Abstract
We study the effective geometric motions of an anisotropic Ginzburg–Landau equation with a small parameter \(\varepsilon >0\) which characterizes the width of the transition layer. For well-prepared initial datum, we show that as \(\varepsilon \) tends to zero the solutions will develop a sharp interface limit which evolves under mean curvature flow. The bulk limits of the solutions correspond to a vector field \({\textbf{u}}(x,t)\) which is of unit length on one side of the interface, and is zero on the other side. The proof combines the modulated energy method and weak convergence methods. In particular, by a (boundary) blow-up argument we show that \({\textbf{u}}\) must be tangent to the sharp interface. Moreover, it solves a geometric evolution equation for the Oseen–Frank model in liquid crystals.
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1 Introduction
In the study of liquid crystals one often encounters elastic energies with anisotropy, i.e. energies with distinct coefficients multiplying the square of the divergence and the curl of the order parameters. Typical examples involve the Oseen–Frank model [30], Ericksen’s model [43, 44] and the Landau–De Gennes model [5]. From a microscopic point of view, the anisotropy of these models can be interpreted as excluded volume potential of molecular interaction, cf. [29]. Anisotropic models also arise in the theory of superconductivity, cf. [10]. The anisotropy brings various new challenges to the studies of both variational problems and their gradient flows of the aforementioned models. In contrast to the convergence analysis of isotropic models, i.e. the (scalar) Allen–Cahn equations (cf. [6, 19, 34, 47, 48, 50, 53, 54]), the powerful analytic tools such as maximum principle and monotonicity formula are not readily established for anisotropic ones.
The attempt of this work is to study an anisotropic system modeling the isotropic-nematic phase transition of a liquid crystal droplet. Let \(d\in \{2,3\}\) be the dimension of the physical domain \(\Omega \) with \(C^3\) boundary \(\partial \Omega \). We consider the anisotropic Ginzburg–Landau type energy
Here \({\textbf{u}}=(u_1,u_2,u_3):\Omega \subset {\mathbb {R}}^d\mapsto {{\mathbb {R}}^3}\) is the order parameter describing the state of the system. The function \(F({\textbf{u}})\) is a double equal-well potential which permits the isotropic-nematic phase transition. More precisely, it attains its global minimum value 0 at \(\{ 0\}\cup {\mathbb {S}}^2\). An example of F is the Chern–Simons–Higgs model \(F({\textbf{u}})= |{\textbf{u}}|^2(1-|{\textbf{u}}|^2)^2\). See for instance [31, 36] for the physics and [9, 27, 28] for the mathematical analysis of related variational problems. The parameter \(\varepsilon >0 \) denotes the relative intensity of elastic and bulk energy, which is usually quite small. The parameter \(\mu >0\) is material dependent which measures the degree of anisotropy.
The energy (1.1) is a simplified case of the full Landau–De Gennes energy (cf. [35, 45]). The variational investigations of the isotropic-nematic phase transition involving (1.1) were first done by Golovaty, Novack, Sternberg and Venkatraman [27, 28] in the static case in 2D. The present paper is concerned with the \(L^2\)-gradient flow of (1.1), i.e. the following system.
where \( D F ({\textbf{u}})\) is the gradient of \(F({\textbf{u}})\) with respect to \({\textbf{u}}\). We shall study the small \(\varepsilon \)-asymptotics of this system with well-prepared initial datum \({\textbf{u}}_\varepsilon ^{in}\) that undergoes a sharp transition across a co-dimensional one interface \(I_0\subset {\mathbb {R}}^d\). We shall show that the energy density \(\frac{\varepsilon }{2} |\nabla {\textbf{u}}_\varepsilon |^2+\frac{1}{\varepsilon }F({\textbf{u}}_\varepsilon )\) will be concentrated on a mean curvature flow \(I:=\bigcup _{t\geqslant 0} I_t\times \{t\}\) starting from \(I_0\), namely
where \({\mathcal {H}}^{d-1}\) is the \((d-1)\) dimensional Hausdorff measure, and \(\sigma \) is a positive constant depending on F. Moreover, we shall derive bulk limit \({\textbf{u}}:=\lim _{\varepsilon \rightarrow 0}{\textbf{u}}_\varepsilon \) away from \(I_t\) and its boundary condition on \(I_t\).
\(I_t\) is the interface, \(\Omega _t^+\) is the nematic phase and \(\Omega _t^-\) is the isotropic phase
System (1.2a) is a vectorial and anisotropic generalization of the scalar parabolic Allen–Cahn equation. In the scalar case, there have been many developments on its co-dimensional one limit to the (two-phase) mean curvature flow during the last two decades. Here we mention two classes of results and postpone the discussions of some others in the sequel. One is the convergence to a Brakke’s flow by Ilmanen [34] using a version of Huisken’s monotonicity formula [32] and tools from geometric measure theory. See also [11, 33, 47, 48, 50, 54] and the references therein for further renovations. Despite of its energetic nature, a major difficulty of such an approach is the control of the so-called discrepancy measure, and almost all existing literatures using this approach rely crucially on a version of Modica’s maximum principle [46]. However, it is not clear whether Modica’s maximum principle holds for elliptic/parabolic systems. Another approach, which relies more on parabolic comparison principle, is the global in time convergence towards the viscosity solution built by Chen–Giga–Goto [13] and independently by Evans–Spruck [20]. Such an approach has been implemented by Evans–Soner–Souganidis [19]. One can also refer to [6, 53] and the references therein for further discussions. These two approaches both give global in time (weak) convergences to weakly defined solutions of the mean curvature flow (up to their extinction times). However, as their technics involve parabolic maximum principle in one way or another, it is not clear how to use them to attack vectorial models in general. It is worth mentioning that for radially symmetric initial datum, Bronsard–Stoth [8] have obtained global in time convergence to the mean curvature flow of planar circles.
To the best of our knowledge, there are mainly two approaches to rigorously justify the convergence of the vectorial Allen–Cahn equations, both assuming that the limiting interface motion has a (local in time) classical solution. Compared with the aforementioned methods, which lead to global in time (weak) convergence, they have quite different natures. The first approach is the asymptotic expansion technics developed by De Mottoni–Schatzman [15] and by Alikakos–Bates–Chen [1]. It has been used recently by Fei–Wang–Zhang–Zhang [22] to study the isotropic-nematic phase transition in liquid crystals, and by Fei–Lin–Wang–Zhang [21] to study matrix-valued Allen–Cahn equations.
The second approach, which also assumes a classical solution of the limiting interface motion (but not the limiting flows in the bulk regions), is the modulated energy method developed by Fischer–Laux–Simon [24]. Such a method is motivated by Jerrard–Smets [37] and Fischer–Hensel [23], and has been generalized to a matrix–valued model by Laux–Liu [40].
In the present work, we shall use the methods employed in [24, 40] to derive the energy convergence (1.3) and the bulk limit \({\textbf{u}}=\lim _{\varepsilon _k\rightarrow 0}{\textbf{u}}_{\varepsilon _k}\) by establishing two modulated energy inequalities. Moreover, the derivation of the anchoring boundary condition of \({\textbf{u}}\) (see (1.18c) below) uses a blow-up argument, which is inspired by a recent work of Lin–Wang [43]. There the authors have studied isotropic-nematic phase transitions in the static case based on an anisotropic Ericksen’s model.
To state the main result, we assume that
starting from a \((d-1)\)-dimensional submanifold \(I_0\subset \Omega \). Here a \((d-1)\)-submanifold refers to an embedded closed smooth surface when \(d=3\) and curve when \(d=2\).
Let \(\Omega ^+_t\) be the domain enclosed by \(I_t\), and \(d_I(x,t)\) be the signed-distance from x to \(I_t\) which takes negative values in \(\Omega ^-_t\), and positive values in \(\Omega ^+_t=\Omega \backslash \overline{\Omega ^-_t}\). Equivalently,
For \(\delta >0\), the (open) \(\delta \)-neighborhood of \(I_t\) is denoted by
Let \(\delta _0\in (0,1)\) be a sufficiently small number so that the nearest point projection
is smooth for any \(t\in [0,T]\), and that the interface (1.4) stays at least \(4\delta _0\) distant away from the physical boundary \(\partial \Omega \). A further description of the geometry can be found in Sect. 2.2 or in [12].
The first step to study the singular limit of (1.2) is to construct a modulated energy which encodes a distance between the energy in (1.1) and an energy corresponding to the moving interface \(I_t\) in (1.4). Following [23, 24, 37], we define an extension of the inward normal vector \({\textbf{n}}(\cdot ,t)\) of \(I_t\) by
where \(\phi \in C_c^2( {\mathbb {R}}; [0,1]) \) is an appropriate cut-off function (see (2.11) below for its precise definition). Now we introduce
where \(\psi _\varepsilon \) is defined by
We shall work with a class of potentials \(F({\textbf{u}})\) under standard assumptions (see e.g. [11, 34]). That is,
where f is a double equal-well potential, namely,
Moreover, the following structural assumptions on f are made:
After an appropriate modification for large |s|, the function \(g(s)=|s||s^2-1|\), which corresponds to the Chern–Simons–Higgs potential, satisfies (1.11).
