Benoit Perthame

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We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system(More)
The Keller-Segel system describes the collective motion of cells which are attracted by a chemical substance and are able to emit it. In its simplest form it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. It is known that, in two space dimensions, for small initial(More)
We present a new formulation of multidimensional scalar conser-<lb>vation laws, which includes both the equation and the entropy criterion. This<lb>formulation is a kinetic one involving an additional variable called velocity by<lb>analogy. We also give some applications of this formulation to new compactness<lb>and regularity results for entropy solutions(More)
We construct a nonlinear kinetic equation and prove that it is welladapted to describe general multidimensional scalar conservation laws. In particular we prove that it is well-posed uniformly in ε the microscopic scale. We also show that the proposed kinetic equation is equipped with a family of kinetic entropy functions analogous to Boltzmann's(More)
Our starting point is a selection-mutation equation describing the adaptive dynamics of a quantitative trait under the influence of an ecological feedback loop. Based on the assumption of small (but frequent) mutations we employ asymptotic analysis to derive a Hamilton-Jacobi equation. Well-established and powerful numerical tools for solving the(More)
We consider the Fisher-KPP equation with a nonlocal saturation effect defined through an interaction kernel φ(x) and investigate the possible differences with the standard Fisher-KPP equation. Our first concern is the existence of steady states. We prove that if the Fourier transform φ̂(ξ) is positive or if the length σ of the nonlocal interaction is short(More)
Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up(More)
We complete the theory of velocity averaging lemmas for transport equations by studying the limiting case of a full space derivative in the source term. Although the compactness of averages does not hold any longer, a speciic estimate remains, which shows compactness of averages in more general situations than those previously known. Our method is based on(More)