Stationary solutions to the spherically symmetric compressible fluid with capillarity effect

TL;DR

This paper proves the existence, uniqueness, and decay properties of stationary solutions to the spherically symmetric Navier--Stokes--Korteweg system with capillarity effects, boundary conditions, and asymptotic analysis.

Contribution

It establishes the existence and decay rates of stationary solutions under small boundary data and analyzes the limit as capillarity vanishes, supported by numerical validation.

Findings

Unique smooth stationary solutions exist under small boundary data.

Impermeable wall solutions decay exponentially to far-field states.

Inflow/outflow solutions decay algebraically, with convergence rates validated numerically.

Abstract

We consider the spherically symmetric Navier--Stokes--Korteweg (NSK) system on the exterior domain $\Omega=\{x\in\mathbb{R}^n~|~|x|>1\}$ with $n\ge2$ when the boundary and far-field data are given. We show that, if the boundary data are sufficiently small, then there exists a unique smooth stationary solution to the spherically symmetric NSK system with impermeable wall, inflow, and outflow boundary conditions. We also establish the decay rate of the stationary solutions. Precisely, the stationary solution for the impermeable wall problem exponentially decays to the far-field states, while that of the inflow/outflow problem algebraically decays. Finally, we investigate the asymptotic convergences of the stationary solution for the impermeable wall problem as the capillarity coefficient vanishes. Numerical results validate that our theoretical convergence rate of the stationary solution is optimal.