Equations In Perforated Domains Research Articles

Stabilization by multiplicative Itô noise for Chafee–Infante equation in perforated domains

The stabilization by noise for parabolic equations in perforated domains, i.e. domains with small holes, is investigated. We show that when the holes are small enough, one can stabilize the unstable equations using suitable multiplicative Itô noise. The results are quantitative, in the sense that we can explicitly estimate the size of the holes and diffusion coefficients for which stabilization by noise takes place. This is done by using the asymptotic behaviour of the first eigenvalue of the Laplacian as the hole shrinks to a point.

Homogenization of the Navier–Stokes equations in perforated domains in the inviscid limit

We study the solution u ɛ to the Navier–Stokes equations in perforated by small particles centered at with no-slip boundary conditions at the particles. We study the behavior of u ɛ for small ɛ, depending on the diameter ɛ α , α > 1, of the particles and the viscosity ɛ γ , γ > 0, of the fluid. We prove quantitative convergence results for u ɛ in all regimes when the local Reynolds number at the particles is negligible. Then, the particles approximately exert a linear friction force on the fluid. The obtained effective macroscopic equations depend on the order of magnitude of the collective friction. We obtain (a) the Euler–Brinkman equations in the critical regime, (b) the Euler equations in the subcritical regime and (c) Darcy’s law in the supercritical regime.

Homogenization and Correctors for Stochastic Hyperbolic Equations in Domains with Periodically Distributed Holes

The goal of this paper is to present new results on homogenization and correctors for stochastic linear hyperbolic equations in periodically perforated domains with homogeneous Neumann conditions on the holes. The main tools are the periodic unfolding method, energy estimates, probabilistic and deterministic compactness results. The findings of this paper are stochastic counterparts of the celebrated work [D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Port. Math. (N.S.) 63 (2006) 467–496]. The convergence of the solution of the original problem to a homogenized problem with Dirichlet condition has been shown in suitable topologies. Homogenization and convergence of the associated energies results recover the work in [M. Mohammed and M. Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, Asymptot. Anal. 97 (2016) 301–327]. In addition to that, we obtain corrector results.

H-convergence of a class of quasilinear equations in perforated domains beyond periodic setting

In this paper, we aim to study the asymptotic behavior (when varepsilon ;rightarrow ; 0) of the solution of a quasilinear problem of the form -mathrm{{div}};(A^{varepsilon }(cdot ,u^{varepsilon }) nabla u^{varepsilon })=f given in a perforated domain Omega backslash T_{varepsilon } with a Neumann boundary condition on the holes T_{varepsilon } and a Dirichlet boundary condition on partial Omega . We show that, if the holes are admissible in certain sense (without any periodicity condition) and if the family of matrices (x,d)mapsto A^{varepsilon }(x,d) is uniformly coercive, uniformly bounded and uniformly equicontinuous in the real variable d, the homogenization of the problem considered can be done in two steps. First, we fix the variable d and we homogenize the linear problem associated to A^{varepsilon }(cdot ,d) in the perforated domain. Once the H^{0}-limit A^{0}(cdot ,d) of the pair (A^{varepsilon },T^{varepsilon }) is determined, in the second step, we deduce that the solution u^{varepsilon } converges in some sense to the unique solution u^{0} in H^{1}_{0}(Omega ) of the quasilinear equation -mathrm{{div}};(A^{0}(cdot ,u^{0})nabla u )=chi ^{0}f (where chi ^{0} is L^{infty } weak ^{star } limit of the characteristic function of the perforated domain). We complete our study by giving two applications, one to the classical periodic case and the second one to a non-periodic one.

Nonlocal and nonlinear evolution equations in perforated domains

In this work we analyze the behavior of the solutions to nonlocal evolution equations of the form ut(x,t)=∫J(x−y)u(y,t)dy−hε(x)u(x,t)+f(x,u(x,t)) with x in a perturbed domain Ωε⊂Ω which is thought as a fixed set Ω from where we remove a subset Aε called the holes. We choose appropriated families of functions hε∈L∞ in order to deal with both Neumann and Dirichlet conditions in the holes setting a Dirichlet condition outside Ω. Moreover, we take J as a non-singular kernel and f as a nonlocal nonlinearity. Under the assumption that the characteristic functions of Ωε have a weak limit, we study the limit of the solutions providing a nonlocal homogenized equation.

