Concentration-cancellation in the Ericksen-Leslie model

We establish the subconvergence of weak solutions to the Ginzburg-Landau approximation to global-in-time weak solutions of the Ericksen-Leslie model for nematic liquid crystals on the torus $\mathbb{T}^2$. The key argument is a variation of concentration-cancellation methods originally introduced by DiPerna and Majda to investigate the weak stability of solutions to the (steady-state) Euler equations.


Introduction
The Ericksen-Leslie model describes the motion of nematic liquid crystal flows [11,19]. The nematic phase of liquid crystals can be thought of as an intermediate state of isotropic flow and a solid crystalline phase where the rod-like molecules do not act freely but tend to align in a certain direction. In order to depict this behaviour, two quantities are used to model the liquid crystal, the velocity v and the unitary director field d which represents the orientation of the molecules in space.
In this article, we study the variant of the Ericksen-Leslie model proposed in [21], which reads (all physical constants set to one) on the space-time domain T 2 × [0, T ] for any given T > 0. Here, p : T 2 × [0, T ] → R denotes the underlying pressure and serves as a Lagrange multiplier subject to the incompressibility condition div v = 0. The system is supplemented by initial data (v 0 , d 0 ) : T 2 → R 2 × S 2 , div v 0 = 0. We refer to [16] for modelling issues of the general Ericksen-Leslie equations as well as analytical aspects. The energetic variational approach is executed in e.g. [7,22].
From the mathematical point of view, (1.1) preserves the major mathematical challenges of the full dynamic Ericksen-Leslie model. Indeed, even on two-dimensional domains, the system (1.1) might form singularities in finite-time as shown by Huang et al. in [18]. Establishing existence or uniqueness of solutions is therefore a non-trivial problem mainly due to the harmonic map heat flow-like equation for the director field d. In order to construct solutions to (1.1), one would like to use an approximation scheme and pass to the limit with the help of a-priori estimates or a more refined analysis. However, the associated energy law, see (3.1) below, does not provide strong enough bounds, and the main problem consists of the limit passage on the right-hand side in the momentum equation. Lin, Lin and Wang [20] and Hong [17] first proved the existence of weak solutions to (1.1) on a two-dimensional bounded domain or R 2 . Both relied on Struwe's [31] construction of partially regular solutions of the harmonic map heat flow in two dimensions. To this end, Hong constructed local-in-time smooth solutions to (1.1) via the Ginzburg-Landau approximation with ǫ → 0 + . System (1.2) depicts a well established approximation to (1.1) (cf. [16]) in order to circumvent the above mentioned difficulties. Lin and Liu [22] first showed existence of weak solutions as well as strong solutions to (1.2) on a bounded domain either in two dimensions or under a smallness condition on the initial data in dimension three for fixed ǫ > 0. The energy related to (1.2) reads where the second term penalizes variations from the constraint |d ǫ | ≡ 1. As ǫ tends to zero, the director field is forced to attain values in the sphere, i.e. |d ǫ | → 1. Thus one expects convergence of d ǫ to solutions of (1.1). Indeed, this fact is proven for strong solutions locally in time in [13,17]. However, an extension of this strong convergence result to larger times is not possible due to blow-up of some solutions to (1.1).
In this work, we actually prove the subconvergence of weak solutions to (1.2) to weak solutions of (1.1) globally in time. In [16, p. 1108] and [2, p. 290] this issue was highlighted as an open problem. This limit passage is also of interest for numerical approximations [34], in the stochastic Ericksen-Leslie system [8] or the flow of magnetoviscoelastic materials (see [29]). The singular limit problem ǫ → 0 + for the harmonic map heat flow into spheres and more general manifolds was first studied by Chen and Struwe in [5,6] (see also [1] for the related Landau-Lifshitz equation). In the Ericksen-Leslie model, the difficulty is to pass to the limit in the stress tensor −div(∇d ⊙ ∇d) as long as one is restricted to the energy estimate (3.1). In general, ∇d ǫ ⊙ ∇d ǫ ⇀ * ∇d ⊙ ∇d + η for a possibly non-vanishing matrix-valued measure η. We will not show η = 0 but use the idea of concentration-cancellation for Euler equations introduced by DiPerna and Majda in [10] (see also [25]) to verify that the weak limit (v, d) fulfills (1.1). This procedure becomes possible since div(∇d ⊙ ∇d) enjoys the same structure as the convective term in the Euler equations While this technique is successful here, we remark that it cannot be used to prove existence of weak solutions to the time-dependent Euler equations in general. One main problem depicts the low regularity of ∂ t v. However, there exist cases where concentration-cancellation occurs, see e.g. the result of Delort [9] for non-negative vorticities bounded in the space of measures (see also [12]), or if certain assumptions on the time-derivative of v [30] or the size of the defect measure are made [10,30]. In contrast to the delicate situation for the Euler equations, enough regularity of ∂ t v in (1.2) is available, see (3.2), such that we can perfom the limit passage without further assumptions. Indeed, it turns out to be sufficient to stick to the initial idea of [10].
A crucial idea of the proof is to fix a time t ∈ [0, T ] and then to carry out a concentrationcancellation argument in the limit passage. In particular, we show that (∇d ǫ (t)) ǫ may concentrate only in a finite number of points. This result is in correspondence with well-known results for approximated harmonic maps, cf. [23,27,35]. From the smallness of the concentration set we conclude that the limit of (1.2) satisfies (1.1) in the weak sense. The method of fixing a time step is inspired by [24], where Lin and Wang consider the three-dimensional liquid crystal flow in the special case of solutions with values in the upper half-sphere d(x, t) ∈ S 2 + . We carry out this program on the space domain T 2 , which allows to use the Fourier expansion.
The structure of the paper is as follows: In the second section, basic notation and the main results, Theorems 2.1 and 2.2, are stated. The third section deals with the proofs of these theorems. The proofs are divided into several steps: Section 3.1 is devoted to a-priori estimates, Sections 3.2 and 3.3 provide an ε 0 -regularity statetment and an estimate on the concentration set in space and Section 3.4 concludes the proofs with the limit passage explained above.

