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arXiv · 1911.04315

Incompressible limit of the Ericksen-Leslie hyperbolic liquid crystal model in compressible flow

Liang Guo·Ning Jiang·Fucai Li·Yi-Long Luo·Shaojun Tang
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Abstract

Ericksen and Leslie proposed a hydrodynamic model for liquid crystals in the format of conservation laws in the 1960s. Their original model includes inertial and compressibility effects, which makes the model a coupled parabolic-hyperbolic system. In this paper we build up the connection between the compressible and incompressible parabolic-hyperbolic liquid crystal model in the framework of classical solutions. We first derive the scaled Ericksen-Leslie system with dimensionless numbers, including Mach, Reynolds, and Ericksen numbers. In particular, we introduce the so-called inertial constant $χ$ which characterizes the inertial effect of the liquid crystal molecular. Next, we establish the energy estimates uniform in the Mach number $ε$ for both the compressible system and its time-derivative system with small data. Then, we pass to the limit $ε\rightarrow 0$ in the compressible system, so that we establish the global classical solution of the incompressible system by the compactness arguments. Moreover, we also obtain the convergence rates associated with $L^2$-norm in the case of well-prepared initial data. This is the first result on the incompressible limit of the compressible parabolic-hyperbolic liquid crystal model and confirms the relations of different parabolic-hyperbolic liquid crystal model rigorously.

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Liang Guo, Ning Jiang, Fucai Li, Yi-Long Luo, Shaojun Tang. 2023-10-31. Incompressible limit of the Ericksen-Leslie hyperbolic liquid crystal model in compressible flow. https://doi.org/10.1007/s00332-023-09972-4

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