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

Abstract

In this paper, we investigate the Cauchy problem for the compressible nematic liquid crystal flows in three-dimensional whole space. First of all, we establish the time decay rates for compressible nematic liquid crystal flows by the method of spectral analysis and energy estimates. Furthermore, we enhance the convergence rates for the higher-order spatial derivatives of density, velocity and director. Finally, the time decay rates of mixed space-time derivatives of solution are also established.

Long‐time behavior of solution for the compressible Navier‐Stokes‐Maxwell equations in

“…These decay rates have also been established by Li and Yu for case q =1. Following the spirit of work, Gao et al verified the decay rate also hold on for k =2. These decay estimates were generalized to the compressible Hall‐MHD equations by Gao and Yao .…”

“…Furthermore, if the initial perturbation is small in L 1 −norm additionally, he also established the following time decay rate for all t ≥0. Recently, Gao et al established the global‐in‐time existence for the compressible nematic liquid crystal flows in three‐dimensional whole space, and time decay rates for the high‐order spatial derivatives of density, velocity and director were obtained by using the Green function, energy estimates, and Fourier splitting method. For more results about the Ericksen‐Leslie system, the readers can refer to previous studies() and references therein.…”

“…Finally, we establish decay rates for the mixed space-time derivatives of solution to the Cauchy problems (2) to (5). (2) to (5) has the time decay rates…”

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

“…The local existence and uniqueness of classical solution were established by Ma,13 and the global existence of classical solutions to the Cauchy problem was shown in Li et al 14 with smooth initial data that has small energy. For more recent results about the compressible nematic liquid crystal flows, the readers can refer to previous works [15][16][17][18][19][20][21][22] and references therein.…”

Uniform regularity and vanishing viscosity limit for the compressible nematic liquid crystal flows

“…The viscous coefficients µ and λ satisfy the physical restrictions µ > 0, 2µ + 3λ > 0, where the parameter ε > 0 is the inverse of the Reynolds number. For more results about the compressible Ericksen-Leslie system (1.1), the readers can refer to [1,2,3,4,5,6,7,8] and references therein. Corresponding to the system (1.1), we impose the following Navier-slip type and Neumann boundary conditions: u ε · n = 0, ((Su ε )n) τ = −(Au ε ) τ , and ∂d ε ∂n = 0, on ∂Ω, (1.2) where A is a given smooth symmetric matrix(see [9]), n is the outward unit vector normal to ∂Ω, (Au ε ) τ represents the tangential component of Au ε .…”