Weak and strong averaging principle for 2D Boussinesq equations with non-Lipschitz Poisson jump noise

TL;DR

This paper investigates the averaging principle for 2D Boussinesq equations influenced by non-Lipschitz Poisson jump noise, establishing convergence results and ergodicity, supported by numerical simulations.

Contribution

It introduces a novel analysis of the averaging principle for 2D Boussinesq equations with non-Lipschitz Poisson noise, including well-posedness, ergodicity, and convergence proofs.

Findings

Vorticity variable converges to the averaged solution as epsilon approaches zero.

Established ergodicity of the temperature variable.

Numerical simulations support theoretical results.

Abstract

In this paper, we study the averaging principle for 2D Boussinesq equations with non-Lipschitz Poisson jump noise. Precisely, we will first explore the well-posedness, regularity estimates and tightness of the vorticity variable. Then, we prove the ergodicity of the temperature variable. Next, we prove that the vorticity variable converge to the solution of the averaged equation in probability and $2p$th-mean, under different conditions, as time scale parameter $\varepsilon$ goes to zero. Finally, we present a specific case study and conduct numerical simulations to substantiate the main conclusions of this paper.