Introduction

In this paper, we will study the simplified Ericksen-Leslie system modeling the hydrodynamics of nematic liquid crystals with variable degrees of orientation and kinematic transports for molecules of various shapes: (u, d, P ) : (1.1) where u(x, t) represents the velocity field of the flow, d(x, t) is the macroscopic averaged orientation field of the nematic liquid crystal modules, and P stands for the pressure function.

Here f (d) = D d F (d) = (|d| 2 -1)d is the gradient of Ginzburg-Landau potential function

represents the Leslie stress tensor and the kinematic transport term respectively. The parameter α ∈ [0, 1] is the shape parameter of the liquid crystal molecule. In particular, α = 0, 1 2 , and 1 corresponds to disc-like, spherical and rod-like molecule shape respectively and (u ⊗ w) ij = u i w j , (∇d ∇d

). For 0 ≤ k ≤ 5, P k denotes the k-dimensional Hausdorff measure on R 3 × R with respect to the parabolic distance: δ((x, t), (y, s)) = max |x -y|, |t -s| , ∀(x, t), (y, s) ∈ R 3 × R.

We let B r (x) denote the ball in R 3 with center x and radius r. For z = (x, t) ∈ R 3 × R + , denote P r (z) := B r (x) × [t -r 2 , t], and

f dxdt for any function f on P r (z).

Since the exact values of ν, λ, γ don't play roles in our analysis, we will assume

With the following identity

the system (1.1) can also be written as

(1.2) subject to the initial condition (u, d)| t=0 = (u 0 , d 0 ) in R 3 .

(1.3)

Definition. A pair of functions (u, d) : R 3 × (0, ∞) → R 3 × R 3 is a weak solution of (1.2) and (1.3)

x )(R 3 ×(0, ∞), R 3 ), and

)

(1.5)

The global and local energy inequalities for (1.2) play the basic roles: for t > 0, ˆR3

). It should be noted that the following cancellation

plays a critical role in the later analysis.

Definition. A weak solution (u, d,

The main theorem of this paper concerns both the existence and partial regularity of suitable weak solutions to the simplified Ericksen-Leslie model. Theorem 1.1. For any u 0 ∈ H, d 0 ∈ H 1 (R 3 , R 3 ) such that F (d 0 ) ∈ L 1 (R 3 ), there exists a global suitable weak solution (u, d, P ) : R 3 × R + → R 3 × R 3 × R of the simplified Ericksen-Leslie system (1.2) and (1.3) such that

where Σ ⊂ R 3 × R + is a closed subset with P 15 7 +σ (Σ) = 0, ∀σ > 0.

A couple of remarks on the presence and size of the singular set Σ are in orders.

Remark 1.2. Mathematically, it is a very challenging problem to ask if the set of singularity Σ is empty or not. Physically, the presence of potential singular set Σ for a solution (u, d) to the hydrodynamic system (1.2) may arise from the 3-D turbulence phenomenons of the underlying fluids (e.g., vortex points, lines, or filaments) as well as the defects of the liquid crystal molecular alignment field d induced by the rotating and stretching effects of fluid velocity field u, see for example Chorin [4]. While Mandelbrot conjectured in [22,23] that the self-similar nature of turbulence of the fluid may result in concentration of possible singularities of u on a set of fractional Hausdorff dimension.

Remark 1.3. The best known result on the set of singularities for the Navier-Stokes equation was obtained by Caffarelli-Kohn-Nirenberg [3], which asserts that it has zero 1-dimensional parabolic Hausdorff measure. For the co-rotational Beris-Edward Q-tensor system for liquid crystals, a result similar to [3] was also obtained by [6] (Q T ) (see the section 5 below). This paper is organized as follows. In section 2, we will derive both the global and local energy inequality for smooth solutions of (1.2) and (1.3). In section 3, we will demonstrate the construction of suitable weak solution. In Section 4, we will prove the ε 0 -regularity criteria for the suitable weak solutions. In section 5, we will finish the proof of the Theorem 1.1.

Global and local energy inequalities

In this section, we will derive both the global and local energy equalities for smooth solutions to (1.2).