Setting and results
Defining A ⊙ B := A ⊤ B, we investigate the initial value problem on T 2 × [0, T ] with T 2 = (R/2πZ) 2 and T > 0 given. The prescribed initial data consist of On T 2 , we may write f ∈ L 2 (T 2 , R 2 ) as Fourier expansion f = k∈Z 2fk e ik·(·) . The homogeneous space of square-integrable functions is denoted byL 2 (T 2 , R 2 ) as well asẆ 1,2 (T 2 , R 2 ) for the homogeneous Sobolev space. We use X div for (weakly) solenoidal functions in the function space X (e.g. for X = L 2 , W 1,p , C ∞ ...). Note that it makes sense to consider v ∈L 2 div as a solution to (2.1) whereas d is rather considered to be an element of the nonhomogeneous space W 1,2 (T 2 , S 2 ) due to the constraint |d| ≡ 1. It is useful to represent f ∈L 2 div (T 2 , R 2 ) as where ∇ ⊥ = (−∂ 2 , ∂ 1 ) ⊤ and g ∈ W 1,2 (T 2 ). That this is possible is easily seen by Fourier expansion (f ∈L 2 div implies k ·f k = 0 for all k ∈Ż 2 , which in turn impliesf k = (−k 2 , k 1 ) ⊤ λ k for some λ k ∈ C and all k ∈Ż 2 = Z 2 \{(0, 0) ⊤ }).
For some Banach space X, the time-dependent Bochner spaces are denoted by L p (0, T ; X) or W 1,p (0, T ; X) respectively and we use · L p (0,T ;X) = · L p t Xx short hand for the norm. Now we are in the position to define a weak solution to (2.1)-(2.5): is called a weak solution to the initial value problem (2.1)-(2.3) subject to the initial conditions Our notion of weak solutions resembles the usual definition of weak harmonic map heat flows (see [6]) and weak Navier-Stokes flows (see [26]). In contrast to previous works, we construct these weak solutions out of weak solutions to the Ginzburg-Landau approximation. The latter reads for 0 < ǫ ≤ 1. Weak solutions (v ǫ , d ǫ ) to (2.6)-(2.8) are known to exist globally in time. Here, we eludicate the limiting behavior as ǫ → 0 + . Formally, (v ǫ , d ǫ ) converge to solutions of (2.1)-(2.5).
We give a precise meaning to this idea with the following result: . Then there exists a weak solution (v, d) in the sense of Definition 2.1 to the system (2.1)-(2.5) that satisfies the energy inequality, i.e.
Remark 2.2 (Uniqueness). As for the harmonic map heat flow, we do not know whether the solution in Theorem 2.2 is a solution in the sense of Struwe in [31]. In particular, the energy is not known to be nonincreasing. For the harmonic map heat flow, Bertsch et al. [3] and Topping [33] proved the existence of infinitely many weak solutions with conserved but increasing energy at certain time steps. The same behaviour may be possible in our case.
Remark 2.3 (Stability with respect to initial data). It can easily be checked that Theorem 2.1 remains true for a sequence of initial data . However, if only weak convergence is given, a result like Theorem 2.1 may not be available in general since various oscillation and concentration effects occur.
3 Proofs of Theorem 2.1 and 2.2

