Publications (40)
Dirac concentrations in Lotka-Volterra parabolic PDEs
Benoit Perthame, Guy Barles
We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a pop…
Motility switching and front-back synchronisation in polarized cells
Gissell Estrada-Rodriguez, Benoit Perthame
The combination of protrusions and retractions in the movement of polarized cells leads to understand the effect of possible synchronisation between the two ends of the cells. This…
Structured Model Conserving Biomass for the Size-spectrum Evolution in Aquatic Ecosystems
Laura Kanzler, Benoit Perthame, Benoit Sarels
Mathematical modelling of the evolution of the size-spectrum dynamics in aquatic ecosystems was discovered to be a powerful tool to have a deeper insight into impacts of human- and…
A Hamilton-Jacobi approach to nonlocal kinetic equations
Nadia Loy, Benoit Perthame
Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, model of BGK type implementing a velocity-jump process. We study both a linear and a nonline…
Asymmetric potentials and motor effect: a large deviation approach
Benoit Perthame, Panagiotis E. Souganidis
We provide a mathematical analysis of appearance of the concentrations (as Dirac masses) of the solution to a Fokker-Planck system with asymmetric potentials. This problem has been…
Existence of solutions of the hyperbolic Keller-Segel model
Benoit Perthame, Anne-Laure Dalibard
We are concerned with the hyperbolic Keller-Segel model with quorum sensing, a model describing the collective cell movement due to chemical signalling with a flux limitation for h…
A simple derivation of BV bounds for inhomogeneous relaxation systems
Benoit Perthame, Nicolas Seguin, Magali Tournus
We consider relaxation systems of transport equations with heterogeneous source terms and with boundary conditions, which limits are scalar conservation laws. Classical bounds fail…
Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies
Alexander Lorz, Tommaso Lorenzi, Michael E. Hochberg +2
Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern. Among the many mechanisms at work in cancers, one of the most important is th…
Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation
Gregoire Nadin, Benoit Perthame, Min Tang
This note investigates the properties of the traveling waves solutions of the nonlocal Fisher equation. The existence of such solutions has been proved recently in \cite{BNPR} but…
On the Inverse Problem for a Size-Structured Population Model
Benoit Perthame, Jorge P. Zubelli
We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population…
Singular Hamilton-Jacobi equation for the tail problem
Sepideh Mirrahimi, Guy Barles, Benoit Perthame +1
In this paper we study the long time-long range behavior of reaction diffusion equations with negative square root -type reaction terms. In particular we investigate the exponentia…
Confined run-and-tumble model with boundary aggregation: long time behavior and convergence to the confined Fokker-Planck model
Jingyi Fu, Jiuyang Liang, Benoit Perthame +1
The motile micro-organisms such as E. coli, sperm, or some seaweed are usually modelled by self-propelled particles that move with the run-and-tumble process. Individual-based stoc…
Data Assimilation for hyperbolic conservation laws. A Luenberger observer approach based on a kinetic description
Anne-Céline Boulanger, Philippe Moireau, Benoit Perthame +1
Developing robust data assimilation methods for hyperbolic conservation laws is a challenging subject. Those PDEs indeed show no dissipation effects and the input of additional inf…
Multiple asymptotics of Kinetic Equations with Internal States
Benoit Perthame, Weiran Sun, Min Tang +1
The run and tumble process is well established in order to describe the movement of bacteria in response to a chemical stimulus. However the relation between the tumbling rate and…
Rare mutations limit of a steady state dispersion trait model
Benoit Perthame, Panagiotis E. Souganidis
The evolution of dispersal is a classical question in evolutionary ecology, which has been widely studied with several mathematical models. The main question is to define the fitte…
Multispecies cross-diffusions: from a nonlocal mean-field to a porous medium system without self-diffusion
Marie Doumic, Sophie Hecht, Benoit Perthame +1
Systems describing the long-range interaction between individuals have attracted a lot of attention in the last years, in particular in relation with living systems. These systems…
Long-term analysis of phenotypically structured models
Alexander Lorz, Benoit Perthame
Phenotypically structured equations arise in population biology to describe the interaction of species with their environment that brings the nutrients. This interaction usually le…
Traveling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion: existence and branching instabilities
Benoit Perthame, Christian Schmeiser, Min Tang +1
How can repulsive and attractive forces, acting on a conservative system, create stable traveling patterns or branching instabilities? We have proposed to study this question in th…
Scalar conservation laws with rough (stochastic) fluxes
Pierre-Louis Lions, Benoit Perthame, Panagiotis E. Souganidis
We develop a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, a special case being stochastic conservation laws with quasilinear…
A Hamilton-Jacobi Approach to Evolution of Dispersal
King-Yeung Lam, Yuan Lou, Benoit Perthame
The evolution of dispersal is a classical question in evolutionary biology, and it has been studied in a wide range of mathematical models. A selection-mutation model, in which the…
Scalar conservation laws with rough (stochastic) fluxes; the spatially dependent case
Pierre-Louis Lions, Benoit Perthame, Panagiotis E. Souganidis
