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

Yachun Li, Ming Mei, Lizhen Zhang

TL;DR

This work analyzes the 3-D stochastic Euler-Poisson equations in a bounded domain with insulating boundary, driven by multiplicative noise. The authors establish global-in-time existence and uniqueness of strong solutions, and develop weighted energy estimates to prove exponential time decay toward a smooth steady state, under small perturbations and structured noise. They further show that the invariant measure of the stochastic dynamics is the Dirac measure at the steady state, i.e. $\delta_{\bar\rho}\times\delta_{0}\times\delta_{\bar\Phi}$, by Krylov–Bogoliubov arguments and energy methods. The results extend the deterministic stability framework to a stochastic setting, with implications for long-time behavior and regularity of stochastic SEP models in semiconductor contexts. The methods combine symmetrization, Itô calculus, BDG inequalities, and weighted energy techniques to handle the lack of temporal regularity induced by Brownian forcing.

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.

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

TL;DR

This work analyzes the 3-D stochastic Euler-Poisson equations in a bounded domain with insulating boundary, driven by multiplicative noise. The authors establish global-in-time existence and uniqueness of strong solutions, and develop weighted energy estimates to prove exponential time decay toward a smooth steady state, under small perturbations and structured noise. They further show that the invariant measure of the stochastic dynamics is the Dirac measure at the steady state, i.e. , by Krylov–Bogoliubov arguments and energy methods. The results extend the deterministic stability framework to a stochastic setting, with implications for long-time behavior and regularity of stochastic SEP models in semiconductor contexts. The methods combine symmetrization, Itô calculus, BDG inequalities, and weighted energy techniques to handle the lack of temporal regularity induced by Brownian forcing.

Abstract

This paper is concerned with -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.
Paper Structure (20 sections, 8 theorems, 228 equations)

This paper contains 20 sections, 8 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Global existence of solutions
  3. Local existence
  4. Estimates up to third-order
  5. Zero-order estimates
  6. First order estimates
  7. Second order estimates
  8. Third-order estimates
  9. Global existence
  10. For stochastic forces under \ref{['2 Condition for stochastic force']} and small perturbation for initial data \ref{['initial conditions assumption']}
  11. For general stochastic forces
  12. Asymptotic stability of solutions
  13. Weighted decay estimates
  14. Zeroth-order weighted estimates
  15. First-order weighted estimates
  16. ...and 5 more sections

Key Result

Proposition 1.1

Let $b(x)>0$ in $\bar{U}$ and $P:\left(0, \infty\right)\rightarrow \left(0, \infty\right)$ be smooth with $P\left(0\right)=0$. Then there exists $\left(\bar{\rho} , \bar{{\bf u}} , \bar{\Phi} \right)$, $\forall \omega\in \Omega$, a unique smooth steady-state solution of the insulating problem with t such that there holds where $\underline{\rho}$ is a constant, and

Theorems & Definitions (20)

  • Proposition 1.1
  • Theorem 1.1: Global existence
  • Theorem 1.2: Convergence to steady state
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.1
  • ...and 10 more