DISCRETE AND CONTINUOUS doi:10.3934/dcds.2016045 DYNAMICAL SYSTEMS Volume 36, Number 10, October 2016 pp. 5579–5594
SERRIN’S REGULARITY RESULTS FOR THE INCOMPRESSIBLE LIQUID CRYSTALS SYSTEM
Xian-Gao Liu School of Mathematic Sciences, Fudan University Shanghai, China Jianzhong Min School of Mathematic Sciences Fudan University/Shanghai University of Medicine and Health Sciences Shanghai, China Kui Wang School of Mathematic Sciences, Soochow University Suzhou, China Xiaotao Zhang School of Mathematic Sciences, Fudan University Shanghai, China
(Communicated by Chongchun Zeng)
Abstract. In this paper, we study the simplified system of the original Erick- sen–Leslie equations for the flow of liquid crystals [10]. Under Serrin criteria [13], we prove a partial interior regularity result of weak solutions for the three- dimensional incompressible liquid crystal system.
1. Introduction. The three-dimensional incompressible liquid crystals system are the following coupled equations ut − ∆u + u · ∇u + ∇P = −div(∇d ∇d), div u = 0, (1.1) dt + u · ∇d − ∆d = f(d), 3 in Q = Ω×[0,T ], for a bounded and smooth domain Ω in R . Where u = (u1, u2, u3) is the velocity field, P is the scalar pressure and d = (d1, d2, d3) is the optical 1 2 molecule direction after penalization, and f(d) = σ2 (|d| − 1)d. Here ∇d ∇d is a symmetric tensor with its component (∇d ∇d)ij given by ∇id · ∇jd. And the initial and boundary conditions are:
u(x, 0) = u0(x), with div(u0) = 0, d(x, 0) = d0(x), for x ∈ Ω
u(x, t) = 0, d(x, t) = d0(x), for (x, t) ∈ ∂Ω × [0,T ] System (1.1) is the simplified system of the original Ericksen–Leslie equations for the flow of liquid crystals. For this system, Lin-Liu [11] proved a regularity result of the suitable weak solution under the C-K-N condition, see [1]. Shortly after,
2010 Mathematics Subject Classification. Primary: 35Q35; Secondary: 76A15. Key words and phrases. Incompressible liquid crystals, Serrin criteria, partial regularity, Calder´on-Zygmund inequality, heat kernel theory.
5579 5580 XIAN-GAO LIU, JIANZHONG MIN, KUI WANG AND XIAOTAO ZHANG
Lin-Lin-Wang [12] also studied the regularity for the nematic liquid crystal system in dimension two, which is similar to system (1.1) with the third equation replaced by 2 dt + u · ∇d − ∆d = |∇d| d. Recently, Huang-Lin-Liu-Wang [7] studied finite time singularity of the nematic liquid crystal flow in dimension three. For more results on the nematic liquid crystal flow, we refer to nice papers [8, 10]. In this article, we mainly focus on the regularity of weak solutions to the incom- pressible liquid crystals system (1.1), which has a lot in common with Naiver-Stokes system. For Naiver-Stokes equations, there are some excellent regularity results, see for example [1,4,5, 15]. Serrin [13] proved that weak solution u is locally regular r,s in the spatial variables if in addition u ∈ Lloc(Q) with 3/r + 2/s < 1. Here and thereafter, u ∈ Lr,s(Q) means Z T Z r s ( |u| dx) r dt < ∞. 0 Ω Shortly, Jones and Riviere [4] extended Serrin’s result [13] to equality case: 3/r + 2/s = 1, see also [5, 14, 15]. We call u satisfying the Serrin condition provided that r,s u ∈ Lloc with 3/r + 2/s = 1. Here, we aim to develop Serrin’s regularity result for liquid crystal system (1.1). To begin with, we recall the definition of weak solutions to system (1.1). Let Q be a region of space-time (R3 × R) and define the space of test functions D by ∞ 3 D := X : X ∈ C0 (Q : R ), div X = 0 . Definition 1.1. A pair (u, d) is called a weak solution of (1.1) in Q if it satisfies system (1.1) in the following distribution sense. ∞ 1. For any ϕ ∈ C0 (Q), Z hu, ∇ϕi dxdt = 0; (1.2) Q 2. for any Φ ∈ D, Z Z hu, Φti + hu, ∆Φi + hu, u · ∇Φi dxdt = − h∇d ∇d, ∇Φi dxdt; (1.3) Q Q ∞ 3 3. for any Ψ ∈ C0 (Q; R ), Z Z hd, Ψti + hd, ∆Ψi + hd, u · ∇Ψi dxdt = − hF (d), Ψi dxdt. (1.4) Q Q Here hu, vi denotes the ordinary product in R3. Remark 1.2. [10, Theorem A] It has been proven that with suitable initial- boundary data, there exits a weak solution (u, d) satisfying u ∈ L2,∞(Q), |∇u| ∈ L2,2(Q), |∇d| ∈ L2,∞(Q) and |∇2d| ∈ L2,2(Q). Motivated by Serrin’s interior regularity of weak solutions of Navier-stokes equa- tions, we prove the following regularity result for the weak solutions of liquid crystals system. Theorem 1.3. Let (u, d) be a weak solution defined in Definition 1.1, satisfying u ∈ L2,∞(Q), |∇u| ∈ L2,2(Q), |∇d| ∈ L2,∞(Q) and |∇2d| ∈ L2,2(Q). SERRIN’S REGULARITY RESULTS 5581
0 s,s 0 Suppose further that u ∈ Lloc (Q) with 1 < s, s < ∞ and 3 2 + ≤ 1. (1.5) s s0 Then both u and d are of class C∞ in the space variables, and each derivative is locally bounded (and hence regular) in Q. Remark 1.4. For Navier-Stokes equations, Iscauriaza, Seregin and Sverak [9] proved Serrin’s result in the critical case s = 3, s0 = ∞. We mention here that Theorem 1.3 is also valid for s = 3, s0 = ∞ provided by an extra assumption that there is an R > 0 such that Z |u(·, t)|3 dx ≤ T BR Ω uniformly for all t and for some absolute constant > 0. The proof is same as that of Theorem 1.3. Remark 1.5. Theorem 1.3 is true for any dimension n, with corresponding Serrin condition n 2 + = 1. s s0 The paper is organized as follows. In section 2, we derive some representation 3 2 formulas of u and d. In section 3, we prove the theorem under condition s + s0 < 1. In the last section, we complete the proof of Theorem 1.3.
