Abstract
This paper considers the initial-boundary value problem of the one-dimensional full compressible nematic liquid crystal flow problem. The initial density is allowed to touch vacuum, and the viscous and heat conductivity coefficients are kept to be positive constants. Global existence of strong solutions is established for any H^{2} initial data in the Lagrangian flow map coordinate, which holds for both vacuum and non-vacuum case. The key difficulty is caused by the lack of the positive lower bound of the density. To overcome such difficulty, it is observed that the ratio of frac{rho _{0(y)}}{rho (t,y)} is proportional to the time integral of the upper bound of temperature and vector director field, along the trajectory. Density weighted Sobolev type inequalities are constructed for both temperature and director field in terms of frac{rho _{0(y)}}{rho (t,y)} and small dependence on their dissipation estimates. Besides this, to deal with cross terms arising due to liquid crystal flow, higher order priori estimates are established by using effective viscous flux.
Highlights
Liquid crystal material can be known as an intermediate phase between solid and liquid
For non-isothermal compressible liquid crystal flow, in 2019, Tang and Sun [55] proved the global existence of strong solutions allowing vacuum, provided that the initial data satisfy some compatibility condition and the heat conductivity satisfies
Motivated by Kazhikhov [35] and Li [50], in this paper we aim to study the global well-posedness of strong solutions to the one-dimensional non-isothermal compressible nematic liquid crystal flow equations, i.e., system (1.5), with constant viscosity and heat conductivity in the presence of vacuum
Summary
Liquid crystal material can be known as an intermediate phase between solid and liquid. For non-isothermal compressible liquid crystal flow, in 2019, Tang and Sun [55] proved the global existence of strong solutions allowing vacuum, provided that the initial data satisfy some compatibility condition and the heat conductivity satisfies. Global strong solution to one-dimensional non-isothermal compressible nematic liquid crystal equations for arbitrary large initial data is not known for constant coefficients and vacuum. Motivated by Kazhikhov [35] and Li [50], in this paper we aim to study the global well-posedness of strong solutions to the one-dimensional non-isothermal compressible nematic liquid crystal flow equations, i.e., system (1.5), with constant viscosity and heat conductivity in the presence of vacuum. In order to control the right-hand side of the above inequality, dissipation estimates are obtained on the director field in terms of L∞ – norm of temperature. The desired estimates follow from (2.40) by applying Proposition 1 and (i) of Proposition 2
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