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 the one-dimensional dissipative Maxwell models introduced in [D. Ben-Avraham, E. Ben-Naim, K. Lindenberg, A. Rosas, Selfsimilarity in random collision processes, Phys. Rev. E 68 (2003) R050103]. This result together with the weak convergence towards the stationary state proven in [L. Pareschi, G. Toscani, Self-similarity and powerlike tails in nonconservative kinetic models, J. Stat. Phys. 124 (2–4) (2006) 747–779] implies the strong convergence in Sobolev norms and in the L1 norm towards it depending on the regularity of the initial
data. As a consequence, the original nonscaled solutions are also proved to be convergent in L1 towards the corresponding self-similar homogeneous cooling state. The proof is based on the (uniform in time) control of the tails of the Fourier transform of the solution, and it holds for a large range of values of the mixing
parameters. In particular, in the case of the one-dimensional inelastic Boltzmann equation, the result does not depend of the degree of inelasticity.