STOCHASTIC HOMOGENIZATION OF PARABOLIC EQUATIONS WITH LOWER-ORDER TERMS

Abstract

The study of homogenization results has long been a central focus in the field of mathematical analysis, particularly for equations without lower-order terms. However, the importance of studying homogenization results for parabolic equations with lower-order terms cannot be understated. In this study, we aim to extend the analysis to homogenization for the general parabolic equation with random coefficients:

∂_𝑡𝑝^𝜖 ー ∇ · (𝐚(𝑥/𝜖, 𝑡/𝜖²) ∇ 𝑝^𝜖) ー 𝐛 (𝑥/𝜖, 𝑡/𝜖²) ∇ 𝑝^𝜖 ー 𝐝 (𝑥/𝜖, 𝑡/𝜖²) 𝑝^𝜖 = 0.

Moreover, we establish the Caccioppoli inequality and Meyers estimate for the generalized parabolic equation. By using the generalized Meyers estimate, we get the weak convergence of 𝑝^𝜖 in 𝐻¹.