Uniqueness and stability of a classical solution to a non cut off Boltzmann equation

An initial value problem for a classical spatially homogeneous Boltzmann equation describes the time evolution of a distribution density for a mono atomic dilute gas of particles, independent of the spatial variable. Existence of a solution to such kind of initial value problem with non cut off collision kernel in the weighted Lebesgue space was proved by L. Arkeryd in 1981 for soft and hard potentials and by C. Villani in 1998 for soft and very soft potentials. These results impose on an initial data the finite entropy and the finite energy conditions. In the recent paper R. Duduchava, R. Kirsch S. Rjasanow 2005 proved existence and uniqueness of a local solution dropping the finite entropy and the finite energy (dropping even the finite impulse) condition. We will report on the uniqueness and stability of a classical solution to Boltzmann equations with non cut off kernels, soft and hard potentials, without the finite entropy constraint. Results on stability known before G.D. Blasio 1974, T. Gustafsson 1988, L. Arkeryd 1988 and B. Wennberg 1994, considered a cut off case only and imposed more constraints on the initial data.