To control the bulk errors, we need another modulated energy:
Here \(\chi (\cdot ,t):={\textbf{1}}_{\Omega _t^+}-{\textbf{1}}_{\Omega _t^-}\), \(h^\pm \) denote the positive/negative parts of a function h respectively, and \(\eta \) is a truncation of the identity function defined by
Note that \(( \eta \circ d_I)~\chi \geqslant 0\) in \(\Omega \) due to our convention on the signed-distance function, and thus the two integrands in (1.12) are both non-negative. We refer the readers to the proof of Theorem 4.1 below for more details on the positivity of (1.12).
Now we state the main result of this work:
Theorem 1.1
Let \(d\in \{2,3\}\), and the assumptions (1.10) and (1.11) be in place. Assume that the moving interface I in (1.4) evolves under mean curvature flow, and the initial datum of (1.2) satisfies the following conditions:
where \(c_1>0\) is independent of \(\varepsilon \). Then there exists \(C_1>0\) independent of \(\varepsilon \) such that
Moreover, up to extraction of a subsequence \(\varepsilon _k\downarrow 0\),
where \({\textbf{u}}\) satisfies the following properties:
Among the conditions in (1.14), the crucial one is (1.14c), which is used to obtain the inequalities in Theorem 3.1 and in Theorem 4.1 below. To construct an initial datum satisfying (1.14), we need the following result.
Proposition 1.1
Let \(I_0\subset \Omega \) be a \((d-1)\)-dimensional submanifold. For any vector field
there exists \({\textbf{u}}_\varepsilon ^{in} \in W^{1,2}_0(\Omega )\cap L^\infty (\Omega )\) such that
and (1.14) holds for a constant \(c_1>0\) which only depends on \(I_0\) and \(\Vert {\textbf{u}}^{in}\Vert _{W^{1,2}(\Omega )}\).
We comment on the conditions in (1.19). When \(d=3\), \(I_0\) is a smooth closed surface in \(\Omega \). Due to topological obstructions, a vector field satisfying (1.19) is usually not smooth. For instance, when \(I_0\) is diffeomorphic to a 2-sphere, due to the hairy ball theorem, \({\textbf{u}}^{in}|_{I_0}\) must have (at least) one pole. One example of such a pole, which is often encountered in the theory of liquid crystal, is given by the hedgehog profile. Locally the tangent vector field near such a pole is \(C^1\)-equivalent to the mapping \({\textbf{h}}(x)=x/|x|: B_1\cap {\mathbb {R}}^2\rightarrow {\mathbb {S}}^1\). Note that \({\textbf{h}}\in W^{\frac{1}{2},2}(B_1\cap {\mathbb {R}}^2)\) but \({\textbf{h}}\notin W^{1,2}(B_1\cap {\mathbb {R}}^2)\). When \(d=2\), there are fewer constraints to arrange a vector field \({\textbf{f}}:I_0\mapsto {\mathbb {S}}^2\subset {\mathbb {R}}^3\) that is orthogonal to the planar curve \(I_0\subset {\mathbb {R}}^2\times \{0\}\). In general, using the extension lemma of Hardt–Lin (cf. [42, Lemma 2.2.10]), any tangent vector field \({\textbf{f}}\in W^{\frac{1}{2}, 2}(I_0;{\mathbb {S}}^2)\) has an extension \({\textbf{u}}^{in}\) satisfying (1.19).
An immediate consequence of Theorem 1.1 is the convergence in (1.3). Indeed, it follows from (1.15) and (2.26b) below that \(\int _{\Omega } \frac{\varepsilon }{2} \mu |{\text {div}}{\textbf{u}}|^2\,dx\xrightarrow {\varepsilon \rightarrow 0}0\), and thus such an energy does not contribute to the surface energy in the limit. However, it forces \({\textbf{u}}\) to satisfy the boundary condition (1.18c). Now applying integration by parts to the last term of (1.7), and then using (1.16) and \(\varvec{\xi }|_{\partial \Omega }=0\), we find
Note that the last step is due to the Green’s formula.
Under additional assumptions, we can show that the limit \({\textbf{u}}\) in (1.17) solves a geometric evolution equation in the bulk region \(\Omega ^+:=\bigcup _{t\in [0,T]}\Omega ^+_t\times \{t\}\).
Theorem 1.2
Let \(d=2\) and the assumptions of Theorem 1.1 be in place. Assume further that
Then there exists a sufficiently small \(\mu >0\) (independent of \(\varepsilon \)) such that the vector field \({\textbf{u}}\) in (1.17) satisfies
for almost every \(t\in (0,T)\) and for every \( \varvec{\Psi }\in C^1_c(\Omega _t^+;{\mathbb {R}}^3)\).
In the above equation \(\wedge \) is the wedge product in \({\mathbb {R}}^3\) and \({\text {rot}}\) is the curl operator. The equation (1.23) is the weak formulation of an Oseen–Frank flow, written as
It can be verified that when \({\textbf{u}}\) is sufficiently regular, then (1.23) implies (1.24). It is worth mentioning that equation of the form (1.24) is the \(L^2\)-gradient flow of the variational problem
where the infimum is taken among mappings \({\textbf{u}}\in W^{1,2}(U;{\mathbb {S}}^2)\) fulfilling certain boundary conditions on \(\partial U\). Note that (1.25) is a special case of the full Oseen–Frank model (cf. [30]).
This work will be organized as follows: In Sect. 2, we shall adapt the modulated energy method of [24] to the vectorial and anisotropic system (1.2), and then derive a differential inequality, i.e. Proposition 2.1. Such an inequality was previously derived in [40] for a matrix-valued equation. When applied to (1.2), it includes a term which does not have an obvious sign due to the additional \({\text {div}}\) term. This problem will be solved in Sect. 3 during the proof of the inequality in Theorem 3.1. This theorem, which leads to the first part of Theorem 1.1, is a major novelty of the present work, and will be employed in Sect. 4 (see Theorem 4.1) to derive the \(L^1\)-estimate of \(\psi _\varepsilon \) in (1.16). Such an estimate will be used in Lemma 4.3 to identify appropriate level sets of \(\psi _\varepsilon \) which converge to \(I_t\) in certain sense. With this key lemma, we derive in Sect. 5 the anchoring boundary condition (1.18c), and thus finish the proof of Theorem 1.1. Section 6 is devoted to the proof of Theorem 1.2. The proof of Proposition 1.1 is quite similar to the construction given in [40]. We present a proof in Appendix A for the convenience of the readers.
2 Preliminaries
2.1 Notation and conventions
We shall adopt the following conventions throughout the paper. Unless specified otherwise, \(C>0\) is a generic constant whose value might change from line to line, and will depend on the geometry of the interface (1.4) but not on \(\varepsilon \) or \(t\in [0,T]\). For two square matrices A and B, their Frobenius inner product is defined by \(A:B:= {\text {tr}}A^{{\textsf{T}}} B\), which induces the norm \(|A|:=\sqrt{{\text {tr}}A^{{\textsf{T}}} A}\). We shall also use the following notation for a vector-valued function \({\textbf{u}}(x,t)=(u_1(x,t),u_2(x,t),u_3(x,t))\) where
To ease computations when \(d=2\), \(\nabla {\textbf{u}}\) will be understood as the matrix \(\begin{pmatrix} \partial _1 u_1&{}\quad \partial _2 u_1 &{}\quad 0 \\ \partial _1 u_2&{}\quad \partial _2 u_2 &{}\quad 0 \\ \partial _1 u_3&{}\quad \partial _2 u_3 &{}\quad 0 \end{pmatrix} \), and
In particular, the latter applies to the normal and the mean curvature vector fields (cf. (2.9) and (2.13) respectively below). For a function of \({\textbf{u}}\), like \(F({\textbf{u}})\), its gradient will be denoted by
We end this section by the following assumptions regarding various constants. Theorem 1.1 will be proved for any fixed constant \(\mu >0\), while Theorem 1.2 is valid for a sufficiently small (fixed) \(\mu \). To simplify the presentation, we shall assume without loss of generality that
Finally we can normalize g (cf. (1.9)) to have
As the \(L^2\)-gradient flow of (1.1), the system (1.2) enjoys the following energy dissipation law:
for arbitrarily large time \({\hat{T}}\). Combining this with the theory of gradient flow and the regularity theory for elliptic system (cf. [4, 45]), one can construct a unique solution to system (1.2) that satisfies
So for almost every \({\hat{t}}\in (0,{\hat{T}})\), we have
Under the assumption (1.11c), the nonlinearity of (1.2a) has a linear growth. So considering the system with initial datum \({\textbf{u}}_\varepsilon (\cdot ,{\hat{t}})\), and using the Hölder estimates for parabolic system (cf. [51]), we deduce that
For initial datum undergoing phase transitions near the initial interface \(I_0\), formal asymptotic analysis suggests that \(\nabla {\textbf{u}}_\varepsilon \) will be singular near \(I_t\). However, the global dissipation law (2.6) is not sufficient to yield the (strong) convergence of \({\textbf{u}}_\varepsilon \), not even in the domain away from \(I_t\). Following a recent work of Fisher et al. [24], we shall establish in this section a differential inequality which modulates the concentration and leads to the compactness of solutions in Sobolev spaces.
