Renyi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations

We investigate the large-time asymptotics of nonlinear diffusion equations $u_t = \Delta u^p$ in dimension $n \ge 1$, in the exponent interval $p > n/(n+2)$, when the initial datum $u_0$ is of bounded second moment. Precise rates of convergence to the Barenblatt profile in terms of the relative R\'enyi entropy are demonstrated for finite-mass solutions defined in the whole space when they are re-normalized at each time $t> 0$ with respect to their own second moment. The analysis shows that the relative R\'enyi entropy exhibits a better decay, for intermediate times, with respect to the standard Ralston-Newton entropy. The result follows by a suitable use of the so-called concavity of R\'enyi entropy power

Opinion modeling on social media and marketing aspects

We introduce and discuss kinetic models of opinion formation on social networks in which the distribution function depends on both the opinion and the connectivity of the agents. The opinion formation model is subsequently coupled with a kinetic model describing the spreading of popularity of a product on the web through a social network. Numerical experiments on the underlying kinetic models show a good qualitative agreement with some measured trends of hashtags on social media websites and illustrate how companies can take advantage of the network structure to obtain at best the advertisement of their products

Condensation phenomena in nonlinear drift equations

We study nonnegative, measure-valued solutions to nonlinear drift type equations modelling concentration phenomena related to Bose-Einstein particles. In one spatial dimension, we prove existence and uniqueness for measure solutions. Moreover, we prove that all solutions blow up in finite time leading to a concentration of mass only at the origin, and the concentrated mass absorbs increasingly the mass converging to the total mass as time goes to infinity. Our analysis makes a substantial use of independent variable scalings and pseudo-inverse functions techniques

First--order continuous models of opinion formation

We study certain nonlinear continuous models of opinion formation derived from a kinetic description involving exchange of opinion between individual agents. These models imply that the only possible final opinions are the extremal ones, and are similar to models of pure drift in magnetization. Both analytical and numerical methods allow to recover the final distribution of opinion between the two extremal ones.Comment: 17 pages, 4 figure

Decay rates for a class of diffusive-dominated interaction equations

We analyse qualitative properties of the solutions to a mean-field equation for particles interacting through a pairwise potential while diffusing by Brownian motion. Interaction and diffusion compete with each other depending on the character of the potential. We provide sufficient conditions on the relation between the interaction potential and the initial data for diffusion to be the dominant term. We give decay rates of Sobolev norms showing that asymptotically for large times the behavior is then given by the heat equation. Moreover, we show an optimal rate of convergence in the $L^1$-norm towards the fundamental solution of the heat equation.Comment: 22 page

Self-similarity and power-like tails in nonconservative kinetic models

In this paper, we discuss the large--time behavior of solution of a simple kinetic model of Boltzmann--Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the nonlinearity and of the time--monotonicity of the temperature, the kinetic model has non trivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution. The same idea is applied to investigate the large-time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and/or decreasing of the mean wealth. In this last case, the large-time behavior of the solution shows a Pareto power law tail. Numerical results confirm the previous analysis

Tracking power system events with accuracy-based PMU adaptive reporting rate

Fast dynamics and transient events are becoming more and more frequent in power systems, due to the high penetration of renewable energy sources and the consequent lack of inertia. In this scenario, Phasor Measurement Units (PMUs) are expected to track the monitored quantities. Such functionality is related not only to the PMU accuracy (as per the IEC/IEEE 60255-118-1 standard) but also to the PMU reporting rate (RR). High RRs allow tracking fast dynamics, but produce many redundant measurement data in normal conditions. In view of an effective tradeoff, the present paper proposes an adaptive RR mechanism based on a real-time selection of the measurements, with the target of preserving the information content while reducing the data rate. The proposed method has been tested considering real-world datasets and applied to four different PMU algorithms. The results prove the method effectiveness in reducing the average data throughput as well as its scalability at PMU concentrator or storage level

On a kinetic model for a simple market economy

In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasi-stationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis

Strong Convergence towards self-similarity for one-dimensional dissipative Maxwell models

We prove the propagation of regularity, uniformly in time, for the scaled solutions of one-dimensional dissipative Maxwell models. This result together with the weak convergence towards the stationary state proven by Pareschi and Toscani in 2006 implies the strong convergence in Sobolev norms and in the L^1 norm towards it depending on the regularity of the initial data. In the case of the one-dimensional inelastic Boltzmann equation, the result does not depend of the degree of inelasticity. This generalizes a recent result of Carlen, Carrillo and Carvalho (arXiv:0805.1051v1), in which, for weak inelasticity, propagation of regularity for the scaled inelastic Boltzmann equation was found by means of a precise control of the growth of the Fisher information.Comment: 26 page