Stochastic Homogenization of Parabolic Equations with Lower-order Terms

TL;DR

This paper extends homogenization theory to parabolic equations with lower-order terms and random coefficients, establishing key inequalities and convergence results for solutions.

Contribution

It introduces homogenization results for parabolic equations with lower-order terms and random coefficients, including new inequalities and convergence analysis.

Findings

Established Caccioppoli inequality for the generalized equation

Proved Meyers estimate for the equation with lower-order terms

Demonstrated weak convergence of solutions in H^1 space

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: \begin{equation*} \partial_{t}p^\epsilon-\nabla\cdot\left(\mathbf{a}\left( \dfrac{x}{\epsilon},\dfrac{t}{\epsilon^2}\right)\nabla p^\epsilon\right)-\mathbf{b}\left( \dfrac{x}{\epsilon},\dfrac{t}{\epsilon^2}\right)\nabla p^\epsilon -\mathbf{d}\left( \dfrac{x}{\epsilon},\dfrac{t}{\epsilon^2}\right) p^\epsilon=0. \end{equation*} 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 $p^\epsilon$ in $H^1$.