A-priori estimates
This section is devoted to establishing the Ginzburg-Landau approximation and the collection of (mostly standard) a-priori estimates. As for most systems arising from physics, this is done by employing the energy law associated to the system and secondary estimates on the time derivatives via duality.
In [22] (see also [29]), Lin and Liu showed that global-in-time weak solutions to the Ginzburg-Landau approximation exist for initial data 1 The result was proven by a Galerkin approximation scheme and carries over to the case Ω = T 2 . More precisely, there is a unique pair solving (2.6)-(2.8) in the weak sense. This regularity suffices to perfom typical calculations yielding energy estimates and a-priori bounds. Multiplying (2.6) by v, (3.1) 1 In [22], d0 ∈ W 3/2,2 (∂Ω) is required for a smooth domain. This is omitted since ∂T 2 = ∅.
Here we benifited from the fact that |d 0 | ≡ 1 almost everywhere. Further, d ǫ enjoys a maximum principle (see [1,24]): is a solution to (2.6)-(2.8). Then d ǫ satisfies Proof. For k ∈ N we define the auxiliary function h k ǫ : By (2.7), we have ≤ ∆h k ǫ in the weak sense. Next we multiply the differential inequality by h k ǫ , and an integration by parts yields (due to the periodicity of T 2 and |d 0 | ≡ 1) This can only be true if h k ǫ = 0 a.e. on T 2 × [0, T ) and therefore the assertion follows.
The energy law (3.1) and Lemma 3.1 yield a-priori bounds x ≤ C. In order to achieve strong convergence we make use of the generalized Aubin-Lions lemma [28,Lemma 7.7] for which some bounds on the time derivatives of (v ǫ , d ǫ ) are needed. The estimates above and (2.7) allow us to deduce ∂ t d ǫ Considering ∂ t v ǫ , we first note that for φ ∈ C ∞ div (T 2 , R 2 ) one has Secondly, we employ the identity div(∇d ǫ ⊙ ∇d ǫ ) = ∇ |∇dǫ| 2 2 + (∇d ǫ ) ⊤ ∆d ǫ . Now testing (2.6) by is valid independently of ǫ > 0. We point out that we benefited from the interplay of solenoidal test functions and the gradient flow structure of the system to improve the control in space of ∂ t v ǫ compared to the standard estimate ∂ t v ǫ ∈ L 2 (0, T ; (W 1,∞ div (T 2 , R 2 )) * ). This is in sharp contrast to the framework of the Euler equations and allows us to use the concentration-cancellation techniques from [10] later on. As a consequence of the above estimates, we can choose a subsequence (ǫ i ) i∈N ⊂ (0, 1] with lim i→∞ ǫ i = 0 + such that v ǫ i → v in L 2 (T 2 × [0, T ], R 2 ) and a.e., (3.3) for any p ∈ (1, ∞) and a.e., (3.6) ) and a.e., (3.7) Additionally, we can choose the subsequence such that

ε 0 -regularity
According to the idea of fixing certain time steps in [0, T ] and passing to the limit, we consider the equation on T 2 for some τ ǫ ⇀ τ in L 2 (T 2 , R 3 ) and u ǫ ∈ W 2,2 (T 2 , R 3 ) being a strong solution for ǫ > 0. This is motivated by equation (2.7) where taking u ǫ = d ǫ (t) (at first formally) for fixed t ∈ [0, T ] leads to this situation. As ǫ → 0 + , we have a singular limit problem and we expect that u ǫ converges to an approximated harmonic map u : T 2 → S 2 , i.e.
Proof. Let x 0 = 0 without loss of generality. The proof is divided into four steps.

The concentration set Σ t
With respect to Theorem 3.1 we need to determine the properties of the set where strong convergence of (u ǫ k ) k = (d ǫ k (t)) k is available. As pointed out previously, strong convergence fails in finitely many (isolated) points. We define the set of singular points at time t ∈ (0, T ] by where ε 0 is given by Theorem 3.1.

Lemma 3.2.
There exists a constant K = K(E 0 ) > 0 independent of t such that Proof. Choose a finite subset for all k ≥ k 0 by construction of Σ t . Thus we have due to the energy estimate (3.1). By the arbitrariness of A N , the set Σ t consists of at most K := E 0 ε 2 0 points.
As a consequence of the previous lemma, we find a subsequence of (d ǫ k (t)) k strongly converging on T 2 \Σ t for every t under consideration.
Then there exists a subsequence 3 such that in L 2 loc (T 2 \Σ t , R 3×2 ). Proof. Let (z j ) j∈N be the set of rational points on T 2 \Σ t and define r j := sup r > 0 : lim inf k→∞ Br(z j ) In general the radii r j might be too large to satisfy which is why we consider 4 5 r j . In view of Theorem 3.1 we want to prove j∈N B r j /5 (z j ) = T 2 \Σ t . Let z ∈ T 2 \Σ t with r z > 0 be such that By density, we choose a z j 0 such that |z j 0 − z| < rz 6 . Thus we have r j 0 ≥ 5 6 r z from (3.17) and therefore |z j 0 − z| < r j 0 5 . Since the covering is countable we use a diagonal argument, Theorem 3.1 and Lemma 3.2 to extract a subsequence which fulfills the assertion.

