Compressible hydrodynamic flow of liquid crystals in 1-D

A blow‐up criterion for incompressible hydrodynamic flow of liquid crystals in dimension two

“…Considering the Dirichlet and Neumann boundary condition for ( u , d ) in dimension one, Ding et al . obtained the existence and uniqueness of global classical solutions with the initial data ( ρ 0 , u 0 , d 0 ) ∈ C 1, α ( I ) × C 2, α ( I ) × C 2, α ( I ) and the initial density away from vacuum, where I = [0,1]. They also addressed both the existence and the uniqueness of global strong solutions for 0 ≤ ρ 0 ∈ H 1 ( I ) and ( u 0 , d 0 ) ∈ H 1 ( I ) × H 2 ( I ).…”

Low Mach number limit of global solutions to 3‐D compressible nematic liquid crystal flows with Dirichlet boundary condition

“…Very recently, the global existence of low‐energy weak solution was obtained in Wu and Tan . The existence and uniqueness of global strong solution for 1‐D case were shown by Ding et al For the case of multidimensional space, the local existence of strong solutions was obtained in Huang et al, and the existence and uniqueness of global strong solutions to the Cauchy problem in critical Besov spaces were proved in Hu and Wu provided that the initial data are close to an equilibrium state. The local existence and uniqueness of classical solution were established by Ma, and the global existence of classical solutions to the Cauchy problem was shown in Li et al with smooth initial data that has small energy.…”

Classical solutions for the compressible liquid crystal flows with nonnegative initial densities

“…In dimension one, Ding et al [10,11] have obtained the global existence for the weak and strong solutions. However, for dimensions at least two, it is reasonable to believe that the local strong solutions may cease to exist globally.…”