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It is well known that the analysis of the large-time asymptotics of Fokker-Planck type equations by the entropy method is closely related to proving the validity of convex Sobolev inequalities. Here we highlight this connection from an applied PDE point of view. In our unified presentation of the theory we present new results to the following topics: an… (More)

We consider the flow of gas in an N-dimensional porous medium with initial density v0(x) ≥ 0. The density v(x, t) then satisfies the nonlinear degenerate parabolic equation vt = ∆vm where m > 1 is a physical constant. Assuming that ∫ (1 + |x|2)v0(x)dx < ∞, we prove that v(x, t) behaves asymptotically, as t → ∞, like the Barenblatt-Pattle solution V(|x|, t).… (More)

- E. GABETTA, G. TOSCANI
- 1997

A class of numerical schemes for nonlinear kinetic equations of Boltzmann type is described. Following Wild’s approach, the solution is represented as a power series with parameter depending exponentially on the Knudsen number. This permits us to derive accurate and stable time discretizations for all ranges of the mean free path. These schemes preserve the… (More)

- Giuseppe Toscani
- 2008

We introduce and discuss certain kinetic models of (continuous) opinion formation involving both exchange of opinion between individual agents and diffusion of information. We show conditions which ensure that the kinetic model reaches non trivial stationary states in case of lack of diffusion in correspondence of some opinion point. Analytical results are… (More)

- G. TOSCANI, C. VILLANI
- 1998

- J. A. Carrillo, G. Toscani
- 2002

In this paper we investigate the large-time behavior of strong solutions to the one-dimensional fourth order degenerate parabolic equation ut = −(uuxxx)x , modeling the evolution of the interface of a spreading droplet. For nonnegative initial values u0(x) ∈ H 1(IR), both compactly supported or of finite second moment, we prove explicit and universal… (More)

- J. A. Carrilloy, G. Toscani, A. Unterreiterk
- 1999

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with connnement by a uniformly convex potential, 2) unconnned scalar equations and 3) unconnned systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is… (More)

- Shi Jin, Lorenzo Pareschi, Giuseppe Toscani
- SIAM J. Numerical Analysis
- 2000

Many transport equations, such as the neutron transport, radiative transfer, and transport equations for waves in random media, have a diiusive scaling that leads to the diiusion equations. In many physical applications, the scaling parameter (mean free path) may diier in several orders of magnitude from the rareeed regimes to the hydrodynamic (diiusive)… (More)

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… (More)

- José A. Carrillo, M. Fornasier, Jesús Rosado, Giuseppe Toscani
- SIAM J. Math. Analysis
- 2010

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)