2.2 The modulated energy
We first set up the geometry of the moving interface I defined in (1.4). Under a local parametrization \(\varvec{\varphi }_t(s):U\subset {\mathbb {R}}^{d-1}\rightarrow I_t\), the mean curvature flow reads
where \( \kappa =\kappa (\varvec{\varphi }_t(s),t)\) is the mean curvature and \({\textbf{n}}={\textbf{n}}(\cdot ,t): I_t\mapsto {\mathbb {S}}^{d-1}\) is the inward normal vector. For any \(t\in [0,T]\) we assume that the nearest-point projection \(P_I(\cdot ,t):B_{4\delta _0}(I_t)\mapsto I_t\) is smooth for some sufficiently small \(\delta _0\in (0,1)\) which only depends on the geometry of I. Analytically we have \(P_I(x,t) =x-\nabla d_I(x,t) d_I(x,t)\). So for each fixed \(t\in [0,T]\), any point \(x\in B_{4\delta _0}(I_t)\) corresponds to a unique pair (r, s) with \(r=d_I(x,t)\) and \(s\in U\), and the identity
holds with independent variables (r, s, t). Differentiating this identity with respect to r and t leads to the following identities:
The significance of these equations is that they extend the normal vector and the normal velocity from \(I_t\) to a neighborhood of it. So we shall also use \({\textbf{n}}\) to denote \(\nabla d_I\) when the latter is smooth. We shall extend \({\textbf{n}}\) to the whole computational domain \(\Omega \) by defining
where \(\phi : {\mathbb {R}}\mapsto {\mathbb {R}}_+\) is an even, smooth function that decreases on [0, 1], and satisfies
To fulfill these requirements, we can simply choose
We proceed with the extension of the mean curvature. Choosing a cut-off function \(\eta _0(x,t)\) such that
we constantly extend the inward mean curvature vector by defining
These combined with (2.10) imply that
Lemma 2.1
There exists a constant \(C>0\) depending only on the geometry of the interface (1.4) such that the following properties hold for every \(t\in [0,T]\):
where \(\nabla {\textbf{H}}:=\{\partial _j H_i\}_{1\leqslant i,j\leqslant 3}\) is a matrix with i being the row index.
Proof
By introducing \(\phi _0(\tau ):=\phi (\frac{\tau }{\delta _0})\), we can rewrite (2.10) as \(\varvec{\xi }=\phi _0 \left( d_I\right) \nabla d_I\). Since \(\phi \) is even, we have \(\phi _0'(0)=0\). This combined with Taylor’s expansion in \(d_I\) implies that
This and (2.13) lead to (2.15a). Using (2.9) and (2.13), we can write (2.8) as the transport equation (2.15b), which leads to the following identities in \(B_{\delta _0}(I_t)\):
These two equations together imply (2.15c). \(\square \)
It will be convenient to introduce
It can be verified using (1.10b) that
By (1.9) we have
Recalling (2.7), we have
Now we define the phase-field analogues of the normal vector and the mean curvature vector respectively by
Note that in (2.20b), the inner product is made with the column vectors of \(\nabla {\textbf{u}}_\varepsilon =(\partial _1 {\textbf{u}}_\varepsilon ,\partial _2 {\textbf{u}}_\varepsilon ,\partial _3 {\textbf{u}}_\varepsilon )\). We deduce from (2.20a) that
Define also the orthogonal projection \(\Pi _{{\textbf{u}}_\varepsilon }\) by
Lemma 2.2
The following equations hold:
Proof
Concerning (2.23a), it suffices to work with the set \(\{ x\mid |{\textbf{u}}_\varepsilon |\notin \{0,1\}\}\) where \(g(|{\textbf{u}}_\varepsilon |)>0\) (cf. (1.10)), for otherwise the equation will follow from (2.17) and (2.19a). On this set we deduce from (2.17) that \( D \textrm{d}^F ({\textbf{u}}_\varepsilon )=\frac{{\textbf{u}}_\varepsilon }{|{\textbf{u}}_\varepsilon |} g(|{\textbf{u}}_\varepsilon |)\ne 0\), and we can rewrite (2.19a) as
This combined with (2.22) implies (2.23a).
Now we turn to the proof of (2.23b). On the set \(\{ x\mid |{\textbf{u}}_\varepsilon |\notin \{0,1\}\}\), we have
and this implies (2.23b) in view of (2.22). \(\square \)
The following lemma establishes coercivity properties of the modulated energy (1.7).
Lemma 2.3
The following estimates hold for every \(t\in [0,T]\):
where \(C=C(\delta _0,\phi )\).
Proof
The case when \(\mu = 0\) has been done in [40], and the proof carries over to the present case. First, it follows from (2.22) that
Combining this with (2.21), we can write
By (2.18) and (2.23a), the second term in the last display is non-negative. Since \(|\varvec{\xi }|\leqslant 1\), we also have (2.26a), (2.26b), (2.26c) and
Combining (2.29) with (2.26a) and the inequality \( 1-\varvec{\xi }\cdot {\textbf{n}}_\varepsilon \leqslant 2\), we obtain (2.26d). Finally, by (2.11) and \(\delta _0\in (0,1)\) we have
This together with (2.26d) implies (2.26e). \(\square \)
The following result was first proved in [24] for the scalar Allen-Cahn equation, and was generalized to the vectorial case in [40].
Proposition 2.1
There exists a generic constant \(C>0\) depending only on the geometry of the interface (1.4) such that
We present a proof of (2.31) in Appendix B for the convenience of the readers.
3 Uniform estimates of solutions
Observe that the second term on the left-hand side of (2.31) does not have an obvious sign. However, we have the following theorem.
Theorem 3.1
Under the assumptions of Theorem 1.1, there exists a constant \(C_0>0\), which depends only on the geometry of the interface (1.4) and \(c_1\) (cf. (1.14c)), such that
It is worth mentioning that \(C_0\) is independent of \(\mu \). The proof of (3.1) relies on the following lemma.
Lemma 3.2
For any function \(\eta _1\) with \(\eta _1(\cdot ,t)\in C_c(B_{4\delta _0}(I_t);{\mathbb {R}}_{\geqslant 0})\), there exists a universal constant \(C>0\) which is independent of t and \(\varepsilon \) such that
Proof
On the set \(\{ x\mid g(|{\textbf{u}}_\varepsilon |)>0\}=\{ x\mid |{\textbf{u}}_\varepsilon |\notin \{ 0, 1\}\}\) we can use (2.23b) and (2.23a) to estimate
On the set \(\{ x\mid |{\textbf{u}}_\varepsilon |=0\}\) we have \(\Pi _{{\textbf{u}}_\varepsilon } \nabla {\textbf{u}}_\varepsilon =0\) by the second case in (2.22). On the open set \( \{ x\mid |{\textbf{u}}_\varepsilon |>0\}\supset \{ x\mid |{\textbf{u}}_\varepsilon |=1\}\) we can write \(\Pi _{{\textbf{u}}_\varepsilon } \nabla {\textbf{u}}_\varepsilon = \nabla |{\textbf{u}}_\varepsilon | \otimes \frac{{\textbf{u}}_\varepsilon }{|{\textbf{u}}_\varepsilon |}\) by the first case in (2.22). This combined with [18, Theorem 4.4] implies that \(\Pi _{{\textbf{u}}_\varepsilon } \nabla {\textbf{u}}_\varepsilon =0\) for a.e. \(x\in \{ x\mid |{\textbf{u}}_\varepsilon |=1\}\). Altogether we have shown that
This together with (2.26d) implies
In \(B_{4\delta _0}(I_t)\) where \({\textbf{n}}=\nabla d_I\), we have the decomposition
Using (2.10) and (2.11), we can estimate the last term by
These inequalities and (2.26e) lead to
Now using (3.6), (2.26d) and (2.26e) we find
The above two estimates together with the formula
yield (3.2). \(\square \)
To proceed we need an \(L^3\)-estimate of \({\textbf{u}}_\varepsilon \).
Lemma 3.3
Under the assumption (1.14b), there exists a constant \(C=C(c_1)>0\) such that
Proof
It follows from (2.18), (2.23a) and the Cauchy–Schwarz inequality that
This and (2.6) lead to (3.8a). To prove (3.8b), we first note that if \(|{\textbf{u}}_\varepsilon |> 2\), then
This combined with Sobolev’s embedding and \(\psi _\varepsilon |_{\partial \Omega }=0\) (cf. (1.2c)) leads to
\(\square \)
Proof of Theorem 3.1
We shall only present the proof in 3D because the 2D case is analogous under the conventions made in Sect. 2.1. We shall employ Einstein summation notation by summing over repeated Latin indices.
We first use (2.31) to get
Observe that the orthogonal projection (2.22) is parallel to \(D \textrm{d}^F({\textbf{u}}_\varepsilon )\) when it does not vanish. So we can write
Substituting this identity into (3.9) we find
To estimate the second term on the left-hand side, we use (1.2a) and (2.20b) to write
Note that the inner product is made with the column vectors of \(\nabla {\textbf{u}}_\varepsilon =(\partial _1 {\textbf{u}}_\varepsilon ,\partial _2 {\textbf{u}}_\varepsilon ,\partial _3 {\textbf{u}}_\varepsilon )\). Using the above formula, we expand the integrands of (3.10) and find
Note that the second term in the last display is non-negative due to Cauchy-Schwarz’s inequality, and this implies that
Adding the above inequality to (3.10) leads to
To estimate the last term, we write \({\textbf{u}}_\varepsilon =(u^\varepsilon _i)_{1\leqslant i\leqslant 3}\) and \({\textbf{H}}=(H_i)_{1\leqslant i\leqslant 3}\). Using integration by parts and (2.14c), we obtain
In view of (2.26b), the first integral in the last display of (3.13) is bounded by
The second integral can be estimated by decomposing \(\nabla u^\varepsilon _j\) and by using (2.14a):
By (2.13) and (2.12), the second integral in the last display can be estimated using (3.2) with \(\eta _1:=|\nabla {\textbf{H}}|^2\). The other two terms can be controlled by \( {(\mu ^{-1} +1)}C\varepsilon ^{-1} E_\varepsilon [ {\textbf{u}}_\varepsilon | I]\) using (2.26b) and (2.26e) respectively. To summarize we deduce from (3.13) and (3.14) that
Combining this with (3.12), we find
Note that due to (2.4) the constant C above can be made independent of \(\mu \). It remains to estimate the last integral in (3.15). By orthogonal decompositionsFootnote 1,
We also need the following identity which follows by taking the wedge product of (1.2a) with \({\textbf{u}}_\varepsilon \).
