STABILITY FOR SOLUTIONS OF A STATIONARY EULER–POISSON PROBLEM

We study the stability of the solutions of a stationary Euler–Poisson problem modeling a plasma diode. The model was presented in a previous paper, where we proved the existence of multiple solutions. Since, physically speaking, we expect only one solution to be present, we propose here two strategies for selecting the physical stationary regime. On the one hand, we study the energy functional associated with the stationary problem and order the different solutions in terms of their energy level and as stationary points for the functional. On the other hand, starting from the remark that a stable stationary state is the state which is reached after an evolution, we solve numerically the corresponding time-dependent Euler–Poisson problem and show that only one of the possible solutions is indeed stable in this sense. These two independent approaches select the same solution that we label therefore as stable.