2. Some representation formulas. In this section we assume that (u, d) is a weak solution characterized in the previous section. To begin with, we recall the representation formula of u given in [13]. 2.1. Representation of u. We denote by w the vorticity of u, that is w(x, t) = curl u(x, t) = ∗∇u(x, t) = ∇ × u(x, t). −4 Let α be the usual mollifier in the space-time space and αh(x, t) = h α(x/h, t/h). It is easy to check that αh is a mollifier supported in Bh(0, 0), where Bh(0, 0) is the round ball centered at the origin with radius h in R3 × R. We then modify u and w by Z uh(x, t) = u(y, s) αh(x − y, t − s) dyds, Z wh(x, t) = w(y, s) αh(x − y, t − s) dyds. Then one can check easily that
wh = curl uh. Since div u(x, t) = 0 in distribution sense, it then follows that
div uh(x, t) = 0. Thereby, we have the following equation via direct calculations
∆(∗uh) = −∇wh. (2.1) Here ∗ denotes the Hodge star operator, for more details one can refer chapter 1 in [2]. From equation (2.1), one can deduce easily the following lemma. 5582 XIAN-GAO LIU, JIANZHONG MIN, KUI WANG AND XIAOTAO ZHANG
Lemma 2.1. [13, Lemma 2] Suppose that as functions of the space variables, u and |∇u| are locally of class L1 in Ω. Then in any compact G in Ω, we have Z ∗ u(x, t) = ∇H(x − ξ) ∧ w(ξ, t) dξ + A(x, t), (2.2) G 1 where H(x) = 4π|x| is the fundamental solution of Laplace equation and A(x, t) is a harmonic 2-form in G for any t. Via direct calculation, one can find (2.2) is equivalent to 1 Z u(x, t) = ∇|x − ξ|−1 × w(ξ, t) dξ + ∗A(x, t). (2.3) 4π G
2.2. Representation of w, ∇d, ∇2d.
2 2.2.1. Deriving equations of w, ∇d, ∇ d. Let√ Gh(x, y, t) be the Green function in the ball centered at the origin with radius h2 − t2 in R3. Then we define a vector T function E = (E1,E2,E3) by Z E(x, t) = √ Gh(x, y, t) ∇αh(y, t) dy, |y|≤ h2−t2 and a scalar function ρh(x, t) by Z ρh(x, t) = √ Gh(x, y, t) αh(y, t) dy. |y|≤ h2−t2 From the properties of Green function (c.f. [3]), it follows that
∆E(x, t) = ∇αh(x, t), (2.4) ∞ and E(x, t) ∈ C0 (Bh(0, 0)). Denote g1 = u ⊗ w − w ⊗ u and g2 = ∆d · ∇d, we then have the following modified equations.
2 Lemma 2.2. For modified functions uh, wh, (∇d)h and (∇ d)h, we have
(uh)t − ∆uh + (div(u ⊗ u)h) + ∇Ph = − div(∇d ∇d)h, (2.5)
1 2 (wh)t − ∆wh = − div gh − curl gh, (2.6)
(∇dh)t − ∆∇dh = −∇(u · ∇d)h + ∇f(d)h, (2.7) and 2 2 2 2 (∇ dh)t − ∆∇ dh = −∇ (u · ∇d)h + ∇ f(d)h, (2.8) where Ph(x, t) is defined by Z Ph(x, t) = (div(u · ∇u) + div div(∇d ∇d)) (ξ, τ) ρh(ξ − x, τ − t) dξdτ. (2.9) Q Proof. To prove equation (2.5), it suffices to show it holds at the origin (0, 0). T Set Φ1 = (αh, 0, 0) , and equation (2.4) gives
Φ1 − ∇E1 ∈ D. SERRIN’S REGULARITY RESULTS 5583
Then using Φ1 − ∇E1 as a testing function in (1.3), we obtain Z hu, (Φ1 − ∇E1)ti + hu, ∆(Φ1 − ∇E1)i + hu, u · ∇(Φ1 − ∇E1)i dxdt Q Z = − h∇d ∇d, ∇(Φ1 − ∇E1)i dxdt. Q Using (1.2), we compute Z Z 1 1 hu, (Φ1 − ∇E1)ti dxdt = u (αh(x, t))t dxdt = −(uh)t(0, 0), Q Q Z 1 hu, ∆(Φ1 − ∇E1)i dxdt = ∆uh(0, 0), Q Z hu, u · ∇(Φ1 − ∇E1)i dxdt Q Z 1 = − (div(u ⊗ u)h)(0, 0) − div(u · ∇u)(x, t) E1(x, t) dxdt Q Z 1 ∂ = − (div(u ⊗ u)h)(0, 0) − div(u · ∇u)(ξ, τ) ρh(ξ − x, τ − t) dξdτ (0, 0), ∂x1 Q and Z h∇d ∇d, ∇(Φ1 − ∇E1)i dxdt Q Z 1 = −div(∇d ∇d)h(0, 0) + div(∇d ∇d)(x, t) ∇E1(x, t) dxdt Q 1 = −div(∇d ∇d)h(0, 0) ∂ Z − divdiv(∇d ∇d)(ξ, τ) ρh(ξ − x, τ − t) dξdτ (0, 0). ∂x1 Q Then we conclude from above identities that
