{\varepsilon }\nonumber \\&+\left( 1-\eta (d(x,I_0))\right) \frac{2S'}{\varepsilon }\sqrt{\frac{f(S)}{3}}-\xi \cdot \mathrm {n}_I\frac{2S'}{\varepsilon }\left( \sqrt{\frac{f(S)}{3}}-\sqrt{\frac{f(S+{\hat{S}}_\varepsilon )}{3}}\right) . \end{aligned}$$
(3.22)

By the exponential decay of (2.19) and (3.15),

$$\begin{aligned}&\frac{\varepsilon \left| \nabla Q_\varepsilon ^{in}\right| }{2}^{2}+\frac{ F(Q_\varepsilon ^{in})}{\varepsilon }-(\xi \cdot \nabla ) d^F(Q_\varepsilon ^{in})\nonumber \\ \leqq&\varepsilon S^2 |\nabla \mathrm {u}^{in}|^2+O(e^{-C/\varepsilon })(|\nabla \mathrm {u}^{in}|^2+1) +\left( 1-\eta (d(x,I_0))\right) \frac{2S'}{\varepsilon }\sqrt{\frac{f(S)}{3}}. \end{aligned}$$
(3.23)

To treat the last term, we first deduce from the exponential decay of (2.19) that

$$\begin{aligned} \left\| \left( \frac{d(x,I_0)}{\varepsilon }\right) ^2S'\left( \frac{d(x,I_0)}{\varepsilon }\right) \right\| _{L^\infty (I_0(\delta _0))}\leqq C \end{aligned}$$
(3.24)

for some C that only depends on \(I_0\). This together with (2.6) implies

$$\begin{aligned}&\left( 1-\eta (d(x,I_0))\right) \frac{2S'}{\varepsilon }\sqrt{\frac{f(S)}{3}}\nonumber \\ =&\varepsilon ^2 \frac{d^2(x,I_0)}{\varepsilon ^2}\frac{2S'}{\varepsilon }\sqrt{\frac{f(S)}{3}} -\eta (d(x,I_0)){\mathbf {1}}_{\{d(x,I_0)> \delta _0/2\}}\frac{2S'}{\varepsilon }\sqrt{\frac{f(S)}{3}}\nonumber \\ \leqq&\varepsilon C(I_0)+Ce^{-\frac{C}{\varepsilon }}. \end{aligned}$$
(3.25)

Substituting the above estimate into (3.23) and use (2.8a), we arrive at

$$\begin{aligned}&\int _{\Omega }\left( \frac{\varepsilon }{2} \left| \nabla Q_\varepsilon ^{in}\right| ^2+\frac{1}{\varepsilon } {F_\varepsilon (Q_\varepsilon ^{in})}-(\xi \cdot \nabla ) d^F(Q_\varepsilon ^{in})\right) \mathrm{d}x\nonumber \\&\quad \leqq \left( \varepsilon +O(e^{-C/\varepsilon })\right) \int _{\Omega }|\nabla \mathrm {u}^{in}|^2 \mathrm{d}x {+\varepsilon ^2|\Omega |}.\end{aligned}$$
(3.26)

On the other hand, by (2.9) and (3.1), we have

$$\begin{aligned} | d^F_\varepsilon (q)-d^F(q)|=|(\phi _{\varepsilon }* d^F)(q)-d^F(q)|\leqq \varepsilon L,~\text {if}~|q|\leqq M, \end{aligned}$$
(3.27)

where \(L=L(M,\phi , F)\). This pointwise estimate implies

$$\begin{aligned}&\left| \int _{\Omega }\left( (\xi \cdot \nabla ) d^F(Q_\varepsilon ^{in})-(\xi \cdot \nabla ) d^F_\varepsilon (Q_\varepsilon ^{in})\right) \mathrm{d}x\right| \nonumber \\&\quad =\left| \int _{\Omega } (\nabla \cdot \xi ) \left( d^F(Q_\varepsilon ^{in})- d^F_\varepsilon (Q_\varepsilon ^{in})\right) \mathrm{d}x \right| \leqq L\varepsilon , \end{aligned}$$
(3.28)

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

$$\begin{aligned} \Vert Q_\varepsilon \Vert _{L^\infty (\Omega \times (0,T))}\leqq c_0. \end{aligned}$$
(3.29)

Proof

On the one hand, by (1.13a), \(|Q_\varepsilon |^2\) fulfills the following identity

$$\begin{aligned} \partial _t |Q_\varepsilon |^2-\Delta |Q_\varepsilon |^2+|\nabla Q_\varepsilon |^2=-\frac{2}{\varepsilon ^2} \left( a|Q_\varepsilon |^2-b{\text {tr}}Q_\varepsilon ^3+c|Q_\varepsilon |^4\right) . \end{aligned}$$
(3.30)

On the other hand, there exists \(\mu >0\) (sufficiently large) such that \(|Q|\geqq \mu \) implies

$$\begin{aligned} a|Q|^2-b{\text {tr}}Q^3+c|Q|^4 > 0. \end{aligned}$$

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. \(\square \)

The Modulated Energy Inequality

As the gradient flow of (1.5), the system (1.13a) has the following energy dissipation law

$$\begin{aligned} E_\varepsilon (Q_\varepsilon (\cdot ,T))+ \int _0^T \int _\Omega \varepsilon |\partial _t Q_\varepsilon |^2 \,\mathrm{d}x \mathrm{d}t=E_\varepsilon (Q_\varepsilon ^{in}(\cdot )),~\text {for all}~ T\geqq 0. \end{aligned}$$
(4.1)

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

$$\begin{aligned}&\mathrm {H}_I(x,t) = {{\tilde{\eta }}}(d(x,I_t))\mathrm {H}_I(P_I(x,t),t)\nonumber \\&\quad ={{\tilde{\eta }}}(d(x,I_t))(\nabla \cdot \mathrm {n}_I)(P_I(x,t),t)\mathrm {n}_I (P_I(x,t),t), \end{aligned}$$
(4.2)

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:

$$\begin{aligned} \partial _t \xi =&-\left( \mathrm {H}_{I} \cdot \nabla \right) \xi -\left( \nabla \mathrm {H}_{I}\right) ^{T} \xi +O(d(x, I_t)), \end{aligned}$$
(4.3a)
$$\begin{aligned} \partial _t |\xi |^{2}=&-\left( \mathrm {H}_{I} \cdot \nabla \right) |\xi |^{2}+O\left( d^{2}(x, I_t)\right) , \end{aligned}$$
(4.3b)

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,

$$\begin{aligned} -\nabla \cdot \xi =&\mathrm {H}_{I} \cdot \xi +O(d(x, I_t)), \end{aligned}$$
(4.4)

and since \(\mathrm {H}_I\) is extended constantly in normal direction, we have

$$\begin{aligned} (\xi \cdot \nabla )\mathrm {H}_I&=0~\text {for all } (x,t) \text { such that }|d(x,I_t)| < \delta _I/2. \end{aligned}$$
(4.5)

Moreover, by the choice of \(\delta _I\) at the beginning of Section 2, we have

$$\begin{aligned} \xi =0~\text {on}~\partial \Omega ~\text {and}~\mathrm {H}_I=0~\text {on}~\partial \Omega . \end{aligned}$$
(4.6)

Finally, we have the following regularity

$$\begin{aligned} |\nabla \xi |&+\left| \mathrm {H}_{I}\right| +\left| \nabla \mathrm {H}_{I}\right| \leqq C(I_0). \end{aligned}$$
(4.7)

We denote the phase-field analogs of the mean curvature and normal vectors by

$$\begin{aligned} \mathrm {H}_\varepsilon (x,t)&{:=}-\left( \varepsilon \Delta Q_\varepsilon -\frac{\nabla _q F(Q_\varepsilon )}{\varepsilon } \right) :\frac{\nabla Q_\varepsilon }{\left| \nabla Q_\varepsilon \right| }, \end{aligned}$$
(4.8a)
$$\begin{aligned} \mathrm {n}_\varepsilon (x,t)&{:=}\frac{\nabla \psi _\varepsilon (x,t)}{|\nabla \psi _\varepsilon (x,t)|}, \end{aligned}$$
(4.8b)

respectively, where \(\psi _\varepsilon \) is defined by (2.8b). Here and throughout we use the convention that  :  denotes the contraction in the indices ij 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)

$$\begin{aligned} \nabla \psi _\varepsilon (x,t)&= \nabla _q d^F_\varepsilon (Q_\varepsilon ) :\nabla Q_\varepsilon (x,t) \qquad \text {for a.e.\ }(x,t)\in \Omega \times (0,T). \end{aligned}$$
(4.9)

