Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces

Gahn, Markus

doi:10.3934/dcdsb.2019151

Cited by 3 publications

(9 citation statements)

References 21 publications

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“…Remark 6.1. It is worth remaking that if we consider a nonlinear term f (u ε ) in the first equation in (1) which satisfies the same assumption than g, we obtain Theorem 1.1 with a additional term |Y * | |Y | f (u) in the first equation in (8).…”

Section: By Periodicityψmentioning

confidence: 99%

“…An error estimate for this model, under extra regularity assumptions on the data, can be found in Amar and Gianni [2]. More recently, in [8], Gahn derivates some general two-scale compactness results for coupled bulk-surface problems and applies these results to an elliptic problem with a non dynamical boundary condition, which involves the Laplace-Beltrami operator, in a multi-component domain.…”

Section: Introduction and Setting Of The Problemmentioning

confidence: 98%

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Existence, Uniqueness and Homogenization of Nonlinear Parabolic Problems with Dynamical Boundary Conditions in Perforated Media

Anguiano

2019

Mediterr. J. Math.

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This paper deals with the homogenization of the reaction-diffusion equations in a domain containing periodically distributed holes of size ε, with a dynamical boundary condition of reactive-diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holeswhere ∆Γ denotes the Laplace-Beltrami operator on the surface of the holes, ν is the outward normal to the boundary, δ > 0 plays the role of a surface diffusion coefficient and g is the nonlinear term. We generalize our previous results (see [3]) established in the case of a dynamical boundary condition of pure-reactive type, i.e., with δ = 0. We prove the convergence of the homogenization process to a nonlinear reaction-diffusion equation whose diffusion matrix takes into account the reactive-diffusive condition on the surface of the holes.

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Section: By Periodicityψmentioning

confidence: 99%

Section: Introduction and Setting Of The Problemmentioning

confidence: 98%

Existence, Uniqueness and Homogenization of Nonlinear Parabolic Problems with Dynamical Boundary Conditions in Perforated Media

Anguiano

2019

Mediterr. J. Math.

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“…Denote by

\overset{\nabla Ψ_{i ε}}{true}

the extension by zero of

\nabla {normalΨ}_{i ε}

inside the holes. From (), we have

\tilde{\nabla {normalΨ}_{i ε}} = \tilde{\nabla_{y} false(w_{i} + y_{i} false)} = \tilde{\nabla_{y} w_{i}} false(y false) + e_{i} χ_{Y^{*}},

and taking into account [4, Corollary 2.10], we have

\begin{matrix} true \\ right \overset{\nabla Ψ_{i ε}}{true} ⇀ \frac{1}{false| Y false|} \int_{Y^{*}} (() e_{i} + \nabla_{y} w_{i} false(y false)) d y normalweakly normalin L^{2} (Ω) . \end{matrix}

Due to that

w_{i} \in {double-struckH}_{normalper} ∖ R

(see [9, Theorem 4.1]), let

\nabla_{normalΓ} γ_{0} false(Ψ_{i ε} false)

be the tangential gradient of

γ_{0} false(Ψ_{i ε} false)

\partial F_{ε}

and we denote by

μ_{h}^{ε}

the above introduced linear form in the particular case in which

h false(\frac{x}{ε} false) = \nabla_{normalΓ} γ_{0} false(Ψ_{i ε} (x) false)

.…”

Section: Homogenized Model: Proof Of the Main Theoremmentioning

confidence: 99%

“…More recently, in [9], Gahn derives some general two-scale compactness results for coupled bulk-surface problems and applies these results to an elliptic problem with a non-dynamical boundary condition, which involves the Laplace-Beltrami operator, in a multi-component domain.…”

Section: Introduction and Setting Of The Problemmentioning

confidence: 99%

Homogenization of parabolic problems with dynamical boundary conditions of reactive‐diffusive type in perforated media

Anguiano

2020

Z Angew Math Mech

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Homogenization of parabolic problems with dynamical boundary conditions. This paper deals with the homogenization of the reaction-diffusion equations in a domain containing periodically distributed holes of size , with a dynamical boundary condition of reactive-diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holes ∇ ⋅ + = Δ Γ − (), where Δ Γ denotes the Laplace-Beltrami operator on the surface of the holes, is the outward normal to the boundary, > 0 plays the role of a surface diffusion coefficient and is the nonlinear term. We generalize our previous results established in the case of a dynamical boundary condition of pure-reactive type, i.e., with = 0. We prove the convergence of the homogenization process to a nonlinear reaction-diffusion equation whose diffusion matrix takes into account the reactive-diffusive condition on the surface of the holes.

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“…For such kind of problems in [4,16] two-scale compactness results are derived for connected surfaces, where in [16] the method of unfolding is used. Compactness results for a coupled bulk-surface problem when the evolution of the trace of the bulk-solution on the surface Γ ǫ is described by a diffusion equation, are treated in [5,12]. In [5] continuity of the traces across the interface is assumed, where in [12] also jumps across the interface are allowed and also compactness results for the disconnected domain Ω 2 ǫ are derived.…”

Section: Introductionmentioning

confidence: 99%

Homogenization of a two-phase problem with nonlinear dynamic Wentzell-interface condition for connected-disconnected porous media

Gahn¹

2020

Preprint

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We investigate a reaction-diffusion problem in a two-component porous medium with a nonlinear interface condition between the different components. One component is connected and the other one is disconnected. The ratio between the microscopic pore scale and the size of the whole domain is described by the small parameter ǫ. On the interface between the components we consider a dynamic Wentzell-boundary condition, where the normal fluxes from the bulk-domains are given by a reaction-diffusion equation for the traces of the bulk-solutions, including nonlinear reactionkinetics depending on the solutions on both sides of the interface. Using two-scale techniques, we pass to the limit ǫ → 0 and derive macroscopic models, where we need homogenization results for surface diffusion. To cope with the nonlinear terms we derive strong two-scale results.

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Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces

Cited by 3 publications

References 21 publications

Existence, Uniqueness and Homogenization of Nonlinear Parabolic Problems with Dynamical Boundary Conditions in Perforated Media

Existence, Uniqueness and Homogenization of Nonlinear Parabolic Problems with Dynamical Boundary Conditions in Perforated Media

Homogenization of parabolic problems with dynamical boundary conditions of reactive‐diffusive type in perforated media

Homogenization of a two-phase problem with nonlinear dynamic Wentzell-interface condition for connected-disconnected porous media

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