Stationary solution to Stochastically Forced Euler-Poisson Equations in Bounded Domain: Part 1. 3-D Insulating Boundary

Abstract

This paper is concerned with $3$-D stochastic Euler-Poisson equations with insulating boundary conditions forced by the Wiener process. We first establish the global existence and uniqueness of the solution to the system, then we prove that the solution converges to its steady-state time-asymptotically. To obtain the converging rate, we need to develop weighted energy estimates, which are not required for the deterministic counterpart of the problem. Moreover, we observe that the invariant measure is just the Dirac measure generated by the steady-state, in which the time-exponential convergence rate to the steady-state plays an essential role.