Functional Inequalities and PDE in the Life Sciences

Paris, January 12-13, 2012

Université Paris-Dauphine

Titles and abstracts

Eric Carlen

Title: Stability for Gagliardo-Nirenberg-Sobolev inequalities and the long-time behavior of Critical Mass Solutions of the Keller-Segel Equation.

Abstract: We prove the existence of essentially optimal basins of attraction for the critical mass Keller-Segel equation, and prove a quantitative bound on the rate convergence to the stationary solutions in each basin. A key role is played by a stability results for a Gagliardo-Nirenberg-Sobolev inequality. Some related open problems will be discussed. This talk will be based on joint work with A. Blanchet and J.A. Carrillo and with A. Figali.

Jose Antonio Carrillo

Title: Keller-Segel, fast-diffusion, and functional inequalities

Abstract: We will show how the critical mass classical Keller-Segel system and the critical displacement convex fast-diffusion equation in two dimensions are related. On one hand, the critical fast diffusion entropy functional helps to show global existence around equilibrium states of the critical mass Keller-Segel system. On the other hand, the critical fast diffusion flow allows to show functional inequalities such as the Logarithmic HLS inequality in simple terms who is essential in the behavior of the subcritical mass Keller-Segel system. HLS inequalities can also be recovered in several dimensions using this procedure. It is crucial the relation to the GNS inequalities obtained by DelPino and Dolbeault. This talk corresponds to two works in collaboration with E. Carlen and A. Blanchet, and with E. Carlen and M. Loss.

Lucilla Corrias

Title: Modeling chemotaxis with systems of parabolic-parabolic type : the state of the art

Abstract: This talk is about PDE models for chemotaxis. Emphasis will be given on the so called doubly parabolic Keller-Segel system. Results on existence of global in time solutions, blow-up in finite and infinite time, long time asymptotic of global solutions, shall be reviewed. Finally, open problems will be highlighted. 

Jérôme Coville 

Title: Nonlocal models in population dynamics

Abstract: I will present results on two problems issue of a nonlocal model in population dynamics. 

In a first part, I will first focus on the analysis of an heterogeneous nonlocal dispersal process using a relative entropy methods. 

Then, I will present some results of large time behavior of a solution of a Lotka-Volterra equation with mutations, following the same line of ideas.

Pierre Degond 

Title: Does self-organization imply breakdown of statistical independence ? 

Abstract: Simulating systems consisting of a large number of interacting agents is numerically intensive. The use of meso or macroscopic models such as kinetic or hydrodynamic models may be more powerful. But these models are valid if the particles become statistically independent when their number becomes large. This is the so-called propagation of chaos. At first sight, systems exhibiting self-organization may fail obeying this property, because self-organization is about building correlations between the particles. For a large class of binary (or multiply) interacting particle models, we show that this intuition is wrong and that these systems actually do satisfy propagation of chaos, at least on the so-called kinetic time-scale. But, this property may be lost at longer time scales, as shown by one counter-example. At such long time scales, whether propagation of chaos is true or not remains open in general. However, our results show at least that correlation build-up requires large time scales.

(joint work with E. Carlen (Rutgers) and B. Wennberg (Göteborg))

Alessio Figalli

Title: Di Perna-Lions' theory, with application to semiclassical limits for the Schrödinger equation

Abstract: At the beginning of the '90, DiPerna and Lions studied in detail the connection between transport equations and ordinary differential equations. In particular, by proving an existence and uniqueness result at the level of the transport equation, they obtained (roughly speaking) existence and uniqueness of solutions for ODEs with Sobolev vector-fields for a.e. initial condition. Ten years later, Ambrosio has been able to extend such a result to BV vector fields. In some recent works we have investigated this theory in a more general setting, which allows us to show the semiclassical convergence of the quantum dynamics to the Liouville dynamics for the linear Schrödinger equations, under very weak regularity assumptions on the potential. In analogy to the classical DiPerna-Lions' theory, the price to pay for allowing singular potential is that the convergence result holds true only for "a.e. initial data", where "a.e." is with respect to a suitable family of reference measures in the space of the initial data. The aim of this talk is to give an overview of these results.

Massimo Fornasier

Title: Particle systems and kinetic equations modeling interacting agents in high dimension

Abstract: In this talk we explore how concepts of high-dimensional data compression via random projections onto lower-dimensional spaces can be applied for tractable simulation of certain dynamical systems modeling complex interactions. In such systems, one has to deal with a large number of agents (typically millions) in spaces of parameters describing each agent of high dimension (thousands or more). Even with today’s powerful computers, numerical simulations of such systems are prohibitively expensive. We propose an approach for the simulation of dynamical systems governed by functions of adjacency matrices in high dimension, by random projections via Johnson-Lindenstrauss embeddings, and recovery by compressed sensing techniques. We show how these concepts can be generalized to work for associated kinetic equations, by addressing the phenomenon of the delayed curse of dimensionality, known in information-based complexity for optimal measure quantization in high dimension.

(joint work with Jan Haskovec and Jan Vybiral)

Gaël Raoul

Title: An integro-differential model for evolution.

Abstract: We consider a population structured by a phenotypic trait (e.g. the resistance of a bacteria to a drug). The growth rate of the population of trait x is a function depending on x, but also on the total population present in the environment. This simple model is used in biology to study darwinian evolutionary phenomenon, such as speciation, long time behaviour of a population, etc...

This model is especially interesting biologically, because it can be used to study the phenotypic diversity of a species, and the interaction between evolution and spatial structure, which are still poorly understood. Those questions lead to challenging mathematical problems.

Clément Sire

Title: Collapse and evaporation dynamics of a canonical self-gravitating gas

Abstract: We review the out of equilibrium properties of a self-gravitating gas of particles in the presence of a strong friction and a random force (canonical gas) so that its bare diffusion constant takes the form $D(\rho)=T\rho^{1/n}$, where $\rho$ is the local particle density. Its equation of state is hence $P(\rho)=D(\rho)\rho$. Depending on the spatial dimension $d$, the index $n$, the temperature $T$, and whether the system is confined to a finite box or not, the system can display an equilibrium state, collapse (and post-collapse) or evaporation phases. The properties of these different phases will be illustrated by analytical and numerical results, and a complete dynamical phase diagram of the system will be presented. 

Giuseppe Toscani

Title: Kinetic and Fokker-Planck models for goods exchange in microeconomics

Abstract: We introduce various kinetic equations able to describe the exchange of goods among a huge population of agents. The leading idea is to describe the exchange by means of some fundamental rules in microeconomics, in particular by using Cobb-Douglas utility functions for the binary exchange, and the Edgeworth box for the description of the common exchange area in which utility is increasing for both agents. In presence of uncertainty, it is shown that the solution develops Pareto tails, where the Pareto index depends on the ratio between the gain and the variance of the uncertainty. Among others, the result holds true for a linearized Boltzmann-like equation, whose solution is shown to converge to the solution of a drift-diffusion equation of a Fokker-Planck type. The research has been done jointly with Stefano Demichelis and Carlo Brugna, from the University of Pavia.