Global regularity for 3D large spherically symmetric degenerate compressible Navier-Stokes equations with superlinear BD viscosity and far-field vacuum
Abstract
Superlinear density-dependent viscosity presents a fundamental obstacle of global regularity to the compressible Navier-Stokes equations in the presence of far-field vacuum: the viscous diffusion vanishes at infinity, so excluding vacuum on bounded sets does not suffice to control the flow globally. For α∈ (1,11.7) and γ∈ [2α−1,3α−1) , we establish global regularity for arbitrarily large, strong, finite-mass initial data with positive density at every finite point and vacuum at infinity. First, we introduce new rescalsed variables regarding the velocity, effective velocity and mass, then we exploit several novel quantities near the center and employ a refined maximum principle to establish the Lipschitz continuity of the velocity at the center-thereby resolving the singularity of the velocity gradient. Another key mechanism is a quantitative control of material volume distortion that remains effective in the vacuum tail. Conservative estimates for the physical and effective velocity gradients yield uniformly vanishing exterior variation tails. These estimates identify a material trajectory with small cumulative Jacobian variation and lead to two-sided bounds for the exterior volume Jacobian, making the evolving density comparable to its initial value along particle paths. On spatial scales determined by the initial density, the resulting degenerate equations admit uniform parabolic estimates that recover control of the physical deformation without a global density lower bound. Combined with compact positivity estimates and a one-sided Glaeser inequality, this closes a homogeneous continuation argument without imposing L2-integrability on the density powers themselves. The solutions conserve mass, preserve far-field vacuum, and remain smooth and strictly positive on every compact spacetime cylinder.
Related
- Finite-Time Vanishing of Vacuum and Large-Time Behavior of the Multi-Dimensional Degenerate Compressible Navier-Stokes Equations with Large Spherically Symmetric Initial Data
- Sharp sign criterion for global existence and blow-up of strong solutions with arbitrarily large initial data in the 3D spherically symmetric compressible Navier-Stokes equations
- Well-posedness, Regularity, and Strong Approximations of Superlinear Stochastic Reaction-Diffusion Equation
- Improved Regularity for Nonlocal Hamilton--Jacobi Equations with Superlinear Hamiltonians
- Spherically symmetric Finsler metrics satisfying the σT-condition
BibTeX
@misc{huang2026global,
title = {{Global regularity for 3D large spherically symmetric degenerate compressible Navier-Stokes equations with superlinear BD viscosity and far-field vacuum}},
author = {Xiangdi Huang and Bingshu Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32062},
eprint = {2610.32062},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32062},
note = {CC0},
}
RIS
TY - UNPB
TI - Global regularity for 3D large spherically symmetric degenerate compressible Navier-Stokes equations with superlinear BD viscosity and far-field vacuum
AU - Xiangdi Huang
AU - Bingshu Li
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32062
UR - https://arxiv.org/abs/2610.32062
AB - Superlinear density-dependent viscosity presents a fundamental obstacle of global regularity to the compressible Navier-Stokes equations in the presence of far-field vacuum: the viscous diffusion vanishes at infinity, so excluding vacuum on bounded sets does not suffice to control the flow globally. For α∈ (1,11.7) and γ∈ [2α−1,3α−1) , we establish global regularity for arbitrarily large, strong, finite-mass initial data with positive density at every finite point and vacuum at infinity. First, we introduce new rescalsed variables regarding the velocity, effective velocity and mass, then we exploit several novel quantities near the center and employ a refined maximum principle to establish the Lipschitz continuity of the velocity at the center-thereby resolving the singularity of the velocity gradient. Another key mechanism is a quantitative control of material volume distortion that remains effective in the vacuum tail. Conservative estimates for the physical and effective velocity gradients yield uniformly vanishing exterior variation tails. These estimates identify a material trajectory with small cumulative Jacobian variation and lead to two-sided bounds for the exterior volume Jacobian, making the evolving density comparable to its initial value along particle paths. On spatial scales determined by the initial density, the resulting degenerate equations admit uniform parabolic estimates that recover control of the physical deformation without a global density lower bound. Combined with compact positivity estimates and a one-sided Glaeser inequality, this closes a homogeneous continuation argument without imposing L^2-integrability on the density powers themselves. The solutions conserve mass, preserve far-field vacuum, and remain smooth and strictly positive on every compact spacetime cylinder.
ER -