Abstract

Objective: This paper investigates the long-time behavior of weak solutions for the three-dimensional compressible nematic liquid crystal flow (NLCF) in the presence of vacuum, with a focus on understanding the energy dissipation mechanism. Methods: On the basis of Ericksen-Leslie dynamical system and compressible Navier-Stokes equations, as well as the harmonic mapping heat flux equation, based on a precise a priori estimate method, the effective viscosity flux method, weak convergent theorem in functional analysis method, overcomes mathematical analysis difficulty and physical modeling difficulty caused by the extremely degenerate momentum equation because of the vacuum state. Results: Under some reasonable initial energy bound and boundary assumptions, we can obtain that the nonlinear coupled system has in time a globally well-posed weak solution with finite energy. At the same time, $t\longrightarrow \infty$, there will be a situation where, on average microscopically, the fluid’s macro-scopic velocity fields tend toward zero. At that same moment, every microscopic orientation vector field, representing the average orientation of the liquid crystals, converges asymptotically to certain steady harmonic mappings that satisfy either homogeneous Neumann boundary conditions or homogeneous Dirichlet boundary conditions. Conclusion: The existence of the vacuum implies that there is an essential absence of local dissipativity and mathematical properties are lost in the system; however, with careful construction of a Lyapunov functional, as well as higher order energy integrals inequalities, it is still possible to characterize the long-term behaviour of the system under energy dissipation. In addition, it can enhance the theoretical understanding of the partial differential equations (PDEs) governing complex fluid dynamics. It also provides a solid theoretical foundation and analytical framework for the engineering applications of nematic liquid crystal materials under high-pressure environments or cavitation conditions.