ASYMPTOTIC STABILITY OF STATIONARY SOLUTIONS TO THE EULER-POISSON EQUATIONS ARISING IN PLASMA PHYSICS

The main concern of the present paper is to analyze a sheath formed around a surface of a material with which plasma contacts. Here, for a formation of the sheath, the Bohm criterion requires the velocity of positive ions should be faster than a certain physical constant. The behavior of positive ions in plasma is governed by the Euler-Poisson equations. Mathematically, the sheath is regarded as a special stationary solution. We first show that the Bohm criterion gives a sufficient condition for an existence of the stationary solution by using the phase plane analysis. Then it is shown that the stationary solution is time asymptotically stable provided that an initial perturbation is sufficiently small in the weighted Sobolev space. Moreover we obtain the convergence rate of the time global solution towards the stationary solution subject to the decay rate of the initial perturbation. These theorems are proved by a weighted energy method.