This motivates the definition of the following projection of \(\partial _i Q_\varepsilon \) onto the span of \(\nabla _q d^F_\varepsilon (Q_\varepsilon )\)

$$\begin{aligned} \Pi _{Q_\varepsilon } \partial _i Q_\varepsilon =\left\{ \begin{array}{ll} \left( \partial _i Q_\varepsilon :\frac{\nabla _q d^F_\varepsilon (Q_\varepsilon )}{|\nabla _q d^F_\varepsilon (Q_\varepsilon )|}\right) \frac{\nabla _q d^F_\varepsilon (Q_\varepsilon )}{|\nabla _q d^F_\varepsilon (Q_\varepsilon )|},&{}~\text {if}~\nabla _q d^F_\varepsilon (Q_\varepsilon )\ne 0,\\ 0,&{}~\text {otherwise}. \end{array} \right. \end{aligned}$$
(4.10)

Hence, (4.9) implies

$$\begin{aligned} |\nabla \psi _\varepsilon |&= |\Pi _{Q_\varepsilon } \nabla Q_\varepsilon | |\nabla _q d^F_\varepsilon (Q_\varepsilon )| \qquad \qquad \text {for a.e.\ }(x,t)\in \Omega \times (0,T), \end{aligned}$$
(4.11a)
$$\begin{aligned} \Pi _{Q_\varepsilon } \nabla Q_\varepsilon&=\frac{|\nabla \psi _\varepsilon |}{|\nabla _q d^F_\varepsilon (Q_\varepsilon )|^2}\nabla _q d^F_\varepsilon (Q_\varepsilon )\otimes \mathrm {n}_\varepsilon \qquad \text {for a.e.\ }(x,t)\in \Omega \times (0,T), \end{aligned}$$
(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 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

$$\begin{aligned} |\nabla _q d^F_\varepsilon (q)| \leqq \sqrt{2 F_\varepsilon (q)},\qquad \forall q\in {\mathcal {Q}},\, |q|\leqq c_0,\, \forall \varepsilon \in (0,\varepsilon _0).\end{aligned}$$
(4.12)

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

$$\begin{aligned} |\nabla _q d^F_\varepsilon (q)|&=\left| \int _{{\mathbb {R}}^5} \phi _\varepsilon (p) \nabla _q d^F(q-p) \mathrm{d}p\right| \\&\leqq \int _{{\mathbb {R}}^5} \sqrt{\phi _\varepsilon (p)}\sqrt{\phi _\varepsilon (p)} \sqrt{2 F(q-p)}\mathrm{d}p \\&\leqq \sqrt{\int _{{\mathbb {R}}^5} \phi _\varepsilon (p) 2 F(q-p)\mathrm{d}p }\\&\leqq \sqrt{\int _{{\mathbb {R}}^5} \phi _\varepsilon (p) \left( 2 F(q)+C_0|p|\right) \mathrm{d}p } \end{aligned}$$

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

$$\begin{aligned} |\nabla _q d^F_\varepsilon (q)|&\leqq \sqrt{2F(q)+C_0 \varepsilon ^K\int _{{\mathbb {R}}^5} \phi _\varepsilon (p) |p|\varepsilon ^{-K}\mathrm{d}p }\leqq \sqrt{2F(q)+C_0 \varepsilon ^K}. \end{aligned}$$

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 \(t\in (0,T)\):

$$\begin{aligned} \int \left( \frac{\varepsilon }{2} \left| \nabla Q_\varepsilon \right| ^2+\frac{1}{\varepsilon } F_\varepsilon (Q_\varepsilon )-|\nabla \psi _\varepsilon | \right) \,&\mathrm{d} x \quad \leqq \,\,\,\, E_\varepsilon [Q_\varepsilon | I](t), \end{aligned}$$
(4.13a)
$$\begin{aligned} \frac{1}{2}\int \left( \sqrt{\varepsilon }\left| \Pi _{Q_\varepsilon }\nabla Q_\varepsilon \right| -\frac{1}{\sqrt{\varepsilon }}\sqrt{2 F_\varepsilon (Q_\varepsilon )} \right) ^{2}\,&\mathrm{d} x \nonumber \\ \quad +\frac{\varepsilon }{2} \int \left( \left| \nabla Q_\varepsilon -\Pi _{Q_\varepsilon }\nabla Q_\varepsilon \right| ^2 \right) \,&\mathrm{d} x \quad \leqq \,\,\,\, E_\varepsilon [ Q_\varepsilon | I](t), \end{aligned}$$
(4.13b)
$$\begin{aligned} \frac{1}{2}\int \left( \sqrt{\varepsilon }\left| \Pi _{Q_\varepsilon }\nabla Q_\varepsilon \right| -\frac{1}{\sqrt{\varepsilon }} |\nabla _q d^F_\varepsilon (Q_\varepsilon )| \right) ^{2}\,&\mathrm{d} x \quad \leqq \,\,\,\, E_\varepsilon [ Q_\varepsilon | I](t), \\ \int \left( \sqrt{\varepsilon }\left| \nabla Q_\varepsilon \right| -\frac{1}{\sqrt{\varepsilon }} |\nabla _q d^F_\varepsilon (Q_\varepsilon )| \right) ^{2} \,&\mathrm{d} x \nonumber \end{aligned}$$
(4.13c)
$$\begin{aligned} \quad +\int \left( 1-\xi \cdot \mathrm {n}_{\varepsilon }\right) \left( {\frac{\varepsilon }{2}}\left| \Pi _{Q_\varepsilon }\nabla Q_\varepsilon \right| ^{2}+\left| \nabla \psi _{\varepsilon }\right| \right) \,&\mathrm{d} x \quad \leqq C E_\varepsilon [ Q_\varepsilon | I](t), \end{aligned}$$
(4.13d)
$$\begin{aligned} \int \left( \frac{\varepsilon }{2} \left| \nabla Q_\varepsilon \right| ^{2}+\frac{1}{\varepsilon }{ F_\varepsilon (Q_\varepsilon )}+|\nabla \psi _\varepsilon |\right) \min \left( d^2(x,I_t),1\right) \,&\mathrm{d} x \quad \leqq C E_\varepsilon [ Q_\varepsilon | I](t). \end{aligned}$$
(4.13e)

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

$$\begin{aligned} E_\varepsilon [ Q_\varepsilon | I]\geqq&\frac{1}{2}\int \left( \sqrt{\varepsilon }\left| \nabla Q_\varepsilon \right| -\frac{1}{\sqrt{\varepsilon }}|\nabla _q d^F_\varepsilon (Q_\varepsilon )| \right) ^{2} \mathrm{d} x\nonumber \\&+\int \left( |\nabla _q d^F_\varepsilon (Q_\varepsilon )||\nabla Q_\varepsilon |-\left| \nabla \psi _\varepsilon \right| \right) \, \mathrm{d} x\nonumber \\&+\int \left( 1-\xi \cdot \mathrm {n}_{\varepsilon }\right) \left| \nabla \psi _{\varepsilon }\right| \, \mathrm{d} x. \end{aligned}$$
(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

$$\begin{aligned} \varepsilon \left| \Pi _{Q_\varepsilon }\nabla Q_\varepsilon \right| ^{2}&=\left| \nabla \psi _{\varepsilon }\right| +\sqrt{\varepsilon }\left| \Pi _{Q_\varepsilon }\nabla Q_\varepsilon \right| \left( \sqrt{\varepsilon }\left| \Pi _{Q_\varepsilon }\nabla Q_\varepsilon \right| -\frac{ |\nabla _q d^F_\varepsilon (Q_\varepsilon )|}{\sqrt{\varepsilon }}\right) \nonumber \\&\leqq \left| \nabla \psi _{\varepsilon }\right| +\frac{\varepsilon }{2} \left| \Pi _{Q_\varepsilon } \nabla Q_\varepsilon \right| ^{2}+\frac{1}{2}\left( \sqrt{\varepsilon }\left| \Pi _{Q_\varepsilon }\nabla Q_\varepsilon \right| -\frac{ |\nabla _q d^F_\varepsilon (Q_\varepsilon )|}{\sqrt{\varepsilon }} \right) ^{2}. \end{aligned}$$
(4.15)

Hence

$$\begin{aligned} \frac{\varepsilon }{2} \left| \Pi _{Q_\varepsilon }\nabla Q_\varepsilon \right| ^{2}&\leqq \left| \nabla \psi _{\varepsilon }\right| +\frac{1}{2} \left( \sqrt{\varepsilon }\left| \Pi _{Q_\varepsilon }\nabla Q_\varepsilon \right| -\frac{ |\nabla _q d^F_\varepsilon (Q_\varepsilon )|}{\sqrt{\varepsilon }} \right) ^2. \end{aligned}$$
(4.16)

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

$$\begin{aligned} 1-\xi \cdot \mathrm {n}_\varepsilon \geqq 1-\eta \geqq \min \left( d^2(x,I_t), \frac{\delta _I^2}{4}\right) \geqq \frac{\delta _I^2}{4} \min (d^2(x,I_t),1). \end{aligned}$$
(4.17)

