Incompressible limit of Cauchy problem to the 3D compressible Euler–Korteweg equations with ill-prepared initial data

The study focuses on incompressible limit of Cauchy problem to the 3D Euler–Korteweg equations with ill-prepared initial data. We first construct the uniform estimate for local strong solutions to the system with the aid of gauge functions. After that, we establish the uniform with respect to Mach number and capillarity coefficient dispersion estimates for the propagators associated with the system, and justify the convergence from the compressible Euler–Korteweg system towards the incompressible Euler equations.