Energy-transport and drift-diffusion limits of nonisentropic Euler–Poisson equations

Two relaxation limits in critical spaces for the scaled nonisentropic Euler–Poisson equations with the momentum relaxation time and energy relaxation time are considered. As the first step of this justification, the uniform (global) classical solutions to the Cauchy problem in Chemin–Lernerʼs spaces with critical regularity are constructed. Furthermore, by the compactness argument, it is rigorously justified that the scaled classical solutions converge to the solutions of energy-transport equations and drift-diffusion equations, respectively, with respect to different time scales.