Jean Louis Woukeng

Université de Dschang · Department of Mathematics and Computer Science

A manuscript on the acoustic properties of almost periodic porous media.

The paper deals with the homogenization of reaction-diffusion equations with large reaction terms in a multi-scale porous medium. We assume that the fractures and pores are equidistributed and that the coefficients of the equations are periodic. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is defin...

We address the homogenization of a semilinear hyperbolic stochastic partial differential equation with highly oscillating coefficients, in the context of ergodic algebras with mean value. To achieve our goal, we use a suitable variant of the sigma-convergence concept that takes into account both the random and deterministic behaviours of the phenom...

The paper is devoted to the homogenization of Richards' type equations in a deterministic multiscale porous medium filled with soft inclusions. The medium consists of a deterministic ensemble of coarse aggregates embedded within a matrix containing a deterministic ensemble of fine aggregates, leading to a network in which the inclusions (of differe...

The current work deals with the global dynamics of 2D stochastic tidal equations in a highly heterogeneous environment. With the help of the stochastic version of the sigma-convergence method in conjunction with the Prokhorov and Skorokhod compactness theorems, we prove that the dynamics at the macroscopic level is of the same type at the microscop...

This paper provide evidence of the existence and uniqueness result for the viscosity solutions of inhomogeneous Vlasov equation. we consider the Cauchy-Dirichlet problem for the relativistic Vlasov equation with near vacuum initial data where the distribution function depends on the time, the position, the momentum and the non-Abelian charge of par...

In this paper, the two dimensional Signorini static contact problem in linear elasticity is presented. We present the weak formulation of frictionless and the frictional contact problems. In the both cases, the boundary integral operators are used to propose a boundary variational formulation whose resolution by the generalized Newton method is pre...

To fight against Ebola virus disease, several measures have been adopted. Among them, isolation, safe burial and vaccination occupy a prominent place. In this paper, we present a model which takes into account these three control strategies as well as the indirect transmission through a polluted environment. The asymptotic behavior of our model is...

The aim of this work is to provide the first strong convergence result of numerical approximation of a general time-fractional second order stochastic partial differential equation involving a Caputo derivative in time of order $\alpha\in(\frac 12; 1)$ and driven simultaneously by a multiplicative standard Brownian motion and additive fBm with Hurs...

This work deals with the homogenization of two dimensions' tidal equations. We study the asymptotic behavior of the sequence of the solutions using the sigma-convergence method. We establish the convergence of the sequence of solutions towards the solution of an equivalent problem of the same type.

The paper deals with the homogenization of linear Boltzmann equations by the means of the sigma-convergence method. Replacing the classical periodicity hypothesis on the coefficients of the collision operator by an abstract assumption covering a great variety of physical behaviours, we prove that the density of the particles converges to the soluti...

This work is concerned with the homogenization of initial boundary value parabolic equations with hysteresis, containing nonlinear monotone operators in the diffusion term. We make use of the sigma‐convergence concept together with the properties of hysteresis operator to derive the homogenized problem. The homogenized operator is obtained in terms...

Since 1976 many outbreaks of Ebola virus disease have occurred in Africa, and up to now, no treatment is available. Thus, to fight against this illness, several control strategies have been adopted. Among these measures, isolation, safe burial and vaccination occupy a prominent place. In this paper therefore, we present a model which takes into acc...

In the current work, we are performing the asymptotic analysis, beyond the periodic setting, of the Cahn-Hilliard-Navier-Stokes system. Under the general deterministic distribution assumption on the microstructures in the domain, we find the limit model equivalent to the heterogeneous one. To this end, we use the sigma-convergence concept which is...

In the current work, we are performing the asymptotic analysis, beyond the periodic setting, of the Cahn–Hilliard–Navier–Stokes system. Under the general deterministic distribution assumption on the microstructures in the domain, we find the limit model equivalent to the heterogeneous one. To this end, we use the sigma-convergence concept which is...

The paper deals with the homogenization of a linear Boltzmann equation by the means of the sigma-convergence method. Under a general deterministic assumption on the coefficients of the equation, we prove that the density of the particles converges to a solution of a drift-diffusion equation. To achieve our goal, we use the Krein-Rutman theorem for...

We address the homogenization of the Vlasov equations using the sigma‐convergence method. Assuming that the electromagnetic field is strong and is highly oscillating in both space and time, we derive the homogenization result. We then study some special cases leading to already known results.

For a class of linear elliptic equations of general type with rapidly oscillating coefficients, we use the sigma-convergence method to prove the homogenization result and a corrector-type result. In the case of asymptotic periodic coefficients we derive the optimal convergence rates for the zero order approximation of the solution with no smoothnes...

For a homogenization problem associated to a linear elliptic operator, we prove the existence of a distributional corrector and we find an approximation scheme for the homogenized coefficients. We also study the convergence rates in the asymptotic almost periodic setting, and we show that the rates of convergence for the zero order approximation, a...