“Strange term” in homogenization of attractors of reaction–diffusion equation in perforated domain

Abstract We consider reaction–diffusion equation in perforated domain, with rapidly oscillating coefficient in boundary conditions. We do not assume any Lipschitz condition for the nonlinear function in the equation, so, the uniqueness theorem for the corresponding initial boundary value problem may not hold for the considered reaction-diffusion equation. We prove that the trajectory attractors of this equation tend in a weak sense to the trajectory attractors of the homogenized reaction-diffusion equation with a “strange term” (potential). Bibliography: 48 titles.

Nonlocal problems in perforated domains

AbstractIn this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form $f(x) = \int \nolimits _{B} J(x-y) (u(y) - u(x)) {\rm d}y$ with x in a perforated domain $\Omega ^\epsilon \subset \Omega $. Here J is a nonsingular kernel. We think about $\Omega ^\epsilon $ 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 latter case we impose that u vanishes in the holes but integrate in the whole ℝN (B = ℝN) and in the former we just consider integrals in ℝN minus the holes ($B={\open R} ^N \setminus (\Omega \setminus \Omega ^\epsilon )$). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of $\Omega ^\epsilon $ has a weak limit, $\chi _{\epsilon } \rightharpoonup {\cal X}$ weakly* in L∞(Ω), we analyse the limit as ε → 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 the order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behaviour of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems.

Homogenization of the discrete diffusive coagulation–fragmentation equations in perforated domains

The asymptotic behavior of the solution of an infinite set of Smoluchowski's discrete coagulation–fragmentation–diffusion equations with non-homogeneous Neumann boundary conditions, defined in a periodically perforated domain, is analyzed. Our homogenization result, based on Nguetseng–Allaire two-scale convergence, is meant to pass from a microscopic model (where the physical processes are properly described) to a macroscopic one (which takes into account only the effective or averaged properties of the system). When the characteristic size of the perforations vanishes, the information given on the microscale by the non-homogeneous Neumann boundary condition is transferred into a global source term appearing in the limiting (homogenized) equations. Furthermore, on the macroscale, the geometric structure of the perforated domain induces a correction in the diffusion coefficients.

Nonlocal evolution equations in perforated domains

In this paper, we study a nonlocal evolution equation posed in perforated domains. We consider problems of the form with x in a perturbed domain . We think about Ωε as a fixed set Ω from where we have removed the subset Aε 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 Ωε has a weak limit, weakly∗ in L∞(Ω) as ε→0, we analyze the limit of the solutions proving a nonlocal homogenized evolution equation.

On the homogenization of the Stokes equations in perforated domains with application to fluid/solid interaction problems

On the homogenization of the Stokes equations in perforated domains with application to fluid/solid interaction problems

The Method of Reflections, Homogenization and Screening for Poisson and Stokes Equations in Perforated Domains

We study the convergence of the Method of Reflections for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains which consist in the exterior of balls. We prove that the method converges if the balls are contained in a bounded region and the density of the electrostatic capacity of the balls is sufficiently small. If the capacity density is too large or the balls extend to the whole space, the method diverges, but we provide a suitable modification of the method that converges to the solution of the Dirichlet problem also in this case. We give new proofs of classical homogenization results for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains using the (modified) Method of Reflections.

Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains

In this paper, we investigate a linear hyperbolic stochastic partial differential equation (SPDE) with rapidly oscillating � -periodic coefficients in a domain with small holes (of size-� ) under Neumann conditions on the boundary of the holes and Dirichlet condition on the exterior boundary. When the number of these holes approach infinity, i.e. their sizes approach zero, the homogenized problem is a hyperbolic SPDE with constant coefficients in the domain without perforations. Moreover the convergence of the associated energy to that of the homogenized system is established.

The hole-filling method and the multiscale computation for the wave equations in perforated domains

In this paper we discuss the multiscale computation of the wave equations with rapidly oscillating coefficients in a perforated domain. The hole-filling method through filling all holes with a very compliant material is introduced. As the material parameters of the weak phase go to zero, the proof of this limiting process is derived. Based on the above hole-filling method, the multiscale asymptotic method and the convergence results for the associated multiphase problem in a domain without holes are presented. The numerical experiments are carried out to validate the theoretical results.