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

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} E_\varepsilon [ Q_\varepsilon | I] +\frac{1}{2\varepsilon }\int \left( \varepsilon ^2 \left| \partial _t Q_\varepsilon \right| ^2-|\mathrm {H}_\varepsilon |^2\right) \mathrm{d}x\nonumber \\&\quad +\frac{1}{2\varepsilon }\int \left| \varepsilon \partial _t Q_\varepsilon -(\nabla \cdot \xi )\nabla _q d^F_\varepsilon (Q_\varepsilon ) \right| ^2 \mathrm{d}x\nonumber \\&\quad +\frac{1}{2\varepsilon }\int \Big | \mathrm {H}_\varepsilon -\varepsilon |\nabla Q_\varepsilon |\mathrm {H}_I \Big |^2 \mathrm{d}x \leqq CE_\varepsilon [ Q_\varepsilon | I]. \end{aligned}$$
(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 energy \(E_\varepsilon [Q_\varepsilon |I ]\):

Lemma 4.4

Under the assumptions of Theorem 2.1, the following identity holds:

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} E \left[ Q_\varepsilon | I\right] +\frac{1}{2\varepsilon }\int \left| \varepsilon \partial _t Q_\varepsilon -(\nabla \cdot \xi ) \nabla _q d^F_\varepsilon (Q_\varepsilon ) \right| ^2 \mathrm{d} x \nonumber \\&\quad +\frac{1}{2\varepsilon }\int \big | \mathrm {H}_\varepsilon -\varepsilon |\nabla Q_\varepsilon | \mathrm {H}_I \big |^2\,\mathrm{d} x\nonumber \\ =&\frac{1}{2\varepsilon } \int \Big | (\nabla \cdot \xi ) |\nabla _q d^F_\varepsilon (Q_\varepsilon )|\mathrm {n}_\varepsilon +\varepsilon |\Pi _{Q_\varepsilon } \nabla Q_\varepsilon | \mathrm {H}_I\Big |^2\,\mathrm{d} x \end{aligned}$$
(4.19a)
$$\begin{aligned}&+\frac{\varepsilon }{2} \int |\mathrm {H}_I|^2\left( |\nabla Q_\varepsilon |^2-|\Pi _{Q_\varepsilon }\nabla Q_\varepsilon |^2\right) \,d x -\int \nabla \mathrm {H}_{I}: (\xi -\mathrm {n}_\varepsilon )^{\otimes 2}\left| \nabla \psi _{\varepsilon }\right| \,\mathrm{d} x \end{aligned}$$
(4.19b)
$$\begin{aligned}&+\int \left( \nabla \cdot \mathrm {H}_I\right) \left( \frac{\varepsilon }{2} |\nabla Q_\varepsilon |^2 +\frac{1}{\varepsilon }F_\varepsilon (Q_\varepsilon ) -|\nabla \psi _\varepsilon | \right) \,\mathrm{d} x\nonumber \\&+\int \left( \nabla \cdot \mathrm {H}_I\right) \left( 1-\xi \cdot \mathrm {n}_\varepsilon \right) |\nabla \psi _\varepsilon |\, \mathrm{d} x+ J_\varepsilon ^1+ J_\varepsilon ^2, \end{aligned}$$
(4.19c)

where we use the notation

$$\begin{aligned} J_\varepsilon ^1&{:=}&\int \nabla \mathrm {H}_{I}: \mathrm {n}_\varepsilon \otimes \mathrm {n}_{\varepsilon }\left( |\nabla \psi _\varepsilon |-\varepsilon |\nabla Q_\varepsilon |^2\right) \mathrm{d}x\nonumber \\&+\varepsilon \int \nabla \mathrm {H}_I:(\mathrm {n}_\varepsilon \otimes \mathrm {n}_\varepsilon )\left( |\nabla Q_\varepsilon |^2-|\Pi _{Q_\varepsilon } \nabla Q_\varepsilon |^2\right) \mathrm{d}x \nonumber \\&-\varepsilon \int \sum _{i,j=1}^3(\nabla \mathrm {H}_I)_{ij} \Big ((\partial _i Q_\varepsilon -\Pi _{Q_\varepsilon } \partial _i Q_\varepsilon ):(\partial _j Q_\varepsilon -\Pi _{Q_\varepsilon } \partial _j Q_\varepsilon )\Big )\mathrm{d}x ,\,\,\,\,\,\,\, \end{aligned}$$
(4.20)
$$\begin{aligned} J_\varepsilon ^2&{:=}&-\int \left( \partial _t \xi +\left( \mathrm {H}_{I} \cdot \nabla \right) \xi +\left( \nabla \mathrm {H}_{I}\right) ^{T} \xi \right) \cdot (\mathrm {n}_\varepsilon -\xi ) |\nabla \psi _\varepsilon |\mathrm{d}x\nonumber \\&-\int \Big (\partial _t \xi +\left( \mathrm {H}_{I} \cdot \nabla \right) \xi \Big )\cdot \xi \, |\nabla \psi _\varepsilon |\mathrm{d}x. \end{aligned}$$
(4.21)

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

$$\begin{aligned} \begin{aligned} \frac{1}{2} \int&\left| \frac{1}{\sqrt{\varepsilon }} (\nabla \cdot \xi ) |\nabla _q d^F_\varepsilon (Q_\varepsilon )|\mathrm {n}_\varepsilon +\sqrt{\varepsilon } |\Pi _{Q_\varepsilon } \nabla Q_\varepsilon | \mathrm {H}_I\right| ^2\mathrm{d} x \\ \leqq&\int \left| (\nabla \cdot \xi ) \left( \sqrt{\varepsilon } |\Pi _{Q_\varepsilon } \nabla Q_\varepsilon | - \frac{1}{\sqrt{\varepsilon }} |\nabla _q d^F_\varepsilon (Q_\varepsilon )| \right) \mathrm {n}_\varepsilon \right| ^2 \mathrm{d} x \\ {}&+\int \left| (\nabla \cdot \xi ) \sqrt{\varepsilon } |\Pi _{Q_\varepsilon } \nabla Q_\varepsilon | (\mathrm {n}_\varepsilon -\xi )\right| ^2 \mathrm{d} x \\ {}&+ \int \left| (\mathrm {H}_I +(\nabla \cdot \xi ) \xi ) \sqrt{\varepsilon } |\Pi _{Q_\varepsilon } \nabla Q_\varepsilon | \right| ^2 \mathrm{d} x. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} |\xi - \mathrm {n}_\varepsilon |^2 \leqq 2 (1-\mathrm {n}_\varepsilon \cdot \xi ). \end{aligned}$$

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 \(\mathrm {n}_\varepsilon =\mathrm {n}_\varepsilon -\xi +\xi \):

$$\begin{aligned} J_\varepsilon ^1\leqq & {} \int \nabla \mathrm {H}_{I}: \left( \mathrm {n}_\varepsilon \otimes (\mathrm {n}_\varepsilon -\xi )\right) \left( |\nabla \psi _\varepsilon |-\varepsilon |\nabla Q_\varepsilon |^2\right) \mathrm{d}x\nonumber \\&+\int \nabla \mathrm {H}_{I}: \mathrm {n}_\varepsilon \otimes \xi \left( |\nabla \psi _\varepsilon |-\varepsilon |\nabla Q_\varepsilon |^2\right) \mathrm{d}x+ C E_\varepsilon [Q_\varepsilon | I]\nonumber \\\leqq & {} \Vert \nabla \mathrm {H}_I\Vert _{L^\infty }\int |\mathrm {n}_\varepsilon -\xi | \left( \varepsilon |\nabla Q_\varepsilon |^2-\varepsilon |\Pi _{Q_\varepsilon }\nabla Q_\varepsilon |^2\right. \nonumber \\&\qquad \quad \qquad \!\qquad \qquad \qquad \qquad \qquad \qquad \left. +\left| \varepsilon |\Pi _{Q_\varepsilon }\nabla Q_\varepsilon |^2-|\nabla \psi _\varepsilon |\right| \right) \mathrm{d}x\nonumber \\&+C\int \min \left( d^2(x,I_t),1\right) \left( |\nabla \psi _\varepsilon |+\varepsilon |\nabla Q_\varepsilon |^2\right) \mathrm{d}x+ C E_\varepsilon [Q_\varepsilon | I].\qquad \end{aligned}$$
(4.22)

Note that in the last step we employed

$$\begin{aligned} \nabla \mathrm {H}_{I}: \mathrm {n}_\varepsilon \otimes \xi =(\xi \cdot \nabla \mathrm {H}_I)\cdot \mathrm {n}_\varepsilon \end{aligned}$$

