Stationary solution to Stochastically Forced Euler-Poisson Equations in Bounded Domain: Part 2. 1-D Ohmic Contact Boundary

TL;DR

This paper proves the asymptotic stability and existence of an invariant measure for a 1-D stochastic Euler-Poisson system with Ohmic contact boundary conditions, highlighting the effects of stochastic forces and boundary conditions on stability.

Contribution

It introduces a novel approach to establish stability and invariant measures for stochastic Euler-Poisson equations with Ohmic contact boundaries, overcoming challenges posed by stochastic forces.

Findings

Global existence of solutions around the steady state.

Exponential decay of perturbed solutions to the steady state.

Existence of an invariant measure corresponding to the steady state.

Abstract

In this paper, we establish the asymptotic stability of the steady-state for a 1-D stochastic Euler-Poisson equations with Ohmic contact boundary conditions forced by the Wiener process. We utilize Banach's fixed point theorem and the a priori energy estimates uniformly in time to ensure the global existence of solutions around the steady state. In contrast to the deterministic case, the presence of stochastic forces lead to the lack of temporal derivatives of momentum, posing challenges for energy estimates. Furthermore, Ohmic contact boundary conditions pose greater challenges for energy estimates compared to systems with insulating boundary conditions. To address this issue, we establish asymptotic stability concerning the spatial derivatives through weighted energy estimates for the estimates of stochastic integrals, employing a technique distinct from that of the deterministic case. Furthermore, we demonstrate the existence of an invariant measure based on the a priori energy estimates. This invariant measure precisely corresponds to the Dirac measure generated by the steady state, due to the exponential decay of perturbed solutions around the steady state.