Using the above two identities, we integrate by parts to obtain
Inserting this identity into (3.15), and using the Cauchy–Schwarz inequality, (3.8b) and (2.26b), we find
Note that in the last step we used integration by parts, the identity
which follows from (2.22), and the identities \(( \partial _k {\textbf{u}}_\varepsilon ) \wedge ( \partial _k {\textbf{u}}_\varepsilon )= 0\) for each fixed \(k\in \{1,2,3\}\). Finally, applying the Cauchy–Schwarz inequality and then (2.26b) and (3.8b) in the last two integrals of (3.16), we find
This combined with (1.14c) and Grönwall’s inequality leads to (3.1). \(\square \)
Using (2.26e) and (3.1), we readily obtain the following corollary.
Corollary 3.4
Under the assumptions of Theorem 1.1, there exists a constant \(C>0\), which depends only on the geometry of the interface (1.4) and \(c_1\), such that
hold for each fixed \(\delta \in (0, \delta _0)\).
Indeed, (3.19b) follows from (3.19a) and the inequality
which is a consequence of (3.1). Another consequence of (3.1) is the following lemma concerning
Lemma 3.5
Under the assumptions of Theorem 1.1, there exists a constant \(C>0\), which depends only on the geometry of the interface (1.4) and \(c_1\), such that
Proof
We first deduce from (3.1) and (2.26b) that
By (3.21) we have the identity \({\textbf{u}}_\varepsilon =|{\textbf{u}}_\varepsilon |{\widehat{{\textbf{u}}}}_\varepsilon \). Using this and (2.22), we can write
Substituting this formula into (3.23), we obtain the estimate of the first integral on the left-hand side of (3.22a). To control the second one, we use the following formula which follows from (2.22):
Note that on the set \(\{ x\mid |{\textbf{u}}_\varepsilon |=0\}\), we have \(\nabla |{\textbf{u}}_\varepsilon |=0\) a.e., and thus the above formula is still valid. This and (3.23) yield the estimate of \({\widehat{{\textbf{u}}}}_\varepsilon \cdot \nabla |{\textbf{u}}_\varepsilon |\) and (3.22a) is proved.
Regarding (3.22b), it suffices to estimate over the set
because the integral over its complement vanishes. By (2.17) and (2.19a), we have \(U_\varepsilon \subset \{ x\mid |{\textbf{u}}_\varepsilon |\notin \{0,1\}\}\) where \(g(|{\textbf{u}}_\varepsilon |)=|D \textrm{d}^F|({\textbf{u}}_\varepsilon )>0\). This combined with (2.19b) and (2.20a) implies that
On the other hand, by the polar decomposition \({\textbf{u}}_\varepsilon =|{\textbf{u}}_\varepsilon |{\widehat{{\textbf{u}}}}_\varepsilon \) and orthogonality \({\hat{{\textbf{u}}}}_\varepsilon \perp \partial _{x_j} {\hat{{\textbf{u}}}}_\varepsilon \), we have
Setting \({\widehat{{\textbf{u}}}}_\varepsilon \cdot {\textbf{n}}_\varepsilon =:\cos \theta _\varepsilon \), we have
This inequality, (2.26a) and (3.26) together imply that
Note that in the last step we have used the identity \(\big |\nabla |{\textbf{u}}_\varepsilon |\big |\sqrt{2F ({\textbf{u}}_\varepsilon )}=|\nabla \psi _\varepsilon |\), which holds on \(U_\varepsilon \). So (3.22b) follows from conjugation. \(\square \)
4 Estimates of level sets
Recalling (2.5), the main result of this section is the following \(L^1\)-estimate of \(\psi _\varepsilon \).
Theorem 4.1
Under the assumptions of Theorem 1.1, there exists \(C>0\) independent of \(\varepsilon \) such that
Proof
We shall denote the positive and negative parts of a function h by \(h^+\) and \(h^-\) respectively. For simplicity we shall suppress \(\, dx\) in a volume integral. By [18, pp. 153], for any \(h\in W^{1,1}(\Omega )\), we have
Our goal is to estimate \(2\psi _\varepsilon -1-\chi \) where \(\chi (x,t)=\pm 1\) in \(\Omega _t^\pm \). Using the formula \(h=h^+-h^-\), we can write
and we shall estimate its difference with \(\chi \). This will be done by establishing differential inequalities for the following energies which add up to (1.12):
where \(\eta (z)\) is defined by (1.13) and \(|\eta |(z)=:\zeta (z)\). It is obvious that the integrand of (4.5a) is non-negative. Since \(\psi _\varepsilon \geqslant 0\), we have \((\psi _\varepsilon -1)^-\in [0,1]\) and thus \([1 -2(\psi _\varepsilon -1)^-]\) ranges in \([-1,1]\). Using the identity \((\eta \circ d_I)~\chi =|\eta \circ d_I|\), we deduce that the integrand of (4.5b) is also non-negative and
Finally, we deduce from (1.14c) that
Step 1: estimates of weighted errors. Using (1.8) and (1.9), we have
Using this and (4.3) we can calculate
By (2.15b), the integrand of the last integral vanishes on \(B_{\delta _0}(I_t)\). Moreover, we can combine the second and the third integrals in the last display using integration by parts. Using also that \(\Vert {\text {div}}{\textbf{H}}\Vert _{L^\infty _{x,t}}\leqslant C\) and (2.26e), we find
Now using (4.7), (3.20) and (3.1), we can apply the Grönwall lemma and obtain \(\sup _{t\in [0,T]}g_\varepsilon (t)\leqslant C \varepsilon \) for some C which is independent of \(\varepsilon \). Concerning \(h_\varepsilon \), for simplicity we introduce \(w_\varepsilon :=\chi -[1- 2(\psi _\varepsilon -1)^-]\). Using the identity \((\partial _i \chi )~\eta \circ d_I \equiv 0\) (in the sense of distribution), we find
So by the same calculation for \(g_\varepsilon \) we obtain
Using (4.7) and (3.20), we can apply the Grönwall lemma and obtain \(\sup _{t\in [0,T]}h_\varepsilon (t)\leqslant C\varepsilon \). Finally, by (4.4) and (4.6), we find
and this proves (4.1).
Step 2: remove the weight. First note that (4.11) implies (4.2) with \(\Omega \) replaced by \(\Omega \backslash B_{\delta _0}(I_t)\). So we shall focus on the estimate on \(B_{\delta _0}(I_t)\). We set \(\chi _\varepsilon :=2\psi _\varepsilon -1\) and abbreviate \(\delta _0\) by \(\delta \). For fixed \(t\in [0,T]\) and \(p\in I_t\) with normal vector \({\textbf{n}}={\textbf{n}}(p)\), applying Hölder’s inequality and Lemma 4.2 below with \(f(r,p,t)=\left| \chi \left( p + r {\textbf{n}}, t\right) -\chi _\varepsilon (p+ r {\textbf{n}}, t )\right| \), we find
In view of (1.8) and (1.2c), we have \(\psi _\varepsilon =0\) on \(\partial \Omega \). So by Sobolev’s embedding \(W^{1,1}\hookrightarrow L^{3/2}\) we obtain
Note that in the last step we employed (3.8a) and (4.11). This gives the desired estimate in \(B_{\delta _0}(I_t)\) and thus the proof of (4.2) is finished. \(\square \)
Lemma 4.2
For any integrable function \(f:[-\delta ,\delta ]\rightarrow {\mathbb {R}}_{\geqslant 0}\), we have
Proof
We write \(x=(x_1,x_2,x_3),y=(y_1,y_2,y_3)\) and \(F(x)=f(x_1)f(x_2)f(x_3)\). By symmetry and the Hölder inequality, we find
\(\square \)
Now we turn to the study of the level sets of \(\psi _\varepsilon \). The main tool is the following estimate, which is a consequence of (2.20a), (2.26d) and (3.1).
Lemma 4.3
For each \(t\in [0,T]\) there exists a null set \({\mathcal {N}}_t^\varepsilon \subset (0,1/8)\) such that the following holds: for every \(\alpha \in (0,1/8)\backslash {\mathcal {N}}_t^\varepsilon \), there exist
such that the sets
are of finite perimeter and
where \(C>0\) is independent of \(t, \varepsilon \) and \(\alpha \).