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

$$\begin{aligned} J_\varepsilon ^1\leqq & {} C\int |\mathrm {n}_\varepsilon -\xi | \left| \varepsilon |\Pi _{Q_\varepsilon }\nabla Q_\varepsilon |^2-|\nabla \psi _\varepsilon |\right| \mathrm{d}x+ C E_\varepsilon [Q_\varepsilon | I]\nonumber \\= & {} C\int |\mathrm {n}_\varepsilon -\xi | \sqrt{\varepsilon } |\Pi _{Q_\varepsilon }\nabla Q_\varepsilon | \left| \sqrt{\varepsilon } |\Pi _{Q_\varepsilon }\nabla Q_\varepsilon |-\frac{|\nabla _q d^F_\varepsilon (Q_\varepsilon )|}{\sqrt{\varepsilon }}\right| \mathrm{d}x\nonumber \\&+ C E_\varepsilon [Q_\varepsilon | I]. \end{aligned}$$
(4.23)

Finally applying the Cauchy-Schwarz inequality and then (4.13d) and (4.13b), we obtain

$$\begin{aligned} J_\varepsilon ^1 \leqq C E_\varepsilon [Q_\varepsilon | I]. \end{aligned}$$
(4.24)

As for \(J_\varepsilon ^2\) (4.21), we employ (4.3a) and (4.3b) and yield

$$\begin{aligned} J_\varepsilon ^2 \leqq C \int \Big ( |\mathrm {n}_\varepsilon -\xi |^2+\min (d^2(x,I_t),1)\Big )|\nabla \psi _\varepsilon | \mathrm{d}x\leqq CE_\varepsilon [Q_\varepsilon | I], \end{aligned}$$
(4.25)

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,

$$\begin{aligned}&\int \nabla \mathrm {H}_{I}: (\xi \otimes \mathrm {n}_{\varepsilon })\left| \nabla \psi _{\varepsilon }\right| \, \mathrm{d} x -\int (\nabla \cdot \mathrm {H}_{I}) \, \xi \cdot \nabla \psi _{\varepsilon } \, \mathrm{d} x\nonumber \\&\quad =\int \nabla \mathrm {H}_{I}: (\xi -\mathrm {n}_\varepsilon ) \otimes \mathrm {n}_{\varepsilon }\left| \nabla \psi _{\varepsilon }\right| \, \mathrm{d} x+\int \mathrm {H}_\varepsilon \cdot \mathrm {H}_I |\nabla Q_\varepsilon |\, \mathrm{d} x \nonumber \\&\qquad +\int \nabla \cdot \mathrm {H}_I \left( \frac{\varepsilon }{2} |\nabla Q_\varepsilon |^2 +\frac{1}{\varepsilon }F_\varepsilon (Q_\varepsilon ) -|\nabla \psi _\varepsilon | \right) \, \mathrm{d} x\nonumber \\&\qquad +\int \nabla \cdot \mathrm {H}_I ( |\nabla \psi _\varepsilon |-\xi \cdot \nabla \psi _\varepsilon )\, \mathrm{d} x\nonumber \\&\qquad -\int \sum _{i,j=1}^3(\nabla \mathrm {H}_I)_{ij}\, \varepsilon \left( \partial _i Q_\varepsilon : \partial _j Q_\varepsilon \right) \, \mathrm{d} x +\int \nabla \mathrm {H}_{I}: (\mathrm {n}_\varepsilon \otimes \mathrm {n}_{\varepsilon })\left| \nabla \psi _{\varepsilon }\right| \, \mathrm{d} x\nonumber \\ \end{aligned}$$
(4.26)

Proof

The LHS of (4.26) can be written as

$$\begin{aligned}&\int \nabla \mathrm {H}_{I}: \left( \xi \otimes \mathrm {n}_{\varepsilon }\right) \left| \nabla \psi _{\varepsilon }\right| \, \mathrm{d} x -\int (\nabla \cdot \mathrm {H}_{I}) \, \xi \cdot \nabla \psi _{\varepsilon } \, \mathrm{d} x\nonumber \\&=\int \nabla \mathrm {H}_{I}: (\xi -\mathrm {n}_\varepsilon ) \otimes \mathrm {n}_{\varepsilon }\left| \nabla \psi _{\varepsilon }\right| \, \mathrm{d} x+\int \nabla \mathrm {H}_{I}: \left( \mathrm {n}_\varepsilon \otimes \mathrm {n}_{\varepsilon }\right) \left| \nabla \psi _{\varepsilon }\right| \, \mathrm{d} x\nonumber \\&\quad -\int (\nabla \cdot \mathrm {H}_{I})\, \xi \cdot \nabla \psi _{\varepsilon } \, \mathrm{d} x. \end{aligned}$$
(4.27)

To treat the second term on the RHS of (4.26), we introduce the energy stress tensor \(T_\varepsilon \)

$$\begin{aligned} \begin{aligned} (T_\varepsilon )_{ij}=&\left( \frac{\varepsilon }{2} |\nabla Q_\varepsilon |^2 +\frac{1}{\varepsilon } F_\varepsilon (Q_\varepsilon ) \right) \delta _{ij} - \varepsilon \partial _i Q_\varepsilon : \partial _j Q_\varepsilon . \end{aligned} \end{aligned}$$
(4.28)

In view of (4.8a), we have the identity

$$\begin{aligned} \nabla \cdot T_\varepsilon =-\varepsilon \nabla Q_\varepsilon : \Delta Q_\varepsilon + \frac{1}{\varepsilon } \nabla _q F_\varepsilon (Q_\varepsilon ) : \nabla Q_\varepsilon =\mathrm {H}_\varepsilon |\nabla Q_\varepsilon |. \end{aligned}$$
(4.29)

Testing this identity by \(\mathrm {H}_I\), integrating by parts and using (4.6), we obtain

$$\begin{aligned} \begin{aligned} \int \mathrm {H}_\varepsilon \cdot \mathrm {H}_I |\nabla Q_\varepsilon |\,\mathrm{d} x&= - \int \nabla \mathrm {H}_I :T_\varepsilon \,\mathrm{d} x\\&=- \int \nabla \cdot \mathrm {H}_I \left( \frac{\varepsilon }{2} |\nabla Q_\varepsilon |^2 +\frac{1}{\varepsilon } F_\varepsilon (Q_\varepsilon ) \right) \mathrm{d}x\\&\ \quad + \int \sum _{i,j=1}^3(\nabla \mathrm {H}_I)_{ij} \, \varepsilon \left( \partial _i Q_\varepsilon : \partial _j Q_\varepsilon \right) \mathrm{d} x. \end{aligned} \end{aligned}$$
(4.30)

Adding zero leads to

$$\begin{aligned} \begin{aligned}&\int \nabla \mathrm {H}_{I}: \mathrm {n}_\varepsilon \otimes \mathrm {n}_{\varepsilon }\left| \nabla \psi _{\varepsilon }\right| \mathrm{d} x\\&\quad =\int \mathrm {H}_\varepsilon \cdot \mathrm {H}_I |\nabla Q_\varepsilon | \mathrm{d}x+\int \nabla \cdot \mathrm {H}_I \left( \frac{\varepsilon }{2} |\nabla Q_\varepsilon |^2 +\frac{1}{\varepsilon } F_\varepsilon (Q_\varepsilon ) -|\nabla \psi _\varepsilon | \right) \,\mathrm{d} x\\&\qquad +\int \nabla \cdot \mathrm {H}_I |\nabla \psi _\varepsilon |\,\mathrm{d} x\\&\qquad -\int \sum _{i,j=1}^3(\nabla \mathrm {H}_I)_{ij}\,\varepsilon \left( \partial _i Q_\varepsilon : \partial _j Q_\varepsilon \right) \mathrm{d} x+\int (\nabla \mathrm {H}_{I}): (\mathrm {n}_\varepsilon \otimes \mathrm {n}_{\varepsilon })\left| \nabla \psi _{\varepsilon }\right| \mathrm{d} x. \end{aligned} \end{aligned}$$
(4.31)

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

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} E_\varepsilon [ Q_\varepsilon | I] +\varepsilon \int |\partial _t Q_\varepsilon |^2\,\mathrm{d} x-\int (\nabla \cdot \xi ) \nabla _q d^F_\varepsilon (Q_\varepsilon ): \partial _t Q_\varepsilon \,\mathrm{d} x\nonumber \\&=\int \left( \mathrm {H}_{I} \cdot \nabla \right) \xi \cdot \nabla \psi _\varepsilon \,\mathrm{d} x +\int \left( \nabla \mathrm {H}_{I}\right) ^{T} \xi \cdot \nabla \psi _\varepsilon \,\mathrm{d} x \\&\quad -\int \left( \partial _t \xi +\left( \mathrm {H}_{I} \cdot \nabla \right) \xi +\left( \nabla \mathrm {H}_{I}\right) ^{T} \xi \right) \cdot \nabla \psi _\varepsilon \,\mathrm{d} x. \end{aligned}$$

Due to the symmetry of the Hessian of \(\psi _\varepsilon \) and the boundary conditions (4.6), we have

$$\begin{aligned} \int \nabla \cdot (\xi \otimes \mathrm {H}_I ) \cdot \nabla \psi _\varepsilon \, \mathrm{d} x = \int \nabla \cdot (\mathrm {H}_I \otimes \xi ) \cdot \nabla \psi _\varepsilon \, \mathrm{d} x. \end{aligned}$$

