Global existence of classical solutions for the multi-dimensional compressible Navier-Stokes-Poisson equations on solid balls for arbitrary spherically symmetric large initial data

TL;DR

This paper proves the global existence of classical solutions for the multi-dimensional compressible Navier-Stokes-Poisson equations under spherical symmetry on solid balls, accommodating large initial data and addressing central singularities.

Contribution

It establishes the first global classical solutions for large initial data in multi-dimensional domains with spherical symmetry, using BD entropy and effective velocity coupling.

Findings

Successfully handled the singularity at the center of the ball.

Established $L^$ estimates for key quantities.

Provided bounds for the density and gravitational potential.

Abstract

Whether the 3D compressible Navier-Stokes-Poisson equations admit global classical solutions for general large initial data has long been a challenging open problem. In this paper, we provide an affirmative answer to this question under spherical symmetry on solid balls . Specifically, we consider the initial-boundary value problem for the multi-dimensional compressible equations with density-dependent viscosity coefficients satisfying the BD-type entropy equality, namely, assuming $\mu=\rho^{\alpha},\ \lambda=(\alpha-1)\rho^{\alpha}$ with $N=2, \alpha\in (\frac{1}{2},1]$ and $N=3, \alpha\in (\frac{5}{6},1]$, we establish the global existence of spherically symmetric classical solutions to the compressible Navier-Stokes-Poisson equations for both gaseous stars and plasmas with arbitrarily large initial data on solid balls. Our key observation lies in successfully handling the singularity at the center of the ball. By controlling the growth orders of the density and the gravitational potential at the central singularity, leveraging the structural advantages of the BD entropy and spherical symmetry, and fully exploiting the coupling between the effective velocity and the velocity, we establish $L^\infty$ estimates for the key quantities, which in turn yield upper and lower bound estimates for the density. This can be regarded as the first result on the existence of global classical solutions for arbitrarily large initial data to the compressible Navier-Stokes-Poisson equations in a truly multi-dimensional domain with high-dimensional features.