Proof
To prove (4.16a), we consider the set
It follows from the co-area formula of BV function [18, section 5.5] that \( S_t^{\varepsilon ,\alpha }\) has finite perimeter for every \(\alpha \in (0,1/8)\backslash \widetilde{{\mathcal {N}}}_t^\varepsilon \) for some null set \(\widetilde{{\mathcal {N}}}_t^\varepsilon \subset (0,1/8)\). Moreover, by (4.13), we have for every \(\alpha \in (0,1/8)\backslash \widetilde{{\mathcal {N}}}_t^\varepsilon \) that
where \(\varvec{\nu }\) is the outward normal of the set \( S_t^{\varepsilon ,\alpha }\), defined on its (measure-theoretic) boundary. Since \(|\varvec{\xi }|\leqslant 1\) on \(\Omega \) and \(\psi _\varepsilon \leqslant 1\) on \(S_t^{\varepsilon ,\alpha }\), we have
where \(|A|={\mathcal {L}}^d (A)\) is the d-Lebesgue measure of a set A. Combining this with (4.18), we find
By the divergence theorem, we have
Inserting these two equations into (4.19), we find
where \(A\triangle B:=(A-B)\cup (B-A)\) is the symmetric difference of two sets A and B. We first estimate \(r_\varepsilon ^+:= ~\Big | \Omega _t^+\triangle {\{x\mid \psi _\varepsilon >\tfrac{1}{2}+\alpha \}}\Big |\).
Now using Chebyshev’s inequality and (4.2), we find \(r_\varepsilon ^+\leqslant C \varepsilon ^{1/4} \). Similar estimates apply to \( |S_t^{\varepsilon ,\alpha }|\) and \(r_\varepsilon ^-:= | \Omega _t^- \triangle {\{x\mid \psi _\varepsilon <\tfrac{1}{2}-\alpha \}} |\). Substituting these estimates into (4.20), we find
So (4.16a) follows from Fubini’s theorem.
To prove (4.16b), we consider the set
Using (4.13) and the co-area formula, we have for every \(\alpha \in (0,1/8)\backslash {\mathcal {N}}_t^\varepsilon \) that
where \( {\mathcal {N}}_t^\varepsilon \supset \widetilde{{\mathcal {N}}}_t^\varepsilon \) is a null set in (0, 1/8) and \(\varvec{\nu }\) is the outward normal of \(\partial Q_t^{\varepsilon ,\alpha }\). Since \(\psi _\varepsilon \leqslant 3\) on \(Q_t^{\varepsilon ,\alpha }\), we have \(\left| \int _{Q_t^{\varepsilon ,\alpha }} ({\text {div}}\varvec{\xi }) \psi _\varepsilon \, dx\right| \leqslant C |Q_t^{\varepsilon ,\alpha }|,\) and thus
Using (2.14c), we have \(\int _{\Omega }({\text {div}}\varvec{\xi })\, dx=0\), and thus
This combined with Chebyshev’s inequality and (4.2) implies that
Substituting this in (4.23) leads to
So (4.16b) follows from Fubini’s theorem. \(\square \)
We end this section with the following result concerning the convergence of \({\textbf{u}}_{\varepsilon }\).
Proposition 4.1
For every sequence \(\varepsilon _k\downarrow 0\) there exists a subsequence, which we will not relabel, such that \({\textbf{u}}_k:={\textbf{u}}_{\varepsilon _k}\) satisfies
where \({\textbf{u}}={\textbf{u}}(x,t)\) satisfies
Furthermore,
Before proving this result, we state the Aubin–Lions–Simon lemma. See [41, Theorem 8.62, Exercise 8.63] or [52, Corollary 8] for the proof.
Lemma 4.4
Let \(I \subset {\mathbb {R}}\) be an open bounded interval, let \(\left( Y_0,\Vert \cdot \Vert _{Y_0}\right) ,\left( Y_1,\Vert \cdot \Vert _{Y_1}\right) \), and \(\left( Y_2,\Vert \cdot \Vert _{Y_2}\right) \) be Banach spaces with \(Y_0 \hookrightarrow Y_1 \hookrightarrow Y_2\). Assume that the embedding \(Y_0 \hookrightarrow Y_1\) is compact. Let \({\mathcal {V}}\) be the Banach space of all functions \(u \in L^\infty \left( I; Y_0\right) \) whose distributional derivative \(\partial _t u\) belongs to \(L^2\left( I; Y_2\right) \) endowed with the norm
Then the embedding \({\mathcal {V}} \hookrightarrow C\left( {\bar{I}}; Y_1\right) \) is compact.
Proof of Proposition 4.1
Define \(\Omega ^\pm :=\bigcup _{t\in (0,T]}\Omega _t^\pm \times \{t\}\). We first deduce from (3.1) and (2.26b) that
for some C independent of \(\varepsilon \). On the other hand, by (2.22) we find
for \(0\leqslant i\leqslant 3\) where \(\partial _0:=\partial _t\). Combining (4.29) and (4.28) with (3.8b), we deduce that
So it follows from the Banach–Alaoglu theorem (cf. [41, A.5.]) that
where
It follows from (3.8b), (3.19a) and (3.19b) that, for any fixed \( \delta \in (0,\delta _0)\), up to extraction of subsequences there exists \(\varepsilon _k=\varepsilon _k(\delta )\xrightarrow {k\rightarrow \infty }0\) such that
By (4.33a) and (4.33b), we have \({\textbf{u}}={\bar{{\textbf{u}}}}_\delta \) a.e. in \(U^\pm (\delta ):=\cup _{t\in [0,T]} \left( \Omega _t^\pm \backslash B_{\delta }(I_t)\right) \times \{t\}\). This combined with (4.33c) and (4.33d) leads to
Furthermore, employing (4.33b)–(4.33d) and Lemma 4.4, we obtain
By passing to a sequential limit \(\delta =\delta _\ell \xrightarrow {\ell \rightarrow 0} 0\) and by a diagonal argument we obtain (4.27) up to extraction of subsequences.
Now we turn to the proof of (4.26). Using (3.19a), (4.35) and Fatou’s lemma, we deduce that
for any fixed \(\delta \in (0,\delta _0)\). This together with (1.10) implies that \( {\textbf{u}}\) ranges in \(\{0\}\cup {\mathbb {S}}^2\) a.e. in \(\Omega \times (0,T)\). This combined with (4.2) and (4.34) yields (4.26c) and
Now we show the integrability of \(\nabla _{x,t} {\textbf{u}}\) up to the boundary. To this aim, we choose a sequence of functions
By (4.31) and (4.27), we deduce that for \(0\leqslant i\leqslant 3\),
By (4.32) and the dominated convergence theorem, we can take \(k\rightarrow \infty \) and get
This and (4.31) lead to (4.25a) and (4.25b). Since \({\textbf{u}}\) maps \(\Omega ^+\) into \({\mathbb {S}}^2\), we have
This and (4.32) improve (4.36) and yield (4.26a) and (4.26b). \(\square \)
5 Proof of Theorem 1.1: anchoring boundary condition
The inequalities (1.15) and (1.16) have been proved in Theorem 3.1 and in Theorem 4.1. The assertions (1.17), (1.18a) and (1.18b) have been proved in Proposition 4.1 (cf. (4.27c) and (4.26)). It remains to verify (1.18c), and this will be done by applying Lemma 4.3 for every \(t\in [0,T]\) and by choosing an appropriate \(\alpha \) outside the null set \({\mathcal {N}}_t^{\varepsilon _k}\subset (0,1/8)\). For simplicity we shall abbreviate \(\psi _{\varepsilon _k}\) and \({\textbf{u}}_{\varepsilon _k}\) by \(\psi _k\) and \({\textbf{u}}_k\) respectively. For any \(k\geqslant 1\) we can choose \(\beta _k\in [1/2,1]\) such that \(\alpha =\alpha _k:= \beta _k{\varepsilon _k^{1/8}}\notin {\mathcal {N}}_t^{\varepsilon _k}\). Then by Lemma 4.3 there exist
such that
and such that the set
Moreover, there exists \(C>0\) which is independent of t and the particular choice of the subsequence \(\varepsilon _k\) such that
Using these level sets, we can prove the following proposition which improves (4.27) to the convergence of \({\textbf{u}}_k\) up to the boundary \(I_t\).
Proposition 5.1
Let \({\textbf{u}}\) be the limit vector field in Proposition 4.1. For a.e. \(t\in [0,T]\), up to extraction of subsequences which we will not relabel, we have
where \({\widehat{{\textbf{u}}}}_k={\widehat{{\textbf{u}}}}_{\varepsilon _k}\) is defined in (3.21).
