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

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.