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

Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming

We perform an asymptotic analysis of general particle systems arising in collective behavior in the limit of large self-propulsion and friction forces. These asymptotics impose a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity variables. The limit models are obtained by averaging with respect to the fast dynamics. We can include all typical effects in the applications: short-range repulsion, long-range attraction, and alignment. For instance, we can rigorously show that the Cucker-Smale model is reduced to the Vicsek model without noise in this asymptotic limit. Finally, a formal expansion based on the reduced dynamics allows us to treat the case of diffusion. This technique follows closely the gyroaverage method used when studying the magnetic confinement of charged particles. The main new mathematical difficulty is to deal with measure solutions in this expansion procedure

Nonlinear stability of flock solutions in second-order swarming models

In this paper we consider interacting particle systems which are frequently used to model collective behavior in animal swarms and other applications. We study the stability of orientationally aligned formations called flock solutions, one of the typical patterns emerging from such dynamics. We provide an analysis showing that the nonlinear stability of flocks in second-order models entirely depends on the linear stability of the first-order aggregation equation. Flocks are shown to be nonlinearly stable as a family of states under reasonable assumptions on the interaction potential. Furthermore, we numerically verify that commonly used potentials satisfy these hypotheses and investigate the nonlinear stability of flocks by an extensive case-study of uniform perturbations.Comment: 22 pages, 1 figure, 1 tabl