Homogenization and Convergence Rates for Periodic Parabolic Equations with Highly Oscillating Potentials

TL;DR

This paper studies the homogenization and convergence rates of periodic parabolic equations with highly oscillating potentials, providing uniform estimates and analyzing different cases based on potential assumptions.

Contribution

It introduces new uniform estimates and convergence rate results for highly oscillating potentials in periodic parabolic equations, extending previous homogenization techniques.

Findings

Homogenized equations are determined for various potential assumptions.

Convergence rates in $L^ abla(0,T;L^2( abla))$ are established.

Uniform estimates for solutions and higher derivatives are obtained.

Abstract

This paper considers a family of second-order periodic parabolic equations with highly oscillating potentials, which have been considered many times for the time-varying potentials in stochastic homogenization. Following a standard two-scale expansions illusion, we can guess and succeed in determining the homogenized equation in different cases that the potentials satisfy the corresponding assumptions, based on suitable uniform estimates of the $L^2(0,T;H^1(\Omega))$-norm for the solutions. To handle the more singular case and obtain the convergence rates in $L^\infty(0,T;L^2(\Omega))$, we need to estimate the Hessian term as well as the t-derivative term more exactly, which may be depend on $\varepsilon$. The difficulty is to find suitable uniform estimates for the $L^2(0,T;H^1(\Omega))$-norm and suitable estimates for the higher order derivative terms.