Hence, the first integral on the RHS above can be rewritten as

$$\begin{aligned} \int \left( \mathrm {H}_{I} \cdot \nabla \right) \,\xi \cdot \nabla \psi _{\varepsilon } \, \mathrm{d} x= & {} \int \nabla \cdot (\xi \otimes \mathrm {H}_I ) \cdot \nabla \psi _\varepsilon \, \mathrm{d} x -\int (\nabla \cdot \mathrm {H}_{I}) \,\xi \cdot \nabla \psi _{\varepsilon } \, \mathrm{d} x\nonumber \\= & {} \int (\nabla \cdot \xi ) \,\mathrm {H}_{I} \cdot \nabla \psi _{\varepsilon } \, \mathrm{d} x +\int (\xi \cdot \nabla ) \,\mathrm {H}_{I} \cdot \nabla \psi _{\varepsilon } \, \mathrm{d} x \nonumber \\&-\int (\nabla \cdot \mathrm {H}_{I}) \,\xi \cdot \nabla \psi _{\varepsilon } \,\mathrm{d} x. \end{aligned}$$

Therefore

$$\begin{aligned} \begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} E_\varepsilon [ Q_\varepsilon | I] +\varepsilon \int |\partial _t Q_\varepsilon |^2\,\mathrm{d} x-\int (\nabla \cdot \xi ) \nabla _q d^F_\varepsilon (Q_\varepsilon ): \partial _t Q_\varepsilon \,\mathrm{d} x \\ =&\int (\nabla \cdot \xi )\, \mathrm {H}_{I} \cdot \nabla \psi _\varepsilon \mathrm{d} x+\int (\xi \cdot \nabla ) \,\mathrm {H}_{I} \cdot \nabla \psi _{\varepsilon } \,\mathrm{d} x -\int (\nabla \cdot \mathrm {H}_{I})\, \xi \cdot \nabla \psi _{\varepsilon } \,\mathrm{d} x\\&+\int \nabla \mathrm {H}_{I}: \left( \xi \otimes \mathrm {n}_{\varepsilon }\right) \left| \nabla \psi _{\varepsilon }\right| \mathrm{d} x-\int \left( \partial _t \xi +\left( \mathrm {H}_{I} \cdot \nabla \right) \xi +\left( \nabla \mathrm {H}_{I}\right) ^{T} \xi \right) \cdot \nabla \psi _\varepsilon \,\mathrm{d} x. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}&E_\varepsilon [ Q_\varepsilon | I] +\varepsilon \int |\partial _t Q_\varepsilon |^2\,\mathrm{d} x-\int (\nabla \cdot \xi ) \nabla _q d^F_\varepsilon (Q_\varepsilon ): \partial _t Q_\varepsilon \,\mathrm{d} x \\&= \int (\nabla \cdot \xi ) \,\mathrm {H}_{I} \cdot \nabla \psi _{\varepsilon }\,\mathrm{d} x +\int (\xi \cdot \nabla )\, \mathrm {H}_{I} \cdot \nabla \psi _{\varepsilon }\,\mathrm{d} x\\&\quad +\int \nabla \mathrm {H}_{I}: (\xi -\mathrm {n}_\varepsilon ) \otimes \mathrm {n}_{\varepsilon }\left| \nabla \psi _{\varepsilon }\right| \mathrm{d} x\\&\quad +\int \mathrm {H}_\varepsilon \cdot \mathrm {H}_I |\nabla Q_\varepsilon |\,\mathrm{d} x + \int \nabla \cdot \mathrm {H}_I \left( \frac{\varepsilon }{2} |\nabla Q_\varepsilon |^2 +\frac{1}{\varepsilon }F_\varepsilon (Q_\varepsilon ) -|\nabla \psi _\varepsilon | \right) \,\mathrm{d} x\\&\quad +\int \nabla \cdot \mathrm {H}_I \left( |\nabla \psi _\varepsilon |-\xi \cdot \nabla \psi _\varepsilon \right) \mathrm{d} x\\&\quad -\int \sum _{i,j=1}^3(\nabla \mathrm {H}_I)_{ij}\, \varepsilon \left( \partial _i Q_\varepsilon : \partial _j Q_\varepsilon \right) \mathrm{d} x +\int \nabla \mathrm {H}_{I}: \mathrm {n}_\varepsilon \otimes \mathrm {n}_{\varepsilon }\left| \nabla \psi _{\varepsilon }\right| \mathrm{d} x\\&\quad -\int \left( \partial _t \xi +\left( \mathrm {H}_{I} \cdot \nabla \right) \xi +\left( \nabla \mathrm {H}_{I}\right) ^{T} \xi \right) \cdot (\mathrm {n}_\varepsilon -\xi ) |\nabla \psi _\varepsilon |\,\mathrm{d} x\\&\quad -\int \Big (\partial _t \xi +\left( \mathrm {H}_{I} \cdot \nabla \right) \xi \Big )\cdot \xi \,|\nabla \psi _\varepsilon | \mathrm{d}x -\int \left( \nabla \mathrm {H}_{I}\right) ^{T} :(\xi \otimes \xi ) |\nabla \psi _\varepsilon |\,\mathrm{d} x. \end{aligned} \end{aligned}$$
(4.32)

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,

$$\begin{aligned} \begin{aligned} -&\int \sum _{i,j=1}^3 (\nabla \mathrm {H}_I)_{ij} \,\varepsilon \left( \partial _i Q_\varepsilon : \partial _j Q_\varepsilon \right) \mathrm{d}x +\int \nabla \mathrm {H}_{I}: \mathrm {n}_\varepsilon \otimes \mathrm {n}_{\varepsilon }\left| \nabla \psi _{\varepsilon }\right| \mathrm{d}x\\&=\int \nabla \mathrm {H}_{I}: \mathrm {n}_\varepsilon \otimes \mathrm {n}_{\varepsilon }\left| \nabla \psi _{\varepsilon }\right| \mathrm{d}x-\varepsilon \int (\nabla \mathrm {H}_I)_{ij}(\Pi _{Q_\varepsilon } \partial _i Q_\varepsilon :\Pi _{Q_\varepsilon } \partial _j Q_\varepsilon ) \mathrm{d}x\\&\quad -\int \sum _{i,j=1}^3(\nabla \mathrm {H}_I)_{ij} \,\varepsilon \Big ((\partial _i Q_\varepsilon -\Pi _{Q_\varepsilon } \partial _i Q_\varepsilon ):(\partial _j Q_\varepsilon -\Pi _{Q_\varepsilon } \partial _j Q_\varepsilon )\Big ) \mathrm{d}x\\&=\int \nabla \mathrm {H}_{I}: \mathrm {n}_\varepsilon \otimes \mathrm {n}_{\varepsilon }\left( |\nabla \psi _\varepsilon |-\varepsilon |\nabla Q_\varepsilon |^2\right) \mathrm{d}x\\&\quad +\varepsilon \int \nabla \mathrm {H}_I:(\mathrm {n}_\varepsilon \otimes \mathrm {n}_\varepsilon ) \ \left( |\nabla Q_\varepsilon |^2-|\Pi _{Q_\varepsilon } \nabla Q_\varepsilon |^2\right) \mathrm{d}x \\&\quad - \int \sum _{i,j=1}^3(\nabla \mathrm {H}_I)_{ij} \,\varepsilon \Big ((\partial _i Q_\varepsilon -\Pi _{Q_\varepsilon } \partial _i Q_\varepsilon ):(\partial _j Q_\varepsilon -\Pi _{Q_\varepsilon } \partial _j Q_\varepsilon )\Big ) \mathrm{d}x =J_\varepsilon ^1. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} E_\varepsilon [ Q_\varepsilon | I]&=-\varepsilon \int |\partial _t Q_\varepsilon |^2 \mathrm{d}x+\int (\nabla \cdot \xi ) \nabla _q d^F_\varepsilon (Q_\varepsilon ): \partial _t Q_\varepsilon \mathrm{d}x\\&\quad + \int (\nabla \cdot \xi ) \,\mathrm {H}_{I} \cdot \nabla \psi _{\varepsilon }\mathrm{d}x+\int \mathrm {H}_\varepsilon \cdot \mathrm {H}_I |\nabla Q_\varepsilon | \mathrm{d}x \nonumber \\&\quad -\int \nabla \mathrm {H}_{I}: (\xi -\mathrm {n}_\varepsilon )^{\otimes 2}\left| \nabla \psi _{\varepsilon }\right| \mathrm{d}x\\&\quad + J_\varepsilon ^1 +\int \left( \nabla \cdot \mathrm {H}_I\right) \Big ( \frac{\varepsilon }{2} |\nabla Q_\varepsilon |^2 +\frac{1}{\varepsilon }F_\varepsilon (Q_\varepsilon ) -|\nabla \psi _\varepsilon | \Big ) \mathrm{d}x\nonumber \\&\quad +\int (\nabla \cdot \mathrm {H}_I) \left( 1-\xi \cdot \mathrm {n}_\varepsilon \right) |\nabla \psi _\varepsilon |\mathrm{d}x+J_\varepsilon ^2. \end{aligned} \end{aligned}$$
(4.33)