Proof
We first claim that there exists a positive constant \(C_3\) depending only on f (cf. (1.11)) such that the following statement holds for any \(\delta \in (0,1/8)\):
Indeed, by (1.11a), f (and also g) is increasing on \((0,s_0)\). If \(|{\textbf{u}}_\varepsilon |\geqslant s_0\), we are done. Otherwise,
which implies (5.6). This combined with (3.22a) and (5.3) implies
for k sufficiently large. This and (5.3) imply that the distributional derivatives of \( {\textbf{v}}_k(\cdot ,t):= {\textbf{1}}_{ {\bar{\Omega }}_t^k} {\widehat{{\textbf{u}}}}_k(\cdot ,t)\) have no Cantor parts, and the absolute continuous parts \(\{{\textbf{1}}_{ {\bar{\Omega }}_t^k} \nabla {\widehat{{\textbf{u}}}}_k\}_{k\geqslant 1}\) is bounded in \(L^2(\Omega )\). Moreover, their jump parts enjoy the estimate
and \(\{{\textbf{v}}_k(\cdot ,t)\}_{k\geqslant 1}\) is bounded in \(L^\infty (\Omega )\). With these properties, it follows from [2] (or [3, Section 4.1]) that \(\{{\textbf{v}}_k(\cdot ,t)\}_{k\geqslant 1}\) is compact in \(SBV(\Omega )\), the class of special functions of bounded variation on \(\Omega \). More precisely, there exists \({\textbf{v}}(\cdot ,t)\in SBV(\Omega )\) s.t. \({\textbf{v}}_k\rightarrow {\textbf{v}}\) weakly-star in \(BV(\Omega )\) as \(k\rightarrow \infty \), and the absolute continuous part of the gradient \(\nabla ^{a} {\textbf{v}}_k={\textbf{1}}_{ {\bar{\Omega }}_t^k} \nabla {\widehat{{\textbf{u}}}}_k\) converges weakly in \(L^1(\Omega )\) to \(\nabla ^{a} {\textbf{v}}\). To identify \({\textbf{v}}\), we use (4.2) to deduce that \({\textbf{1}}_{{\bar{\Omega }}_t^k}\rightarrow {\textbf{1}}_{\Omega _t^+}\) in \(L^1(\Omega )\) as \(k\rightarrow \infty \). This and (4.27c) yield \({\textbf{v}}(\cdot ,t)={\textbf{1}}_{ \Omega _t^+}~{\textbf{u}}(\cdot ,t)\) a.e. in \(\Omega \), and thus (5.5a) and (5.5b) are proved. Finally by (5.5a), the compact embedding of BV functions and the \(L^\infty \) bound we get (5.5c). \(\square \)
To proceed we define the following measures for Borel sets \(A\subset \Omega \):
Lemma 5.1
For a.e. \(t\in [0,T]\),
Proof
We define truncation functions
By (5.2), we have \(T_k\xrightarrow {k\rightarrow \infty }T\) uniformly on \({\mathbb {R}}\). Moreover,
Indeed, by (2.7) and (2.17) we know that \(\psi _k(\cdot ,t)\in C^1(\Omega )\). Also by (5.3) we have \(T_k'\circ \psi _k={\textbf{1}}_{\Omega _t^k}\) for a.e. \(x\in \Omega \). Therefore, (5.12a) follows from the chain rule (cf. [26, Proposition 3.24]), while (5.12b) follows from (4.2) and the dominated convergence theorem. By (4.13) we have for any \(g\in C^1_c(\Omega )\) that
Recalling that \(\varvec{\xi }\) is the inward normal of \(I_t\), we use (5.12b) to pass to the limit in the above equations and obtain
for any \(g \in C^1_c(\Omega )\). By approximation, one can pass from \(C^1_c(\Omega )\) to \(C^0_c(\Omega )\), and this proves (5.9). \(\square \)
Now we finish the proof of Theorem 1.1 by verifying (1.18c). The proof here is inspired by the blow-up argument in [43]. See also [25] for the applications of such a method in the study of quasi-convex functionals.
Proof of (1.18c) For any \(x_0 \in I_t\) and any \(R>0\), it follows from (5.5c), (5.12b) and the dominated convergence theorem that
We can use spherical coordinate to rewrite the above two integrals in the form of \(\int _0^R\int _{\partial B_r(x_0)}(\cdot ) \,d{\mathcal {H}}^{d-1}dr\), and then apply Fubini’s theorem. Therefore, there exists \(r_{j} \downarrow 0\) such that for each j we have
where \(\varvec{\nu }\) is the outward normal of \(\partial B_{r_j}(x_0)\). Moreover, we can arrange \(r_j\) such that \(\theta (\partial B_{r_{j}}(x_0))=0\). This combined with (5.9) implies that
To proceed, we use convexity to write, for some \(a_m,c_m\in {\mathbb {R}}\), that
(cf. [3, Proposition 2.31]). For \(\theta -a.e. ~x_0\in \textrm{supp}(\theta )=I_t\), we have for each \(m\geqslant 1\) that
Note that in the last step we also used (5.14). It remains to compute the integral in the last display of (5.16) under the limit \(k\rightarrow \infty \) for fixed j, m. To this aim, we use (5.12a) and integration by parts to find
Note that the integrand of \(A_k\) is uniformly bounded in \(L^\infty \). To compute the limit of \(A_k\), we first deduce from (5.10) that \(T_k\circ \psi _k =0\) on the set \(\{x\in \Omega \mid \psi _k(x,t)=b_k\}\) which has finite perimeter (cf. (5.4b)). So we employ (5.3) to find
The limit of the first integral is given in (5.13), and that of the second vanishes in the limit \(k\rightarrow \infty \) by (5.4a). So we conclude that
Concerning the integral \(B_k\), by (5.5b) the sequence \(\{{\textbf{1}}_{{\bar{\Omega }}_t^k}{\text {div}}{\widehat{{\textbf{u}}}}_k\}_{k\geqslant 1}\) converges weakly in \(L^1(\Omega )\). Moreover, \(\{T_k\circ \psi _k\}_{k\geqslant 1}\) is uniformly bounded in \(L^\infty \), and converges a.e. in \(\Omega \) to \( \tfrac{1}{2} {\textbf{1}}_{\Omega _t^+}\), due to (5.12b). Therefore, applying the Product Limit Theorem (cf. [16] or [49, pp. 169]), we obtain
Using (5.19) and (5.20), we can compute the limit in (5.17) and find
where in the last step we used \(\varvec{\xi }=-\varvec{\nu }\) on \(\partial \Omega ^+_t\). Note that \(\varvec{\xi }\) is the inward normal of \(I_t\) according to (2.10), and \(\Omega _t^+\) is the region enclosed by \(I_t\) with outward normal \(\varvec{\nu }\). Substituting (5.21) into (5.16) and then dividing the resulting inequality by \(\theta \left( B_{r_{j}}(x_0)\right) \) and taking \(j\rightarrow \infty \), we find
This together with (5.15) implies that \(({\textbf{u}}\cdot \varvec{\xi })^2(x_0)=0\) for \({\mathcal {H}}^{d-1}\)-a.e. \(x_0\in I_t\). \(\square \)
6 Proof of Theorem 1.2: Oseen–Frank limit in the bulk
The method here is inspired by [17, 38], which has a 2D nature. We set \( \varvec{\tau }_\varepsilon :=\partial _t {\textbf{u}}_\varepsilon \) and write (1.2a) as
By Corollary 3.4 and Proposition 4.1 (cf. (4.27c)), for a.e. \(t_0\in (0,T)\) and for any compact set \(K\subset \subset \Omega _{t_0}^+\), we have
where \({\hat{c}}={\hat{c}}(K,t_0)>1\) is independent of \(\mu \) and \(\varepsilon \).
Proposition 6.1
Let K be a compact set of \( \Omega _{t_0}^+\) and assume that (6.2a) and (6.2b) hold. There exists an absolute constant \(\Lambda \in (0,1)\) with the following property: under the assumptions
there exists a subsequence \(\varepsilon _k\downarrow 0\), which we will not relabel, such that
We shall need the following inequality due to the special choice of f in (1.22):
for some \(C_4>1\). In the sequel \(C_4>1\) will also be used as a generic constant that might change from line to line due to the use of the Sobolev embeddings and elliptic estimates. Note that \(C_4\) is independent of \(\mu \) and r.
Lemma 6.1
Under the assumptions (6.2a) and (6.3b) with a sufficiently small \({\hat{\epsilon }}\) (defined in (6.11) below), we have
Proof
Without loss of generality, we assume \(x_0=0\). For brevity we write \(B_r(0)\) as \(B_r\). Since all arguments are made at \(t=t_0\), we shall suppress the time dependence of \({\textbf{u}}_\varepsilon \).
Step 1. There exists \({\hat{C}}>1\) depending on \({\hat{c}}\) such that for any \(x_1 \in B_r\) we have
To prove (6.7), let \({\hat{{\textbf{u}}}}_\varepsilon (z)={\textbf{u}}_\varepsilon (x_1+\varepsilon z): B_2\rightarrow {\mathbb {R}}^3\). Then we can write (6.1) as
It follows from (6.2a) and a change of variable that \(\{\varepsilon ^2\varvec{\tau }_\varepsilon (x_1+\varepsilon \cdot )\}_{\varepsilon >0}\) is uniformly bounded in \(L^2(B_2)\). Using (6.5), we can estimate
Altogether, we prove that the terms on the right-hand side of (6.8) is bounded in \(L^2(B_2)\). Invoking the interior estimate for elliptic system (cf. [26, Theorem 4.9]), we obtain
Note that \(C_4\) is independent of \(\mu \). Now we estimate the last term by
Substituting this estimate in (6.10) and using Morrey’s embedding \(W^{2,2}\hookrightarrow C^{1/2}\), we obtain \(\Vert {\hat{{\textbf{u}}}}_\varepsilon \Vert _{C^{1/2}({\bar{B}}_1)}\leqslant C_4{\hat{c}}\). Rescaling back, we find (6.7) with
Step 2: We claim that with the choice
we have either (6.6) or
Indeed, if neither of them were valid, then
Since \(16 {\hat{\epsilon }}<1\), it follows from (6.7) that
Using this and (1.22), we deduce one of the following two cases for \(x\in B_{16 {\hat{\epsilon }}\varepsilon }(x_1)\):
a) If \(|{\textbf{u}}_\varepsilon (x_1)|>3\), then \(|{\textbf{u}}_\varepsilon (x)|>2\) and thus \(f( |{\textbf{u}}_\varepsilon (x)|)\geqslant 1\).
b) If \(|{\textbf{u}}_\varepsilon (x_1)|\in (1/4,3/4)\cup (5/4,3]\), then \(f(|{\textbf{u}}_\varepsilon (x_1)|)\geqslant 1/{16}\). By the third condition in (1.22) and (6.14), we have \(f(|{\textbf{u}}_\varepsilon (x)|)> 1/{32}\).
