Homogenization in algebras with mean value

Woukeng, Jean Louis

doi:10.48550/arxiv.1207.5397

Cited by 4 publications

(9 citation statements)

References 24 publications

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“…The following results are justified in [38]. We refer the reader to the above-mentioned paper for their justification.…”

Section: Fundamentals Of Algebras With Mean Valuementioning

confidence: 81%

“…We refer the reader to [38] for details regarding some of the results of this section. A closed subalgebra A of the C*-algebra of bounded uniformly continuous functions BUC(R N ) is an algebra with mean value on R N [14,27,41] if it contains the constants, is translation invariant (u(• + a) ∈ A for any u ∈ A and each a ∈ R N ) and is such that any of its elements possesses a mean value, that is, for any u ∈ A, the sequence (u ε ) ε>0 (defined by…”

Section: Fundamentals Of Algebras With Mean Valuementioning

confidence: 99%

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Introverted algebras with mean value and applications

Woukeng¹

2014

Nonlinear Analysis: Theory, Methods & Applications

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Let A be an introverted algebra with mean value. We prove that its spectrum ∆(A) is a compact topological semigroup, and that the kernel K(∆(A)) of ∆(A) is a compact topological group over which the mean value on A can be identified as the Haar integral. Based on these facts and also on the fact that K(∆(A)) is an ideal of ∆(A), we define the convolution over ∆(A). We then use it to derive some new convergence results involving the convolution product of sequences. These convergence results provide us with an efficient method for studying the asymptotics of nonlocal problems. The obtained results systematically establish the connection between the abstract harmonic analysis and the homogenization theory. To illustrate this, we work out some homogenization problems in connection with nonlocal partial differential equations.

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“…The following results are justified in [38]. We refer the reader to the above-mentioned paper for their justification.…”

Section: Fundamentals Of Algebras With Mean Valuementioning

confidence: 81%

Section: Fundamentals Of Algebras With Mean Valuementioning

confidence: 99%

Introverted algebras with mean value and applications

Woukeng¹

2014

Nonlinear Analysis: Theory, Methods & Applications

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“…We will summarize in this section several results about the two scale convergence that we will use throughout the paper. For the results stated without proofs, see [1], [4] or [27]. First we establish some notations of spaces of periodic functions.…”

Section: Two Scale Convergencementioning

confidence: 99%

“…We mention the paper [25] where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived. A comprehensive theory for solving stochastic homogenization problems has been constructed recently in [27] and [19], where a Σ-convergence method adapted to stochastic processes was developed. An application of the method to the homogenization of a stochastic Navier-Stokes type equation with oscillating coefficients in a bounded domain (without holes) has been provided.…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition

Bessaih

Maris

2015

Applicable Analysis

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The two dimensional Navier-Stokes equation in a perforated domain with a dynamical slip boundary condition is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the holes. We consider a scaling (ε 2 for the viscosity and 1 for the density) that will lead to a time dependent limit problem. However, the noncritical scaling (ε β , β > 1) is considered in front of the nonlinear term. The homogenized system in the limit is obtained as a Darcy's law with memory with two permeabilities and an extra term that is due to the stochastic perturbation on the boundary of the holes. We use the two-scale convergence method. Due to the stochastic integral, the pressure that appears in the variational formulation does not have enough regularity in time. This fact made us rely only on the variational formulation for the passage to the limit on the solution. We obtain a variational formulation for the limit that is solution of a Stokes system with two pressures. This two-scale limit gives rise to three cell problems, two of them give the permeabilities while the third one gives an extra term in the Darcy's law due to the stochastic perturbation on the boundary of the holes.

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“…Algebras with mean value -An overview. We refer the reader to [5,23,36,39] for an extensive presentation of the concept of algebras with mean value (algebras wmv, in short).…”

Section: 1mentioning

confidence: 99%

Nonlocal Stokes-Vlasov system: Existence and deterministic homogenization results

Nguetseng,

Soh,

Woukeng

2014

Preprint

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Our work deals with the systematic study of the coupling between the nonlocal Stokes system and the Vlasov equation. The coupling is due to a drag force generated by the fluid-particles interaction. We establish the existence of global weak solutions for the nonlocal Stokes-Vlasov system in dimensions two and three without resorting to assumptions on higher-order velocity moments of the initial distribution of particles. We then study by the means of the sigma-convergence method, the asymptotic behavior in the general deterministic framework, of the sequence of solutions to the nonlocal Stokes-Vlasov system. In guise of illustration, we provide several physical applications of the homogenization result including periodic, almost-periodic and weakly almost-periodic settings.

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Homogenization in algebras with mean value

Cited by 4 publications

References 24 publications

Introverted algebras with mean value and applications

Introverted algebras with mean value and applications

Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition

Nonlocal Stokes-Vlasov system: Existence and deterministic homogenization results

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