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

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 $p(\rho )={\rho }^{\gamma }$, the viscosity coefficients $\mu (\rho )$ and $\lambda (\rho )$ satisfy either $\mu (\rho )=\tilde{\mu },\lambda (\rho )=\tilde{\lambda }{\rho }^{\alpha }$ or $\mu (\rho )=\tilde{\mu }{\rho }^{\alpha },\lambda (\rho )=\tilde{\lambda }{\rho }^{\alpha }$, and the capillarity coefficient $\kappa (\rho )=\tilde{\kappa }{\rho }^{\beta }$, where $\alpha ,\beta ,\gamma \in \mathbb{R}$ are parameters, and $\tilde{\mu },\tilde{\lambda },\tilde{\kappa }$ are given real constants. Under suitable restrictions on the parameters $\alpha ,\beta$ and $\gamma$, 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}.