New smooth self-similar implosion profiles for compressible Euler equations are constructed with explicit exponents and proven stable under radial and certain non-radial perturbations.
Global Regular Solutions of the Degenerate Compressible Navier-Stokes Equations with Large Initial Data of Spherical Symmetry
3 Pith papers cite this work. Polarity classification is still indexing.
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abstract
A fundamental open problem in the theory of the compressible Navier-Stokes equations is whether regular spherically symmetric flows can develop singularities, such as cavitation or implosion, in finite time. A formidable challenge lies in how the well-known coordinate singularity at the origin can be overcome to control the lower or upper bound of the density. In this paper, when the viscosity coefficients are degenerately density-dependent (as in the shallow water equations), we prove that, for general large spherically symmetric initial data with bounded positive density, solutions remain globally regular and cannot undergo cavitation or implosion in two and three spatial dimensions. Moreover, the far-field vacuum is allowed for the data under consideration here. Our results hold for all adiabatic exponents $\gamma\in(1,\infty)$ in two dimensions, and for physical adiabatic exponents $\gamma\in (1, 3)$ in three dimensions, without any restriction on the size of the initial data.
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math.AP 3years
2026 3verdicts
UNVERDICTED 3representative citing papers
Global well-posedness of regular solutions to barotropic compressible Navier-Stokes with density-dependent viscosities ρ^δ (δ ∈ (1/2,1)) for large spherical symmetric data vanishing at infinity in 2 and 3 dimensions.
Global well-posedness of spherically symmetric classical solutions is established for degenerate compressible Navier-Stokes equations in 2D and 3D with large initial data for alpha above approximately 0.54-0.68 and gamma in specified ranges.
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Smooth and stable Euler implosions
New smooth self-similar implosion profiles for compressible Euler equations are constructed with explicit exponents and proven stable under radial and certain non-radial perturbations.
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Global Regular Solutions of the Compressible Navier-Stokes Equations with Nonlinear Density-Dependent Viscosities and Large Initial Data of Spherical Symmetry
Global well-posedness of regular solutions to barotropic compressible Navier-Stokes with density-dependent viscosities ρ^δ (δ ∈ (1/2,1)) for large spherical symmetric data vanishing at infinity in 2 and 3 dimensions.
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Global Well-Posedness of Classical Solutions to the Multi-Dimensional Degenerate Compressible Navier-Stokes Equations with Large Spherically Symmetric Initial Data
Global well-posedness of spherically symmetric classical solutions is established for degenerate compressible Navier-Stokes equations in 2D and 3D with large initial data for alpha above approximately 0.54-0.68 and gamma in specified ranges.