1 1 1 ∂ 1 (uh)t − ∆uh + (div(u ⊗ u)h) + Ph = −div(∇d ∇d)h, ∂x1 holds at (0, 0) which immediately implies (2.5). To prove (2.6), taking the vorticity of equation (2.5) yields
(wh)t − ∆wh = −curl div(u ⊗ u)h − curl div(∇d ∇d)h. Thus (2.6) is valid if one can show
1 2 curl div(u ⊗ u)h = div gh and curl div(∇d ∇d)h = curl gh. (2.10) In fact if u and d are smooth, then one can show the following identities curl div(u ⊗ u) = div(u ⊗ w − w ⊗ u) and curl div(∇d ∇d) = curl (∆d · ∇d). For general case, (2.10) holds via approximations. Similarly it follows from (1.4) that
(dh)t + (u · ∇d)h − ∆(dh) = f(d)h, 2 and taking the ∇ and ∇ , equations (2.7) and (2.8) hold respectively. 5584 XIAN-GAO LIU, JIANZHONG MIN, KUI WANG AND XIAOTAO ZHANG
2.2.2. Obtaining the representations of w, ∇d, ∇2d. Let K be the fundamental so- lution of heat equation, i.e. ( 2 − 3 − |x| K(x, t) = Ct 2 e 4t , t > 0, 0, t ≤ 0. By Young inequality, one can easily prove the following lemma. Lemma 2.3. [13, Lemma 1] Let k = ∂K/∂x and h(x, t) = R k(x−ξ, x−t) g(ξ, τ) dξ dτ. Then in any region Q = Ω × (T1,T2) we have
k h kLr,r0 (Q)≤ Const · k g kLq,q0 (Q) (2.11) provided that 1 1 1 1 1 ≤ q ≤ r, 1 ≤ q0 ≤ r0, 3( − ) + 2( − ) < 1. q r q0 r0
The constant depends only on T = T2 − T1 and upon the exponents. Now we use the equations in Lemma 2.2 to deduce representation formulas. Lemma 2.4. Assume that u ∈ L2,∞(Q), |∇u| ∈ L2,2(Q), |∇d| ∈ L2,∞(Q) and |∇2d| ∈ L2,2(Q).
Then in any compact region Q0 = G × (0,T ) of Q, we have Z Z 1 2 w = ∇K(x−ξ, t−τ)· g (ξ, τ) dξdτ + ∇K(x−ξ, t−τ)×g (ξ, τ) dξdτ +B1, Q0 Q0 (2.12) Z ∇d = ∇K(x − ξ, t − τ) ⊗ (u(ξ, τ) · ∇d(ξ, τ)) dξdτ Q0 Z − ∇K(x − ξ, t − τ) ⊗ f(ξ, τ) dξdτ + B2, (2.13) Q0 and Z ∇2d = ∇K(x − ξ, t − τ) ⊗ (∇u · ∇d)(ξ, τ) dξdτ (2.14) Q0 Z + ∇K(x − ξ, t − τ) ⊗ (u · ∇2d)(ξ, τ) dξdτ Q0 Z − ∇K(x − ξ, t − τ) ⊗ ∇f(ξ, τ) dξdτ + B3, Q0 where Bi is a i-form satisfying the heat equation, i = 1, 2, 3. To avoid misrepresen- ∂uj ∂di tation, we point out that (∇u · ∇d)ik = . ∂xk ∂xj Proof. Let Z Z 1 2 B1,h = wh − ∇K(x−ξ, t−τ)· gh(ξ, τ) dξdτ − ∇K(x−ξ, t−τ)×gh(ξ, τ) dξdτ. Q0 Q0 (2.15) Then it follows from equation (2.6) that
(B1,h)t − ∆B1,h = 0. 1 2 1 2 By the assumption of u and d, we see that gh and gh converge to g and g 1,1 R in L (Q0) respectively. Then from Young inequality we deduce ∇K(x − Q0 SERRIN’S REGULARITY RESULTS 5585
ξ, t − τ) · g1(ξ, τ) dξdτ and R ∇K(x − ξ, t − τ) × g2(ξ, τ) dξdτ also converge h Q0 h to R ∇K(x − ξ, t − τ) · g1(ξ, τ) dξdτ and R ∇K(x − ξ, t − τ) × g2(ξ, τ) dξdτ in Q0 Q0 1,1 L (Q0) respectively. Meanwhile, it is obviously that B1,h is uniformly bounded 1,1 1,1 in L (Q0). Therefore there exists B1 in L (Q0) satisfying the heat equation and 1,1 B1,h converges to B1 weakly in L (Q0). Since B1,h satisfies the heat equation, then B1 also satisfies the heat equation. Finally letting h goes to zero in (2.15), we obtain formula (2.12). By analogous strategies, formulas (2.13) and (2.14) hold true.
3 2 3 2 3. The case: s + s0 < 1. In this section, we mainly consider the case s + s0 < 1 and use Serrin’s approach in [13] to show Theorem 3.1. Let u be a weak solution defined in Definition 1.1, satisfying u ∈ L2,∞(Q), |∇u| ∈ L2,2(Q), |∇d| ∈ L2,∞(Q) and |∇2d| ∈ L2,2(Q).
