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
Sobolev norm which implies the L^2 convergence of derivatives up to a certain order k depending on the regularity of the initial condition. We prove these results through the study of the linearized coagulation equation in self-similar variables, for which we show a spectral gap in a scale of weighted Sobolev spaces. We also take advantage of the fact that the Laplace or Fourier transforms of this equation can be explicitly solved in this case.
This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffus
ion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation.
We consider a space-homogeneous gas of {it inelastic hard spheres}, with a {it diffusive term} representing a random background forcing (in the framework of so-called {em constant normal restitution coefficients} $alpha in [0,1]$ for the inelasticity
). In the physical regime of a small inelasticity (that is $alpha in [alpha_*,1)$ for some constructive $alpha_* in [0,1)$) we prove uniqueness of the stationary solution for given values of the restitution coefficient $alpha in [alpha_*,1)$, the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solution.
