Quantitative periodic homogenization of parabolic equations with large drift and potential

TL;DR

The paper investigates quantitative homogenization for parabolic equations with large drift and potential in periodic media on bounded domains. It introduces a factorization of the solution into a time-exponential term, a ground-state cell eigenfunction, and a homogenized parabolic component, enabling a rate-based comparison between the $\varepsilon$-problem and its homogenized limit. The authors derive L^2 convergence rates for the transformed problem: $O(\varepsilon^{1/4})$ with merely bounded coefficients, and $O(\varepsilon^{1/2})$ under additional regularity, using a parabolic smoothing operator and corrector constructions rather than invariant-measure techniques. The work extends to non-divergence-like parabolic forms and discusses decay of the original solution, the role of eigenvalue parameters, and potential future improvements such as boundary-layer correctors and optimal $\varepsilon$-rates.

Abstract

This work aims to study the rates in the context of periodic homogenization of parabolic problems with large lower order terms (both drift and potential). We demonstrate that the solution is a product of three terms: (i) a function of time, (ii) the ground-state of an exponential cell eigenvalue problem and (iii) the solution to a parabolic equation with zero effective drift. For the latter, we derive $\mathrm L^2$ rates in the homogenization limit.

Theorems & Definitions (23)

• Remark 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Theorem 2.1
• Lemma 2.2
• proof