[en] We prove the stability and existence of global solutions of Boltzmann equations under general assumptions on the collision operators and the initial conditions. In particular, we introduce a new formulation of the equation and we observe some weak continuity properties of the collision operator

[en] For the simplest of the discrete models of the Boltzmann equation: the Broadwell model, exact solutions have been obtained by Cornille in the form of bisolitons. In the present Note, we build exact solutions for more complex models

No abstract available

[en] We have constructed 2+1 dimensional exact solutions (space x, y time t) for three models: i) the square velocity model and the cubic model (with eight velocities oriented towards the eight corners) which satisfy the same type of nonlinear equations, ii) the Broadwell model. The main difficulty is the positivity condition N_i>0 for the densities N_i. We have obtained three classes of exact solutions: i) densities relaxing towards noniform Maxwellians ii) shock wave iii) semi-periodic solutions

[en] A new and relativistic formulation of Boltzmann's equation is presented

[en] We construct a model of discrete Boltzmann distribution with 2n velocities. In certain cases, the exact solutions we can calculate enable us to resolve the Cauchy's problem and obtain global solutions

No abstract available

[en] We review recent results obtained for inhomogeneous distributions, solutions of d > 1 dimensional non linear Boltzmann equations with both Maxwell particles intermolecular forces and outside forces. We find exact solutions with a structure similar to the homogeneous Bobylev-Krook-Wu distributions. However, on the one hand, unlike the Bobylev inhomogeneous solution which corresponds to a gas in expansion, here these solutions can relax towards absolute Maxwellians. On the other hand, we construct also a class of exact inhomogeneous distributions relaxing towards different oscillating Maxwellians. Two types of external forces are handled: linear spatial forces (mainly harmonic potentials) and linear velocity forces plus uniform source term

[en] We consider a four-velocity discrete and unidimensional Boltzmann model. The mass, momentum and energy conservation laws being satisfied we can define a temperature. We report the exact positive solutions which have been found: periodic in the space and propagating or not when the time is growing, shock waves similarity solutions and (1 + 1)-dimensional solutions

[en] The case of multiple collisions in the discrete Boltzmann equation is considered. We establish sufficient conditions to be verified by the discrete models of the Boltzmann equation, so that the one-dimensional initial value problem possesses a global solution in time, under the constraint that the initial densities and initial mass are positive and bounded