Low Mach number limit for the Navier--Stokes--Korteweg equations with a stationary force

In this paper, we investigate the low Mach number limit for the three-dimensional compressible Navier--Stokes--Korteweg equations in the whole space under a small stationary external force. We first construct a family of small stationary solutions uniformly with respect to the Mach number $\eps$ and prove that both the stationary density fluctuation and the compressible component of the stationary velocity are of order $\eps^2$. For ill-prepared nonstationary perturbations around these stationary solutions, we establish the existence of unique global strong solutions by combining uniform high-order energy estimates with a low-frequency Besov estimate and a Kawashima-type compensating functional. The main difficulty is that the Korteweg term not only changes the elliptic structure of the stationary problem but also modifies the dispersive mechanism of the acoustic modes. In Korteweg-symmetric variables, the associated spectral projections are uniformly bounded zero-order Fourier multipliers, while the acoustic--capillary phase is wave-like at low frequencies and Schr\"odinger-like at high frequencies. Since the source terms generated by the stationary coefficients are not generally integrable in time, we decompose the Duhamel source according to its time-integrability and frequency behavior. Dyadic dispersive estimates, high-frequency damping estimates, and maximal regularity for the heat equation yield the global-in-time convergence rate $\eps^{\min\{1/r,\,1/2-1/p\}}$ in the mixed Besov norms \(L^r(0,\infty;\dot B^s_{p,1})\). As a consequence, Besov embeddings also yield quantitative convergence in the mixed Lebesgue norms \(L^r(0,\infty;L^p)\).