Finite-Time Vanishing of Vacuum and Large-Time Behavior of the Multi-Dimensional Degenerate Compressible Navier-Stokes Equations with Large Spherically Symmetric Initial Data
Abstract
In this paper, we study the two- and three-dimensional barotropic compressible Navier-Stokes equations with degenerate density-dependent viscosity coefficients in the whole space or in a ball for arbitrarily large spherically symmetric initial data. For initial data allowing vacuum, we establish the global existence of weak solutions and derive uniform-in-time a priori estimates. As a consequence, we prove that the vacuum state of weak solutions will vanish in finite time. The key ingredients include a treatment of the pressure terms based on separate estimates in the low- and high-density regions, the Bresch-Desjardins entropy estimates, weighted radial estimates, and a coupled control of the velocity and effective velocity. In the three-dimensional case, this approach also allows us to treat the endpoint case of the adiabatic exponent. For sufficiently regular initial data with strictly positive density, we establish the global existence and large-time behavior of classical solutions.
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BibTeX
@misc{lei2026finitetime,
title = {{Finite-Time Vanishing of Vacuum and Large-Time Behavior of the Multi-Dimensional Degenerate Compressible Navier-Stokes Equations with Large Spherically Symmetric Initial Data}},
author = {Qinghao Lei and Zhilei Liang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37860},
eprint = {2609.37860},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37860},
}
RIS
TY - UNPB
TI - Finite-Time Vanishing of Vacuum and Large-Time Behavior of the Multi-Dimensional Degenerate Compressible Navier-Stokes Equations with Large Spherically Symmetric Initial Data
AU - Qinghao Lei
AU - Zhilei Liang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37860
UR - https://arxiv.org/abs/2609.37860
AB - In this paper, we study the two- and three-dimensional barotropic compressible Navier-Stokes equations with degenerate density-dependent viscosity coefficients in the whole space or in a ball for arbitrarily large spherically symmetric initial data. For initial data allowing vacuum, we establish the global existence of weak solutions and derive uniform-in-time a priori estimates. As a consequence, we prove that the vacuum state of weak solutions will vanish in finite time. The key ingredients include a treatment of the pressure terms based on separate estimates in the low- and high-density regions, the Bresch-Desjardins entropy estimates, weighted radial estimates, and a coupled control of the velocity and effective velocity. In the three-dimensional case, this approach also allows us to treat the endpoint case of the adiabatic exponent. For sufficiently regular initial data with strictly positive density, we establish the global existence and large-time behavior of classical solutions.
ER -