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

TL;DR

This paper proves the global existence and uniqueness of spherically symmetric strong solutions for a multidimensional compressible Navier-Stokes-Korteweg system with large initial data in an unbounded domain, under specific parameter conditions.

Contribution

It establishes the first global existence and uniqueness results for large initial data solutions to this complex system with variable coefficients and capillarity effects.

Findings

Proved global existence of solutions under certain parameter restrictions.

Established uniqueness of spherically symmetric strong solutions.

Applied a weighted energy method combined with Kanel's technique.

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}.