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 $\gamma >1$. 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 $\gamma$. Furthermore in case of $\gamma \ge 2$ 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.