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Global Existence for Slightly Compressible Hydrodynamic Flow of Liquid Crystals in Two Dimensions

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Global Existence for Slightly Compressible Hydrodynamic Flow of Liquid Crystals in Two Dimensions Editorial Board Supported by NSFC Honorary Editor General ZHOU GuangZhao (Zhou Guang Zhao) Editor General ZHU ZuoYan Institute of Hydrobiology, CAS, China Editor-in-Chief YUAN YaXiang Academy of Mathematics and Systems Science, CAS, China Associate Editors-in-Chief CHEN YongChuan Tianjin University, China GE LiMing Academy of Mathematics and Systems Science, CAS, China SHAO QiMan The Chinese University of Hong Kong, China XI NanHua Academy of Mathematics and Systems Science, CAS, China ZHANG WeiPing Nankai University, China Members BAI ZhaoJun LI JiaYu WANG YueFei University of California, Davis, USA University of Science and Technology Academy of Mathematics and Systems of China, China Science, CAS, China CAO DaoMin Academy of Mathematics and Systems LIN FangHua WU SiJue Science, CAS, China New York University, USA University of Michigan, USA CHEN XiaoJun LIU JianYa WU SiYe The Hong Kong Polytechnic University, Shandong University, China The University of Hong Kong, China China LIU KeFeng XIAO Jie University of California, Los Angeles, USA Tsinghua University, China CHEN ZhenQing Zhejiang University, China University of Washington, USA XIN ZhouPing LIU XiaoBo The Chinese University of Hong Kong, CHEN ZhiMing Peking University, China China Academy of Mathematics and Systems University of Notre Dame, USA Science, CAS, China XU Fei MA XiaoNan Capital Normal University, China CHENG ChongQing University of Denis Diderot-Paris 7, Nanjing University, China France XU Feng University of California, Riverside, USA DAI YuHong MA ZhiMing Academy of Mathematics and Systems XU JinChao Academy of Mathematics and Systems Pennsylvania State University, USA Science, CAS, China Science, CAS, China MOK NgaiMing XU XiaoPing DONG ChongYing The University of Hong Kong, China Academy of Mathematics and Systems University of California, Santa Cruz, USA PUIG Lluis Science, CAS, China CNRS, Institute of Mathematics of Jussieu, YAN Catherine H. F. DUAN HaiBao France Academy of Mathematics and Systems Texas A&M University, USA Science, CAS, China QIN HouRong YANG DaChun Nanjing University, China E WeiNan Beijing Normal University, China Princeton University, USA RINGEL Claus M. University of Bielefeld, Germany YE XiangDong Peking University, China University of Science and Technology FAN JianQing SHANG ZaiJiu of China, China Princeton University, USA Academy of Mathematics and Systems Science, CAS, China YU XingXing Georgia Institute of Technology, USA FENG Qi SHEN ZhongMin Academy of Mathematics and Systems Indiana University-Purdue University ZHANG James J. Science, CAS, China Indianapolis, USA University of Washington, USA FU JiXiang SHU Chi-Wang ZHANG JiPing Fudan University, China Brown University, USA Peking University, China GAO XiaoShan SIU Yum-Tong ZHANG Ping Academy of Mathematics and Systems Harvard University, USA Academy of Mathematics and Systems Science, CAS, China SUN LiuQuan Science, CAS, China GE GenNian Academy of Mathematics and Systems ZHANG PingWen Zhejiang University, China Science, CAS, China Peking University, China GUO XianPing SUN XiaoTao ZHANG ShouWu Sun Yat-sen University, China Academy of Mathematics and Systems Columbia University, USA Science, CAS, China HE XuMing ZHANG Xu University of Michigan, USA TAN Lei Sichuan University, China University of Angers, France HONG JiaXing ZHANG YiTang Fudan University, China TANG ZiZhou University of New Hampshire, USA Beijing Normal University, China HSU Elton P. ZHOU XiangYu Northwestern University, USA TEBOULLE Marc Academy of Mathematics and Systems Tel Aviv University, Israel JI LiZhen Science, CAS, China University of Michigan, USA WANG FengYu ZHU XiPing Beijing Normal University, China Sun Yat-sen University, China JING Bing-Yi The Hong Kong University of Science WANG HanSheng ZONG ChuanMing and Technology, China Peking University, China Peking University, China E ditorial Staff CHAI Zhao YANG ZhiHua ZHANG RuiYan [email protected] [email protected] [email protected] SCIENCE CHINA Mathematics . ARTICLES . November 2013 Vol. 56 No. 11: 2233–2250 doi: 10.1007/s11425-013-4620-2 Global existence for slightly compressible hydrodynamic flow of liquid crystals in two dimensions DING ShiJin1, HUANG JinRui1,∗ & LIN JunYu2 1School of Mathematical Sciences, South China Normal University, Guangzhou 510631,China; 2Department of Mathematics, South China University of Technology, Guangzhou 510640,China Email: [email protected], [email protected], [email protected] Received June 19, 2012; accepted August 13, 2012; published online April 25, 2013 Abstract In two dimensions, we study the compressible hydrodynamic flow of liquid crystals with periodic boundary conditions. As shown by Ding et al. (2013), when the parameter λ →∞, the solutions to the compressible liquid crystal system approximate