0 Suppose further that u ∈ Ls,s (Q) with 3 2 + < 1. (3.1) s s0 Then u and d are of class C∞ in the space variables, and each derivative is bounded locally in Q. Proof. We divide the proof into three steps.
Step 1. Prove ∇d ∈ L∞,∞(Q). Since d is bounded (e.g. [10]) and f(d) is smooth with respect to d, then f(d) ∈ L∞,∞(Q). We assert from Lemma 2.3 that Z ∞,∞ ∇K(x − ξ, t − τ) ⊗ f(ξ, τ) dξdτ ∈ L (Q0). Q0 Now we only need to consider the first term Z ∇K(x − ξ, t − τ) ⊗ (u(ξ, τ) · ∇d(ξ, τ)) dξdτ Q0 ρ,ρ0 in the representation (2.13) of ∇d. Suppose for the moment that ∇d ∈ L (Q0) with ρ, ρ0 ≥ 2. It is evident that q,q0 u · ∇d ∈ L (Q0), which follows from H¨olderinequality and where the exponents q, q0 are given through 1 1 1 1 1 1 = + , = + . q s ρ q0 s0 ρ0 Due to condition (3.1), we define a positive κ by 1 3 2 κ = 1 − ( + ) , 6 s s0 and set 1 1 1 1 = − κ and = − κ, r ρ r0 ρ0 if κρ ≥ 1, we set r = ∞, and similarly for r0. Clearly, we have 1 1 1 1 3( − ) + 2( − ) < 1. q r q0 r0 5586 XIAN-GAO LIU, JIANZHONG MIN, KUI WANG AND XIAOTAO ZHANG
Thus using Lemma 2.3 again, we assert Z r,r0 ∇K(x − ξ, t − τ) ⊗ (u(ξ, τ) · ∇d(ξ, τ)) dξdτ ∈ L (Q0). Q0
0 Since r and r0 are larger than ρ and ρ0, then we have that ∇d is in a higher class Lr,r than originally supposed. Repeating this step until ρ, ρ0 ≥ κ−1, we then conclude ∞,∞ ∇d ∈ L (Q0).
2 ∞,∞ ∞,∞ Step 2. Prove w, ∇u, ∇ d ∈ L (Q0). Recall from step 1 that ∇d ∈ L (Q0). Therefore it is left to consider terms Z ∇K(x − ξ, t − τ) ⊗ (∇u · ∇d)(ξ, τ) dξdτ Q0 Z and ∇K(x − ξ, t − τ) ⊗ (u · ∇2d)(ξ, τ) dξdτ Q0 in the representation (2.14) of ∇2d, and Z Z ∇K(x − ξ, t − τ) · g1(ξ, τ) dξdτ and ∇K(x − ξ, t − τ) × g2(ξ, τ) dξdτ Q0 Q0 in the representation (2.12) of w. 2 ρ,ρ0 0 Similarly as in step 1, we assume that ∇ d, w, ∇u ∈ L (Q0) with ρ, ρ ≥ 2. Using H¨olderinequality, we deduce that
1 2 2 q,q0 g , g , ∇u · ∇d, u · ∇ d ∈ L (Q0) with 1 1 1 1 1 1 = + and = + . q s ρ q0 s0 ρ0 By the same process as in step 1, we obtain Z 1 1 k ∇K(x − ξ, t − τ) · g (ξ, τ) dξdτ k r,r0 ≤ C k g k q,q0 , L (Q0) L (Q0) Q0
Z 2 2 k ∇K(x − ξ, t − τ) × g (ξ, τ) dξdτ k r,r0 ≤ C k g k q,q0 , L (Q0) L (Q0) Q0
Z k ∇K(x − ξ, t − τ) · (∇u · ∇d)(ξ, τ) dξdτ k r,r0 ≤ C k ∇u · ∇d k q,q0 , L (Q0) L (Q0) Q0 and Z 2 2 k ∇K(x − ξ, t − τ) · (u · ∇ d)(ξ, τ) dξdτ k r,r0 ≤ C k u · ∇ d k q,q0 L (Q0) L (Q0) Q0
0 r,r0 2 with same exponents r, r as in step 1. Then we have w ∈ L (Q0) and ∇ d ∈ r,r0 r,r0 L (Q0). From the representation (2.3) of u, it follows that u ∈ L (Q0) and r,r0 2 ∇u ∈ L (Q0). Thus we can repeat this process, and finally we obtain w, ∇ d ∈ ∞,∞ L (Q0). SERRIN’S REGULARITY RESULTS 5587
Step 3. Prove that d and u are of class C∞ in the space variables. We 2 ∞,∞ 2 ∞,∞ already obtain in step 2 that ∇d, ∇ d ∈ L (Q0), then g ∈ L (Q0). Recall from Lemma 2.1 that 1 Z u(x, t) = ∇|x − ξ|−1 × w(ξ, t) dξ + ∗A(x, t). (3.2) 4π G ∞,∞ −1 1 Since w ∈ L (Q0) and ∇|x − ξ| ∈ L (Q0), then Young inequality yields ∞,∞ 1 ∞,∞ ∞,∞ u ∈ L (Q0), hence g ∈ L (Q0). It follows from |∇d| + |∆d| ∈ L (Q0) 2 ∞,∞ that g = ∆d · ∇d ∈ L (Q0). Therefore from formula (2.12), we deduce w is spatially H¨oldercontinuous in Q0, with arbitrary exponent θ < 1. Now from (3.2), ∇u is also H¨odercontinuous, then ∇u · ∇d ∈ L∞,∞ and with the formula (2.14) 2 ∞,∞ 2 and u · ∇ d, ∇f(d) ∈ L , we have ∇ d is spatially H¨oldercontinuous in Q0 with arbitrary exponent θ < 1. Thus g1, g2, u · ∇d, u · ∇2d, ∇u · ∇d, f(d),∇f(d) ∞,∞ ∈ L (Q0) are H¨oldercontinuous. And again by the potential theory of the heat convolution we have that wx, (∇d)x and (∆d)x are H¨oldercontinuous. Repeating ∞ this argument, we conclude that d, u ∈ C (Q0) in the space variables.
