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Local smooth solutions of the free-boundary compressible primitive equations with physical vacuum
Abstract
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.
Related
- Global weak solutions with vacuum for isentropic compressible Navier-Stokes and Navier-Stokes-Poisson equations in the unit disk under Dirichlet boundary conditions
- On the intrinsic hydrostatic energy structure of the barotropic compressible primitive equations
- Uniqueness of Leray-Hopf weak solutions to the two-dimensional viscous Primitive Equations
- Global well-posedness of the primitive equations with stochastic wind-driven boundary conditions
- Uniqueness of solutions to the 2D compressible Navier-Stokes equations for bounded densities
BibTeX
@misc{zoechling2026local,
title = {{Local smooth solutions of the free-boundary compressible primitive equations with physical vacuum}},
author = {Tarek Zoechling},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18387},
eprint = {2609.18387},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18387},
}
RIS
TY - UNPB
TI - Local smooth solutions of the free-boundary compressible primitive equations with physical vacuum
AU - Tarek Zoechling
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18387
UR - https://arxiv.org/abs/2609.18387
AB - 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.
ER -