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

Abstract

We establish the existence and uniqueness, up to translation, of monotone viscous-dispersive shock profiles for the one-dimensional Navier-Stokes-Fourier-Korteweg system, provided the two end states satisfy the Rankine-Hugoniot and Lax entropy conditions and the shock strength is small. Taking the shock strength as a singular perturbation parameter, we reduce the profile equations by Fenichel's geometric singular perturbation theory to a scalar equation on a slow manifold and obtain the profile as a heteroclinic orbit. We also give pointwise estimates of the profiles and their derivatives up to fourth order, with explicit leading-order terms for the first derivatives.