that of the incompressible one. Furthermore, Ding et al. (2013) proved that the regular incompressible limit solution is global in time with small enough initial data. In this paper, we show that the solution to the compressible liquid crystal flow also exists for all time, provided that λ is sufficiently large and the initial data are almost incompressible. Keywords liquid crystals, slightly compressible, global existence MSC(2010) 76N10, 35Q30, 35R35 Citation: Ding S J, Huang J R, Lin J Y. Global existence for slightly compressible hydrodynamic flow of liquid crystals in two dimensions. Sci China Math, 2013, 56: 2233–2250, doi: 10.1007/s11425-013-4620-2 1 Introduction 1 The compressible hydrodynamic flow of liquid crystals with respect to the parameter λ = M ,whereM is the Mach number of the fluid defined as the ratio of typical fluid velocities to typical sound speeds, is given by the energetic variational approach as follows (see [6] and furthermore [33] for a Ginzburg-Landau approximation model), ⎧ λ λ λ ⎪ρt + ∇·(ρ u )=0, ⎪ ⎪ ⎪ P(ρλ) ⎨⎪uλ + uλ ·∇uλ + λ2 ∇ρλ t ρλ (1.1) ⎪ λ λ λ λ |∇ λ|2 ⎪ μ λ μ + η ∇ ∇· λ − ν ∇· ∇ λ ∇ λ − n ⎪ = λ Δu + λ ( u ) λ n n Id , ⎪ ρ ρ ρ 2 ⎩⎪ λ λ λ λ λ λ 2 λ nt + u ·∇n = θ (Δn + |∇n | n ), where x ∈ Ω, a domain in Rd,andt 0. The unknowns are the density of fluid ρλ :Ω× [0, +∞) → R1, the velocity field of the fluid uλ ∈ Ω × [0, +∞) → Rd, and the macroscopic average of the nematic liquid crystal orientation field nλ :Ω× (0, +∞) → S2, which is a unit vector. P = P(ρ)isthesmooth ∗Corresponding author c Science China Press and Springer-Verlag Berlin Heidelberg 2013 math.scichina.com www.springerlink.com 2234 Ding S J et al. Sci China Math November 2013 Vol. 56 No. 11 pressure-density function with P(ρ) > 0forρ>0. The symbol ⊗ is the usual Kronecker product, d e.g., (a ⊗ b)ij = aibj for a, b ∈ R . The notation ∇n ∇n denotes the d × d matrix whose (i, j)-th λ λ λ λ entry is given by ∇in ·∇j n. Id is the unit matrix of size d. The constants μ , η , ν and θ represent the shear viscosity, the bulk viscosity, the competition between kinetic energy and potential energy, and microscopic elastic relaxation time for the molecular orientation field, respectively, satisfying μλ, νλ, θλ > 0, and 2μλ + dηλ 0. Without loss of generality, we assume in the following that μλ ≡ μ, ηλ ≡ η, νλ ≡ ν and θλ ≡ θ are uniform constants independent of λ. As λ goes to infinite, (ρλ,uλ,nλ) → (1,U,N) in some sense (see [6] with periodic boundary condition), where (U, N) solves the incompressible hydrodynamic flow equations of liquid crystals [5,9,20]: ⎧ ⎪ ⎨⎪∇·U =0, Ut + U ·∇U + ∇p = μΔU − ν∇·(∇N ∇N) , (1.2) ⎪ ⎩ 2 Nt + U ·∇N = θ(ΔN + |∇N| N), λ λ 2 ∇ 2 P (ρ ) ∇ λ − ν ∇ |∇n | where p is the limit of the term λ ρλ ρ ρλ ( 2 ). In a series of papers, Lin [23] and Lin and Liu [25–27] addressed the existence and partial regularity theory of suitable weak solution to the incompressible hydrodynamic flow of liquid crystals (1.2) of variable length. More precisely, they considered the approximate equations of incompressible hydrodynamic flow 2 2 (1−|N| )N of liquid crystals (|∇N| N is replaced by 2 ), and proved, the local existence of classical solutions and the global existence of weak solutions in dimensions two and three [25]. For any fixed >0, they also showed the existence and uniqueness of global classical solutions either in dimensions two or three when the fluid viscosity μ is sufficiently large; Lin and Liu [26] extended the classical theorem by Caffarelli et al. [1] on the Navier-Stokes equations that asserts the one-dimensional parabolic Hausdorff measure of the singular set of any suitable weak solution is zero. See also [4,28] for relevant results. For the system (1.2), Lin et al. [24] proved that there exists a global weak solution which is regular with the exception of at most finitely many time-slices in dimension two. For the density-dependent incompressible flow of liquid crystals, on one hand, Liu and Zhang [32] obtained the global weak solutions in dimension three with 2 the initial density ρ0 ∈ L for the model of variable length. Jiang and Tan [16] improved the condition ∈ γ 3 → of ρ0 to ρ0 L , γ>2 . However, the estimates depend on , and thus they cannot take the limit 0. On the other hand, considering the original term |∇N|2N, Wen and Ding [35] proved the local existence and uniqueness of the strong solutions to the model in a bounded domain in Rd (d = 2 or 3), provided that the initial density ρ0 0. Furthermore, they got the global existence and uniqueness of the strong solutions with small initial data and infx∈Ω ρ0 > 0 in dimension two. Very recently, Li and Wang [22] proved the existence and uniqueness of the local strong solutions with large initial data and the global strong solutions with small data in Besov space for the initial density away from vacuum in dimension three. For an incompressible non-isothermal model, we refer to [10]. The study for the compressible hydrodynamic flow (1.1) (when λ is a fixed positive constant) began in recent years.
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