) be a solution to the simplified Ericksen-Leslie system (1.2). Then it holds that

Proof. The proof is standard. See for instance [24,25].

Proof. Multiplying the u equation in (1.2) by uφ, integrating over R 3 , and by integration by parts we obtain

By taking derivatives of d equation in (1.2), we have

Then multiplying this equation by ∇dφ, integrating over R 3 , we get

Moreover, multiplying the d equations by f (d)φ, integrating over R 3 , we get

Hence, by adding (2.3), (2.4), (2.5) together, and applying (1.8), we get (1.7).

Existence of suitable weak solutions

In this section, we will follow the same scheme in [3,6] to construct a suitable weak solution to (1.2). We introduce the so-called retarded mollifier Ψ θ for f :

and the mollifying function

It is easy to verify that for θ ∈ (0, 1] and 0

Now with the mollifier Ψ

), we introduce the approximate system of (1.2):

subject to the initial and boundary condition (1.3).

For a fixed large integer N ≥ 1, set θ = T N ∈ (0, 1], we want to find (u θ , d θ , P θ ) solving (3.1). This amounts to solving a coupling system of a Stokes-like system for u and a semi-linear parabolic-like equation for d with smooth coefficients. For m = 0, we have Ψ θ [u θ ] = Ψ θ [d θ ] = 0, and the system (3.1) reduces to a decoupled system

which can be solved easily by the standard theory. Suppose now that the (3.1) has been solved for some 0 ≤ k < N -1. We are going to solve (3.1) in the time interval [kθ, (k + 1)θ] with an initial data (u, d)

Then one can solve the coupling system (3.1) using the Faedo-Galerkin method. In fact, for a pair of smooth test functions (φ, ψ)

)

We can solve the ODE system (3.4)-(3.5) with test function (ψ, φ) chosen to be the basis of

Next we need a uniform bound on (u θ , d θ , P θ ) to pass the limit θ → 0 to get a suitable weak solution. First by direct calculations we can show that

(3.7)

From (3.6), we can obtain that sup

Combining (3.7) and (3.8), we get

From (3.8) and (3.9), we have that u θ is uniformly bounded in

. Therefore, after passing to a subsequence, there exist u ∈

By the Sobolev-interpolation inequality, we have that ∇d θ ∈ L 10 t L 30 13

x , d θ ∈ L 10 t L 10 x , and ˆT 0 ∇d θ 10

By the lower semicontinuity and (3.6), we have, for

holds for a.e. 0 ≤ t ≤ T . Now we want to estimate the pressure function P θ . Taking the divergence of u θ equation in (3.1) gives

(3.13)

For P θ , we claim that P θ in L 5 3 (R 3 × [0, T ]) and

In fact, by Calderon-Zgymund's L p -theory, we have

This uniform estimate implies that there exists P ∈ L 5 3 (R 3 × [0, T ]) such that as θ → 0,

Recalling the u θ equation, we get

and

Similarly, we can show

and

Hence by the Sobolev embedding and Aubin-Lions' compactness Lemma, we can conclude that as θ → 0,

Furthermore, (u θ , d θ , P θ ) satisfies the local energy inequality. In fact, if we multiply the u θ equation in (3.1) by u θ φ, take derivative of the d θ equation in (3.1) and multiply by ∇d θ φ, multiply the d θ equation in (3.1) by f (d θ ), and perform calculations similar to the previous section, we can get

With the convergence (3.14), (3.15), it is easy to check that the limit (u, d) is a weak solution to (1.2) and (1.3). Taking the limit in (3.16) as θ → 0, by the lower semicontinuity we obtain

Putting all those together we show that the local energy inequality (1.7) holds. Therefore (u, d, P ) is a suitable weak solution to (1.2) and (1.3).