Similarly as Navier-Stokes system, for time regularity of a weak solution we need extra assumptions on time derivatives (see the explanations in [13]). For liquid crystal system we prove
2,p Corollary 3.2. Assume additionally that ut, dt ∈ L (Q) with p ≥ 1. Then the space derivatives of u, d are absolutely continuous functions of time. Moreover, there exists a strongly differentiable function P = P (x, t) such that
ut − ∆u + u · ∇u = − div(∇d ∇d) − ∇P (3.3) almost everywhere in Q. Proof. First of all, observe that (2.6) implies
1 2 wt = ∆w − div g − curl g in the weak sense that already been proved. Differentiation of (3.2) yields Z 1 −1 ut(x, t) = ∇|x − ξ| × wt(ξ, t) dξ + ∗At(x, t). 4π G ∞ p Thus ut is of class C in the spatial variables, and each derivative is of class L in time by the assumption. Thereby
Z t2 m m m m 1− 1 p p |D u(x, t2)−D u(x, t1)| ≤ |D ut(x, t)| dt ≤k D ut(x, t) kL (t1,t2) |t2−t1| , t1 proving that the space derivatives of u are absolutely continuous functions of time. Recall from (1.1) that
dt = ∆d − u · ∇d + f(d),
∞ thus dt is of class C in the spatial variables and then the space derivatives of d are absolutely continuous functions of time. We note that (3.3) follows by letting h tend to zero in (2.6), and then P (x, t) ∈ p ∞ L ((0,T ); C (Ω)), completing the proof. 5588 XIAN-GAO LIU, JIANZHONG MIN, KUI WANG AND XIAOTAO ZHANG
3 2 4. The equality case: s + s0 = 1. From previous sections, we know w, ∇d and ∇2d weakly satisfy the following equations respectively
wt − ∆w + div(u ⊗ w − w ⊗ u) = −curl(∆d · ∇d), (4.1)
∇dt − ∆∇d + ∇(u · ∇d) = ∇f(d), (4.2) and 2 2 2 2 ∇ dt − ∆∇ d + ∇(∇u · ∇d + u · ∇ d) = ∇ f(d), (4.3) see Lemma 2.4. Firstly, we use equation (4.2) to improve regularity of ∇d. More precisely we prove Lemma 4.1. With the same assumption as in Theorem 1.3, then for each integer 0 2l 0 l ≥ 1 and any sufficient small compact subset Q b Q, we have ∇d ∈ L (Q ) with 0 k ∇d kL2l(Q0)≤ C(dist(Q , ∂Q), ε) k f(d) kL2l(Q) + k ∇d kL2(Q) where C(dist(Q0, ∂Q), ε) denotes a constant depending on dist(Q0, ∂Q) and ε. Proof. Step 1. Assume firstly that u is uniformly bounded. ∞ 2l−2 2 Let ϕ ∈ C0 (Q) be a cut-off function. Multiplying (4.2) by |∇d| ϕ ∇d (l ≥ 1) and integrating, we then have Z 2l |∇d| 2 2 2 2l−2 2 2 2 2l−4 2 ∂t( ϕ ) + |∇ d| |∇d| ϕ + |∇|∇d| | |∇d| ϕ dxdt Q 2l Z 1 2l 2l−2 2 = |∇d| ϕ ∂tϕ − |∇d| ϕ ∇|∇d| · ∇ϕ dxdt Q l Z + (u · ∇d) · div(∇d|∇d|2l−2ϕ2) − f(d) · div(∇d|∇d|2l−2ϕ2) dxdt. Q Therefore standard estimates implies Z Z sup (|∇d|lϕ)2 dx + |∇(|∇d|lϕ)|2 dxdt t Ω Q Z Z ≤ C(ϕ) |∇d|2l dxdt + C |u|2|∇d|2lϕ2 + |u||∇d|2l|ϕ||∇ϕ| dxdt Q Q Z (4.4) + C |f(d)|2|∇d|2l−2ϕ2 + |f(d)||∇d|2l−1|ϕ||∇ϕ| dxdt Q Z Z Z ≤ C(ϕ) |∇d|2l dxdt + C |u|2|∇d|2lϕ2 dxdt + C(ϕ) |f(d)|2l dxdt, Q Q Q where C(ϕ) is a constant depending on ϕ. Therefore estimates (4.4) holds provided by w, f ∈ L2l(Q) and u is bounded in Q. The last step used Young’s inequality. It follows from H¨olderinequality and Sobolev’s embedding theorem that |∇d|sϕ ∈ π,ρ 3 2 3 L (Q) with π + ρ = 2 , and moreover Z Z l 2 l 2 l 2 k |∇d| ϕ kLπ,ρ(Q)≤ C sup |∇d| ϕ| dx + C |∇(|∇d| ϕ)| dxdt, (4.5) Ω Q (see for example inequality (3.3) in [14]). Using H¨olderinequality again, we obtain Z 2 l 2 2 l 2 |u| (|∇d| ϕ) dxdt ≤k u kLs,s0 (supp ϕ) k |∇d| ϕ kLp∗,q∗ (Q), (4.6) Q SERRIN’S REGULARITY RESULTS 5589 with 1 1 1 1 1 1 = − and = − . p∗ 2 s q∗ 2 s0 Since 3/p∗ + 2/q∗ ≥ 3/2 from assumption 3/s + 2/s0 ≤ 1, then combining (4.4), (4.5) and (4.6) we obtain that l 2 l 2 2l k |∇d| ϕ kLp∗,q∗ (Q) ≤ C(ϕ) k |∇d| kL2(Q) +C(ϕ) k f(d) kL2l(Q) 2 l 2 +C k u kLs,s0 (supp ϕ)k |∇d| ϕ kLp∗,q∗ (Q) . (4.7) By absolutely continuity of the Lebesgue integral for finite p, q, for any sufficiently 2 small we can take supp ϕ small enough such that k u k 0 ≤ ε. Then we Ls,s (supp ϕ) assert from (4.7) that l 2 l 2 2l k |∇d| ϕ kLp∗,q∗ (Q)≤ C(ϕ) k |∇d| kL2(Q) +C(ϕ) k f(d) kL2l(Q) . (4.8) Choosing π = ρ = 10/3 in (4.5), then we obtain from (4.4)-(4.8) that 5 |∇d|lϕ ∈ L2β(Q), β = > 1. 3
Letting l0 = 1, lk+1 = βlk for k ∈ N and Q0 = Q, Qk+1 = {(x, t)|ϕk(x, t) ≥ 1} and ∞ iterating (4.8) with suitable functions ϕk+1 ∈ C0 (Qk) (that means ϕk is supported in Qk−1 and ϕk ≥ 1 in the Qk), we finally get 0 k ∇d kL2l(Q0)≤ C(dist(Q , ∂Q), ε) k f(d) kL2l(Q) + k ∇d kL2(Q) (4.9) for any compact subdomain Q0 and l < ∞. Step 2. Remove the assumption that u is uniformly bounded. We now do approximations to obtain (4.9) for general u . Let Q00 is a subdomain k of Q, but we still sign it as Q for convenient. And let u = u ∗ αhk , where αhk is the usual mollifier and hk → 0 as k → ∞. Clearly, uk → u as k → ∞ in L2(Q), and for all k k k u kLs,s0 (Q)≤ C k u kLs,s0 (Q), where C is an absolutely constant. Let Dk be the solution to the equations k Dt − ∆D + ∇(u · D) = ∇f(d), (4.10) with the initial and boundary data
Dk = ∇d on ∂Ω × [0,T ] = Σ and Ω × {0}. By the well-known Courant-Lebesgue lemma (see also Lemma 2.4 in [14]), we can assume that 2,2 2 ∇d|Σ ∈ L (Σ), ∇d|Ω×{0} ∈ L (Ω × {0}). Besides
||∇d||L2(Ω×{0}) + ||∇d||L2(Σ) ≤ C||∇d||L2(Q) 0 Here, for any small , we can take suitable domain Q b Q such that k k u kLs,s0 (Q0)≤ C k u kLs,s0 (Q0)< Cε. Let w∗ be the solution of the backward heat equation ∗ ∗ ∗ − ∂tw − ∆w = f in Q (4.11) 5590 XIAN-GAO LIU, JIANZHONG MIN, KUI WANG AND XIAOTAO ZHANG with initial and boundary data w∗ = 0 on ∂Ω × [0,T ] ∪ Ω × {T }, where f ∗ ∈ L2(Q : R3 × R3). Then it is well-known that ∗ ∗ 2 ∗ ∗ sup k ∇w (t) kL2(Ω) + k ∂tw kL2(Q) + k ∇ w kL2(Q)≤ C k f kL2(Q) . (4.12) 0≤t≤T Similarly as inequality (4.5), we deduce from H¨olderinequality and Sobolev’s em- bedding theorem that ∇w∗ ∈ Lp∗,q∗(Q) with ∗ ∗ k ∇w kLp∗,q∗(Q)≤ C k f kL2(Q), (4.13) where 1/s + 1/p∗ = 1/2 and 1/s0 + 1/q∗ = 1/2. Like ∇d, by Courant-Lebesgue theorem, we obtain ∗ ∗ ∗ ∗ ||w ||L2(Ω×{0}) + ||w ||L2(Σ) ≤ C||w ||L2(Q) ≤ C||f ||L2(Q), Thus we conclude Z Z T Z ∗ ∂ ∗ ∗ ∇d : w dx − ∇d : w dσdt ≤ C(||∇d||L2,2 ) k f kL2(Q) . (4.14) Ω×{0} 0 ∂Ω ∂n Here for two matrixes A, B ∈ R3 × R3 we define A : B = tr(AB). Integrating by parts, we estimate Z ∗ Dk : f dxdt Q Z ∗ = Dk :(−∂t − ∆)w dxdt Q Z Z T Z ∗ ∂ ∗ = (∇f(d) − ∇(uk · Dk)) : w dxdt − ∇d : w dσdt Q 0 ∂Ω ∂n Z (4.15) + ∇d : w∗ dx Ω×{0} ∗ ∗ ≤ k f(d) kL2(Q)k ∇w kL2(Q) + k Dk kL2(Q)k uk kLs,s0 (Q)k ∇w kLp∗,q∗(Q) ∗ + C(||∇d||L2,2 ) k f kL2(Q) ∗ ∗ ≤C k f(d) kL2(Q)k f kL2(Q) +C k Dk kL2(Q)k uk kLs,s0 (Q)k f kL2(Q) ∗ + C(||∇d||L2,2 ) k f kL2(Q), where we used (4.13-4.15). Since inequality (4.15) holds for all f ∗ ∈ L2(Q), then
k Dk kL2(Q)≤ Cε k Dk kL2(Q) +C k f(d) kL2(Q) +C(||∇d||L2,2 ), 2 and thereby Dk is uniformly bounded in L (Q). Exactly same proof shows Dk is 2 unique. Hence Dk converges to D weakly in L (Q). 2l 0 From step 1, the L (Q ) norm of Dk is uniformly bounded by
k Dk kL2l(Q0)≤ C k f(d) kL2l(Q) +C(||∇d||L2,2 ) 1 ≤ l < ∞, (4.16) 0 for any compact Q b Q. Thus we complete the proof. Now we are in position to prove the main theorem.
