Existence and stability of stationary solutions to the full compressible Navier-Stokes-Korteweg system

TL;DR

This paper investigates the existence, uniqueness, and nonlinear stability of stationary solutions to the full compressible Navier-Stokes-Korteweg system in three-dimensional space, using advanced mathematical techniques.

Contribution

It establishes the existence, uniqueness, and stability of stationary solutions for the full compressible Navier-Stokes-Korteweg system with external forces in D space, employing weighted-L2 and energy methods.

Findings

Existence and uniqueness of stationary solutions proven.

Nonlinear stability of these solutions demonstrated.

Methodology combines weighted-L2 and energy estimates.

Abstract

This paper is concerned with the existence, uniqueness and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier-Stokes-Korteweg system effected by external force of general form in $\mathbb{R}^3$. Based on the weighted-$L^2$ method and some elaborate $L^\infty$ estimates of solutions to the linearized problem, the existence and uniqueness of stationary solution are obtained by the contraction mapping principle. The proof of the stability result is given by an elementary energy method and relies on some intrinsic properties of the full compressible Navier-Stokes-Korteweg system.