Local smooth solutions of the free-boundary compressible primitive equations with physical vacuum

Local existence and uniqueness of smooth solutions are established for the three-dimensional compressible primitive equations with constant viscosity and a physical vacuum free boundary. We employ a Nash-Moser approach to address two distinct derivative losses, one arising from the free boundary geometry in the transformed viscosity and the other from the diagnostic reconstruction of the vertical velocity. We handle the first through Alinhac’s good velocity unknown and the second through a modified height. Combining these two corrections with weighted energy estimates adapted to the vacuum degeneracy yields the tame estimates required for the iteration.