Higher order convergence rates in theory of homogenization III: Viscous Hamilton–Jacobi equations

Kim, Sunghan; Lee, Ki-Ahm

doi:10.1016/j.jde.2018.07.003

Cited by 5 publications

(2 citation statements)

References 11 publications

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“…This is the extended follow-up result of our earlier works [16,17,15], where we wish to estimate the differences between micro-macro concentrations and micro-macro concentration gradients in the energy space of perforated domains. It is also in the same line with the theoretical findings in [20,5,3,2,27,28,12]. Our preceding works show that the macroscopic problem can be self-linear, albeit the semilinear microscopic problem.…”

supporting

confidence: 89%

On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit

Khoa¹,

Thieu²,

Ijioma³

2021

DCDS-B

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In this paper, we consider a microscopic semilinear elliptic equation posed in periodically perforated domains and associated with the Fouriertype condition on internal micro-surfaces. The first contribution of this work is the construction of a reliable linearization scheme that allows us, by a suitable choice of scaling arguments and stabilization constants, to prove the weak solvability of the microscopic model. Asymptotic behaviors of the microscopic solution with respect to the microscale parameter are thoroughly investigated in the second theme, based upon several cases of scaling. In particular, the variable scaling illuminates the trivial and non-trivial limits at the macroscale, confirmed by certain rates of convergence. Relying on classical results for homogenization of multiscale elliptic problems, we design a modified two-scale asymptotic expansion to derive the corresponding macroscopic equation, when the scaling choices are compatible. Moreover, we prove the high-order corrector estimates for the homogenization limit in the energy space H 1 , using a large amount of energy-like estimates. A numerical example is provided to corroborate the asymptotic analysis.

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confidence: 89%

On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit

Khoa¹,

Thieu²,

Ijioma³

2021

DCDS-B

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“…Uniform ellipticity and concavity are some typical properties that are preserved, as shown in [12]. More recently, the authors have proved in the series of papers [18][19][20] that higher regularity of F in the matrix variable is also inherited to F, i.e., if…”

Section: Introductionmentioning

confidence: 98%

Uniform Estimates in Periodic Homogenization of Fully Nonlinear Elliptic Equations

Kim

Lee

2022

Arch Rational Mech Anal

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This article is concerned with uniform $$C^{1,\alpha }$$ C 1 , α and $$C^{1,1}$$ C 1 , 1 estimates in periodic homogenization of fully nonlinear elliptic equations. The analysis is based on the compactness method, which involves linearization of the operator at each approximation step. Due to the nonlinearity of the equations, the linearized operators involve the Hessian of correctors, which appear in the previous step. The involvement of the Hessian of the correctors deteriorates the regularity of the linearized operator, and sometimes even changes its oscillating pattern. These issues are resolved with new approximation techniques, which yield a precise decomposition of the regular part and the irregular part of the homogenization process, along with a uniform control of the Hessian of the correctors in an intermediate level. The approximation techniques are even new in the context of linear equations. Our argument can be applied not only to concave operators, but also to certain class of non-concave operators.

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Higher order convergence rates in theory of homogenization II: Oscillatory initial data

Kim

Lee

2020

Advances in Mathematics

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We establish the higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic Cauchy problems, accompanied with rapidly oscillating initial data. The higher order convergence rates are achieved by constructing the higher order initial layer correctors and the higher order interior correctors that describe the oscillatory behavior of the solutions to the given ε-problems. In order to construct these correctors, we establish a regularity theory in the slow variables, that is, the non-oscillatory physical variables, of solutions to either the spatially periodic Cauchy problems or the cell problems, based on the classical theory for viscosity solutions.

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Higher order convergence rates in theory of homogenization III: Viscous Hamilton–Jacobi equations

Cited by 5 publications

References 11 publications

On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit

On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit

Uniform Estimates in Periodic Homogenization of Fully Nonlinear Elliptic Equations

Higher order convergence rates in theory of homogenization II: Oscillatory initial data

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