We continue the development of the theory of pathwise stochastic entropy solutions for scalar conservation laws in with quasilinear multiplicative ''rough path'' dependence…
Incompressible limit of mechanical model of tumor growth with viscosity
Benoit Perthame, Nicolas Vauchelet
Various models of tumor growth are available in the litterature. A first class describes the evolution of the cell number density when considered as a continuous visco-elastic mate…
A Structured Population Model of Cell Differentiation
Marie Doumic, Anna Marciniak-Czochra, Benoit Perthame +1
We introduce and analyze several aspects of a new model for cell differentiation. It assumes that differentiation of progenitor cells is a continuous process. From the mathematical…
Large-Time Behavior of Periodic Entropy Solutions to Anisotropic Degenerate Parabolic-Hyperbolic Equations
Gui-Qiang Chen, Benoit Perthame
We are interested in the large-time behavior of periodic entropy solutions in to anisotropic degenerate parabolic-hyperbolic equations of second-order. Unlike the pure h…
Existence, regularity and stability in a strongly degenerate nonlinear diffusion and haptotaxis model of cancer invasion
Benoit Perthame, Chiara Villa
We consider a mathematical model of cancer cell invasion of the extracellular matrix (ECM), comprising a strongly degenerate parabolic partial differential equation for the cell vo…
Asymptotic analysis of a selection model with space
Sepideh Mirrahimi, Benoit Perthame
Selection of a phenotypical trait can be described in mathematical terms by 'stage structured' equations which are usually written under the form of integral equations so as to exp…
Time fluctuations in a population model of adaptive dynamics
Sepideh Mirrahimi, Benoit Perthame, Panagiotis E. Souganidis
We study the dynamics of phenotypically structured populations in environments with fluctuations. In particular, using novel arguments from the theories of Hamilton-Jacobi equation…
Stochastic averaging lemmas for kinetic equations
Pierre-Louis Lions, Benoit Perthame, Panagiotis E. Souganidis
We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale. Compared to t…
Multilinear multiplier theorems and their applications to the Jacobian and the Hessian determinant
Hoai-Minh Nguyen, Benoit Perthame
We establish several variants of the multilinear multiplier theorem of Coifman and Meyer. We also present examples that are not covered by existing theories. Our motivation comes f…
A multilayer Saint-Venant system with mass exchanges for Shallow Water flows. Derivation and numerical validation
Emmanuel Audusse, Marie-Odile Bristeau, Benoit Perthame +1
The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges betwee…
Modelling the spatial organization of cell proliferation in the developing central nervous system
Jean Clairambault, Vladimir Flores, Benoit Perthame +3
How far is neuroepithelial cell proliferation in the developing central nervous system a deterministic process? Or, to put it in a more precise way, how accurately can it be descri…
Competition and boundary formation in heterogeneous media: Application to neuronal differentiation
Benoit Perthame, Cristobal Quiñinao, Jonathan Touboul
We analyze an inhomogeneous system of coupled reaction-diffusion equations representing the dynamics of gene expression during differentiation of nerve cells. The outcome of this d…
Mathematical Analysis of a System for Biological Network Formation
Jan Haskovec, Peter Markowich, Benoit Perthame
Motivated by recent physics papers describing rules for natural network formation, we study an elliptic-parabolic system of partial differential equations proposed by Hu and Cai. T…
An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations
Jean Clairambault, Stephane Gaubert, Benoit Perthame
For monotone linear differential systems with periodic coefficients, the (first) Floquet eigenvalue measures the growth rate of the system. We define an appropriate arithmetico-geo…
Free boundary problems for Tumor Growth: a Viscosity solutions approach
Inwon C. Kim, Benoit Perthame, Panagiotis E. Souganidis
The mathematical modeling of tumor growth leads to singular stiff pressure law limits for porous medium equations with a source term. Such asymptotic problems give rise to free bou…
Transversal instability for the thermodiffusive reaction-diffusion system
Michal Kolwalczyk, Benoit Perthame, Nicolas Vauchelet
The propagation of unstable interfaces is at the origin of remarkable patterns that are observed in various areas of science as chemical reactions, phase transitions, growth of bac…
Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration
Emeric Bouin, Vincent Calvez, Nicolas Meunier +4
Invasion fronts in ecology are well studied but very few mathematical results concern the case with variable motility (possibly due to mutations). Based on an apparently simple rea…
On a voltage-conductance kinetic system for integrate and fire neural networks
Benoit Perthame, Delphine Salort
The voltage-conductance kinetic equation for integrate and fire neurons has been used in neurosciences since a decade and describes the probability density of neurons in a network.…
Energy concentration and Sommerfeld condition for Helmholtz equation with variable index at infinity
Benoit Perthame, Luis Vega
We consider the Helmholtz equation with a variable index of refraction , which is not necessarily constant at infinity but can have an angular dependency like $n(x)\to n\_\in…
Notes on a PDE System for Biological Network Formation
Jan Haskovec, Peter Markowich, Benoit Perthame +1
We present new analytical and numerical results for the elliptic-parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological…