Well-Posedness of Degenerate Initial-Boundary Value Problems to a Hyperbolic-Parabolic Coupled System Arising from Nematic Liquid Crystals

Abstract

This paper is focused on the local well-posedness of initial-boundary value and Cauchy problems to a one-dimensional quasilinear hyperbolic-parabolic coupled system with boundary or far field degenerate initial data. The governing system is derived from the theory of nematic liquid crystals, which couples a hyperbolic equation describing the crystal property and a parabolic equation describing the liquid property of the material. The hyperbolic equation is degenerate at the boundaries or spatial infinity, which results in the classical methods for the strictly hyperbolic-parabolic coupled systems being invalid. We introduce admissible weighted function spaces and apply the parametrix method to construct iteration mappings for these two degenerate problems separately. The local existence and uniqueness of classical solutions of the degenerate initial-boundary value and Cauchy problems are established by the contraction mapping principle in their selected function spaces. Moreover, the solutions have no loss of regularity and their existence times are independent of the spatial variable.