ε 0 -Regularity criteria

In this section we will establish the partial regularity for suitable weak solutions (u, d, P ) of (1.2) in R 3 × (0, ∞). The argument is based on a blowing up argument, motivated by that of Lin [18] on the Navier-Stokes equation. Recently, this type of argument has been employed by Du-Hu-Wang [6] for the partial regularity in the co-rotational Beris-Edwards system in dimension three. However, the kinematic transport effects in (1.2) destroy the maximum principle for d, which is necessary to apply the argument by [18] and [6]. To overcome this new difficulty, we adapt some ideas from Giaquinta-Giusti [10] to control the mean oscillation of d in L 6 . More precisely, we have Lemma 4.1. For any M > 0, there exist

, and

and

In the absence of maximum principle for the director field d, the L 6 -norm of the mean oscillation of d plays the role in obtaining the (local) boundedness of (u, ∇d) ∈ (4.15). By closely examining the proof of Lemma 4.1, the L 6 -norm can be relaxed to the L p -norm of the mean oscillation of d as long as p > 5. However, this does not seem to improve the estimate of the dimension of the singular set Σ of (u, ∇d), since we can

, which can yield the boundedness of L 20 3 -norm of the mean oscillation of d (see (5.4)

below).

Proof. We prove it by contradiction. Suppose that the conclusion were false. Then there exists M 0 > 0 such that for any τ ∈ (0, 1 2 ), there exist ε i → 0, C i → ∞, and r i > 0, and

and

From (4.6), we see that

Define the blowing-up sequence

where

It follows from (4.4), (4.5) that

Furthermore, ( u i , d i , P i ) is a suitable weak solution of the blowing-up version of (1.2):

(4.10)

From (4.8), we assume that there exists

such that, after passing to a subsequence,

It follows from (4.8) and the lower semicontinuity that

We claim that

We choose a cut-off function φ ∈ C ∞ 0 (P 1 (0)) such that 0 ≤ φ ≤ 1, φ ≡ 1 on P 1 2 (0), and

Replacing φ by φ 2 i in (1.7), by Young's inequality we can show sup

By rescaling and using the estimates (4.7), (4.8), and (4.9), we can show that sup

This yields (4.13). Hence we may assume that

Thus by the same interpolation as in (3.11), we have

and there exists a constant d ∈ R 3 , with |d| ≤ M 0 , such that, after passing to subsequence,

(4.17)

Hence ( u, d, P ) :

By Lemma 4.3 and (4.12), we have that (

and ∃α 0 ∈ (0, 1) such that

We now claim that

In fact, from the equation for u i and d i in (4.10) we can conclude that

x +L

Thus (4.21) follows from Aubin-Lions' compactness Lemma. This implies that for any

),

where lim i→∞ o(1) = 0. Now we need to estimate the pressure P i . By taking divergence of the u i equation in (4.18) we see that

We claim that τ -foot_1 ˆPτ(0)

does not necessarily have a small L 2 -norm in P1 2 (0), to achieve (4.25) we will show the following strong convergence in L 2 :

In order to prove (4.26), first observe that by subtracting the equation (4.10) from the equations (4.18), we see that

solves the following system of equations in P1 2 (0):

div u i = 0,

(4.27)

Since ( u i , d i , P i ) is a suitable weak solution of (4.10) and Lemma 4.2 guarantees the smoothness of ( u, d, P ), it is not hard to see that (4.27) also enjoys a local energy inequality which leads to (4.26). In fact, multiplying the u i equation by u i φ, and ∇ d i equation by ∇ d i φ, integrating the resulting equation over R 3 × [0, T ], and applying the integration by parts, we obtain that

Therefore we can add (4.28) and (4.29) to obtain that

(4.32)

From the convergence (4.16), we know that lim i→∞

This, together with (4.17), implies that as i → ∞,

I k → 0 and

Therefore

and (4.26) holds.

) 2 ≤ t ≤ 0, define P

i (•, t) : R 3 → R by

i (•, t) = ( P i -P

i )(•, t). Then

i , by the Calderon-Zgymund theory we have that

Hence we have

From the standard theory on linear elliptic equations, P

∈ C ∞ (B 5 16 (0)) satisfies that for any 0 < τ < 9 32 , ) and sufficiently large i 0 , depending on τ 0 , such that for any i ≥ i 0 , it holds that

This contradicts (4.8). The proof of Lemma 4.1 is completed. Now we will establish the smoothness of the limit equation (4.18) in the following lemma.