Proof of Theorem 1.3. Assuming that both u and ∇d are uniformly bounded, we prove higher regularity of w and ∇2d via equations (4.1) and (4.3) respectively. SERRIN’S REGULARITY RESULTS 5591
Similarly we choose test functions as w|w|2l−2ϕ2 and ∇2d|∇2d|2l−2ϕ2 for the equations (4.1) and (4.3) respectively. Then we obtain Z 2 2l |∇ d| 2 2 2 2l−2 2 2 2 2 2 2l−4 2 ∂t( ϕ) + |∇∇ d| |∇ d| ϕ + |∇|∇ d| | |∇ d| ϕ dxdt Q 2l Z 1 2 2l 2 2 2 2l−2 = |∇ d| ϕ∂tϕ − ∇|∇ d| |∇ d| ϕ · ∇ϕ Q l Z + (∇u · ∇d) · (∇(∇2d|∇2d|2l−2ϕ2)) + (u · ∇2d) · (∇(∇2d|∇2d|2l−2ϕ2)) Q Z + ∇f(d) · (∇(∇2d|∇2d|2l−2ϕ2)), Q and Z 2l 2 |w| ϕ 2 2l−2 2 1 2 2 2l−4 2 ∂t + |∇w| |w| ϕ + (l − 1)|∇|w| | |w| ϕ Q 2l 2 Z 2l |w| 2 2l−2 2l−2 2 = ϕ∂tϕ − ∇|w| |w| ϕ∇ϕ + (u ⊗ w − w ⊗ u): ∇(w|w| ϕ ) Q l Z − curl(∆d · ∇d)(w|w|2l−2ϕ2). Q Then by standard estimate Z Z sup (|∇2d|lϕ)2 dx + |∇(|∇2d|lϕ)|2 dxdt t Ω Q Z 2 2l 2 2 2 2l−2 2 2 2 2 2 2l−2 2 ≤C |∇ d| ϕ|∂tϕ| + |∇u| |∇d| |∇ d| ϕ + |u| |∇ d| |∇ d| ϕ Q Z + C (|∇u||∇d| + |u||∇2d|)|∇2d|2l−1ϕ|∇ϕ| Q Z (4.17) + C(ϕ) |∇f(d)|2|∇2d|2l−2 + |∇f(d)||∇2d|2l−1 Q Z Z ≤C(ϕ) |∇2d|2l + C |∇d|2(|∇u|2l + |∇2d|2l)ϕ2 Q Q Z Z + C |u|2|∇2d|2lϕ2 dxdt + C(ϕ) |∇f(d)|2l Q Q and Z Z sup (|w|lϕ)2 + |∇(|w|lϕ)|2 t Ω Q Z 2l 2l 2 2l 2 ≤C |w| |ϕ∂tϕ| + |u||w| ϕ|∇ϕ| + |u| |w| ϕ Q Z (4.18) + C |∆d · ∇d|2|w|2l−2ϕ2 Q Z Z Z ≤C(ϕ) |w|2l + C |u|2(|w|lϕ)2 + C (|∇d|l)2(|∆d|lϕ)2. Q Q Q From inequality (4.8), we deduce Z Z 2 l 2 2 l 2 2 l 2 k |∇ d| ϕ kLπ,ρ(Q)≤ C sup |∇ d| ϕ| dx + C |∇(|∇ d| ϕ)| dxdt, (4.19) t Ω Q 5592 XIAN-GAO LIU, JIANZHONG MIN, KUI WANG AND XIAOTAO ZHANG and Z Z l 2 l 2 l 2 k |w| ϕ kLπ,ρ(Q)≤ C sup | |w| ϕ| dx + C |∇(|w| ϕ)| dxdt. (4.20) t Ω Q Recall that 1 Z u(x, t) = ∇|x − ξ|−1 × w(ξ, t) dξ + A(x, t), 4π Ω where A is a harmonic function, and the other term is classical singular operator 0 applied to w. Suppose w ∈ Ls,s (Q) for any s, s0 ≥ 2. Since u ∈ L2,∞(Q), then from the representation of u and the Lp estimate of harmonic function, we conclude that k ∇A kLs,s0 (Q0)≤ C k w kLs,s0 (Q) + k u kL2,∞(Q) .
