arXiv subjects

Harsha Hutridurga

Publications and source records attributed to Harsha Hutridurga.

24 records · Page 2Linked to original sources

arXiv2017

Diffusion limit for Vlasov-Fokker-Planck Equation in bounded domains

We derive a diffusion approximation for the kinetic Vlasov-Fokker-Planck equation in bounded spatial domains with specular reflection type boundary conditions. The method of proof involves the construction of a particular class of test functions to be chosen in the weak formulation of the kinetic model. This involves the analysis of the underlying Hamiltonian dynamics of the kinetic equation coupled with the reflection laws at the boundary. This approach only demands the solution family to be weakly compact in some weighted Hilbert space rather than the much tricky $\mathrm L^1$ setting.

math.AP↗

arXiv2016

Simultaneous diffusion and homogenization asymptotic for the linear Boltzmann equation

This article is on the simultaneous diffusion approximation and homogenization of the linear Boltzmann equation when both the mean free path $\varepsilon$ and the heterogeneity length scale $η$ vanish. No periodicity assumption is made on the scattering coefficient of the background material. There is an assumption made on the heterogeneity length scale $η$ that it scales as $\varepsilon^β$ for $β\in(0,\infty)$. In one space dimension, we prove that the solutions to the kinetic model converge to the solutions of an effective diffusion equation for any $β\le2$ in the $\varepsilon\to0$ limit. In any arbitrary phase space dimension, under a smallness assumption of a certain quotient involving the scattering coefficient in the $H^{-\frac{1}{2}}$ norm, we again prove that the solutions to the kinetic model converge to the solutions of an effective diffusion equation in the $\varepsilon\to0$ limit.

math.AP↗

arXiv2016

Convergence along mean flows

We develop a technique of multiple scale asymptotic expansions along mean flows and a corresponding notion of weak multiple scale convergence. These are applied to homogenize convection dominated parabolic equations with rapidly oscillating, locally periodic coefficients and $\mathcal{O}(\eps^{-1})$ mean convection term. Crucial to our analysis is the introduction of a fast time variable, $τ=\frac{t}{\eps}$, not apparent in the heterogeneous problem. The effective diffusion coefficient is expressed in terms of the average of Eulerian cell solutions along the orbits of the mean flow in the fast time variable. To make this notion rigorous, we use the theory of ergodic algebras with mean value.

math.AP↗

arXiv2016

On the Maxwell-Stefan diffusion limit for a mixture of monatomic gases

Multi-species Boltzmann equations for gaseous mixtures, with analytic cross sections and under Grad's angular cutoff assumption, are considered under diffusive scaling. In the limit, we formally obtain an explicit expression for the binary diffusion coefficients in the Maxwell-Stefan equations.

math.AP↗

arXiv2014

Upscaling nonlinear adsorption in periodic porous media - Homogenization approach

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 pore boundaries, 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 [G. Allaire et al, Chemical Engineering Science, 65 (2010), pp.2292-2300], [G. Allaire et al, SIAM J. Math. Anal., 42 (2010), pp.125-144], [Allaire et al, IMA J Appl Math., 77 (2012), pp.788-815]. Under a technical assumption of equal drift velocities in the bulk and on the pore boundaries, we obtain a nonlinear monotone diffusion equation as the homogenized model. Our main technical tool is the method of two-scale convergence with drift [Marusic-Paloka et al, Journal of London Math. Soc., 72 (2005), pp.391-409]. We provide some numerical test cases in two space dimensions to support our theoretical analysis.

math.AP↗

arXiv2014

On the homogenization of multicomponent transport

This paper 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.

math.AP↗