Generalized multiscale finite element methods for problems in perforated heterogeneous domains

Complex processes in perforated domains occur in many real-world applications. These problems are typically characterized by physical processes in domains with multiple scales. Moreover, these problems are intrinsically multiscale and their discretizations can yield very large linear or nonlinear systems. In this paper, we investigate multiscale approaches that attempt to solve such problems on a coarse grid by constructing multiscale basis functions in each coarse grid, where the coarse grid can contain many perforations. In particular, we are interested in cases when there is no scale separation and the perforations can have different sizes. In this regard, we mention some earlier pioneering works, where the authors develop multiscale finite element methods. In our paper, we follow Generalized Multiscale Finite Element Method (GMsFEM) and develop a multiscale procedure where we identify multiscale basis functions in each coarse block using snapshot space and local spectral problems. We show that with a few basis functions in each coarse block, one can approximate the solution, where each coarse block can contain many small inclusions. We apply our general concept to (1) Laplace equation in perforated domains; (2) elasticity equation in perforated domains; and (3) Stokes equations in perforated domains. Numerical results are presented for these problems using two types of heterogeneous perforated domains. The analysis of the proposed methods will be presented elsewhere.

Effective macroscopic interfacial transport equations in strongly heterogeneous environments for general homogeneous free energies

We study phase field equations in perforated domains for arbitrary free energies. These equations have found numerous applications in a wide spectrum of both science and engineering problems with homogeneous environments. Here, we focus on strongly heterogeneous materials with perforations such as porous media. To the best of our knowledge, we provide the first derivation of upscaled equations for general free energy densities. In view of the versatile applications of phase field equations, we expect that our study will lead to new modelling and computational perspectives for interfacial transport and phase transformations in strongly heterogeneous environments.

Multiscale asymptotic expansions methods and numerical algorithms for the wave equations in perforated domains

In this paper, we are concerned with the wave equations in perforated domains with a homogeneous Neumann condition on the boundary of the holes. The multiscale asymptotic expansion of the solution for the problem are constructed and associated explicit convergence rates are obtained. A multiscale numerical method is introduced. Finally, we present some numerical results which support strongly the convergence theorem.

Viscosity method for homogenization of parabolic nonlinear equations in perforated domains

In this paper, we develop a viscosity method for homogenization of Nonlinear Parabolic Equations constrained by highly oscillating obstacles or Dirichlet data in perforated domains. The Dirichlet data on the perforated domain can be considered as a constraint or an obstacle. Homogenization of nonlinear eigen value problems has been also considered to control the degeneracy of the porous medium equation in perforated domains. For the simplicity, we consider obstacles that consist of cylindrical columns distributed periodically and perforated domains with punctured balls. If the decay rate of the capacity of columns or the capacity of punctured ball is too high or too small, the limit of u ϵ will converge to trivial solutions. The critical decay rates of having nontrivial solution are obtained with the construction of barriers. We also show the limit of u ϵ satisfies a homogenized equation with a term showing the effect of the highly oscillating obstacles or perforated domain in viscosity sense.

Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains

In this paper, we study the spectral properties of Dirichlet problems for second order elliptic equation with rapidly oscillating coefficients in a perforated domain. The asymptotic expansions of eigenvalues and eigenfunctions for this kind of problem are obtained, and the multiscale finite element algorithms and numerical results are proposed.

Homogenization in perforated domains with rapidly pulsing perforations

The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the period of the holes goes to zero. Since standard conservation laws do not hold in this model, a rst diculty is to get ap rioriestimates of the solutions. We obtain them in a weighted space where the weight is the principal eigenfunction of an \adjoint periodic time-dependent eigenvalue problem. This problem is not a classical one, and its investigation is an important part of this work. Then, by using the multiple scale method, we construct the leading terms of a formal expansion (with respect to ) of the solution and give the limit \homogenized problem. An interesting peculiarity of the model is that, depending on the geometry of the holes, a large convection term may appear in the limit equation. Mathematics Subject Classication. 35B27, 74Q10, 76M50.

Homogenization of a parabolic equation in perforated domain with Dirichlet boundary condition

In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains $$\begin{gathered} \partial _t b(\tfrac{x}{{d_\varepsilon }},u_\varepsilon ) - div a(u_\varepsilon , \nabla u_\varepsilon ) = f(x,t) in \Omega _\varepsilon x (0, T), \hfill \\ u_\varepsilon ) = 0 on \partial \Omega _\varepsilon x (0, T), \hfill \\ u_\varepsilon (x, 0) = u_0 (x) in \Omega _\varepsilon . \hfill \\ \end{gathered} $$ . Here, Ωɛ = ΩS ε is a periodically perforated domain andd ε is a sequence of positive numbers which goes to zero. We obtain the homogenized equation. The homogenization of the equations on a fixed domain and also the case of perforated domain with Neumann boundary condition was studied by the authors. The homogenization for a fixed domain and $$b(\frac{x}{{d_\varepsilon }},u_\varepsilon ) \equiv b(u_\varepsilon )$$ has been done by Jian. We also obtain certain corrector results to improve the weak convergence.