The work deals with a study of a nonlinear parabolic equation with hysteresis, containing a nonlinear monotone operator in the diffusion term. The well-posedness of the model equation is addressed by using an implicit time discretization scheme in conjunction with the piecewise monotonicity of the hysteresis operator, and a fundamental inequality d...

In the deterministic homogenization of nonlinear monotone elliptic PDEs, we prove the existence of a distributional corrector and we find an approximation scheme for the homogenized coefficient. The obtained results represent an important step towards the numerical implementation of the results from the deterministic homogenization theory beyond th...

We carry out the deterministic homogenization of nonlinear elliptic operators beyond periodicity. To proceed with, we prove the existence of nonlinear correctors in the usual distributional sense. This lays the foundation for the study of regularity results in the nonlinear deterministic homogenization theory beyond the periodic setting.

We address the homogenization of a semilinear hyperbolic stochastic partial differential equation with highly oscillating coefficients, in the context of ergodic algebras with mean value. To achieve our goal, we use a suitable variant of the sigma-convergence concept that takes into account both the random and deterministic behaviours of the phenom...

In this paper we proceed with the multiscale analysis of semilinear damped stochastic wave motions. The analysis is made by combining the well-known sigma convergence method with its stochastic counterpart, associated to some compactness results such as the Prokhorov and Skorokhod theorems. We derive the equivalent model, which is of the same type...

For a family of linear hyperbolic damped stochastic wave equations with rapidly oscillating coefficients, we establish the homogenization result by using the sigma-convergence method. This is achieved under an abstract assumption covering special cases like the periodicity, the almost periodicity and some others. © 2017 Institute of Mathematics, Ac...

In this paper we study the acoustic properties of porous media saturated by an incompressible viscoelastic fluid. The model considered here consists of a linear deformable porous skeleton having memory that is surrounded by a viscoelastic Oldroyd fluid. Assuming the microstructures to be almost periodically distributed and under the almost periodic...

The paper deals with the homogenization of reaction-diffusion equations with large reaction terms in a multi-scale porous medium. We assume that the fractures and pores are equidistributed and that the coefficients of the equations are periodic. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is defin...

In most of the linear homogenization problems involving convolution terms so far studied, the main tool used to derive the homogenized problem is the Laplace transform. Here we propose a direct approach enabling one to tackle both linear and nonlinear homogenization problems that involve convolution sequences without using Laplace transform. To ill...

The sigma convergence method was introduced by G. Nguetseng for studying deterministic homogenization problems beyond the periodic setting and extended by him to the case of deterministic homogenization in general deterministic perforated domains. Here we show that this concept can also model such problems in more general domains. We illustrate thi...

Our work deals with the systematic study of the coupling between the nonlocal Stokes system and the Vlasov equation. The coupling is due to a drag force generated by the fluid-particles interaction. We establish the existence of global weak solutions for the nonlocal Stokes-Vlasov system in dimensions two and three without resorting to assumptions...

We study the flow generated by an incompressible viscoelastic fluid in a fractured porous medium. The model consists of a fluid flow governed by Stokes-Volterra equations evolving in a periodic double-porosity medium. Using the multiscale convergence method associated to some recent tools about the convergence of convolution sequences, we show that...

Let AA be an introverted algebra with mean value. We prove that its spectrum Δ(A)Δ(A) is a compact topological semigroup, and that the kernel K(Δ(A))K(Δ(A)) of Δ(A)Δ(A) is a compact topological group over which the mean value on AA can be identified as the Haar integral. Based on these facts and also on the fact that K(Δ(A))K(Δ(A)) is an ideal of Δ...

We study the flow generated by an incompressible viscoelastic fluid in a fractured porous medium. The model consists of a fluid flow governed by Stokes-Volterra equations evolving in a periodic double-porosity medium. Using the multiscale convergence method associated to some recent tools about the convergence of convolution sequences, we show that...

We study homogenization for nonstationary Navier–Stokes systems in a fissured medium of general deterministic type. Assuming that the blocks of the porous medium consist of deterministically distributed inclusions and the elasticity tensors satisfy general deterministic hypotheses, we prove that the macroscopic problem is a Navier-Stokes type equat...

Let A be an introverted algebra with mean value. We prove that its spectrum \Delta (A) is a compact topological semigroup, and that the kernel K(\Delta (A)) of \Delta (A) is a compact topological group over which the mean value on A can be identified as the Haar integral. Based on these facts and also on the fact that K(\Delta (A)) is an ideal of \...

In several works, the theory of strongly continuous groups is used to build a framework for solving stochastic homogenization problems. Following this idea, we construct a detailed and comprehensive theory of homogenization. This enables to solve homogenization problems in algebras with mean value, regardless of whether they are ergodic or not, the...