Now we complete squares for the first four terms on the RHS of (4.33): Reordering terms, we have

$$\begin{aligned} -&\varepsilon |\partial _t Q_\varepsilon |^2+ (\nabla \cdot \xi ) \nabla _q d^F_\varepsilon (Q_\varepsilon ): \partial _t Q_\varepsilon + (\nabla \cdot \xi ) \mathrm {H}_{I} \cdot \nabla \psi _{\varepsilon }+ \mathrm {H}_\varepsilon \cdot \mathrm {H}_I |\nabla Q_\varepsilon | \nonumber \\&= -\frac{1}{2\varepsilon } \Big ( |\varepsilon \partial _t Q_\varepsilon |^2 -2(\nabla \cdot \xi ) \nabla _q d^F_\varepsilon (Q_\varepsilon ): \varepsilon \partial _t Q_\varepsilon +(\nabla \cdot \xi )^2 | \nabla _q d^F_\varepsilon (Q_\varepsilon )|^2 \Big ) \nonumber \\&\quad - \frac{1}{2\varepsilon } |\varepsilon \partial _t Q_\varepsilon |^2 + \frac{1}{2\varepsilon }(\nabla \cdot \xi )^2 | \nabla _q d^F_\varepsilon (Q_\varepsilon )|^2 + (\nabla \cdot \xi ) \mathrm {H}_{I} \cdot \nabla \psi _{\varepsilon } \nonumber \\ {}&\quad - \frac{1}{2\varepsilon } \Big ( |\mathrm {H}_\varepsilon |^2 - 2\mathrm {H}_\varepsilon \cdot \varepsilon |\nabla Q_\varepsilon | \mathrm {H}_I + \varepsilon ^2 |\nabla Q_\varepsilon |^2 |\mathrm {H}_I|^2\Big ) \nonumber \\&\quad + \frac{1}{2\varepsilon } \Big ( |\mathrm {H}_\varepsilon |^2 + \varepsilon ^2 |\nabla Q_\varepsilon |^2 |\mathrm {H}_I|^2\Big ) \nonumber \\ {}&= -\frac{1}{2\varepsilon } \Big |\varepsilon \partial _t Q_\varepsilon - (\nabla \cdot \xi ) \nabla _q d^F_\varepsilon (Q_\varepsilon ) \Big |^2 - \frac{1}{2\varepsilon } \Big |\mathrm {H}_\varepsilon - \varepsilon |\nabla Q_\varepsilon | \mathrm {H}_I \Big |^2 \nonumber \\&\quad - \frac{1}{2\varepsilon } |\varepsilon \partial _t Q_\varepsilon |^2 +\frac{1}{2\varepsilon } |\mathrm {H}_\varepsilon |^2 \nonumber \\ {}&\quad + \frac{1}{2\varepsilon } \Big ( (\nabla \cdot \xi )^2 |\nabla _q d^F_\varepsilon (Q_\varepsilon )|^2 + 2\varepsilon (\nabla \cdot \xi ) \nabla \psi _\varepsilon \cdot \mathrm {H}_I + |\varepsilon \Pi _{Q_\varepsilon }\nabla Q_\varepsilon |^2 |\mathrm {H}_I|^2 \Big ) \nonumber \\ {}&\quad +\frac{\varepsilon }{2} \left( |\nabla Q_\varepsilon |^2- | \Pi _{Q_\varepsilon }\nabla Q_\varepsilon |^2\right) |\mathrm {H}_I|^2. \end{aligned}$$
(5.1)

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). \(\square \)

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

$$\begin{aligned}&{{\text {ess~sup}}}_{t\in [0,T]} \int _\Omega \left( \left| \nabla Q_\varepsilon -\Pi _{Q_\varepsilon }\nabla Q_\varepsilon \right| ^2 \right) \mathrm{d}x\nonumber \\&\quad + \int _0^T\int _\Omega \left( \left| \partial _t Q_\varepsilon -\Pi _{Q_\varepsilon }\partial _t Q_\varepsilon \right| ^2 \right) \, \mathrm{d}x\mathrm{d}t\leqq e^{(1+T)C(I_0)}. \end{aligned}$$
(5.2)

Moreover, for any fixed \(\delta \in (0, \delta _I)\), if holds that

$$\begin{aligned} \mathrm{{ess~sup}}_{t\in [0,T]}\int _{\Omega ^\pm (t)\backslash I_t(\delta )}\left( |\nabla Q_\varepsilon |^2+\frac{{F(Q_\varepsilon )+\varepsilon ^3}}{\varepsilon ^2}\right) \mathrm{d}x\leqq & {} \delta ^{-2}e^{(1+T)C(I_0)}, \qquad \quad \end{aligned}$$
(5.3)
$$\begin{aligned} \int _0^T\int _{\Omega ^\pm (t)\backslash I_t(\delta )} |\partial _t Q_\varepsilon |^2\,\mathrm{d}x \mathrm{d}t\leqq & {} \delta ^{-2}e^{(1+T)C(I_0)}. \end{aligned}$$
(5.4)

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

$$\begin{aligned}&\mathrm{{ess~sup}}_{t\in [0,T]} \frac{1}{\varepsilon }E_\varepsilon [ Q_\varepsilon | I] (t)+\frac{1}{\varepsilon ^2}\int _0^T\int _\Omega \left| \varepsilon \partial _t Q_\varepsilon -\nabla _q d^F_\varepsilon (Q_\varepsilon ) (\nabla \cdot \xi ) \right| ^2\, \mathrm{d}x\mathrm{d}t \nonumber \\&+\frac{1}{\varepsilon ^2}\int _0^T\int _\Omega \left( \varepsilon ^2 \left| \partial _t Q_\varepsilon \right| ^2-|\mathrm {H}_\varepsilon |^2+ \left| \mathrm {H}_\varepsilon -\varepsilon \mathrm {H}_I |\nabla Q_\varepsilon \right| |^2\right) \, \mathrm{d}x\mathrm{d}t\nonumber \\&\qquad \leqq \frac{e^{(1+T)C(I_0)}}{\varepsilon }E_\varepsilon [Q_\varepsilon | I](0) \leqq e^{(1+T)C(I_0)}. \end{aligned}$$
(5.5)

On the other hand, using the orthogonal projection (4.10), we obtain

$$\begin{aligned} \left| \varepsilon \partial _t Q_\varepsilon -\nabla _q d^F_\varepsilon (Q_\varepsilon ) (\nabla \cdot \xi ) \right| ^2&=\left| \varepsilon \partial _t Q_\varepsilon -\varepsilon \Pi _{Q_\varepsilon } \partial _t Q_\varepsilon \right| ^2\\&\quad +\left| \varepsilon \Pi _{Q_\varepsilon } \partial _t Q_\varepsilon -\nabla _q d^F_\varepsilon (Q_\varepsilon ) (\nabla \cdot \xi ) \right| ^2. \end{aligned}$$

This, together with (5.4), yields

$$\begin{aligned}&\frac{1}{\varepsilon ^2}\int _0^T\int _\Omega \left| \varepsilon \partial _t Q_\varepsilon -\varepsilon \Pi _{Q_\varepsilon } \partial _t Q_\varepsilon \right| ^2\nonumber \\&\quad +\frac{1}{\varepsilon ^2}\int _0^T \int _\Omega \left| \varepsilon \Pi _{Q_\varepsilon } \partial _t Q_\varepsilon -\nabla _q d^F_\varepsilon (Q_\varepsilon ) (\nabla \cdot \xi ) \right| ^2 \leqq e^{(1+T)C(I_0)}. \end{aligned}$$
(5.6)

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

$$\begin{aligned} \frac{1}{\varepsilon ^2}\int _0^T\int _\Omega \left( \varepsilon ^2 \left| \partial _t Q_\varepsilon \right| ^2-|\mathrm {H}_\varepsilon |^2+ \left| \mathrm {H}_\varepsilon -\varepsilon \mathrm {H}_I |\nabla Q_\varepsilon |\right| ^2\right) \, \mathrm{d}x\mathrm{d}t\leqq e^{(1+T)C(I_0)}.\nonumber \\ \end{aligned}$$
(5.7)

