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

TL;DR

This work analyzes the weak and strong averaging principles for a two-scale 2D stochastic Boussinesq system driven by non-Lipschitz Poisson jump noise on the torus. It develops a rigorous framework establishing well-posedness and regularity for the coupled vorticity and temperature fields, proves ergodicity of the frozen temperature dynamics, and derives both weak and strong convergence of the vorticity to an averaged equation as the fast-slow time-scale ratio $\varepsilon$ vanishes. The averaged model is built from coefficients obtained by averaging against the invariant measure of the frozen equation, and its well-posedness is shown under non-Lipschitz, concave-growth assumptions. A concrete example with numerical simulations confirms the theoretical results, demonstrating convergence of $j^{\varepsilon}$ to the averaged solution and supporting the practical validity of the averaging approach for stochastic fluid-thermal systems with jump noise.

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.

Figures (2)

• Figure 1: 100 samples of error($\varepsilon$)
• Figure 2: mean squared error of $\varepsilon$

Theorems & Definitions (29)

• Remark 2.1
• Definition 3.1
• Theorem 3.2
• proof
• Remark 3.1
• Theorem 3.3
• proof
• Lemma 3.1
• proof
• Theorem 3.4