Contraction of viscous-dispersive shocks: Zero viscosity-capillarity limits

We prove the contraction property of any large solution perturbed from a viscous-dispersive shock wave of the Navier--Stokes--Korteweg (NSK) system. The contraction holds up to a dynamical shift, since the contraction is measured by the relative entropy that is locally $L^2$. We use the contraction property to show the global existence of large solution perturbed from a viscous-dispersive shock wave. To prove the contraction property, we first employ the effective velocity to transform the NSK system into the system of two degenerate parabolic equations, then apply the method of $a$-contraction with shifts. The contraction property does not depend on the strengths of viscosity and capillarity. Based on this uniformity, we show the existence of zero viscosity-capillarity limits of solutions to the NSK system, on which Riemann shocks are unique and stable up to shifts.