Regularized Boltzmann operators

In this paper, we propose two regularization approaches for the Boltzmann collision operator. The constructed operators preserve the mass, momentum and energy; their equilibrium states are Maxwellians and they satisfy the H-theorem. In the first approach, the regularization consists in allowing microscopic collisions which do not exactly preserve energy and momentum. However, the limit of the mollified operator when the cut-off parameter tends to 0 is not the usual Boltzmann operator unless a certain condition on the distribution function is satisfied. In the second approach, the regularization relies on a smoothing of the masses of the particles and leads to a regularized operator which formally tends to the Boltzmann operator for any arbitrary distribution function, when the cut-off parameter tends to zero.