Cahn–Hillard and Keller–Segel systems as high-friction limits of Euler–Korteweg and Euler–Poisson equations

We consider a combined system of Euler–Korteweg and Euler–Poisson equations with friction and exponential pressure with exponent \(\gamma > 1\) $\gamma >1$ . We show the existence of dissipative measure-valued solutions in the cases of repulsive and attractive potential in Euler–Poisson system. The latter case requires additional restriction on \(\gamma \) $\gamma$ . Furthermore in case of \(\gamma \ge 2\) $\gamma \ge 2$ we show that the strong solutions to the Cahn–Hillard–Keller–Segel system are a high-friction limit of the dissipative measure-valued solutions to Euler–Korteweg–Poisson equations.