scholarly journals Regularity Criteria for a Coupled Navier-Stokes and Q-Tensor System

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Jishan Fan ◽  
Tohru Ozawa
Keyword(s):  
Liquid Crystal ◽  
Nematic Liquid Crystal ◽  
Stokes System ◽  
Type System ◽  
Parabolic Type ◽  
Strong Solutions ◽  
Regularity Criteria ◽  
Navier Stokes ◽  
Liquid Crystal Flow ◽  
Local Strong Solutions

We study a system describing the evolution of a nematic liquid crystal flow. The system couples a forced Navier-Stokes system describing the flow with a parabolic-type system describing the evolution of the nematic crystal director fields (Q-tensors). We prove some regularity criteria for the local strong solutions. However, we do not provide estimates on the rates of increase of high norms.

scholarly journals Global solution to the compressible non-isothermal nematic liquid crystal equations with constant heat conductivity and vacuum

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tariq Mahmood ◽  
Mei Sun
Keyword(s):  
Liquid Crystal ◽  
Nematic Liquid Crystal ◽  
Heat Conductivity ◽  
Initial Density ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sobolev Type ◽  
Director Field ◽  
Liquid Crystal Flow

AbstractThis paper considers the initial-boundary value problem of the one-dimensional full compressible nematic liquid crystal flow problem. The initial density is allowed to touch vacuum, and the viscous and heat conductivity coefficients are kept to be positive constants. Global existence of strong solutions is established for any $H^{2}$ H 2 initial data in the Lagrangian flow map coordinate, which holds for both vacuum and non-vacuum case. The key difficulty is caused by the lack of the positive lower bound of the density. To overcome such difficulty, it is observed that the ratio of $\frac{\rho _{0(y)}}{\rho (t,y)}$ ρ 0 ( y ) ρ ( t , y ) is proportional to the time integral of the upper bound of temperature and vector director field, along the trajectory. Density weighted Sobolev type inequalities are constructed for both temperature and director field in terms of $\frac{\rho _{0(y)}}{\rho (t,y)}$ ρ 0 ( y ) ρ ( t , y ) and small dependence on their dissipation estimates. Besides this, to deal with cross terms arising due to liquid crystal flow, higher order priori estimates are established by using effective viscous flux.

scholarly journals Continuation Criterion for the 2D Liquid Crystal Flows

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Jishan Fan ◽  
Tohru Ozawa
Keyword(s):  
Liquid Crystal ◽  
Stokes System ◽  
Blow Up ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Sobolev Inequalities ◽  
Wave Maps ◽  
Logarithmic Sobolev Inequalities ◽  
Blow Up Criterion

We consider the 2D liquid crystal systems, which consists of Navier-Stokes system coupled with wave maps or biharmonic wave maps, respectively. By logarithmic Sobolev inequalities, we obtain a blow-up criterion ∇d,∂td∈L1(0,T;B˙∞,∞0(ℝ2)) for the case with wave maps, and we prove the existence of a global-in-time strong solutions for the case with biharmonic wave maps.

scholarly journals A Regularity Criterion for Compressible Nematic Liquid Crystal Flows

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Jishan Fan ◽  
Tohru Ozawa
Keyword(s):  
Liquid Crystal ◽  
Bounded Domain ◽  
Nematic Liquid Crystal ◽  
Blow Up ◽  
Strong Solutions ◽  
Regularity Criterion ◽  
Flow Modeling ◽  
Nematic Liquid Crystal Flows ◽  
Local Strong Solutions ◽  
Compressible Nematic Liquid Crystal

We prove a blow-up criterion for local strong solutions to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials in a bounded domain.

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