Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model

We investigate the long time behavior of the critical mass Patlak-Keller-Segel equation. This equation has a one parameter family of steady-state solutions $\rho_\lambda$, $\lambda>0$, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional ${\mathcal H}_\lambda$ coming from the critical fast diffusion equation in $\R^2$. We construct solutions of Patlak-Keller-Segel equation satisfying an entropy-entropy dissipation inequality for ${\mathcal H}_\lambda$. While the entropy dissipation for ${\mathcal H}_\lambda$ is strictly positive, it turns out to be a difference of two terms, neither of which need to be small when the dissipation is small. We introduce a strategy of "controlled concentration" to deal with this issue, and then use the regularity obtained from the entropy-entropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards $\rho_\lambda$. In the present paper, we do not provide any estimate of the rate of convergence, but we discuss how this would result from a stability result for a certain sharp Gagliardo-Nirenberg-Sobolev inequality.Comment: This version of the paper improves on the previous version by removing the small size condition on the value of the second Lyapunov functional of the initial data. The improved methodology makes greater use of techniques from optimal mass transportation, and so the second and third sections have changed places, and the current third section completely rewritte

Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system with $d\ge3$ and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass $M_c$ such that if $M \in (0,M_c]$ solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for $M \in (0,M_c)$. While characterising the eventual infinite time blowing-up profile for $M=M_c$, we observe that the long time asymptotics are much more complicated than in the classical Patlak-Keller-Segel system in dimension two

Convergence of the mass-transport steepest descent scheme for the sub-critical Patlak-Keller-Segel model

Variational steepest descent approximation schemes for the modified Patlak-Keller-Segel equation with a logarithmic interaction kernel in any dimension are considered. We prove the convergence of the suitably interpolated in time implicit Euler scheme, defined in terms of the Euclidean Wasserstein distance, associated to this equation for sub-critical masses. As a consequence, we recover the recent result about the global in time existence of weak-solutions to the modified Patlak-Keller-Segel equation for the logarithmic interaction kernel in any dimension in the sub-critical case. Moreover, we show how this method performs numerically in one dimension. In this particular case, this numerical scheme corresponds to a standard implicit Euler method for the pseudo-inverse of the cumulative distribution function. We demonstrate its capabilities to reproduce easily without the need of mesh-refinement the blow-up of solutions for super-critical masses

Ground States for Diffusion Dominated Free Energies with Logarithmic Interaction

Replacing linear diffusion by a degenerate diffusion of porous medium type is known to regularize the classical two-dimensional parabolic-elliptic Keller-Segel model. The implications of nonlinear diffusion are that solutions exist globally and are uniformly bounded in time. We analyse the stationary case showing the existence of a unique, up to translation, global minimizer of the associated free energy. Furthermore, we prove that this global minimizer is a radially decreasing compactly supported continuous density function which is smooth inside its support, and it is characterized as the unique compactly supported stationary state of the evolution model. This unique profile is the clear candidate to describe the long time asymptotics of the diffusion dominated classical Keller-Segel model for general initial data.Comment: 30 pages, 2 figure

Inclusion of persons with disabilities in systems of social protection: a population-based survey and case-control study in Peru.

OBJECTIVE: This study aims to assess the needs of people with disabilities and their level of inclusion in social protection programmes. DESIGN: Population based-survey with a nested case-control study. SETTING: Morropon, a semiurban district located in Piura, northern Peru. PARTICIPANTS: For the population survey, a two-stage sampling method was undertaken using data from the most updated census available and information of each household member aged ≥5 years was collected. In the nested case-control study, only one participant, case or control, per household was included in the study. PRIMARY AND SECONDARY OUTCOME MEASURES: Disability was screened using the Washington Group short questionnaire. A case, defined as an individual aged ≥5 years with disabilities, was matched with one control without disabilities by sex and age (±5 years). Information was collected on socioeconomic status, education, health and rehabilitation and social protection participation. RESULTS: The survey included 3684 participants, 1848 (50.1%) females, mean age: 36.4 (SD: 21.7). A total of 290 participants (7.9%; 95% CI 7.0% to 8.7%) were classified as having disability. Adults with disabilities were more likely to be single (OR=3.40; 95% CI 1.54 to 7.51) and not to be working (OR=4.36; 95% CI 2.26 to 8.40), while those who did work were less likely to receive the national minimum wage (ie, 750 PEN or about US$265; p=0.007). People with disabilities were more likely to experience health problems. There was no difference between those enrolled in any social protection programme among participants with and without disabilities. CONCLUSIONS: People with disabilities were found to have higher needs for social protection, but were not more likely to be enrolled in social protection programmes. The Peruvian social protection system should consider adding disability status to selection criteria in their cash transfer programmes as well as implementing disability-specific interventions

Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions

"Vegeu el resum a l'inici del document del fitxer adjunt"

Infinite time aggregation for the critical Patlak-Keller-Segel model in R2

We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R2. Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of "free-energy solutions", namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free-energy solutions with initial data as before for the critical mass 8 Π/Χ. Actually, we prove that solutions blow-up as a delta dirac at the center of mass when t→∞ keeping constant their second moment at any time. Furthermore, all moments larger than 2 blow-up as t→∞ if initially bounded

Measuring gravitational effects on antimatter in space

A direct measurement of the gravitational acceleration of antimatter has never been performed to date. Recently, such an experiment has been proposed, using antihydrogen with an atom interferometer and an antihydrogen confinament has been realized at CERN. In alternative we propose an experimental test of the gravitational interaction with antimatter by measuring the branching fraction of the CP violating decay of KL in space. In fact, even if the theoretical Standard Model explains the CPV with the presence of pure phase in the KMC Kobaiashi-Maskava-Cabibbo matrix, ample room is left for contributions by other interactions and forces to generate CPV in the mixing of the neutral K and B mesons. Gravitation is a good candidate and we show that at the altitude of the International Space Station, gravitational effects may change the level of CP violation such that a 5 sigma discrimination may be obtained by collecting the KL produced by the cosmic proton flux within a few years

Quantum Resonant Leptogenesis and Minimal Lepton Flavour Violation

It has been recently shown that the quantum Boltzmann equations may be relevant for the leptogenesis scenario. In particular, they lead to a time-dependent CP asymmetry which depends upon the previous dynamics of the system. This memory effect in the CP asymmetry is particularly important in resonant leptogenesis where the asymmetry is generated by the decays of nearly mass-degenerate right-handed neutrinos. We study the impact of the non-trivial time evolution of the CP asymmetry in the so-called Minimal Lepton Flavour Violation framework where the charged-lepton and the neutrino Yukawa couplings are the only irreducible sources of lepton-flavour symmetry breaking and resonant leptogenesis is achieved. We show that significant quantitative differences arise with respect to the case in which the time dependence of the CP asymmetry is neglected.Comment: 23 pages, 7 figure

From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity

This article reviews some aspects in the current relationship between mathematical and numerical General Relativity. Focus is placed on the description of isolated systems, with a particular emphasis on recent developments in the study of black holes. Ideas concerning asymptotic flatness, the initial value problem, the constraint equations, evolution formalisms, geometric inequalities and quasi-local black hole horizons are discussed on the light of the interaction between numerical and mathematical relativists.Comment: Topical review commissioned by Classical and Quantum Gravity. Discussion inspired by the workshop "From Geometry to Numerics" (Paris, 20-24 November, 2006), part of the "General Relativity Trimester" at the Institut Henri Poincare (Fall 2006). Comments and references added. Typos corrected. Submitted to Classical and Quantum Gravit