# Registered Data

## [00154] Homogenization of PDEs in domains with oscillating boundaries or interfaces

**Session Date & Time**:- 00154 (1/2) : 5B (Aug.25, 10:40-12:20)
- 00154 (2/2) : 5C (Aug.25, 13:20-15:00)

**Type**: Proposal of Minisymposium**Abstract**: Homogenization is a mathematical way of understanding microscopic structure via macroscopic medium and hence has enormous applications in science and engineering fields, including material science, as heat diffusion, fluid flows, deformations, and biological applications as electrical conduction in tissues like nerve or hearth fibers. In this symposium, we consider two types of closely related homogenization problems: complex domains consisting of a fixed part and a rapidly oscillating part, and domains with oscillating interfaces with jump-conditions. The aim of this minisymposium is to present recent results in these two important subjects by known specialists worldwide, and allow discussions opening new directions.**Organizer(s)**: Patrizia Donato, Akambadath K. Nandakumaran**Classification**:__35B27__,__74A50__,__74KXX__,__76M50__,__35BXX__**Speakers Info**:- Micol Amar ( Sapienza Università di Roma)
- Juan Casado-Díaz (University of Sevilla)
- Editha C. Jose (University of The Philippines Los Baños)
- Bidhan Chandra Sardar (IIT Ropar)
- Manuel Luna-Laynez (University of Sevilla)
- Maria Neuss-Radu (Universität Erlangen-Nürnberg)
- Carmen Perugia (Università del Sannio)
- Klas Pettersson (Chalmers University of Technology)

**Talks in Minisymposium**:**[00392] Asymptotic analysis of a parabolic problem with a rough fast oscillating interface****Author(s)**:- Patrizia Donato (Univ Rouen Normandie, France)
**Editha Carreon Jose**(University of the Philippines Los Banos)- Daniel Onofrei (University of Houston)

**Abstract**: In this talk, we will discuss the well posedness and prove several homogenization results for a parabolic problem with an imperfect contact on the rough fast oscillating interface separating a domain occupied by heterogeneous materials. The complexity of the domain geometry and the imperfect contact on the interface create interesting multiscale phenomena with different macroscale behaviors depending on model parameters.

**[00399] Homogenization by unfolding of a Bingham fluid in a thin domain with rough boundary****Author(s)**:**Carmen Perugia**(Department of Science and Technology, University of Sannio)- Manuel Villanueva-Pesqueira (Universidad Pontificia Comillas)
- Giuseppe Cardone (University of Naples "Federico II")

**Abstract**: We consider a Bingham flow in a thin domain with rough boundary and with no-slip boundary condition on the whole boundary of the domain. By using an adapted linear unfolding operator we perform a detailed analysis of the asymptotic behavior of the Bingham flow when thickness of the domain and roughness periodicity tends to zero. We obtain the homogenized problem for the velocity and the pressure, which preserves the nonlinear character of the flow.

**[00401] Homogenization of Stokes system with Neumann condition on highly oscillating boundary****Author(s)**:**Bidhan Chandra Sardar**(Indian Institute of Technology Ropar)

**Abstract**: We consider the steady Stokes system in a $n$-dimensional domain $\Omega_{\varepsilon}$ with a rapidly oscillating $(n - 1)$ dimensional boundary prescribed with Neumann boundary condition and periodicity along the lateral sides is considered. We aim to study the limiting analysis $(as\; \varepsilon\to 0)$ of the steady Stokes problem and identify the limit problem in a fixed domain. Finally, show the weak convergences of velocities are improved to strong convergence by introducing corrector terms.

**[00402] Heat conduction in composite media involving imperfect contact conditions****Author(s)**:**Micol Amar**(Sapienza Università di Roma)- Daniele Andreucci (Sapienza Università di Roma)
- Claudia Timofte (University of Bucharest)

**Abstract**: We present a model which exhibit simultaneously jumps in the solution and in the flux, involving also the mean average of the physical fields representing the different phases. The starting model is a composite made by a hosting medium containing a periodic array of inclusions of small size, coated by a thin layer made by two different materials, one encapsulated in the other. The smallness of the thin layer leads us to perform first a two-step concentration procedure. The periodic structure leads to a homogenization limit, in order to achieve a macroscopic description.

**[00407] Fluids with a non-slip condition on a non-periodic oscillating boundary****Author(s)**:**Juan Casado-Díaz**(University of Seville)- Manuel Luna-Laynez (University of Seville)

**Abstract**: It is known that a viscous fluid with a non-slip condition on a rough boundary behaves as if an adherence condition is imposed. This is the case for a boundary given by $x_3=\delta_\varepsilon\psi(x_1/\varepsilon,x_2/\varepsilon)$, $\psi$ periodic, and $\delta_\varepsilon/\varepsilon^{3\over 2}$ tending to infinity. The rugosity has not effect if it tends to zero. In the critical case a new zero order term appears in the boundary condition. Here, we extend these results to non-periodic boundaries.

**[00412] Derivation of coupled Stokes-Plate-Equations for fluid flow through thin porous elastic layers****Author(s)**:**Maria Neuss-Radu**(Friedrich-Alexander-Universität Erlangen-Nürnberg)- Markus Gahn (University Heidelberg)
- Willi J\"ager (University Heidelberg)

**Abstract**: We consider two fluid-filled bulk domains separated by a thin porous elastic layer with thickness and periodicity $\epsilon$. The fluid flow is described by an instationary Stokes-system, and the solid via linear elasticity. By rigorous homogenization and dimension reduction methods, we derive for $\epsilon \to 0$ an effective model consisting of the Stokes-equations in the bulk domains coupled to a time dependent plate equation with homogenized coefficients on the effective interface separating the bulk regions.

**[00416] Homogenization of a two-component domain with an oscillating thick interface****Author(s)**:**Klas Pettersson**(Freelance)- Patrizia Donato (Univ Rouen Normandie, France)

**Abstract**: The homogenization of an elliptic boundary value problem is considered in a finite cylindrical domain that consists of two components separated by a periodically oscillating thick interface. On the interface, the flux is assumed to be continuous, and the jump of the solution to be proportional to the flux through the interface. By means of the periodic unfolding method, we derive the homogenized system, and prove the convergence of the solutions and their energies.

**[00417] A decomposition result for thin domains with rough boundary****Author(s)**:- Juan Casado-Díaz (University of Seville)
**Manuel Luna-Laynez**(University of Seville)- Francisco Javier Suárez-Grau (University of Seville)

**Abstract**: We prove a decomposition result for the pressure of a fluid in a thin domain. This result extends to the linear elasticity framework, providing better estimates for the Korn constant and fine decompositions for the elastic deformations. We show how these results, which give additional compactness properties, apply to study the asymptotic behaviour of some problems in fluid mechanics and elasticity, posed in thin domains with rough boundaries.