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

$$\begin{aligned}&~\varepsilon ^2 \left| \partial _t Q_\varepsilon \right| ^2-|\mathrm {H}_\varepsilon |^2+ \left| \mathrm {H}_\varepsilon -\varepsilon \mathrm {H}_I |\nabla Q_\varepsilon |\right| ^2\\ =&~\varepsilon ^2 \left| \partial _t Q_\varepsilon \right| ^2+\varepsilon ^2 |\mathrm {H}_I|^2 |\nabla Q_\varepsilon |^2+2\varepsilon ^2 (\mathrm {H}_I\cdot \nabla ) Q_\varepsilon :\partial _t Q_\varepsilon \\ \geqq&~\varepsilon ^2 \left| \partial _t Q_\varepsilon \right| ^2+\varepsilon ^2 |(\mathrm {H}_I\cdot \nabla ) Q_\varepsilon |^2+2\varepsilon ^2 (\mathrm {H}_I\cdot \nabla ) Q_\varepsilon :\partial _t Q_\varepsilon \\ =&~\varepsilon ^2|\partial _t Q_\varepsilon +(\mathrm {H}_I \cdot \nabla ) Q_\varepsilon |^2. \end{aligned}$$

This implies

$$\begin{aligned} \int _0^T\int _\Omega |\partial _t Q_\varepsilon +(\mathrm {H}_I \cdot \nabla ) Q_\varepsilon |^2\, \mathrm{d}x \mathrm{d}t\leqq e^{(1+T)C(I_0)}, \end{aligned}$$
(5.8a)

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

$$\begin{aligned}&\left[ \partial _t Q_{\varepsilon _k},Q_{\varepsilon _k} \right] =\left[ \partial _t Q_{\varepsilon _k}-\Pi _{Q_{\varepsilon _k}}\partial _t Q_{\varepsilon _k},Q_{\varepsilon _k} \right] \nonumber \\&\quad \xrightarrow {k\rightarrow \infty } {\bar{S}}_0(x,t)~\text {weakly in}~ L^2(0,T;L^2(\Omega )), \end{aligned}$$
(5.8b)
$$\begin{aligned}&\left[ \partial _i Q_{\varepsilon _k},Q_{\varepsilon _k} \right] =\left[ \partial _i Q_{\varepsilon _k}-\Pi _{Q_{\varepsilon _k}}\partial _i Q_{\varepsilon _k},Q_{\varepsilon _k} \right] \nonumber \\&\quad \xrightarrow {k\rightarrow \infty } {\bar{S}}_i(x,t)~\text {weakly-star in}~ L^\infty (0,T;L^2(\Omega )) \end{aligned}$$
(5.9a)

for \(1\leqq i\leqq d\). Moreover,

$$\begin{aligned} \partial _t Q_{\varepsilon _k}\xrightarrow { k\rightarrow \infty } \partial _t Q&,~\text {weakly in}~ L^2(0,T;L^2_{loc}(\Omega ^\pm (t))), \end{aligned}$$
(5.9b)
$$\begin{aligned} \nabla Q_{\varepsilon _k}\xrightarrow {k\rightarrow \infty } \nabla Q&,~\text {weakly in}~ L^\infty (0,T;L^2_{loc}(\Omega ^\pm (t))), \end{aligned}$$
(5.9c)
$$\begin{aligned} Q_{\varepsilon _k}\xrightarrow {k\rightarrow \infty } Q&,~\text {strongly in}~ C([0,T];L^2_{loc}(\Omega ^\pm (t))), \end{aligned}$$
(5.10)

where \(Q=Q(x,t)\) is represented as

$$\begin{aligned}&Q (x,t)=s^\pm \left( \mathrm {u}(x,t) \otimes \mathrm {u}(x,t)-\frac{1}{3}I_3\right) ~\text {a.e.}~(x,t)\in \Omega ^\pm _T\end{aligned}$$
(5.11)

for some unit vector field

$$\begin{aligned} \mathrm {u}\in L^\infty (0,T;H^1(\Omega ^+(t);{{\mathbb {S}}^2}))&\cap&H^1(0,T;L^2(\Omega ^+(t);{{\mathbb {S}}^2}))\nonumber \\&\cap&C([0,T];L^2(\Omega ^+(t);{{\mathbb {S}}^2})). \end{aligned}$$
(5.12)

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

$$\begin{aligned}&\frac{\partial d^F_\varepsilon (Q)}{\partial Q}=\sum _{k=1}^3\frac{\partial g}{\partial \lambda _k }\frac{\partial \lambda _k }{\partial Q}=\sum _{k=1}^3\frac{\partial g}{\partial \lambda _k }\mathrm {n}_k(Q)\otimes \mathrm {n}_k(Q),~\nonumber \\&\quad \text {in a neighborhood of}~Q_0. \end{aligned}$$
(5.13)

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

$$\begin{aligned} \left[ \nabla _q d^F_\varepsilon (Q), Q\right] =0, \end{aligned}$$
(5.14)

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

$$\begin{aligned} {[}\Pi _{Q_\varepsilon }\partial _t Q_\varepsilon (x,t),Q_\varepsilon (x,t)]= & {} 0,\nonumber \\ {[}\Pi _{Q_\varepsilon }\partial _i Q_\varepsilon (x,t),Q_\varepsilon (x,t)]= & {} 0~a.e. ~(x,t)\in \Omega _T \end{aligned}$$
(5.15)

for \(1\leqq i\leqq d\). This together with (3.29) and (5.1) implies

$$\begin{aligned}&\Vert \left[ \partial _t Q_\varepsilon ,Q_\varepsilon \right] \Vert _{L^2(0,T;L^2(\Omega ))}+\Vert \left[ \nabla Q_\varepsilon ,Q_\varepsilon \right] \Vert _{L^\infty (0,T;L^2(\Omega ))}\nonumber \\&=\Vert \left[ \partial _t Q_\varepsilon -\Pi _{Q_\varepsilon }\partial _t Q_\varepsilon ,Q_\varepsilon \right] \Vert _{L^2(0,T;L^2(\Omega ))}\nonumber \\&\quad +\Vert \left[ \nabla Q_\varepsilon -\Pi _{Q_\varepsilon }\nabla Q_\varepsilon ,Q_\varepsilon \right] \Vert _{L^\infty (0,T;L^2(\Omega ))}\leqq C \end{aligned}$$
(5.16a)

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

$$\begin{aligned} \partial _t Q_{\varepsilon _k}\xrightarrow { k\rightarrow \infty } \partial _t {\bar{Q}}_{\delta }&,~\text {weakly in}~ L^2(0,T;L^2(\Omega ^\pm (t)\backslash I_t(\delta ))), \end{aligned}$$
(5.16b)
$$\begin{aligned} \nabla Q_{\varepsilon _k}\xrightarrow {k\rightarrow \infty } \nabla {\bar{Q}}_{\delta }&,~\text {weakly-star in}~ L^\infty (0,T;L^2(\Omega ^\pm (t)\backslash I_t(\delta ))), \end{aligned}$$
(5.16c)
$$\begin{aligned} Q_{\varepsilon _k}\xrightarrow {k\rightarrow \infty } {\bar{Q}}_{\delta }&,~\text {weakly-star in}~ L^\infty (\Omega \times (0,T)), \end{aligned}$$
(5.16d)
$$\begin{aligned} Q_{\varepsilon _k}\xrightarrow {k\rightarrow \infty } {\bar{Q}}_{\delta }&,~\text {strongly in}~ C([0,T];L^2(\Omega ^\pm (t)\backslash I_t(\delta ))). \end{aligned}$$
(5.17)

By a diagonal argument, we infer there exists

$$\begin{aligned} Q\in L^2(0,T;H^1_{loc}(\Omega ^\pm (t)))\cap L^\infty (\Omega ^\pm ), ~\text {with}~\partial _t Q\in L^2(0,T;L^2_{loc}(\Omega ^\pm (t)))\nonumber \\ \end{aligned}$$
(5.18)

such that the convergence (5.2) as well as

$$\begin{aligned} Q(x,t)={\bar{Q}}_\delta (x,t) ~\text { in }~L^\infty (0,T;H^1( \Omega ^\pm (t)\backslash I_t(\delta )) \end{aligned}$$

hold for every \(\delta >0\) and every \(t\in [0,T]\). Moreover, by (5.17), the interpolation theory and (5.16c), we have

$$\begin{aligned} Q\in C([0,T];L^2(\Omega ^\pm (t))\cap L^\infty (\Omega \times (0,T)). \end{aligned}$$

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

$$\begin{aligned} F(Q)\in C([0,T];L^2(\Omega ^\pm (t)). \end{aligned}$$

We use (5.9c), (5.2), and Fatou’s lemma to deduce that

$$\begin{aligned} F(Q(x,t))=0,~\forall t\in [0,T]~\text { and a.e. in } x\in \Omega ^\pm (t). \end{aligned}$$
(5.19)

This, together with (1.10), implies

$$\begin{aligned} |Q|(x,t)\in \{0,s^+\sqrt{\tfrac{2}{3}}\},~\forall t\in [0,T]~\text { and a.e. in } x\in \Omega ^\pm (t). \end{aligned}$$
(5.20)