To summarize, we have the following inequality:
Integrating this inequality over \(B_{16{\hat{\epsilon }}\varepsilon }(x_1)\) and using the assumption \(\varepsilon < r/4\), we find
However, this would contradict (6.3b) since \(B_{ 16{\hat{\epsilon }}\varepsilon }(x_1)\subset B_{2r}(x_0)\). So (6.13) is not valid and the claim is proved.
Step 3: We shall rule out (6.12).
Assuming (6.12), we deduce from (1.22) that \(F({\textbf{u}}_\varepsilon )=|{\textbf{u}}_\varepsilon |^2\). By (6.1) we have
For \(z\in B_1\), we introduce \({\widetilde{{\textbf{u}}}}_\varepsilon (z):={\textbf{u}}_\varepsilon (rz)\) and \({\widetilde{\varvec{\tau }}}_\varepsilon (z):=\varvec{\tau }_\varepsilon (rz)\). Then
By the interior estimate for elliptic system, we have
Indeed, one can adapt the proof of [26, Theorem 4.9] to gain the term \(r^2\varepsilon ^{-2}\Vert {\widetilde{{\textbf{u}}}}_\varepsilon \Vert _{L^2(B_{1/2 })}\). By (6.18), (6.2a) and the conclusion in step 2, we find
This implies that \({\textbf{u}}_\varepsilon \rightarrow 0\) strongly in \(L^2(B_{r/2})\), which contradicts (4.2) since \(B_{r/2}\subset K\subset \subset \Omega _t^+\). Therefore, we rule out (6.12) and obtain (6.6). \(\square \)
By (6.6), we have polar decomposition \({\textbf{u}}_\varepsilon =\rho _\varepsilon {\textbf{v}}_\varepsilon \) where
We set
and define the projection
for a vector field \({\textbf{a}}\).
Lemma 6.2
Under the assumptions \(\varepsilon \leqslant r/4\), (6.2a) and (6.3b) for \({\hat{\epsilon }}\) defined in (6.11), \(\rho _\varepsilon \) satisfies the following equation in \(B_r(x_0)\).
where \({\mathcal {N}}_{1,\varepsilon }(\cdot ,\cdot ):{\mathbb {R}}^{4\times 3}\times {\mathbb {R}}^{4\times 3}\mapsto {\mathbb {R}}\) is bilinear with uniformly bounded coefficients. Also, \({\textbf{v}}_\varepsilon \) satisfies the following equation in \(B_r(x_0)\).
where \({\mathcal {N}}_{2,\varepsilon }(\cdot ,\cdot ):{\mathbb {R}}^{4\times 3}\times {\mathbb {R}}^{4\times 3}\mapsto {\mathbb {R}}^3\) is bilinear with uniformly bounded coefficients.
Proof
To simplify the presentation we will suppress the subscript \(\varepsilon \). By (6.19) we have \(|{\textbf{v}}|^2\equiv 1\) and thus
Substituting (6.19) into (6.1), we find
Testing (6.25) with \({\textbf{v}}\) and using (6.24), we obtain
The terms in the last line are bilinear with respect to \(\nabla {\textbf{w}}=(\nabla {\textbf{v}},\nabla \rho )\), and we denote their sum by \(-{\mathcal {N}}_{1,\varepsilon }(\nabla {\textbf{w}},\nabla {\textbf{w}})\). By (6.6), it has bounded coefficients and thus (6.22) is proved.
To derive (6.23), we shall use the following identities.
These combined with (6.24) lead to
Now applying \((\cdot )_{\Vert }\) to the equation in (6.25), and using (6.27) and (6.28), we obtain
The terms in the second line of the above equation are bilinear with respect to \(\nabla {\textbf{w}}\), and we denote their sum by \(-{\mathcal {N}}_{2,\varepsilon }(\nabla {\textbf{w}},\nabla {\textbf{w}})\). By (6.6), it has bounded coefficients, and thus (6.23) is proved. \(\square \)
Proof of Proposition 6.1
We first show that, by choosing \({\hat{\epsilon }}\) and \(\mu \) sufficiently small, we have
Recalling (6.20), we deduce from (6.3b) and (6.6) that
Recalling that \(r<1\), let \(\chi \) be a \(C^2\) cut-off function such that
and let \({\bar{{\textbf{w}}}}_\varepsilon :=({\bar{{\textbf{v}}}}_\varepsilon ,{\bar{\rho }}_\varepsilon )\) with
Multiplying (6.23) by \(\chi \) and using the linearity of \({\textbf{a}}_{\Vert }\) about \({\textbf{a}}\) (cf. (6.21)), we find
and \({\bar{{\textbf{v}}}}_\varepsilon |_{\partial B_1(x_0)}=0\). For brevity we denote \(L^p(B_1(x_0))\) by \(L^p\). Note that the commutators in (6.34) involve at most first order derivatives of \({\textbf{w}}_\varepsilon =({\textbf{v}}_\varepsilon ,\rho _\varepsilon )\), which satisfies (6.31). Now applying the \(L^p\)-estimate for elliptic equation [39, pp. 109] (componentwise) in (6.34), and invoking (6.31) and (6.6), we have
Note that the prefactors \(r^{-1}\) and \(r^{-2}\) are due to the differentiation of \(\chi \) (cf. (6.32)), and that \(C_4\) is independent of r. To estimate the last term, we employ the bi-linearity of \({\mathcal {N}}_{2,\varepsilon }\), (6.31) and (6.6):
Note that in the last step we used the Sobolev’s embedding \(W^{1,4/3}(B_1)\subset L^4(B_1)\). Combining (6.36) with (6.35), we obtain
Now we turn to the estimate of \(\rho _\varepsilon \). Using (6.6) and (1.22), we have \(f'(\rho _\varepsilon )=2(\rho _\varepsilon -1)\) in \(B_r(x_0)\). Now multiplying (6.22) by \(\chi \) and using the linearity of (6.21), we find
In the same way as we did for (6.37), we find
Combining this with (6.37) and (6.31) we discover
Note that before Lemma 6.1, we have assumed that \(C_4>1\) and \({\hat{c}}>1\). Recall also the choice of \({\hat{\epsilon }}\) in (6.11). By choosing
we find \(C_4 \max \{ {\hat{\epsilon }}, \mu \}<1/2\). This combined with (6.38) yields
In view of (6.32) and (6.33), this implies (6.30).
Now using (6.2b), we have \(\rho _{\varepsilon _k}(\cdot ,t_0)\xrightarrow {k\rightarrow \infty } |{\textbf{u}}|(\cdot ,t_0)=1\) strongly in \(L^2(B_r(x_0))\). Thus, using (6.6) we find
These together with (6.30) and the Gagliardo-Nirenberg interpolation inequality yield
Finally, using (6.6) and (6.39) we find
and finish the proof of (6.4). \(\square \)
Proof of Theorem 1.2
We employ the covering argument in [14]. For any test function \(\varvec{\Psi }\in C_c^1(\Omega _t^+;{\mathbb {R}}^3)\), we choose \(K=\overline{\textrm{supp} (\varvec{\Psi })}\subset \subset \Omega _t^+\), and we define the singular set at time \(t \in (0, T]\) by
We claim that \(\Sigma _t\) is discrete. Indeed, choose an arbitrary finite set \(\{y_j\}_{j=1}^J\subset \Sigma _t\) with mutually disjoint balls \(\{B_{2r_j}(y_j)\}_{j=1}^J\) inside K with \(r_j<1/2\). Since J is finite, there exists \(k_J>0\) such that for any \(k\geqslant k_J\) we have
Combined with (6.2a), this implies
As a result, \(J\leqslant 4{\hat{c}}^2{\hat{\epsilon }}^{-2}\) and thus \(\Sigma _t\) is discrete. Therefore w.l.o.g. we can assume that \(\Sigma _t=\{x_0\}\) and \(B_{2r}(x_0)\subset K\). Let \(\eta \in C_c^1(B_2(0))\) be a cut-off function which \(\equiv 1\) in \(B_1(0)\). Then
It is obvious that \(\varvec{\Psi }_\delta =0\) in \(B_{\delta }(x_0)\). By (6.40) and Proposition 6.1, we have
Using these properties, we can apply \( \wedge {\textbf{u}}_{\varepsilon _k} \cdot \varvec{\Psi }_\delta \) to both sides of (1.2a), integrate by parts and then send \(k\rightarrow \infty \):
Note that we have also used \(\partial _t {\textbf{u}}_{\varepsilon _k} \wedge {\textbf{u}}_{\varepsilon _k}\xrightarrow {k\rightarrow \infty } \partial _t {\textbf{u}}\wedge {\textbf{u}}\) weakly in \(L^2(0,T;L^{6/5}(\Omega ))\), which is due to Proposition 4.1. By (6.43) and the regularity of \({\textbf{u}}\) (cf. (4.26a) and (4.26b)), we can send \(\delta \rightarrow 0\) in the first and the second integrals in (6.45) using the dominated convergence theorem. Concerning the third one, we have
We claim that the second integral on the right-hand side vanishes as \(\delta \rightarrow 0\). Indeed, by the Cauchy–Schwarz inequality we have
Now using \(\lim _{\delta \rightarrow 0}\eta (\tfrac{ x-x_0}{\delta })= 0\) for any \(x\ne x_0\), we can send \(\delta \rightarrow 0\) in (6.46) and obtain
By the same argument we can compute the fourth integral in (6.45) and find
Using the above two formulas, we can send \(\delta \rightarrow 0\) in (6.45) and obtain (1.23). \(\square \)
Notes
For a square matrix A, the decomposition \(A=\frac{A+A^{{\textsf{T}}}}{2}+\frac{A-A^{{\textsf{T}}}}{2}\) is orthogonal under the Frobenius inner product \(A:B\triangleq {\text {tr}}(A^{{\textsf{T}}} B)\).
