Papers by Jean Louis Woukeng
Mathematical Problems in Engineering, Apr 24, 2023
In this paper, the two-dimensional Signorini static contact problem in linear elasticity is prese... more In this paper, the two-dimensional Signorini static contact problem in linear elasticity is presented. We present the weak formulation of the frictional contact problems, and the boundary integral operators are used to propose a boundary variational formulation whose resolution by the generalized Newton method is presented. Moreover, a particular formulation by the fxed point method associated with the augmented Lagrangian is proposed for efcient analysis of contact problems with Coulomb friction, and powerful algorithms are constructed. Te discretization is carried over by using the Galerkin method. Te resulting linear system is solved by using a preconditioned conjugate gradient (CG) iterative solver.
Journal of nonlinear science, Mar 15, 2024
Starting from a classic non-local (in space) Cahn-Hilliard-Stokes model for twophase flow in a th... more Starting from a classic non-local (in space) Cahn-Hilliard-Stokes model for twophase flow in a thin heterogeneous fluid domain, we rigorously derive by mathematical homogenization a new effective mixture model consisting of a coupling of a non-local (in time) Hele-Shaw equation with a non-local (in space) Cahn-Hilliard equation. We then analyse the resulting model and prove its well-posedness. A key to the analysis is the new concept of sigma-convergence in thin heterogeneous domains allowing to pass to the homogenization limit with respect to the heterogeneities and the domain thickness simultaneously.
arXiv (Cornell University), Jun 21, 2012
The paper deals with the existence and almost periodic homogenization of some model of generalize... more 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 and an external force depending nonlinearly on the velocity. Next, the density of the fluid being non constant, we combine some compactness arguments with the sigma-convergence method to study the asymptotic behavior of the velocity field.
arXiv (Cornell University), Jun 27, 2015
The paper deals with the homogenization of reaction-diffusion equations with large reaction terms... more 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 defined on a fixed domain and is of convection-diffusionreaction type.
Mathematics
In a thin heterogeneous porous layer, we carry out a multiscale analysis of Smoluchowski’s discre... more In a thin heterogeneous porous layer, we carry out a multiscale analysis of Smoluchowski’s discrete diffusion–coagulation equations describing the evolution density of diffusing particles that are subject to coagulation in pairs. Assuming that the thin heterogeneous layer is made up of microstructures that are uniformly distributed inside, we obtain in the limit an upscaled model in the lower space dimension. We also prove a corrector-type result very useful in numerical computations. In view of the thin structure of the domain, we appeal to a concept of two-scale convergence adapted to thin heterogeneous media to achieve our goal.
Communications in Nonlinear Science and Numerical Simulation
The aim of this work is to provide the first strong convergence result of numerical approximation... more 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 α ∈ (1 2 ; 1) and driven simultaneously by a multiplicative standard Brownian motion and additive fBm with Hurst parameter H ∈ (1 2 , 1), more realistic to model the random effects on transport of particles in medium with thermal memory. We prove the existence and uniqueness results and perform the spatial discretization using the finite element and the temporal discretization using a fractional exponential integrator scheme. We provide the temporal and spatial convergence proofs for our fully discrete scheme and the result shows that the convergence orders depend on the regularity of the initial data, the power of the fractional derivative, and the Hurst parameter H.
The aim of this work is to provide the first strong convergence result of numerical approximation... more 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 α ∈ (1 2 ; 1) and driven simultaneously by a multiplicative standard Brownian motion and additive fBm with Hurst parameter H ∈ (1 2 , 1), more realistic to model the random effects on transport of particles in medium with thermal memory. We prove the existence and uniqueness results and perform the spatial discretization using the finite element and the temporal discretization using a fractional exponential integrator scheme. We provide the temporal and spatial convergence proofs for our fully discrete scheme and the result shows that the convergence orders depend on the regularity of the initial data, the power of the fractional derivative, and the Hurst parameter H.
Banach Journal of Mathematical Analysis, 2015
In several works, the theory of strongly continuous groups is used to build a framework for solvi... more 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, thereby responding affirmatively to the question raised by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators. Matem. Zametki, 33 (1983) 571-582 (english transl.: Math. Notes, 33 (1983) 294-300)] to know whether it is possible to homogenize problems in nonergodic algebras. We also state and prove a compactness result for Young measures in these algebras. As an important achievement we study the homogenization problem associated with a stochastic Ladyzhenskaya model for incompressible viscous flow, and we present and solve a few examples of homogenization problems related to nonergodic algebras.