By taking the \(L^2\)-norm, we obtain two continuous functions:

$$\begin{aligned}f^\pm (t){:=}\Vert Q(\cdot ,t)\Vert _{L^2(\Omega ^\pm (t))}\in C([0,T];\{0,s^+\sqrt{\tfrac{2}{3}|\Omega ^\pm (t)|}\}).\end{aligned}$$

On the other hand, by the choice of the initial condition (2.21) and the convergence (5.16d), we deduce that

$$\begin{aligned} Q(x,0)= {\mathbf {1}}_{\Omega ^+(0)}s^+ \ \left( \mathrm {u}^{in}(x)\otimes \mathrm {u}^{in}(x)-\frac{1}{3} I_3\right) ,~\text {a.e.~in}~\Omega ^\pm (0)\backslash I_0(\delta ) \end{aligned}$$

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

$$\begin{aligned} f^+(t)=s^+\sqrt{\tfrac{2}{3}|\Omega ^+(t)|},f^-(t)=0,\quad \forall t\in [0,T]. \end{aligned}$$

This, together with (5.19), implies

$$\begin{aligned} Q(x,t)=0,~&\forall t\in [0,T]~\text { and a.e. in } x\in \Omega ^-(t), \end{aligned}$$
(5.21)
$$\begin{aligned} Q(x,t)\in {\mathcal {N}},~&\forall t\in [0,T]~\text { and a.e. in } x\in \Omega ^+(t), \end{aligned}$$
(5.22)

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

$$\begin{aligned} \mathrm {u}\in L^\infty (0,T;H^1_{loc}(\Omega ^+(t);{{\mathbb {S}}^2})) \text { with }\partial _t \mathrm {u} \in L^2(0,T;L^2_{loc}(\Omega ^+(t);{{\mathbb {S}}^2})). \end{aligned}$$
(5.23)

It remains to improve the integrability of \(\nabla _{x,t} \mathrm {u}\). To this end, we choose a sequence

$$\begin{aligned} \psi _\ell (x,t)\in C_c^\infty (\Omega ^+_T)~\text { such that }~\psi _\ell (x,t)\xrightarrow {\ell \rightarrow \infty } {\mathbf {1}}_{\Omega ^+_T}(x,t). \end{aligned}$$
(5.24)

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

$$\begin{aligned} \psi _\ell {\bar{S}}_i= \psi _\ell \left[ \partial _i Q,Q \right] =s_+^2\psi _\ell \left( \partial _i \mathrm {u}\otimes \mathrm {u}-\mathrm {u}\otimes \partial _i \mathrm {u}\right) ,~0\leqq i\leqq d, \end{aligned}$$
(5.25)

where \(\partial _0{:=}\partial _t\). Note that for each fixed \(i\in \{0,\ldots ,3\}\),

$$\begin{aligned} \partial _i \mathrm {u}\otimes \mathrm {u}-\mathrm {u}\otimes \partial _i \mathrm {u}=\begin{pmatrix} 0&{} (\partial _i \mathrm {u}\wedge \mathrm {u})_3 &{}-(\partial _i \mathrm {u}\wedge \mathrm {u})_2\\ -(\partial _i \mathrm {u}\wedge \mathrm {u})_3 &{} 0 &{} (\partial _i \mathrm {u}\wedge \mathrm {u})_1\\ (\partial _i \mathrm {u}\wedge \mathrm {u})_2 &{} -(\partial _i \mathrm {u}\wedge \mathrm {u})_1 &{} 0 \end{pmatrix},\end{aligned}$$
(5.26a)

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

$$\begin{aligned}&\partial _t \mathrm {u}\wedge \mathrm {u}\in L^\infty (0,T;L^2(\Omega ^+(t))),\\&\partial _i \mathrm {u}\wedge \mathrm {u}\in L^2(0,T;L^2(\Omega ^+(t))), ~\text { for } i\in \{1,\ldots ,d\}. \end{aligned}$$
(5.26b)

Retaining that \(\mathrm {u}\) maps into \({{\mathbb {S}}^2}\), we deduce

$$\begin{aligned} |\partial _t \mathrm {u}|^2 =|\partial _t \mathrm {u}\wedge \mathrm {u}|^2,\qquad |\partial _i \mathrm {u}|^2=|\partial _i \mathrm {u}\wedge \mathrm {u}|^2~a.e.~\text {in}~\Omega ^+_T,~1\leqq i\leqq d,\end{aligned}$$

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

$$\begin{aligned} \Phi (x,t)=\begin{pmatrix} 0&{}\varphi _3 &{}-\varphi _2\\ -\varphi _3 &{} 0 &{} \varphi _1\\ \varphi _2 &{} -\varphi _1 &{} 0 \end{pmatrix} \end{aligned}$$
(5.27)

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

$$\begin{aligned} \int _{\Omega _T}\left[ \partial _t Q_{\varepsilon _k}, Q_{\varepsilon _k}\right] :\Phi \, \mathrm{d}x\mathrm{d}t + \int _{\Omega _T} \sum _{j=1}^3[\partial _j Q_{\varepsilon _k}, Q_{\varepsilon _k}]:\partial _j \Phi \, \mathrm{d}x\mathrm{d}t =0. \end{aligned}$$
(5.28)

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

$$\begin{aligned}&\sum _{\pm }\int _0^T\int _{\Omega ^\pm (t)\backslash I_t(\delta )}\left( \left[ \partial _t Q_{\varepsilon _k}, Q_{\varepsilon _k}\right] :\Phi + \sum _{j=1}^3[\partial _j Q_{\varepsilon _k}, Q_{\varepsilon _k}]:\partial _j \Phi \, \right) \, \mathrm{d}x \mathrm{d}t\nonumber \\&+\int _0^T\int _{ I_t(\delta )}\left( \left[ \partial _t Q_{\varepsilon _k}, Q_{\varepsilon _k}\right] :\Phi + \sum _{j=1}^3[\partial _j Q_{\varepsilon _k}, Q_{\varepsilon _k}]:\partial _j \Phi \, \right) \, \mathrm{d}x\mathrm{d}t=0. \end{aligned}$$
(5.29)

Using (5.2), (5.2) and (5.10), we can pass \(k\rightarrow \infty \) and yield

$$\begin{aligned}&\int _0^T\int _{\Omega ^+(t)\backslash I_t(\delta )}\left( \left[ \partial _t Q, Q\right] :\Phi + \sum _{j=1}^3[\partial _j Q, Q]:\partial _j \Phi \right) \, \mathrm{d}x\mathrm{d}t\nonumber \\&\qquad +\int _0^T\int _{ I_t(\delta )}\left( {\bar{S}}_0 :\Phi + \sum _{j=1}^3 {\bar{S}}_j:\partial _j \Phi \right) \, \mathrm{d}x\mathrm{d}t=0. \end{aligned}$$
(5.30)

By \(|\mathrm {u}|=1\) a.e., (5.10), (5.27) and (5.25), we obtain the following identities :

$$\begin{aligned}{}[\partial _t Q, Q]:\Phi&=s_+^2\left( \partial _t \mathrm {u}\otimes \mathrm {u}-\mathrm {u}\otimes \partial _t \mathrm {u}\right) : \Phi =2s_+^2\partial _t \mathrm {u}\wedge \mathrm {u}\cdot \varphi \\ [\partial _j Q, Q]:\partial _j\Phi&=s_+^2\left( \partial _j \mathrm {u}\otimes \mathrm {u}-\mathrm {u}\otimes \partial _j \mathrm {u}\right) : \partial _j\Phi =2s_+^2\partial _j \mathrm {u}\wedge \mathrm {u}\cdot \partial _j \varphi \end{aligned}$$

Thus we obtain

$$\begin{aligned}&2s_+^2 \int _0^T\int _{\Omega ^+(t)\backslash I_t(\delta )}\left( \partial _t \mathrm {u}\wedge \mathrm {u}\cdot \varphi +\sum _{j=1}^3(\partial _j \mathrm {u}\wedge \mathrm {u})\cdot \partial _j \varphi \right) \, \mathrm{d}x\mathrm{d}t\nonumber \\ {}&\qquad + \int _0^T\int _{ I_t(\delta )}\left( {\bar{S}}_0 :\Phi + \sum _{j=1}^3{\bar{S}}_j:\partial _j \Phi \right) \, \mathrm{d}x\mathrm{d}t=0. \end{aligned}$$
(5.31)

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

$$\begin{aligned} \int _0^T\int _{\Omega ^+(t)}\partial _t \mathrm {u}\wedge \mathrm {u}\cdot \varphi \, \mathrm{d}x\mathrm{d}t +\int _0^T\int _{\Omega ^+(t)}\sum _{j=1}^3(\partial _j \mathrm {u}\wedge \mathrm {u})\cdot \partial _j \varphi \, \mathrm{d}x\mathrm{d}t =0.\end{aligned}$$
(5.32)

This concludes the proof of Theorem 2.1. \(\square \)