Research

My research interests fall under the broad category of Partial Differential Equations. Below, I shall present some of my research activities. The research topics are catalogued under two rubriques:

HOMOGENISATION:

1. Homogenisation of reactive flows

2. Homogenisation of nonlinear reaction-diffusion systems

3. Homogenisation of transport in strong convection regime

KINETIC THEORY:

4. Trend to Equilibrium in kinetic theory

5. Diffusion approximation in kinetic theory

6. Regularity issues in kinetic theory

Homogenisation of reactive flows

This project debuted in 2010 with my PhD advisor Grégoire Allaire. The objective of works in this rubrique has been to understand the effective behaviour of transport of solutes in porous media in presence of reactions. To begin with, we have worked on the surface reactions at the fluid-solid interface in porous media.

Linear Isotherm:

In this work, we have undertaken the study of convection-diffusion process in a porous medium in the presence of a chemical reaction on the pore surfaces. The reaction we have considered is a linear adsorption phenomenon on the pore surfaces. Mathematically, this system is described in terms of a solution to a system of convection-diffusion equation in the medium and a system of surface convection-diffusion on the pore surfaces. These two systems are then coupled through a reaction term on the boundaries. We have obtained the homogenized problem for our microscopic model in moving frame. We have justified the upscaling using two-scale convergence with drift. Some Numerical tests were also undertaken using package FreeFem++ to study the effect of the variation of the rate constant and the surface molecular diffusion on the effective coefficients.

You can find the preprint of the article here: lin_iso

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Nonlinear Isotherm:

We consider the homogenization of a model of reactive flows through periodic porous media involving a single solute which can be absorbed and desorbed on the pore boundaries. This is a system of two convection-diffusion equations, one in the bulk and one on the solid surfaces, coupled by an exchange reaction term. The novelty of our work is to consider a nonlinear reaction term, a so-called Langmuir isotherm, in an asymptotic regime of strong convection. We therefore generalize previous works on a similar linear model. Under a technical assumption of equal drift velocities in the bulk and on the pore surfaces, we obtain a nonlinear monotone diffusion equation as the homogenized model. Our main technical tool is the method of two-scale convergence with drift. We provide some numerical test cases in two space dimensions to support our theoretical analysis.

You can find the preprint of the article here: iso_nonlin

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Multicomponent transport:

This work is devoted to the homogenization of weakly coupled cooperative parabolic systems in strong convection regime with purely periodic coefficients. Our approach is to factor out oscillations from the solution via principal eigenfunctions of an associated spectral problem and to cancel any exponential decay in time of the solution using the principal eigenvalue of the same spectral problem. We employ the notion of two-scale convergence with drift in the asymptotic analysis of the factorized model as the lengthscale of the oscillations tends to zero. This combination of the factorization method and the method of two-scale convergence is applied to upscale an adsorption model for multicomponent flow in an heterogeneous porous medium.

You can find the preprint of the article here: multi_comp

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Trend to Equilibrium in kinetic theory

Degenerate Vlasov-Fokker-Planck equation:

This project debuted in 2014 with Helge Dietert, Frédéric Hérau and Clément Mouhot. The objective of this work has been to study the long-time behaviour of the solutions to Vlasov-Fokker-Planck equation in the presence of degeneracy. We impose a ‘uniform geometric control condition’ on the coefficient function which guarantees the exponential decay of the semi-group. The preprint of this note shall follow soon.

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Diffusion approximation in kinetic theory

In this project, initiated with Claude Bardos and Clément Mouhot, the objective is to study the linear Boltzmann equation under diffusive scaling in various asymptotic regimes. This work tries to fill in the gaps regarding the diffusive limit of linear Boltzmann equation that has been studied since the 1980’s.

Vlasov-Fokker-Planck equation:

This project debuted in 2015 with Ludovic Cesbron concerns the study of the diffusion limit for the Vlasov-Fokker-Planck equation in bounded domains with specular reflection B.C. It employs a new approach to address the diffusion approximation problems in kinetic theory using Hamiltonian dynamics of the transport operator. The preprint of this article shall follow soon.

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Linear Boltzmann equation:

This project debuted in 2014 with Claude Bardos shall address the simultaneous diffusion and homogenisation limit of the linear Boltzmann equation. This is a work in progress and uses some classical tools from Functional analysis…

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