ArXiv, 2022
The aim of this work is to provide the first strong convergence result of numerical approximation... more 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 α ∈ ( 2 ; 1) and driven simultaneously by a multiplicative standard Brownian motion and additive fBm with Hurst parameter H ∈ ( 2 , 1), more realistic to model the random effects on transport of particles in medium with thermal memory. We prove the existence and uniqueness results and perform the spatial discretization using the finite element and the temporal discretization using a fractional exponential integrator scheme. We provide the temporal and spatial convergence proofs for our fully discrete scheme and the result shows that the convergence orders depend on the regularity of the initial data, the power of the fractional derivative, and the Hurst parameter H .
In this paper, we show that the concept of sigma-convergence associated to stochastic processes c... more 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 Skorokhod theorems, we prove that the sequence of solutions of this problem converges in probability towards the solution of an equation of the same type. To proceed with, we use a suitable version of sigma-convergence method, the sigma-convergence for stochastic processes, which takes into account both the deterministic and random behaviours of the solutions of the problem. We apply the homogenization result to some concrete physical situations such as the periodicity, the almost periodicity, the weak almost periodicity, and others.
Homogenization of a stochastic nonlinear reaction-diffusion equation with a large non- linear ter... more 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 equation, namely, the stochastic non- linear convection-diffusion equation which we explicitly derive in terms of appropriated functionals. We study some particular cases such as the periodic framework, and many others. This is achieved under a suitable generalized concept of sigma-convergence for stochastic processes.
In this paper we proceed with the multiscale analysis of semilinear damped stochastic wave motion... more 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 as the micro-model.
Let A be an introverted algebra with mean value. We prove that its spectrum \Delta (A) is a compa... more 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 \Delta (A), we define the convolution over \Delta (A). We then use it to derive some new convergence results involving the convolution product of sequences. These convergence results provide us with an efficient method for studying the asymptotics of nonlocal problems. The obtained results systematically establish the connection between the abstract harmonic analysis and the homogenization theory. To illustrate this, we work out some homogenization problems in connection with nonlocal partial differential equations.
Homogenization of Wilson-Cowan type of nonlocal neural field models is investigated. Motivated by... more 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 deterministic setting. Key ingredients in this study are the notion of algebras with mean value and the related concept of sigma-convergence.
In several works, the theory of strongly continuous groups is used to build a framework for solvi... more 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, thereby responding affirmatively to the question raised by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators. Matem. Zametki, 33 (1983) 571-582 (english transl.: Math. Notes, 33 (1983) 294-300)] to know whether it is possible to homogenize problems in nonergodic algebras. We also state and prove a compactness result for Young measures in these algebras. As an important achievement we study the homogenization problem associated with a stochastic Ladyzhenskaya model for incompressible viscous flow, and we present and solve a few examples of homogenization problems related to nonergodic al...
Our work deals with the systematic study of the coupling between the nonlocal Stokes system and t... more 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 on higher-order velocity moments of the initial distribution of particles. We then study by the means of the sigma-convergence method, the asymptotic behavior in the general deterministic framework, of the sequence of solutions to the nonlocal Stokes-Vlasov system. In guise of illustration, we provide several physical applications of the homogenization result including periodic, almost-periodic and weakly almost-periodic settings.
In the current work, we are performing the asymptotic analysis, beyond the periodic setting, of t... more 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 suitable for the passage to the limit.
We study, beyond the classical periodic setting, the homogenization of linear and nonlinear parab... more 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 homogenization structures by the first author, and our homogenization approach falls under the two-scale convergence method. Various concrete examples are worked out with a view to pointing out the wide scope of our approach and bringing the role of homogenization structures to light.
Corrector Problem and Homogenization of Nonlinear Elliptic Monotone PDE
In the deterministic homogenization of nonlinear monotone elliptic PDEs, we prove the existence o... more 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 the periodic setting.
Since 1976 many outbreaks of Ebola virus disease have occurred in Africa, and up to now, no treat... more 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 account these three control strategies as well as the difference in immunological responses of infected by distinguishing individuals with moderate and severe symptoms. The sensitivity analysis of the control reproduction number suggests that, the vaccination response works better than the two other control measures at the beginning of the disease, and there is no need to vaccinate people in order to overcome Ebola. The global asymptotic dynamic of the model with controls is completely achieved exhibiting a sharp threshold behavior. Consequently, the global stability of the endemic equilibrium when the control reproduction number is greater t...
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Papers by Jean Louis Woukeng