Boundary homogenization in perforated domains for adsorption problems with an advection term

Brillard, Alain; Gómez, D.; Лобо, М.; Pérez, Eugenia; Шапошникова, Т. А.

doi:10.1080/00036811.2016.1153631

Cited by 13 publications

(21 citation statements)

References 17 publications

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“…On the other hand, if (x) ≡ 1 in Ω, the limit problem (3) given by Theorem 1 is the nonlocal Dirichlet equation (4), and we have the strong convergence…”

Section: Proof Of Theorem 1 and Their Corollariesmentioning

confidence: 82%

Nonlocal evolution equations in perforated domains

Pereira

2018

Math Methods in App Sciences

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In this paper, we study a nonlocal evolution equation posed in perforated domains. We consider problems of the form u t (t,We think about Ω as a fixed set Ω from where we have removed the subset A that we call the holes.Moreover, we take J as a nonsingular kernel. Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of Ω has a weak limit, ⇀  weakly * in L ∞ (Ω) as → 0, we analyze the limit of the solutions proving a nonlocal homogenized evolution equation.KEYWORDS asymptotic analysis, Neumann problem, nonlocal equations, perforated domains 6368

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“…On the other hand, if (x) ≡ 1 in Ω, the limit problem (3) given by Theorem 1 is the nonlocal Dirichlet equation (4), and we have the strong convergence…”

Section: Proof Of Theorem 1 and Their Corollariesmentioning

confidence: 82%

Nonlocal evolution equations in perforated domains

Pereira

2018

Math Methods in App Sciences

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“…Many techniques and methods have been developed in order to understand the effect of the holes in perforated domains on the solutions of PDE problems with different boundary values. From pioneering works to recent ones we can still mention [1,6,8,11,17,29,30,36,33] and references therein that are concerned with elliptic and parabolic equations, nonlinear operators, as well as Stokes and Navier-Stokes equations from fluid mechanics. Note that this kind of problem is an "homogenization" problem, since the heterogeneous domain Ω is replaced by a homogeneous one, Ω, in the limit.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal problems in perforated domains

Pereira

Rossi

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper we analyze nonlocal equations in perforated domains. We consider nonlocal problems of the form f (x) = B J(x − y)(u(y) − u(x))dy with x in a perforated domain Ω ⊂ Ω. Here J is a non-singular kernel. We think about Ω as a fixed set Ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside Ω. In the later case we impose that u vanishes in the holes but integrate in the whole R N (B = R N ) and in the former we just consider integrals in R N minus the holes (B = R N \ (Ω \ Ω )). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of Ω has a weak limit, χ X weakly * in L ∞ (Ω), we analyze the limit as → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls we obtain that the critical radius is of order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behavior of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems.

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“…Let us mention that stationary problems associated with our model have been addressed in Gómez et al and Brillard et al, where an extensive bibliography on the model and previous works is provided. In this paper, we give results that extend and complement those in Brillard et al and Gómez et al to arbitrary shapes of the obstacles and to the dimension n > 3, respectively. Further specifying, Brillard et al address the convergence in the most critical situation of the time‐independent scalar problem for a domain perforated by balls.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics for models of non‐stationary diffusion in domains with a surface distribution of obstacles

Gómez

Лобо

Pérez‐Martínez

2018

Math Methods in App Sciences

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We consider a time-dependent model for the diffusion of a substance through an incompressible fluid in a perforated domain Ω , Ω ⊂ Ω ⊂ R n with n = 3, 4. The fluid flows in a domain containing a periodical set of "obstacles" (Ω∖Ω ) placed along an inner (n − 1)-dimensional manifold Σ ⊂ Ω. The size of the obstacles is much smaller than the size of the characteristic period . An advection term appears in the partial differential equation linking the fluid velocity with the concentration, while we assume a nonlinear adsorption law on the boundary of the obstacles. This law involves a monotone nonlinear function of the concentration and a large adsorption parameter. The "critical adsorption parameter" depends on the size of the obstacles , and, for different sizes, we derive the time-dependent homogenized models. These models contain a "strange term" in the transmission conditions on Σ, which is a nonlinear function and inherits the properties of . The case in which the fluid velocity and the concentration do not interact is also considered for n ≥ 3. CorrespondenceMaría-Eugenia Pérez-Martínez, ETSI Caminos, Canales y Puertos. Av. de los Castros s/n.,

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Boundary homogenization in perforated domains for adsorption problems with an advection term

Cited by 13 publications

References 17 publications

Nonlocal evolution equations in perforated domains

Nonlocal evolution equations in perforated domains

Nonlocal problems in perforated domains

Asymptotics for models of non‐stationary diffusion in domains with a surface distribution of obstacles

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