Currently displaying 1 – 20 of 22

Showing per page

Order by Relevance | Title | Year of publication

PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic

Benoît Perthame — 2004

Applications of Mathematics

Modeling the movement of cells (bacteria, amoeba) is a long standing subject and partial differential equations have been used several times. The most classical and successful system was proposed by Patlak and Keller & Segel and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on...

Concentration in the Nonlocal Fisher Equation: the Hamilton-Jacobi Limit

Benoît PerthameStephane Génieys — 2010

Mathematical Modelling of Natural Phenomena

The nonlocal Fisher equation has been proposed as a simple model exhibiting Turing instability and the interpretation refers to adaptive evolution. By analogy with other formalisms used in adaptive dynamics, it is expected that concentration phenomena (like convergence to a sum of Dirac masses) will happen in the limit of small mutations. In the present work we study this asymptotics by using a change of variables that leads to a constrained Hamilton-Jacobi equation. We prove the convergence analytically...

Regularity in kinetic formulations via averaging lemmas

Pierre-Emmanuel JabinBenoît Perthame — 2002

ESAIM: Control, Optimisation and Calculus of Variations

We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity...

Regularity in kinetic formulations via averaging lemmas

Pierre-Emmanuel JabinBenoît Perthame — 2010

ESAIM: Control, Optimisation and Calculus of Variations

We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to...

A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation

Emmanuel AudusseMarie-Odile BristeauBenoît PerthameJacques Sainte-Marie — 2011

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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 between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighboring layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure...

Global stability of steady solutions for a model in virus dynamics

Hermano FridPierre-Emmanuel JabinBenoît Perthame — 2003

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider a simple model for the immune system in which virus are able to undergo mutations and are in competition with leukocytes. These mutations are related to several other concepts which have been proposed in the literature like those of shape or of virulence – a continuous notion. For a given species, the system admits a globally attractive critical point. We prove that mutations do not affect this picture for small perturbations and under strong structural assumptions. Based on numerical...

Line-energy Ginzburg-Landau models : zero-energy states

Pierre-Emmanuel JabinFelix OttoBenoÎt Perthame — 2002

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider a class of two-dimensional Ginzburg-Landau problems which are characterized by energy density concentrations on a one-dimensional set. In this paper, we investigate the states of vanishing energy. We classify these zero-energy states in the whole space: They are either constant or a vortex. A bounded domain can sustain a zero-energy state only if the domain is a disk and the state a vortex. Our proof is based on specific entropies which lead to a kinetic formulation, and on a careful...

High frequency limit of the Helmholtz equations.

Jean-David BenamouFrançois CastellaTheodoros KatsaounisBenoit Perthame — 2002

Revista Matemática Iberoamericana

We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term ( which does not share the...

Stochastic averaging lemmas for kinetic equations

Pierre-Louis LionsBenoît PerthamePanagiotis E. Souganidis

Séminaire Laurent Schwartz — EDP et applications

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 the deterministic case and as far as we work in L 2 , the nature of regularity on averages is not changed in this stochastic kinetic equation and stays in the range of fractional Sobolev spaces at the price of an additional expectation. However all the exponents are changed; either time decay rates are slower (when...

High Frequency limit of the Helmholtz Equations

Jean-David BenamouFrançois CastellaThodoros KatsaounisBenoît Perthame

Séminaire Équations aux dérivées partielles

We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term (which does not share the...

A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation

Emmanuel AudusseMarie-Odile BristeauBenoît PerthameJacques Sainte-Marie — 2011

ESAIM: Mathematical Modelling and Numerical Analysis

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 between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighboring layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic...

Global Stability of Steady Solutions for a Model in Virus Dynamics

Hermano FridPierre-Emmanuel JabinBenoît Perthame — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a simple model for the immune system in which virus are able to undergo mutations and are in competition with leukocytes. These mutations are related to several other concepts which have been proposed in the literature like those of shape or of virulence – a continuous notion. For a given species, the system admits a globally attractive critical point. We prove that mutations do not affect this picture for small perturbations and under strong structural assumptions. Based on numerical...

Page 1 Next

Download Results (CSV)