14,407 research outputs found
Renyi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations
We investigate the large-time asymptotics of nonlinear diffusion equations
in dimension , in the exponent interval , when the initial datum 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 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
Existence of Compactly Supported Global Minimisers for the Interaction Energy
The existence of compactly supported global minimisers for continuum models
of particles interacting through a potential is shown under almost optimal
hypotheses. The main assumption on the potential is that it is catastrophic, or
not H-stable, which is the complementary assumption to that in classical
results on thermodynamic limits in statistical mechanics. The proof is based on
a uniform control on the local mass around each point of the support of a
global minimiser, together with an estimate on the size of the "gaps" it may
have. The class of potentials for which we prove existence of global minimisers
includes power-law potentials and, for some range of parameters, Morse
potentials, widely used in applications. We also show that the support of local
minimisers is compact under suitable assumptions.Comment: Final version after referee reports taken into accoun
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