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Acknowledgements
Y. Liu is partially supported by NSF of China under Grant 11971314. We thank an anonymous referee for helpful comments.
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Appendices
Appendix A: Proof of Proposition 1.1
Proof of Proposition 1.1
We first recall that \(\sigma =1\) (cf. (2.5)), \(I_0\subset \Omega \) is the initial interface and \(\eta _0\) is the cut-off function in (2.12). Then we define
where \(\theta (z)\) is the solution of the ODE
We note that \(d_{I_0}\) is Lipschitz continuous in \(\Omega \), and thus by Rademacher’s theorem we have \(|\nabla d_{I_0}|\leqslant 1\) a.e. in \(\Omega \). Recalling (1.19), we define
One can verify that \({\textbf{u}}_\varepsilon ^{in}\in W^{1,2}_0(\Omega )\cap L^\infty (\Omega )\), \(\Vert {\textbf{u}}_\varepsilon ^{in}\Vert _{L^\infty (\Omega )}\leqslant 1\), and
So the condition (1.14a) is verified. To verify the others, we first compute the modulated energy in (1.7) for the initial datum \({\textbf{u}}_\varepsilon ^{in} \). We write (A.1) as
where \( {\hat{s}}_\varepsilon (x):=\left( 1-\eta _0\left( x \right) \right) \left( {\textbf{1}}_{\Omega ^+_0} -\theta \left( \frac{d_{I_0}(x)}{\varepsilon }\right) \right) \). Invoking (2.12) and the exponential convergence of \(\theta (z)\) as \(z\rightarrow \pm \infty \) (cf. (A.2)), we deduce that
for some constant \(C>0\) that only depends on \(I_0\). By a Taylor’s expansion, we find
Combining (A.3), (A.5) with (A.6), we obtain
Note that we have also employed the identities \(\partial _{x_i} {\textbf{u}}^{in}\cdot {\textbf{u}}^{in}=0\) a.e. in \(\Omega \). Recalling (1.8), we have
So we can compute
It follows from (2.10) that \(1-\varvec{\xi }\cdot {\textbf{n}}_{I_0}=O(d_I^2)\). So we have
Note that the last term can be written as
Substituting the above two equations into (A.8), we find
Note that the integrand of the first integral on the right-hand side of (A.9) vanishes due to the identity \(\theta '^2(z)=2f(\theta (z))\), which follows from (A.2). Now we turn to the first term in (1.7). Using (A.6) we can estimate
By the exponential decay of \(\theta '(z)\) as \(z\rightarrow \pm \infty \), we deduce that
Using this, (1.19) and Hardy’s inequality (cf. [7]), we find
Combining this with (A.10) and (A.9) we derive \(E_\varepsilon [{\textbf{u}}_\varepsilon ^{in} | I_0]\leqslant C\varepsilon \). Recalling (1.21), we have also obtained (1.14b). To verify (1.14c), we shall compute (1.12) at \(t=0\). By (A.7), we see that
We shall only give the estimate in \(B_{\delta _0}(I_0)\cap \Omega _0^+\) because the one in \(B_{\delta _0}(I_0)\cap \Omega _0^-\) follows in the same way, and the one in \(\Omega \backslash B_{\delta _0}(I_0)\) is due to (A.6) and the exponential convergence of \(\theta (z)\) at \(\pm \infty \).
where the last step is due to the exponential decay of \(Q(z):=z\int _{\theta (z)}^1\sqrt{2f(s)}\, ds\) as \(z\uparrow \infty \). \(\square \)
Appendix B: Proof of Proposition 2.1
Lemma B.1
The following identity holds:
Proof
We introduce the stress tensor \( (T_\varepsilon )_{ij}:= \left( \frac{\varepsilon }{2} |\nabla {\textbf{u}}_\varepsilon |^2 +\frac{1}{\varepsilon } F ({\textbf{u}}_\varepsilon ) \right) \delta _{ij} - \varepsilon \partial _i {\textbf{u}}_\varepsilon \cdot \partial _j {\textbf{u}}_\varepsilon .\) By (2.20b), we have the identity \(\nabla \cdot T_\varepsilon ={\textbf{H}}_\varepsilon |\nabla {\textbf{u}}_\varepsilon |.\) Testing this identity with \({\textbf{H}}\), integrating by parts and using (2.14c), we obtain
So adding zero leads to
which yields (B.1). \(\square \)
Lemma B.2
Under the assumptions of Theorem 1.1, the following identity holds:
where
Proof
We shall employ the Einstein summation convention by summing over repeated indices. Using the energy dissipation law in (2.6) and adding zero, we find
By the symmetry of \(\nabla ^2\psi _\varepsilon \) and the boundary conditions in (2.14c), we have
Hence, the first integral on the right-hand side of (B.5) can be rewritten as
Therefore,
Now using (B.1) to replace the third and the fourth integrals on the right-hand side of the above equation, we find
We shall show that \(J_\varepsilon ^1\) arises from the second and the third to last integrals by proving the following identity:
where \((n_\varepsilon ^\ell )_{1\leqslant \ell \leqslant 3}={\textbf{n}}_\varepsilon \). Such an identity holds obviously on the set \(\{ x\mid {\textbf{u}}_\varepsilon =0\}\) by (2.22). It also holds on \(\{ x \mid g(|{\textbf{u}}_\varepsilon |)>0\}\) due to the following identity which follows from (2.22) and (2.23a):
On the open set \(\{ x\mid |{\textbf{u}}_\varepsilon |>0\}\) which includes \(\{ x\mid |{\textbf{u}}_\varepsilon |=1\}\), we deduce from (2.22) and (2.19a) that \(\Pi _{{\textbf{u}}_\varepsilon } \partial _j {\textbf{u}}_\varepsilon =(\partial _j |{\textbf{u}}_\varepsilon |) ~{\textbf{u}}_\varepsilon \). By [18, Theorem 4.4] we have \(\partial _j |{\textbf{u}}_\varepsilon |=0\) a.e. on \(\{ x\mid |{\textbf{u}}_\varepsilon |=1\}\). We thus complete the proof of (B.7).
Now by (B.7) and adding zero, we find
Using the identities \(\nabla \psi _\varepsilon ={\textbf{n}}_\varepsilon |\nabla \psi _\varepsilon |\) and \( \nabla {\textbf{H}}:(\varvec{\xi }\otimes \varvec{\xi })=0\) (due to (2.14b)), we merge the second and the third integrals on the right-hand side of (B.6):
Now we complete squares for the first four terms on the right-hand side of (B.8). Reordering terms, we have
Substituting this identity into (B.8), we arrive at (B.2). \(\square \)
Proof of Proposition 2.1
The proof here is the same as the case \(\mu =0\), done in [40, Lemma 4.4]. This is because the form of the energy dissipation law (2.6) remains unchanged in the presence of the divergence term in (1.2a).
We first estimate the right-hand side of (B.2) by \(E_\varepsilon [{\textbf{u}}_\varepsilon | I]\) up to a constant that only depends on \(I_t\). Concerning (B.2a), it follows from the triangle inequality that
The first integral on the right-hand side of the above inequality is controlled using (2.26c). Due to the elementary inequality \(|\varvec{\xi }- {\textbf{n}}_\varepsilon |^2 \leqslant 2 (1-{\textbf{n}}_\varepsilon \cdot \varvec{\xi })\), the second integral is controlled by (2.26d). The third integral can be treated using the relation \({\textbf{H}}=({\textbf{H}}\cdot \varvec{\xi }) \varvec{\xi }+O(d_I(x,t))\) and (2.15a). So it can be controlled by (2.26e).
The integrals in (B.2b) can be controlled using (2.26c) and (2.26d). The one in (B.2c) is controlled by (2.26a). The first term in (B.2d) can be controlled using (2.26d). It remains to estimate (B.3) and (B.4). The integrals of the last two terms defining \(J_\varepsilon ^1\) can be controlled by (2.26b). Therefore,
The first and the third integrals in the last display can be estimated using (2.26b) and (2.26e) respectively. Then we employ (2.23a) to find
Finally applying the Cauchy-Schwarz inequality and then (2.26c) and (2.26d), we obtain \(\int J_\varepsilon ^1 \,dx\leqslant C E_\varepsilon [{\textbf{u}}_\varepsilon | I].\) As for \(J_\varepsilon ^2\) (B.4), we employ (2.15c) and (2.26e) to obtain \(\int J_\varepsilon ^2\,dx \leqslant C E_\varepsilon [{\textbf{u}}_\varepsilon | I].\) All in all, we have proved that the right-hand side of (B.2) is bounded by \(E_\varepsilon [{\textbf{u}}_\varepsilon | I]\) up to a multiplicative constant which only depends on \(I_t\). \(\square \)
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