Benoı̂t Perthame

Numerical modeling of avalanches based on Saint Venant equations using a kinetic scheme

[1] Numerical modeling of debris avalanche is presented here. The model uses the long-wave approximation based on the small aspect ratio of debris avalanches as in classical Saint Venant model of shallow water. Depth-averaged equations using this approximation are derived in a reference frame linked to the topography. Debris avalanche is treated here as a single-phase, dry granular flow with Coulomb-type behavior. The numerical finite volume method uses a kinetic scheme based on the description of the microscopic behavior of the system to define numerical fluxes at the interfaces of a finite element mesh. The main advantage of this method is to preserve the height positivity. The originality of the presented scheme stands in the introduction of a Dirac distribution of particles at the microscopic scale in order to describe the stopping of a granular mass when the driving forces are under the Coulomb threshold. Comparisons with analytical solutions for dam break problems and experimental results show the efficiency of the model in dealing with significant discontinuities and reproducing the flowing and stopping phase of granular avalanches. The ability of the model to describe debris avalanche behavior is illustrated here by schematic numerical simulation of an avalanche over simplified topography. Coulomb-type behavior with constant and variable friction angle is compared in the framework of this simple example. Numerical tests show that such an approach not only provides insights into the flowing and stopping stage of the granular mass but allows observation of interesting behavior such as the existence of a fluid-like zone behind a stopped solid-like granular mass in specific situations, suggesting the presence of horizontal surfaces in the deposited mass.

Global existence to the BGK model of Boltzmann equation

Abstract We present an existence and a stability proof for solutions to the BGK model of Boltzmann Equation δ,f+v·▽ x f+f=M[f], t⩽0,x∈ R N ,v∈ R N M[f]=( p (2φT) N 2 ) exp ( −|v−u| 2 2T ), (p,pu,p|u| 2 +pT)(1,x)=∞ R N (1,v,|v| 2 )f(t,x,v)dv. It relies on the strong compactness of ϱ, u , T and on a new estimate on the third moment of ƒ: ∝ ¦v¦ ƒ dv . We also prove the entropy relation for (1).

A chemotaxis model motivated by angiogenesis

Abstract We consider a simple model arising in modeling angiogenesis and more specifically the development of capillary blood vessels due to an exogenous chemo-attractive signal (solid tumors for instance). It is given as coupled system of parabolic equations through a nonlinear transport term. We show that, by opposition to some classical chemotaxis model, this system admits a positive energy. This allows us to develop an existence theory for weak solutions. We also show that, in two dimensions, this system admits a family of self-similar waves. To cite this article: L. Corrias et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Gelation and mass conservation in coagulation-fragmentation models

Abstract The occurrence of gelation and the existence of mass-conserving solutions to the continuous coagulation–fragmentation equation are investigated under various assumptions on the coagulation and fragmentation rates, thereby completing the already known results. A non-uniqueness result is also established and a connection to the modified coagulation model of Flory is made.

Boltzmann equation with infinite energy: renormalized solutions and distributional solutions for small initial data and initial data close to a Maxwellian

We prove new existence results for the Boltzmann equation with an initial data with infinite energy. In the framework of renormalized solutions we assume

L1 contraction property for a Boltzmann equation with Pauli statistics

(|x|^\alpha + |x-v|^2) \, f_0 \in L^1

Global solutions of some chemotaxis and angiogenesis systems in high space dimensions

instead of

A new model of Saint Venant and Savage–Hutter type for gravity driven shallow water flows

(|x|^2 + |v|^2) \, f_0 \in L^1

Gelation in Coagulation and Fragmentation Models

, and we show new a priori estimates. In the framework of distributional solutions we treat small initial data compared to a Maxwellian of the type

General entropy equations for structured population models and scattering

\exp ( - |x-v|^2/2)