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

RESUMO

We establish a sharp criterion for global-in-time existence versus finite-time blow-up of strong solutions to the 3D spherically symmetric compressible Navier-Stokes equations on a solid ball, based solely on the initial sign of effective velocity. The viscosity coefficients are assumed to satisfy the Bresch-Desjardins structure $\mu=\rho^{\alpha}$, $\lambda=(\alpha-1)\rho^{\alpha}$, with the effective velocity given by $v=u+\alpha \rho^{\alpha-2}\rho_r$. Previously, global existence results for strong solutions in higher dimensions were restricted to the case $\alpha\le 1$ with the endpoint $\alpha=1$ corresponding to the viscous Saint-Venant (shallow water) system. In this paper, we extend the global existence theory beyond this threshold to the supercritical regime $\alpha>1$. Specifically, whenever the initial effective velocity is nonnegative on the boundary of the ball, we prove the global-in-time existence of strong solutions for arbitrarily large initial data. The main difficulty lies in deriving a uniform lower bound for the density, which requires handling singular estimates at the center of the ball. To overcome this, we exploit several novel quantities near the center and employ a refined maximum principle to establish, for the first time, the Lipschitz continuity of the velocity at the center. Subsequently, we derive Dini-Gronwall-type inequalities for suitably constructed functions, which close the density lower bound estimate. Conversely, we can construct a family of initial data for which the initial effective velocity is negative on the boundary, such that the corresponding strong solutions blow up in finite time, with vacuum appearing exactly on the boundary at the blow-up time. Our results establish a clean dichotomy between global regularity and singularity formation, governed purely by the initial sign of the effective velocity.