Global spherically symmetric solutions to the multidimensional isentropic compressible Navier--Stokes--Korteweg system with large initial data

Abstract

In this paper, we investigate the global existence of spherically symmetric strong solutions with large initial data to an initial-boundary value problem of the multidimensional isentropic compressible Navier-Stokes-Korteweg system in an unbounded exterior domain. We consider the case when the pressure $p(\ddot{I})=\ddot{I}{}^3$, the viscosity coefficients $\hat{I}¼(\ddot{I})$ and $\hat{I}»(\ddot{I})$ satisfy either $\hat{I}¼(\ddot{I})=\tilde{\hat{I}}¼,\hat{I}»(\ddot{I})=\tilde{\hat{I}}»\ddot{I}{}^{\hat{I}}\pm$ or $\hat{I}¼(\ddot{I})=\tilde{\hat{I}}¼\ddot{I}{}^{\hat{I}}\pm ,\hat{I}»(\ddot{I})=\tilde{\hat{I}}»\ddot{I}{}^{\hat{I}}\pm$, and the capillarity coefficient $\hat{I}º(\ddot{I})=\tilde{\hat{I}}º\ddot{I}{}^2$, where $\hat{I}\pm ,{\hat{I}}^2,{\hat{I}}^3\in \mathbb{R}$ are parameters, and $\tilde{\hat{I}}¼,\tilde{\hat{I}}»,\tilde{\hat{I}}º$ are given real constants. Under suitable restrictions on the parameters $\hat{I}\pm ,{\hat{I}}^2$ and ${\hat{I}}^3$, we establish the global existence and uniqueness of spherically symmetric strong solutions. The proof relies on the radically weighted energy method combined with the technique developed by Y.~Kanel'\cite{28}.