Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one

Abstract

We consider the equation modeling the compressible hydrodynamic flow of liquid crystals in one dimension. In this paper, we establish the existence of a weak solution (ρ, u, n) of such a system when the initial density function 0 ≤ ρ 0 ∈ L γ for γ > 1, u 0 ∈ L 2 , and n 0 ∈ H 1 . This extends a previous result by [12], where the existence of a weak solution was obtained under the stronger assumption that the initial density function 0 < c ≤ ρ

Strong solutions to the density-dependent incompressible nematic liquid crystal flows

“…Therefore, many researchers attempt to obtain global strong solutions under some additional assumptions. Wen and Ding [18] also established the global existence and uniqueness of solutions for two dimensional case if the initial density is away from vacuum and the initial data is of small norm. Global existence of strong solutions with small initial data to three dimensional liquid crystal equations are obtained by Li and Wang in [20] for constant density case, Li and Wang in [21] for nonconstant but positive density case, and Ding, Huang and Xia in [22].…”

“…Recently, Hu and Wu [17] proved the decay of the velocity field for arbitrary large regular initial data with the initial density being away from vacuum in two dimensional domain with smooth boundary. As for the case of |∇d| 2 d, Wen and Ding [18] obtained the local existence and uniqueness of strong solutions to the Dirichlet problem in bounded domain with initial density being allowed to have vacuum. Since the strong solutions of a harmonic map can blow up in finite time [19], one cannot expect to get a global strong solution with general initial data.…”

Long-time behavior of solution for the compressible nematic liquid crystal flows in R3

“…Recently, Liu and Zhang [11], for the density dependent model, obtained the global weak solutions in dimension three with the initial density ρ 0 ∈ L 2 , which was improved by Jiang and Tan [12] for the case ρ 0 ∈ L γ (γ > 3 2 ). Under the constraint d ∈ S 2 , Wen and Ding [13] established the local existence for the strong solution and obtained the global solution under the assumptions of small energy and positive initial density. Later, Hong [14] and Lin, Lin and Wang [15] showed independently the global existence of a weak solution in two-dimensional space.…”

Time‐periodic solution to the compressible nematic liquid crystal flows in periodic domain

“…For the compressible case, the simplified Ericksen‐Leslie system becomes more complicate, which is a strongly coupling system between the compressible Navier‐Stokes equation and the transported harmonic map heat flow to S 2 . The authors in Ding et al first consider the solutions to the initial‐boundary value problem in with nonnegative initial density. They obtained the global existence and uniqueness for classical, weak, or strong solutions in dimension 1.…”