[Paper Review] Higher Order Convergence Rates in Theory of Homogenization II: Oscillatory Initial Data

Sunghan Kim, Ki-Ahm Lee|arXiv (Cornell University)|Jan 12, 2017

Advanced Mathematical Modeling in Engineering29 references3 citations

TL;DR

This paper establishes higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic Cauchy problems with rapidly oscillating initial data. By constructing higher order initial layer and interior correctors using a macroscopic regularity theory and exponential decay estimates, the authors derive an asymptotic expansion that captures oscillatory behavior near the initial time and in the interior, even for fully nonlinear operators where correctors lose periodicity over time.

ABSTRACT

We establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic Cauchy problems accompanied with rapidly oscillating initial data. Such result is new even for linear problems. Here we construct higher order initial layer and interior correctors, which describe the oscillatory behavior near the initial and interior time zone of the domain. To construct higher order correctors, we develop a regularity theory in macroscopic scales, and prove an exponential decay estimate for initial layer correctors. The higher order expansion requires an iteration process: successively correcting the initial layer, then the interior. This leads to a more complicated asymptotic expansion, as compared to the non-oscillating data case, and this complexity is present even in the linear case. A notable observation for fully nonlinear operators is that even if the given operator is space-time periodic, the interior correctors become aperiodic in the time variable as we proceed with the iteration process. Moreover, each interior corrector of higher order is paired with a space-time periodic version, and the difference between the two decays exponentially fast with time.

Motivation & Objective

• To establish higher order convergence rates in periodic homogenization for fully nonlinear uniformly parabolic equations with highly oscillating initial data.
• To address the challenge of constructing higher order correctors when initial data are oscillatory, which introduces complex coupling between initial layer and interior correctors.
• To develop a regularity theory in macroscopic scales to handle the nonlinear and oscillatory structure of the problem.
• To prove exponential decay of initial layer correctors and show that higher-order interior correctors, though aperiodic in time, differ exponentially fast from their periodic counterparts.
• To extend the theory beyond linear problems, demonstrating that even for linear equations, the results are new and nontrivial due to the oscillatory initial condition.

Proposed method

• Construct initial layer correctors that capture the rapid oscillations near the initial time, using a viscosity solution framework and exponential decay estimates.
• Develop a macroscopic regularity theory for solutions in slow variables to control the growth of derivatives in the homogenized limit.
• Introduce an iterative scheme to successively correct for initial layer and interior effects, accounting for nonlinear coupling between correctors.
• Define higher order interior correctors that are not space-time periodic, even when the original operator is periodic, due to the nonlinear structure.
• Pair each higher-order interior corrector with a periodic version and prove that their difference decays exponentially in time.
• Use the method of asymptotic expansions with multiple scales, incorporating both fast (ε⁻¹x, ε⁻²t) and slow (x,t) variables, to derive error estimates.

Experimental results

Research questions

• RQ1 What is the optimal rate of convergence in homogenization of fully nonlinear parabolic equations with oscillatory initial data?
• RQ2 How can higher order correctors be systematically constructed when the initial data are rapidly oscillating, leading to complex coupling between initial and interior layers?
• RQ3 What happens to the periodicity of correctors in the interior when the original operator is space-time periodic but the equation is fully nonlinear?
• RQ4 Can exponential decay estimates be established for initial layer correctors in the context of fully nonlinear equations?
• RQ5 How does the nonlinearity of the operator affect the structure and decay properties of higher-order correctors compared to the linear case?

Key findings

• The paper establishes a higher order asymptotic expansion for solutions to fully nonlinear parabolic Cauchy problems with oscillatory initial data, achieving convergence rates of order ε^m for any m ≥ 2.
• Initial layer correctors are constructed and shown to decay exponentially in time, which is crucial for controlling the error near t = 0.
• Higher order interior correctors become aperiodic in the time variable, even when the original operator is space-time periodic, due to the nonlinear structure.
• Each higher-order interior corrector is paired with a periodic version, and their difference decays exponentially in time, enabling effective error control.
• The method yields a quantitative error estimate of order ε for the m-th order expansion, uniformly in x ∈ ℝⁿ and t ∈ [0,T], for ε ≤ 1/2.
• The construction is valid for both linear and fully nonlinear equations, and the results are new even in the linear case due to the oscillatory initial data.