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In this paper we investigate the large-time behavior of strong solutions to the one-dimensional fourth order degenerate parabolic equation u t = −(uu xxx) x , mode-ling the evolution of the interface of a spreading droplet. For nonnegative initial values u 0 (x) ∈ H 1 (IR), both compactly supported or of finite second moment, we prove explicit and universal(More)
We review the state-of-the-art in the modelling of the aggregation and collective behavior of interacting agents of similar size and body type, typically called swarming. Starting with individual-based models based on " particle "-like assumptions, we connect to hydrody-namic/macroscopic descriptions of collective motion via kinetic theory. We emphasize the(More)
In this paper, we analyse the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [16], which describes the collective behavior of an ensemble of organisms, animals or devices. This kinetic version introduced in [24] is here obtained starting from a Boltzmann-type equation. The large-time behavior of the(More)
We quantify the long-time behavior of a system of (partially) inelas-tic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures. Existence, uniqueness, boundedness of moments and regularity of a steady state are derived from this basic property. The solutions of the kinetic model are proved(More)
We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with con®nement by a uniformly convex potential, 2) uncon®ned scalar equations and 3) uncon®ned systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is(More)
We introduce and discuss Grad's moment equations for dilute granular systems of hard spheres with dissipative collisions and variable coefficient of restitution, under the assumption of weak inelasticity. An important byproduct is that in this way we obtain an hydrodynamic description of a system of nearly elastic particles by a direct procedure from the(More)
We construct self-similar solutions of the Dafermos regularization of a system of conservation laws near structurally stable Riemann solutions composed of Lax shocks and rarefactions, with all waves possibly large. The construction requires blowing up a manifold of gain-of-stability turning points in a geometric singular perturbation problem, as well as a(More)
In this paper we investigate the large-time behavior of solutions to the first initial-boundary value problem for the non-linear diffusion u t = (u m) xx , m > 0. In particular, we prove exponential decay of u(x, t) towards its own steady state in L 1-norm for long times and we give an explicit upper bound for the rate of decay. The result is based on a new(More)
We prove that the reciprocal of Fisher information of a log-concave probability density X in R n is concave in t with respect to the addition of a Gaussian noise Z t = N (0, tI n). As a byproduct of this result we show that the third derivative of the entropy power of a log-concave probability density X in R n is nonnegative in t with respect to the(More)