arXiv subjects

M Hillairet

Publications and source records attributed to M Hillairet.

4 recordsLinked to original sources

arXiv2026

Analysis of the steady solutions of the Fisher's infinitesimal model; a Hilbertian approach

We provide an asymptotic analysis of a nonlinear integro-differential equation which describes the evolutionary dynamics of a population which reproduces sexually and which is subject to selection and competition. The sexual reproduction is modeled via a nonlinear integral term, known as the Fisher's 'infinitesimal model'. We consider a small segregational variance regime, where a parameter in the infinitesimal model, which measures the deviation between the trait of the offspring and the mean parental trait, is small with respect to the selection variance. In this regime, we characterize the steady states of the problem and analyze their stability. Our method relies on a spectral analysis involving Hermite polynomials, highlighting the specific structure of the nonlinear reproduction term. We expect that the framework developed in this article will contribute to progress on several related problems that were out of reach with previous methods.

math.AP↗

arXiv2023

A moment-based approach for the analysis of the infinitesimal model in the regime of small variance

We provide an asymptotic analysis of a nonlinear integro-differential equation describing the evolutionary dynamics of a population which reproduces sexually and which is subject to selection and competition. The sexual reproduction is modeled via a nonlinear integral term, known as the ''infinitesimal model''. We consider a regime of small segregational variance, where a parameter in the infinitesimal operator, which measures the deviation between the trait of the offspring and the mean parental trait, is small. We prove that, in this regime, the phenotypic distribution remains close to a Gaussian profile with a fixed small variance and we characterize the dynamics of the mean phenotypic trait via an ordinary differential equation. While similar properties were already proved for a closely related model using a Hopf-Cole transformation and perturbative analysis techniques, we provide an alternative proof which relies on a direct study of the dynamics of the moments of the phenotypic distribution and a contraction property of the Wasserstein distance.

math.AP↗

arXiv2016

A compressible multifluid system with new physical relaxation terms

In this paper, we rigorously derive a new compressible multifluid system from compressible Navier-Stokes equations with density-dependent viscosity in the one-dimensional in space setting. More precisely, we propose and mathematically derive a generalization of the usual one velocity Baer-Nunziato model with a new relaxation term in the PDE governing the volume fractions. This new relaxation term encodes the change of viscosity and pressure between the different fluids. For the reader's convenience, we first establish a formal derivation in the bifluid setting using a WKB decomposition and then we rigorously justify the multifluid homogenized system using a kinetic formulation via Young measures characterization.

math.AP↗

arXiv2015

Justification of lubrication approximation: an application to fluid/solid interactions

We consider the stationary Stokes problem in a three-dimensional fluid domain $\mathcal F$ with non-homogeneous Dirichlet boundary conditions. We assume that this fluid domain is the complement of a bounded obstacle $\mathcal B$ in a bounded or an exterior smooth container $Ω$. We compute sharp asymptotics of the solution to the Stokes problem when the distance between the obstacle and the container boundary is small.

math.AP↗