Convergence rates in homogenization of higher-order parabolic systems

Niu, Weisheng; Xu, Yao

doi:10.3934/dcds.2018183

Cited by 8 publications

(3 citation statements)

References 25 publications

(34 reference statements)

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“…It does not work for higher-order parabolic systems, as more regularity in y is required. Later, in [20] when dealing with higher-order systems, flux correctors were constructed by lifting the regularity w.r.t. space only.…”

Section: Introductionmentioning

confidence: 99%

Convergence rates in homogenization of parabolic systems with locally periodic coefficients

Xu¹

2020

Preprint

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In this paper we study the quantitative homogenization of second-order parabolic systems with locally periodic (in both space and time) coefficients. The O(ε) scale-invariant error estimate in L 2 (0, T ;cylinders under nearly minimum smoothness conditions on the coefficients. This process relies on the good use of smoothing operators and fractional derivatives on intervals. We also develop a construction of flux correctors in the parabolic manner and a sharp estimate for temporal boundary layers. The paper improves the corresponding result of [21].

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Section: Introductionmentioning

confidence: 99%

Convergence rates in homogenization of parabolic systems with locally periodic coefficients

Xu¹

2020

Preprint

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“…The uniform interior and boundary Lipschitz estimates were established in [11] and [15], while the convergence rates have been studied in different contexts in [12,32,23]. See also [20,22,1,13] for more related results. Very recently, quantitative estimates in periodic homogenization of the following parabolic operator with non-self-similar scales ∂ t − div A(x/ε, t/ε ℓ )∇ , 0 < ℓ < ∞, (1.7) were studied by the first author and Shen in [14].…”

Section: Introductionmentioning

confidence: 99%

Homogenization of Parabolic Equations with Non-self-similar Scales

Geng¹,

Shen²

2019

Arch Rational Mech Anal

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This paper is concerned with quantitative homogenization of second-order parabolic systems with periodic coefficients varying rapidly in space and time, in different scales. We obtain large-scale interior and boundary Lipschitz estimates as well as interior C 1,α and C 2,α estimates by utilizing higher-order correctors. We also investigate the problem of convergence rates for initial-boundary value problems.MSC2010: 35B27, 35K40.

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“…In 2016, Shen [18] proved the L 2 convergence rate for the mixed Dirichlet-Neumann boundary value problems. In 2018, Niu and Xu [11] got the L 2 convergence rate for 2mthorder equations with periodic oscillating coefficients.…”

Section: Introductionmentioning

confidence: 99%

Convergence rates in homogenization of p-Laplace equations

Zhao

Wang

2019

Bound Value Probl

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This paper is concerned with homogenization of p-Laplace equations with rapidly oscillating periodic coefficients. The main difficulty of this work is due to the nonlinear structure in this field concerning p-Laplace equations itself. Utilizing the layer and co-layer type estimates as well as homogenization techniques, we establish the desired error estimates. As a consequence, we obtain the rates of convergence for solutions in W 1,p 0 as well as L p . Meanwhile, our convergence rate results do not involve the higher derivatives any more. This may be viewed as rather surprising. The novelty of this work is that it seems to find a new analysis method in quantitative homogenization. MSC: Primary 35B27; secondary 35J15

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Convergence rates in homogenization of higher-order parabolic systems

Cited by 8 publications

References 25 publications

Convergence rates in homogenization of parabolic systems with locally periodic coefficients

Convergence rates in homogenization of parabolic systems with locally periodic coefficients

Homogenization of Parabolic Equations with Non-self-similar Scales

Convergence rates in homogenization of p-Laplace equations

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