Analysis of the linear navier stokes korteweg model with neumann boundary conditions in three dimensional half-space

This study discusses the solution of the Navier-Stokes Korteweg model, which describes two-phase fluid flow with capillary effects, with Neumann boundary conditions in the half-space. The main objective is to detail the resolution process of the resolvent equation system in the half-space related to the Navier-Stokes Korteweg model with Neumann boundary conditions. The resolution is carried out in several steps. First, the resolvent equation system is reduced using even and odd extensions. Then, a partial Fourier transform is applied, resulting in a simpler ordinary differential equation. The findings of this research indicate the existence of a solution operator for the resolvent equation of the Navier-Stokes Korteweg model with Neumann boundary conditions in the half-space. This solution applies for two cases involving the coefficients, depending on certain conditions related to the fluid properties.