The paper deals with the existence and almost periodic homogenization of some model of generalized Navier-Stokes equations. We first establish an existence result for non-stationary Ladyzhenskaya equations with a given non constant density. The external force depends nonlinearly on the velocity. Next, to proceed with homogenization, as the density...

Homogenization of Wilson-Cowan type of nonlocal neural field models is investigated. Motivated by the presence of a convolution terms in this type of models, we first prove some general convergence results related to convolution sequences. We then apply theses results to the homogenization problem of the Wilson-Cowan type model in a general determi...

We study in this paper the periodic homogenization problem related to a strongly nonlinear reaction-diffusion equation. Owing to the large reaction term, the homogenized equation has a rather quite different form which puts together both the reaction and convection effects. We show in a special case that, the homogenized equation is exactly of a co...

In this paper, we show that the concept of sigma-convergence associated to stochastic processes can tackle the homogenization of stochastic partial differential equations. In this regard, the homogenization problem for a stochastic nonlinear partial differential equation is studied. Using some deep compactness results such as the Prokhorov and Skor...

Homogenization of a stochastic nonlinear reaction-diffusion equation with a large non- linear term is considered. Under a general Besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equati...

Motivated by the fact that in nature almost all phenomena behave randomly in some scales and deterministically in some other scales, we build up a framework suitable to tackle both deterministic and stochastic homogenization problems simultaneously, and also separately. Our approach, the stochastic sigma-convergence, can be seen either as a multisc...

We study the multiscale homogenization of a nonlinear hyperbolic equation in a periodic setting. We obtain an accurate homogenization result. We also show that as the nonlinear term depends on the microscopic time variable, the global homogenized problem thus obtained is a system consisting of two hyperbolic equations. It is also shown that in spit...

The purpose of the present work is to introduce a framework which enables us to study nonlinear homogenization problems. The starting point is the theory of algebras with mean value. Very often in physics, from very simple experimental data, one gets complicated structure phenomena. These phenomena are represented by functions which are permanent i...

In this paper we discuss the concept of stochastic two-scale conver-gence, which is appropriate to solve coupled -periodic and stochastic-homoge-nization problems. This concept is a combination of both well-known two-scale convergence and stochastic two-scale convergence in the mean schemes, and is a generalization of the said previous methods. By...

In order to widen the scope of the applications of deterministic homogenization, we consider here the homogenization problem for a family of integral functionals. The homogenization procedure tending to be classical, the choice focused on the convex integral functionals is made just to simplify the presentation of the paper. We use a new approach b...

We redefine the homogenization algebras without requiring the separability assumption. We show that this enables one to treat more complicated homogenization problems than those solved by the previous theory. In particular we exhibit an example of algebra which, contrary to the algebra of almost periodic functions, induces no homogenization algebra...

In this letter we show that in contrast to what has been done so far in the deterministic homogenization theory, we can solve nonlinear homogenization problems in a general way by leaning solely on a single assumption. We also show that this assumption is suitable for finding particular solutions (such as almost periodic ones) of some partial diffe...

We study the reiterated homogenization of nonlinear parabolic differential equations associated with monotone operators. Contrary to what is usually done in the deterministic homogenization theory, we present a new approach based on a deterministic assumption on the coefficients of the operators, which allows us to consider the concrete homogenizat...

The paper deals with the homogenization problem beyond the periodic setting, for a degenerated nonlinear non-monotone elliptic type operator on a perforated domain Ω ε in ℝ N with isolated holes. While the space variable in the coefficients a 0 and a is scaled with size ε (ε>0 a small parameter), the system of holes is scaled with ε 2 size, so that...

Multiscale homogenization of nonlinear non-monotone degenerated parabolic operators is investigated. Under a periodicity assumption on the coefficients of the operators under consideration, we obtain by means of multiscale convergence method, an accurate homogenization result. It is also shown that in spite of the presence of several time scales th...

Reiterated deterministic homogenization problem for nonlinear pseudo monotone parabolic type operators is considered beyond the usual periodic setting. We present a new approach based on the generalized Besicovitch type spaces, which allows to consider general assumptions on the coefficients of the operators under consideration. In particular we so...

This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in ℝ N with isolated holes of size ɛ2 (ɛ > 0 a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homoge...

Deterministic homogenization has been till now applied to the study of monotone operators, the determination of the limiting problem being systematically based on the monotonicity of the operator under consideration. Here we mean to show that deterministic homogenization also tackle non-monotone operators. More precisely, under an abstract general...

We discuss the nonstochastic homogenization of nonlinear parabolic differential operators in an abstract setting framed to bridge the gap between periodic and stochastic homogenization theories. Instead of the classical periodicity hypothesis, we have here an abstract assumption covering a great variety of concrete behaviours in both space and time...

We study, beyond the classical periodic setting, the homogenization of linear and nonlinear parabolic differential equations associated with monotone operators. The usual periodicity hypothesis is here substituted by an abstract deterministic assumption characterized by a great relaxation of the time behaviour. Our main tool is the recent theory of...