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Regularized Boltzmann operators. (English) Zbl 0910.76073

Summary: 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 operators 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 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.

MSC:

76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
45K05 Integro-partial differential equations
82B40 Kinetic theory of gases in equilibrium statistical mechanics
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