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 $p(\rho)=\rho^\gamma$, 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}.