Time-asymptotic stability of viscous shocks for the outflow problem of one-dimensional compressible fluids of Korteweg type

We study the time-asymptotic stability of viscous-dispersive shock waves for the outflow problem of the barotropic Navier–Stokes–Korteweg equations, which describe viscous fluids with internal capillarity. Assuming that the far-field state is subsonic or transonic and that the velocity at the boundary is larger than the far-field velocity, we prove that the solution converges to the corresponding viscous-dispersive shock wave as $t\to +\mathrm{\infty }$, provided that the shock amplitude and the initial perturbation are sufficiently small. The proof is based on the method of $a$-contraction with shifts (for viscous equations) introduced in \cite{KV17,KV21,KVW23}. A main difficulty comes from controlling the boundary effect of the viscous-dispersive shock wave, as well as the influence of capillarity near the boundary.