) and

) is a weak solution of the linear system (4.18), then ( u, d) ∈ C ∞ (P1 4 (0)), and the following estimate

holds for any τ ∈ (0,

).

Proof. The smoothness of the limit equation (4.18) doesn't follow from the standard theory of linear equations, since the source term of u equations involve terms depending on the third order derivatives of d. It is based on higher order energy methods, for which the cancellation property, as in the derivation of local energy inequality for suitable weak solution to (1.2), plays a critical role. This strategy has been adapted by Huang-Lin-Wang in [11,Lemma 3.2] for the full Ericksen-Leslie system in 2D. However, it is more delicate here due to the low temporal integrability of pressure. To address this issue, we split the pressure into two parts P (1) and P (2) , where P (1) solves the Poisson equation involving ∆ d which belongs to L 2 , and P (2) , while is only L 3 2 in time, is harmonic in space. In fact, if we take the divergence of the equation (4.18) 1 , then we have P satisfies the following Poisson equation: Define P (1) (•, t) : R 3 → R,

and P (2) (•, t) := ( P -P (1) )(•, t). For P (1) , by Calderon-Zygmund's singular integral estimate we have

Hence we can integrate the inequality above in time to get ˆP1

For P (2) , it is easy to see that

By the standard regularity theory of harmonic function we have ˆP 5

16

Once again, we have the cancellation

(4.47)

We have the following estimates:

For the pressure P , taking divergence of the equation (4.18) 1 yields that for any -

It turns out that we can extend the energy method above to arbitary order. Here we sketch the proof. For nonnegative multiple indices β, γ and δ such that γ = β + δ and δ is of order

(4.53)

By differentiating ( P (1) , P (2) ) (k -1) times we can estimate ˆP1

Thus, we get sup

From Sobolev's interpolation inequality, we have

Substituting this inequality in (4.60) and by suitable adjusting of the radius, we can show that sup

) , P

) .

(4.61) With (4.61), we can apply the regularity for both the linear Stokes equations and the linear heat equation (c.f. [16,21]) to conclude that ( u, d)

). Furthermore, applying the elliptic estimate for the pressure equation (4.40), we see that P ∈ C ∞ (P 1

4

). Therefore ( u, d, P ) ∈ C ∞ (P 1

4

) and the estimate (4.39) holds. The proof is completed.

The oscillation Lemma admits the following iterations.

Lemma 4.4. Let (u, d, P ), M, ε 0 (M ), τ 0 (M ), C 0 (M ), z 0 be as in Lemma 4.1. Then there exist r 0 = r 0 (M ),

then for any k = 1, 2, . . . , we have

Proof. We prove it by an induction on k. By translational invariance we may assume that z 0 = 0, and we abbreviate d r to be d 0,r for simplicity.

For k = 1, the conclusion follows from Lemma 4.1, if we choose ε 1 such that ε 1 < ε 0 . Suppose the conclusion is true for all k ≤ k 0 , k 0 ≥ 1, we show it remains true for k = k 0 + 1. By the inductive hypothesis

Then

If we choose sufficiently small r 0 = r 0 (M ), ε 1 = ε 1 (M ), we see

It follows directly from Lemma 4.1 with r replaced by τ k 0 r that

This completes the proof.

The local boundedness of the solutions can be obtained by utilizing the Riesz potential estimates between Morrey spaces as in the following lemma. Lemma 4.5. For any M > 0, there exists

then for any

(u, ∇d) L p (P r 0 4

where ε 1 is the constant in Lemma 4.4.

