Strong solutions of the compressible nematic liquid crystal flow

We study strong solutions of the simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows in a domain $\Omega \subset\mathbb R^3$. We first prove the local existence of unique strong solutions provided that the initial data $\rho_0, u_0, d_0$are sufficiently regular and satisfy a natural compatibility condition. The initial density function $\rho_0$ may vanish on an open subset (i.e., an initial vacuum may exist). We then prove a criterion for possible breakdown of such a local strong solution at finite time in terms of blow up of the quantities $\|\rho\|_{L^\infty_tL^\infty_x}$ and $\|\nabla d\|_{L^3_tL^\infty_x}$.

[1]  P. Gennes,et al.  The physics of liquid crystals , 1974 .

[2]  F. Lin,et al.  Nonparabolic dissipative systems modeling the flow of liquid crystals , 1995 .

[3]  David Kinderlehrer,et al.  Existence and partial regularity of static liquid crystal configurations , 1986 .

[4]  Changyou Wang,et al.  Heat Flow of Harmonic Maps Whose Gradients Belong to $$L^{n}_{x}L^{\infty}_{t}$$ , 2008 .

[5]  Jing Li,et al.  Serrin-Type Criterion for the Three-Dimensional Viscous Compressible Flows , 2010, SIAM J. Math. Anal..

[6]  Tosio Kato,et al.  Remarks on the breakdown of smooth solutions for the 3-D Euler equations , 1984 .

[7]  Daisuke Fujiwara,et al.  An L_r-theorem of the Helmholtz decomposition of vector fields , 1977 .

[8]  Huanyao Wen,et al.  Blow Up Criterion for Compressible Nematic Liquid Crystal Flows in Dimension Three , 2011, 1104.5685.

[9]  Haim Brezis,et al.  Remarks on the Euler equation , 1974 .

[10]  F. Lin,et al.  The analysis of harmonic maps and their heat flows , 2008 .

[11]  Hi Jun Choe,et al.  Strong solutions of the Navier-Stokes equations for isentropic compressible fluids , 2003 .

[12]  J. Ericksen,et al.  Hydrostatic theory of liquid crystals , 1962 .

[13]  Fanghua Lin,et al.  Partial regularity of the dynamic system modeling the flow of liquid crystals , 1995 .

[14]  Fanghua Lin,et al.  Liquid Crystal Flows in Two Dimensions , 2010 .

[15]  Jacques Simon,et al.  Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure , 1990 .

[16]  Xian-gao Liu,et al.  A blow-up criterion for the compressible liquid crystals system , 2010, 1011.4399.

[17]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[18]  Hi Jun Choe,et al.  Unique solvability of the initial boundary value problems for compressible viscous fluids , 2004 .

[19]  Eduard Feireisl,et al.  Dynamics of Viscous Compressible Fluids , 2004 .

[20]  Changyou Wang,et al.  Compressible hydrodynamic flow of liquid crystals in 1-D , 2011 .

[21]  F. Lin,et al.  On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals , 2010 .

[22]  Mariarosaria Padula,et al.  Existence of global solutions for 2-dimensional viscous compressible flows , 1986 .

[23]  F. M. Leslie.  Some constitutive equations for liquid crystals , 1968 .

[24]  Gui-Qiang G. Chen,et al.  A Study of the Navier-Stokes Equations with the Kinematic and Navier Boundary Conditions , 2008, 0901.0147.

[25]  Pierre-Louis Lions,et al.  Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models , 1998 .

[26]  A. Chambolle,et al.  Crack Initiation in Brittle Materials , 2008 .

[27]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[28]  Chao Wang,et al.  A Beale-Kato-Majda Blow-up criterion for the 3-D compressible Navier-Stokes equations , 2010, 1001.1247.

[29]  Zhouping Xin,et al.  Blowup Criterion for Viscous Baratropic Flows with Vacuum States , 2010, 1004.5469.

[30]  W. Wahl.  Estimating ∇u by div u and curl u , 1992 .

[31]  Modelling of Nematic Liquid Crystals in Electromagnetic Fields , 2009 .

[32]  Zhifei Zhang,et al.  Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows , 2012 .

[33]  Min-Chun Hong.  Global existence of solutions of the simplified Ericksen–Leslie system in dimension two , 2011 .

[34]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[35]  Changyou Wang,et al.  Blow up Criterion for Nematic Liquid Crystal Flows , 2011, 1104.5683.

[36]  V. Solonnikov.  Lp-Estimates for Solutions to the Initial Boundary-Value Problem for the Generalized Stokes System in a Bounded Domain , 2001 .

[37]  Changyou Wang,et al.  WEAK SOLUTION TO COMPRESSIBLE HYDRODYNAMIC FLOW OF LIQUID CRYSTALS IN DIMENSION ONE , 2010 .

[38]  G. Ponce.  Remarks on a paper by J. T. Beale, T. Kato, and A. Majda , 1985 .

[39]  F. Lin.  Nonlinear theory of defects in nematic liquid crystals; Phase transition and flow phenomena , 1989 .