Hydrodynamic limit from nonlinear Fokker--Planck to barotropic Euler equations

The hydrodynamic limit to the barotropic Euler equations, including power-law pressure $P(\rho )={\rho }^{\gamma }$, for a kinetic nonlinear Fokker–Planck equation with degenerate diffusion is established. This extends the well-known result of the derivation of isothermal Euler equations via Fokker–Planck equation with linear diffusion. We establish the asymptotic analysis using the relative entropy method by quantifying error estimates for pressures and employing the generalized Log-Sobolev inequality for degenerate diffusion.