Limit passage ǫ → 0 +
Finally, exploiting the results from the previous section, we want to conclude the desired limit passage as ǫ → 0 + . As mentioned in the introduction, the idea is to look at fixed times t and use concentration-cancellation arguments from [10] for the stress tensor term div(∇d ⊙ ∇d) in the momentum equation.
The inspiration for this argument is taken from [15], which in turn relies on [31]. Well-known arguments from [6] (see also [1]) then allow to pass from (2.7) to (2.3).
To begin with, we consider the weak formulation of (2.7). Note that for ξ ∈ W 1,2 ( for a, b, c : T 2 → R 3 being weakly differentiable. From the convergence statements (3.3), (3.6)-(3.9) and the bound from the maximum principle |d ǫ | ≤ 1 a.e. for all ǫ > 0, we conclude that the limit of the weak formulation is Here we have |d| ≡ 1 a.e. therefore all derivatives (in particular the first term involving ∂ t d and ∂ x j d for j = 1, 2) are perpendicular to d a.e. Using this fact and setting for the wedge-product. This shows that the limits (v, d) satisfy the director equation (2.3). The remaining part is to show that (v, d) fulfill the momentum equation. As ∇d ǫ (t) ⇀ ∇d(t), Theorem 3.1 and Proposition 3.2 show that no oscillations occur in the limit and the set of concentrations is finite at least for a subsequence. Recall from (3.10) and (3.11) that (v ǫ , d ǫ )(·, t) strongly converges to the limit for a.e. t ∈ [0, T ]. We set τ ǫ := ∂ t d ǫ + (v ǫ · ∇)d ǫ and τ := ∂ t d + (v · ∇)d. Due to the a-priori bounds from Section 3.1, the set has full measure by Fatou's lemma, i.e. |A| = T . Without loss of generality, let A be such that (v ǫ , d ǫ )(t) → (v, d)(t) as in (3.10) and (3.11) for every t ∈ A. Fix t ∈ A. Now there exists a subsequence for which where we identified the limit in t by the strong convergence of (v ǫ j (t), d ǫ j (t)) j∈N in L 2 . Since this is true for any subsequence, the full sequence ((∂ t v ǫ , ∇v ǫ , ∇d ǫ , τ ǫ )(t)) ǫ converges weakly. Next, we take a test function φ ∈ C ∞ div (T 2 , R 2 ). Since φ is solenoidal, it is for some η ∈ C ∞ (T 2 ) (see Section 2) and the constant vanishes if T 2 φ = 0. Testing the momentum equation (2.6) at time t ∈ A\{0} by φ we obtain By Lemma 3.3, there exists a subsequence 5 (v ǫ , d ǫ ) ǫ , which in general depends on t, such that where Σ t is finite according to Lemma 3.2. By density, it suffices to show the weak formulation (3.18) in the limit for all functions η(x) = e ik·x with k ∈Ż 2 . First note that the only problematic terms are the ones related to ∂ t v and ∇d ⊙ ∇d. However, choosing a smooth cut-off function ψ which vanishes in a neighborhood of Σ t , we may pass to the limit with the test function ∇ ⊥ η(x) = ∇ ⊥ e ik·x ψ(x) , i.e.
It remains to "fill" the holes and we do so by considering every point in Σ t separately. To this end, observe that the equations (2.1) and (2.6) are covariant under rotations. To be more specific, let 0 = x 0 ∈ Σ t , without loss of generality, be the invariant point of the rotation. Taking a test function ∇ ⊥ β with supp β ⊂ B r , r > 0, the rotation of coordinates Qy = x for Q ⊤ = Q −1 yields by a change of variables. Similar identities hold for all other terms, i.e.

not relabeled
For r > 0 sufficiently small we know by Lemma 3.3 that (∇d ǫ ⊙ ∇d ǫ )(·, t) only concentrates in x 0 = 0 ∈ B r and so does (∇ y d ǫ ⊙ ∇ y d ǫ )(Q(·), t) by the same token. We imitate the key idea of [10]. Because of the rotational covariance, it is enough to consider test functions h(x) = h(x 1 ) in a neighborhood of the concentration point x 0 = 0. To cut off the concentration point, define h n for n ∈ N large enough by the elliptic ODE h ′′ n = (1 − 1 (−1/n,1/n) )h ′′ , h n (−r) = h(−r), h n (r) = h(r).