Homogenization of locally periodic parabolic operators with non-self-similar scales

Abstract

We investigate quantitative estimates in homogenization of the locally periodic parabolic operator with multiscales $${\partial }_t-\text{div}(A(x,t,x/\epsilon ,t/{\kappa }^2)\nabla ),\epsilon >0,\kappa >0.$$ Under proper assumptions, we establish the full-scale interior and boundary Lipschitz estimates. These results are new even for the case $\kappa =\epsilon$, and for the periodic operators ${\partial }_t-\text{div}(A(x/\epsilon ,t/{\epsilon }^{\ell })\nabla ),$ $0<\epsilon ,\ell <\infty ,$ of which the large-scale Lipschitz estimate down to $\epsilon +{\epsilon }^{\ell /2}$ was recently established by the first author and Shen in Arch. Ration. Mech. Anal. 236(1): 145--188 (2020). Due to the non-self-similar structure, the full-scale estimates do not follow directly from the large-scale estimates and the blow-up argument. As a byproduct, we also derive the convergence rates for the corresponding initial-Dirichlet problems, which extend the results in the aforementioned literature to more general settings.

arXiv comment40 pages

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Cited by in corpus (1)

• Parabolic Homogenization with an Interface