Meanwhile,

Hence we get that

Then we deduce from Lemma 4.4 that for any k = 1, 2, . . . ,

By Lebesgue's differentiation theorem, we have |d| ≤ M a.e. in P r 0 2 (z 0 ). Furthermore, we have

), it holds for any 0 < s < r 0 2 and z ∈ P r 0 2 (z 0 ),

By the Campanato theory, d ∈ C θ (P r 0 4 (z 0 )) and (4.65) holds. Now for φ ∈ C ∞ 0 (P r 0 2 )(z 0 ), from (2.3), (2.4) we can derive the following local energy inequality:

Let φ ∈ C ∞ 0 (P 2s (z)) be a cut-off function of P s (z). Replacing φ by φ 2 in (4.68), we can show that for 0 < s < r 0 2 ,

Now we are ready to perform the Riesz potential estimate. For any open set

It follows from (4.67) and (4.69) that there exists α ∈ (0, 1) such that

2 ). Then

We can check that F ∈ M where I β is the parabolic Riesz potential of order β on R 4 , 0 ≤ β ≤ 5, defined by

Applying the Riesz potential estimates [12], we conclude that ∇w ∈ M

Since lim

= ∞, we conclude that for any 1 < p < ∞, ∇w ∈ L p (P r 0 (z 0 )) and

, it follows from the theory of heat equations that ∇(d -w) ∈ L ∞ (P r 0 4 (z 0 )). Therefore for any 1 < p < ∞, d ∈ L p (P r 0 4 (z 0 ), and ∇d L p (P r 0 4 (z 0 )) ≤ C(p)(r 0 + ε 1 ).

We now proceed with the estimation of u. Let v : R 3 × (0, ∞) → R 3 solve the Stokes equation:

By using the Oseen kernel, an estimate of v can be given by

where

As above, we can check that X ∈ M Hence we conclude that v ∈ M

), and

, we have that uv ∈ L ∞ (P r 0 4 (z 0 )). Therefore for any 1 < p < ∞, u ∈ L p (P r 0 4 (z 0 )) and u L p (P r 0

For the rest of this section, we will establish the higher order regularity of (1.2). Again we prove it via a high order energy method which has been employed by Huang-Lin-Wang [11] for general Ericksen-Leslie systems in dimension two, and Du-Hu-Wang [6] for co-rotational Beris-Edwards model in dimension three. Lemma 4.6. Under the same assumption as Lemma 4.5, we have that for any k ≥ 0,

(z 0 ) and the following estimates hold

(4.78)

In particular, ( u, d) is smooth in P r 0 4 (z 0 ). Proof. For simplicity, assume z 0 = (0, 0) and r 0 = 2. (4.78) can be proved by an induction on k. It is clear that when k = 0, (4.78) follows directly from the local energy inequality (4.68). Here we indicate to how to proof (4.78) for k ≥ 1. Suppose that (4.78) holds for k ≤ l -1, we want to show that (4.78) also holds for k = l. From the induction hypothesis, we have that for 0 Also, for 1 ≤ j ≤ l -1, we have

By Lemma 4.5 we also have that any i ∈ N + and 1 < p < ∞,

Notice that ∇ l-1 P satisfies

, and

and P (foot_5) (•, t) := (P -P (1) )(•, t). For P (1) , we have that We see that P (2) satisfies

Then we derive from the regularity of harmonic function that for 1

Now take l-th order spatial derivative of the equation (1.2) 1 , we have 2

(4.89)

Now we have the following estimate:

For I 5 , set A l α := S α [∇ l ∆d, d], and B l α := ∇ l S α [∆d, d] -A l α , then we have

Then we get

Now we take (l + 1)-th order spartial derivative of the equation (1.2) 3 , we have

Then we have the following estimates:

Now we estimate

Combine all estimate above, and with the cancellation I 51 = K 21 , we arrive at

ˆB2

|∇ l+1 d| 2 η 2 dx.

For lower order terms, we have that for 1 ≤ j ≤ l -1, ˆB2 |∇ l-1 u| 2 |∇ j u| 2 η 2 dx ≤ ∇ l-1 uη This yields that the conclusion holds for k = l. Thus the proof is complete.

Partial regularity

As a consequence of Lemma 4.6, we get the following regularity criteria for (1. Then there exists δ 1 > 0 such that (u, d) ∈ C ∞ (P δ 1 (z)).

The following Lemma is well-known, see [10]. p (Q T ), 1 ≤ p < ∞, contains all f 's satisfying From the global energy estimate (1.6) and the Sobolev embedding theorem, we have