  We consider a set of bipolar Euler-Poisson equations and study two asymptotic
limiting processes. The first is the zero-electron-mass limit, which formally
results in a non-linear adiabatic electron system. In a second step, we analyse
the combined zero-electron-mass and quasi-neutral limits, which together lead
to the compressible Euler equations. Using the relative energy method, we
rigorously justify these limiting processes for weak solutions of the
two-species Euler-Poisson equations that dissipate energy, as well as for
strong solutions of the limit systems that are bounded away from vacuum. This
justification is valid in the regime of initial data for which strong solutions
exist. To deal with the electric potential, in the first case we use elliptic
theory, whereas in the second case we employ the theory of Riesz potentials and
properties of the Neumann function.
