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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.
We consider the spatially homogeneous Boltzmann equation for ballistic annihilation in dimension d 2. Such model describes a system of ballistic hard spheres that, at the moment of interaction, either annihilate with probability $alpha$ $in$ (0, 1) o
We show that solutions to Smoluchowskis equation with a constant coagulation kernel and an initial datum with some regularity and exponentially decaying tail converge exponentially fast to a self-similar profile. This convergence holds in a weighted
Continuous-time random walks are generalisations of random walks frequently used to account for the consistent observations that many molecules in living cells undergo anomalous diffusion, i.e. subdiffusion. Here, we describe the subdiffusive continu
We consider the self-similar fragmentation equation with a superquadratic fragmentation rate and provide a quantitative estimate of the spectral gap.
These notes are devoted to the problem of finite-dimensional reduction for parabolic PDEs. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mane projection