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Serrin's Regularity Results for the Incompressible 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 8 < ut − ∆u + u · ru + rP = −div(rd rd); div u = 0; (1.1) : dt + u · rd − ∆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 (jdj − 1)d. Here rd rd is a symmetric tensor with its component (rd rd)ij given by rid · rjd. And the initial and boundary conditions are: u(x; 0) = u0(x); with div(u0) = 0; d(x; 0) = d0(x); for x 2 Ω u(x; t) = 0; d(x; t) = d0(x); for (x; t) 2 @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 · rd − ∆d = jrdj 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 2 Lloc(Q) with 3=r + 2=s < 1. Here and thereafter, u 2 Lr;s(Q) means Z T Z r s ( juj dx) r dt < 1: 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 2 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 1 3 D := X : X 2 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 1. For any ' 2 C0 (Q), Z hu; r'i dxdt = 0; (1.2) Q 2. for any Φ 2 D, Z Z hu; Φti + hu; ∆Φi + hu; u · rΦi dxdt = − hrd rd; rΦi dxdt; (1.3) Q Q 1 3 3. for any Ψ 2 C0 (Q; R ), Z Z hd; Ψti + hd; ∆Ψi + hd; u · rΨi 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 2 L2;1(Q); jruj 2 L2;2(Q); jrdj 2 L2;1(Q) and jr2dj 2 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 2 L2;1(Q); jruj 2 L2;2(Q); jrdj 2 L2;1(Q) and jr2dj 2 L2;2(Q): SERRIN'S REGULARITY RESULTS 5581 0 s;s 0 Suppose further that u 2 Lloc (Q) with 1 < s; s < 1 and 3 2 + ≤ 1: (1.5) s s0 Then both u and d are of class C1 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 = 1. We mention here that Theorem 1.3 is also valid for s = 3; s0 = 1 provided by an extra assumption that there is an R > 0 such that Z ju(·; t)j3 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) = r × 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 jruj are locally of class L1 in Ω. Then in any compact G in Ω, we have Z ∗ u(x; t) = rH(x − ξ) ^ w(ξ; t) dξ + A(x; t); (2.2) G 1 where H(x) = 4πjxj 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) = rjx − ξj−1 × w(ξ; t) dξ + ∗A(x; t): (2.3) 4π G 2.2. Representation of w; rd; r2d. 2 2.2.1. Deriving equations of w; rd; r d. Letp 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) = p Gh(x; y; t) rαh(y; t) dy; jy|≤ h2−t2 and a scalar function ρh(x; t) by Z ρh(x; t) = p Gh(x; y; t) αh(y; t) dy: jy|≤ h2−t2 From the properties of Green function (c.f. [3]), it follows that ∆E(x; t) = rαh(x; t); (2.4) 1 and E(x; t) 2 C0 (Bh(0; 0)). Denote g1 = u ⊗ w − w ⊗ u and g2 = ∆d · rd; we then have the following modified equations. 2 Lemma 2.2. For modified functions uh, wh, (rd)h and (r d)h, we have (uh)t − ∆uh + (div(u ⊗ u)h) + rPh = − div(rd rd)h; (2.5) 1 2 (wh)t − ∆wh = − div gh − curl gh; (2.6) (rdh)t − ∆rdh = −∇(u · rd)h + rf(d)h; (2.7) and 2 2 2 2 (r dh)t − ∆r dh = −∇ (u · rd)h + r f(d)h; (2.8) where Ph(x; t) is defined by Z Ph(x; t) = (div(u · ru) + div div(rd rd)) (ξ; τ) ρh(ξ − x; τ − t) dξdτ: (2.9) Q Proof.
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