Existence and uniqueness of viscous-dispersive shock profiles for the Navier-Stokes-Fourier-Korteweg system

We establish the existence and uniqueness, up to translation, of monotone viscous-dispersive shock profiles for the one-dimensional Navier-Stokes-Fourier-Korteweg (NSFK) system, provided the two constant end states satisfy the Rankine-Hugoniot and Lax entropy conditions and the shock strength is small. Treating the small shock strength as a singular perturbation parameter, we construct the profiles via Fenichel’s geometric singular perturbation theory. We further provide pointwise estimates for the profiles, which are used in the nonlinear stability analysis of generic Riemann solutions to the NSFK system in a companion work.