Zero-electron-mass and quasi-neutral limits for bipolar Euler–Poisson systems

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 nonlinear adiabatic electron system. In a second step, we analyze 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.