115 research outputs found
On Rosenau-Type Approximations to Fractional Diffusion Equations
Owing to the Rosenau argument in Physical Review A, 46 (1992), pag. 12-15,
originally proposed to obtain a regularized version of the Chapman-Enskog
expansion of hydrodynamics, we introduce a non-local linear kinetic equation
which approximates a fractional diffusion equation. We then show that the
solution to this approximation, apart of a rapidly vanishing in time
perturbation, approaches the fundamental solution of the fractional diffusion
(a L\'evy stable law) at large times
One-Dimensional Fokker-Planck Equations and Functional Inequalities for Heavy Tailed Densities
We present and discuss connections between the problem of trend to equilibrium for one-dimensional Fokker-Planck equations modeling socio-economic problems, and one-dimensional functional inequalities of the type of Poincare, Wirtinger and logarithmic Sobolev, with weight, for probability densities with polynomial tails. As main examples, we consider inequalities satisfied by inverse Gamma densities, taking values on R+, and Cauchy-type densities, taking values on R
On unconditional well-posedness of modified KdV
Bourgain(1993) proved that the periodic modified KdV equation (mKdV) is
locally well-posed in Sobolev spave H^s(T), s >= 1/2, by introducing new
weighted Sobolev spaces X^s,b, where the uniqueness holds conditionally, namely
in the intersection of C([0, T]; H^s) and X^s,b. In this paper, we establish
unconditional well-posedness of mKdV in H^s(T), s >= 1/2, i.e. we in addition
establish unconditional uniqueness in C([0, T]; H^s), s >= 1/2, of solutions to
mKdV. We prove this result via differentiation by parts. For the endpoint case
s = 1/2, we perform careful quinti- and septi-linear estimates after the second
differentiation by parts.Comment: 18 pages, small changes in Section 1. (Remark 1.2 added), to appear
in Int. Math. Res. No
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
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