The relaxed friction-type approximation of the Navier-Stokes-Cahn-Hilliard system has a hyperbolic first-order subsystem in 1D, proved via a convex entropy-entropy flux pair.
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A Note on Hyperbolic Relaxation of the Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flow
The relaxed friction-type approximation of the Navier-Stokes-Cahn-Hilliard system has a hyperbolic first-order subsystem in 1D, proved via a convex entropy-entropy flux pair.