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

2021 ◽  
Vol 12 (03) ◽  
Author(s):  
Mogtaba Mohammed ◽  
Waseem Asghar Khan
Keyword(s):  
Partial Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Original Problem ◽  
Hyperbolic Equations ◽  
Dirichlet Condition ◽  
Energy Estimates ◽  
Perforated Domains ◽  
Periodic Unfolding Method ◽  
Unfolding Method ◽  
Linear Hyperbolic Equations

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.

Homogenization and correctors of Robin problem for linear stochastic equations in periodically perforated domains

2020 ◽  
Vol 120 (1-2) ◽  
pp. 123-149
Author(s):  
Mogtaba Mohammed ◽  
Noor Ahmed
Keyword(s):  
Parabolic Equations ◽  
Original Problem ◽  
Stochastic Equation ◽  
Dirichlet Condition ◽  
Perforated Domains ◽  
Linear Parabolic Equations ◽  
Convergence Method ◽  
Periodic Unfolding Method ◽  
Additional Regularity ◽  
Unfolding Method

In this paper, we present homogenization and corrector results for stochastic linear parabolic equations in periodically perforated domains with non-homogeneous Robin conditions on the holes. We use the periodic unfolding method and probabilistic compactness results. Homogenization results presented in this paper are stochastic counterparts of some fundamental work given in [Cioranescu, Donato and Zaki in Port. Math. (N.S.) 63 (2006), 467–496]. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a parabolic stochastic equation in fixed domain with Dirichlet condition on the boundary. In contrast to the two scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems.

Homogenization and correctors for linear stochastic equations via the periodic unfolding methods

2019 ◽  
Vol 19 (05) ◽  
pp. 1950040 ◽  
Author(s):  
Mogtaba Mohammed
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Stochastic Equations ◽  
Partial Differential ◽  
Convergence Method ◽  
Oscillating Coefficients ◽  
Periodic Unfolding Method ◽  
Additional Regularity ◽  
Unfolding Method

In this paper, we use the periodic unfolding method and Prokhorov’s and Skorokhod’s probabilistic compactness results to obtain homogenization and corrector results for stochastic partial differential equations (PDEs) with periodically oscillating coefficients. We show the convergence of the solutions of the original problems to the solutions of the homogenized problems. In contrast to the two-scale convergence method, the corrector results obtained in this paper are without any additional regularity assumptions on the solutions of the original problems

Homogenisation of the Stokes problem with a pure non-homogeneous slip boundary condition by the periodic unfolding method

2011 ◽  
Vol 22 (4) ◽  
pp. 333-345 ◽  
Author(s):  
ANCA CAPATINA ◽  
HORIA ENE
Keyword(s):  
Boundary Condition ◽  
Porous Medium ◽  
Stokes System ◽  
Stokes Problem ◽  
Slip Boundary Condition ◽  
Perforated Domains ◽  
Slip Boundary ◽  
Periodic Unfolding Method ◽  
The Right ◽  
Unfolding Method

We study the homogenisation of the Stokes system with a non-homogeneous Fourier boundary condition on the boundary of the holes, depending on a parameter γ. Such systems arise in the modelling of the flow of an incompressible viscous fluid through a porous medium under the influence of body forces. At the limit, by using the periodic unfolding method in perforated domains, we obtain, following the values of γ, different Darcy's laws of typeMu= −N∇p+Fwith suitable matricesMandNwithFdepending on the right-hand side in the bulk term and in the boundary condition.

scholarly journals The periodic unfolding method for perforated domains and Neumann sieve models

2008 ◽  
Vol 89 (3) ◽  
pp. 248-277 ◽  
Author(s):  
D. Cioranescu ◽  
A. Damlamian ◽  
G. Griso ◽  
D. Onofrei
Keyword(s):  
Perforated Domains ◽  
Periodic Unfolding Method ◽  
Unfolding Method

scholarly journals Comparison of some solution concepts for linear first-order hyperbolic differential equations with non-smooth coefficients

2008 ◽  
Vol 84 (98) ◽  
pp. 123-157 ◽  
Author(s):  
Simon Haller ◽  
Günther Hörmann
Keyword(s):  
Method Of Characteristics ◽  
Lipschitz Continuity ◽  
Hyperbolic Equations ◽  
Energy Estimates ◽  
Solution Concepts ◽  
First Order ◽  
Logical Independence ◽  
Smooth Coefficients ◽  
Hyperbolic Differential Equations ◽  
Linear Hyperbolic Equations

We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on refined techniques concerning energy estimates. We provide a series of examples both as simple illustrations of the notions and conditions involved but also to show logical independence among the concepts.

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