Existence of strong solutions to the stationary compressible Navier–Stokes–Korteweg equations with large external force

Abstract

In this study, we show the existence of strong solution to the boundary value problem of the steady compressible Navier–Stokes–Korteweg equation with large external forces in bounded domain, provided that the Mach number is appropriately small. Moreover, the Mach number limit of the strong solutions is rigorously verified. The main idea in the proof is to split the original equation into two parts: (i) a system of stationary incompressible Navier–Stokes–Korteweg equations with large forces, (ii) a system of stationary compressible Navier–Stokes–Korteweg equations with small forces. Introducing the “modified pressures”, (i) is reduced to a system of stationary incompressible Navier–Stokes equations with large forces coupled with an elliptic equation, (ii) is reduced to a system of stationary compressible Navier–Stokes equations with small forces coupled with an elliptic equation. Based on the known results for linear incompressible Navier–Stokes equation, linear transport equation and elliptic equation, we establish uniformity in the Mach number a priori estimates. Further, using the Schauder fixed point theorem, we present the existence of a strong solution. At the same time, from the uniform a priori estimates, we show the zero Mach number limit of the strong solutions, which converge to the solutions of the corresponding incompressible Navier–Stokes–Korteweg equations.