Convergence of compressible Euler–Poisson system to incompressible Euler equations

In this paper, we study the quasi-neutral limit of compressible Euler–Poisson equations in plasma physics in the torus TdTd. For well prepared initial data the convergence of solutions of compressible Euler–Poisson equations to the solutions of incompressible Euler equations is justified rigorously by an elaborate energy methods based on studies on an λ-weighted Lyapunov-type functional. One main ingredient of establishing uniformly a priori estimates with respect to λ is to use the curl–div decomposition of the gradient.