0 where Q b Q. Then 1 Z ∇u = ∇(∇|x − ξ|−1 × w(ξ, t)) dξ + ∇A(x, t). 4π Ω By the Calder´on-Zygmund theorem (e.g. [6])
k (∇u)ϕ kLs,s0 (Q)≤ C(n, p) k w kLs,s0 (Q) +C(ϕ)C(||u||2,∞). By H¨older’sinequality, we estimate Z 2 l 2 2 l 2 |∇d| (|∇u| ϕ) dxdt ≤k ∇d kLs,s0 (supp ϕ)k |∇u| ϕ kLp∗,q∗ (Q) Q 2 l 2 ≤ C k ∇d kLs,s0 (supp ϕ) (k |w| kLp∗,q∗ (Q) +C(ϕ)C(||u||L2,∞(Q))), (4.21)
Z 2 2 l 2 2 2 l 2 |∇d| (|∇ d| ϕ) dxdt ≤k ∇d kLs,s0 (supp ϕ)k |∇ d| ϕ kLp∗,q∗ (Q), (4.22) Q Z 2 2 l 2 2 2 l 2 |u| (|∇ d| ϕ) dxdt ≤k u kLs,s0 (supp ϕ)k |∇ d| ϕ kLp∗,q∗ (Q), (4.23) Q Z 2 l 2 2 l 2 |u| (|∇u| ϕ) dxdt ≤k u kLs,s0 (supp ϕ)k |∇u| ϕ kLp∗,q∗ (Q) Q 2 l 2 ≤ C||u||Ls,s0 (supp ϕ)(k |w| kLp∗,q∗ (Q) +C(ϕ)C(k u kL2,∞(Q))), (4.24) and Z 2l 2 l 2 l 2 2 l 2 |∇d| (|∇ d| ϕ) dxdt ≤k |∇d| kLs,s0 (supp ϕ)k |∇ d| ϕ kLp∗,q∗ (Q) . (4.25) Q Where 1 1 1 1 1 1 = − , = − , p∗ 2 s q∗ 2 s0 and 3 2 3 3 2 + = + 1 − + . p∗ q∗ 2 s s0 SERRIN’S REGULARITY RESULTS 5593
In the hypothesis 1 < s, s0 < ∞ and 3/s + 2/s0 ≤ 1, then p∗ = 4/(s − 2) + 2 < 6 and 3/p∗ + 2/q∗ ≥ 3/2. So (4.17–4.25) yield that 2 l 2 l 2 k |∇ d| ϕ kLp∗,q∗(Q) + k |w| ϕ kLp∗,q∗(Q) 2 l 2 l 2 2 l 2 ∗ ∗ ≤C(ϕ) k |∇ d| kL2(Q) +C(ϕ) k |w| ϕ kL2(Q) +C k ∇d kLs,s0 (supp ϕ)k |w| kLp ,q (Q) 2 2 l 2 2 2 l 2 ∗ ∗ ∗ ∗ + C k ∇d kLs,s0 (supp ϕ)k |∇ d| ϕ kLp ,q (Q) +C k u kLs,s0 (supp ϕ)k |∇ d| ϕ kLp ,q (Q) 2 l 2 l 2 2 l 2 ∗ ∗ ∗ ∗ + C k u kLs,s0 (supp ϕ)k |w| kLp ,q (Q) +C k |∇d| kLs,s0 (supp ϕ)k |∇ d| ϕ kLp ,q (Q) 2l 2 2 + C(ϕ) k ∇f(d) kL2l +C(ϕ)C(k u kL2,∞(Q))(k u kLs,s0 (supp ϕ) + k ∇d kLs,s0 (supp ϕ)).
By absolutely continuity of the Lebesgue integral for finite s, s0, we can take 2 2 supp(ϕ) small enough such that k u k 0 ≤ ε, k ∇d k 0 ≤ ε and Ls,s (supp ϕ) Ls,s (supp ϕ) l 2 k |∇d| k 0 ≤ ε, where ε is sufficiently small. Then we obtain Ls,s (supp ϕ) 2 l 2 l 2 k |∇ d| ϕ kLp∗,q∗(Q) + k |w| ϕ kLp∗,q∗(Q) 2 l 2 l 2 2l ≤ C(ϕ) k |∇ d| kL2(Q) + k |w| kL2(Q) + k ∇f(d) kL2l(Q) +C(k u kL2,∞(Q))
2 l 2 l 2 2l ≤ C(ϕ) k |∇ d| kL2(Q) + k |w| kL2(Q) + k ∇d kL2l(Q) +C(k u kL2,∞(Q)) . Therefore taking π = ρ = 10/3 in (4.19) and (4.20), we obtain 5 |∇2d|l, |∇u|l ∈ L2β(Q0), with β = . 3
3+ε0,∞ Then we do iterate similarly to ∇d to obtain that w ∈ Lloc (Q) for some ε0 > 0. For the general case, we do approximations similarly as in the step 2 of Lemma 4.1.
Acknowledgments. The first author’s research is partially supported by NSFC No.11131005.
REFERENCES
[1] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771–831. [2] B. Chow, P. Lu and L. Ni, Hamilton’s Ricci Flow, volume 77 of Graduate Studies in Mathe- matics, American Mathematical Society, Providence, RI; Science Press, New York, 2006. [3] L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998. [4] E. B. Fabes, B. F. Jones and N. M. Rivi`ere, The initial value problem for the Navier-Stokes equations with data in Lp, Arch. Rational Mech. Anal., 45 (1972), 222–240. [5] Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations, 62 (1986), 186–212. [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. [7] T. Huang, F. Lin, C. Liu and C. Wang, Finite time singularity of the nematic liquid crys- tal flow in dimension three, Archive for Rational Mechanics and Analysis, (2016), 1–32, arXiv:1504.0108. [8] T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows, Communications in Partial Differential Equations, 37 (2012), 875–884. [9] L. Iskauriaza, G. A. Ser¨eginand V. Shverak, L3,∞-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3–44. [10] F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501–537. 5594 XIAN-GAO LIU, JIANZHONG MIN, KUI WANG AND XIAOTAO ZHANG
[11] F.-H. Lin and C. Liu, Partial regularity of the dynamic system modeling the flow of liquid crystals, Discrete Contin. Dynam. Systems, 2 (1996), 1–22. [12] F. Lin, J. Lin and C. Wang, Liquid crystal flows in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297–336. [13] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187–195. [14] M. Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437–458. [15] W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Aspects of Mathematics, E8. Friedr. Vieweg & Sohn, Braunschweig, 1985. Received August 2015; revised November 2015. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]