Work

Finite-Time Vanishing of Vacuum and Large-Time Behavior of the Multi-Dimensional Degenerate Compressible Navier-Stokes Equations with Large Spherically Symmetric Initial Data

Year: 2026

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.37860

Authors: Lei, Qinghao, Liang, Zhilei

Institutions: Chinese Academy of Sciences, Institute of Applied Mathematics, University of Chinese Academy of Sciences, Southwestern University of Finance and Economics, Academy of Mathematics and Systems Science

Language: English

Topic: Navier-Stokes equation solutions

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Natural Science Foundation of China

Award: G6334870596

OpenAlex ID: W7214979414

Cites: 5

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Related to: 10