Global Strong Solutions to the One-Dimensional Isentropic Compressible Liquid Crystal Equations with a Vacuum Free Boundary and Large Initial Data

In this paper, we establish the global existence and uniqueness of strong solutions to the one-dimensional vacuum free boundary problem for the isentropic compressible liquid crystal equations under the influence of gravity with large initial data, where the density is allowed to vanish continuously at the free boundary. The main difficulty is the degeneracy of the momentum equation caused by the vanishing of the density, which prevents standard energy methods from giving pointwise control of the velocity gradient. Working in Lagrangian coordinates, we derive a time-uniform pointwise lower bound and a finite-time pointwise upper bound on the Jacobian ηx, together with a finite L∞-bound on the velocity gradient vx on any finite time interval [0,T], which, in particular, guarantees that the free boundary is a well-defined C1 curve. This appears to be the first global strong solution result for this problem; the earlier work of Huang and Ding establishes global weak solutions, for which the free boundary is not addressed in a pointwise sense.