Compactness of Linearized Kinetic Operators

Abstract

International audienceThis article reviews various results on the compactness of the linearized Boltzmann operator and of its generalization to mixtures of non-reactive monatomic gases

The Maxwell–Stefan diffusion limit for a kinetic model of mixtures with general cross sections

“…use (6) and (8) again, and the fact that Θ is an isometry. Note that, in both cases, the Jacobian of the changes of variables is 1.…”

“…The linearized Boltzmann operator L can then be dened as in [5,8], so that L can be considered as an operator L 2 (R 3 ) I → L 2 (R 3 ) I with the standard Lebesgue product measure (without any weight). For any i, the i th component of Lg writes…”

On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

“…Besides this, in [10] a new multi-species Carleman's representation and a new Povznertype inequality was proved, due to the loss of symmetry arisen from different masses. In [6,7], compactness of one part of the linearized multi-species operator was studied, moreover, in [4] it was shown that in the diffusive limit, the multi-species Boltzmann equation converges to the Maxwell-Stefan system. In [2], the Chapman-Enskog asymptotics for a mixture of gases was presented.…”

“…We consider the multi-species Boltzmann equation describing the evolution of a multi-species mono-atomic nonreactive gaseous mixture with additional uncertainty coming from the initial data and collision kernel, which was studied analytically in the deterministic setting in [1,3,5,7,8,10,14]. Compared to the single-species deterministic analysis of the Boltzmann equation, dealing with different conserved quantities due to different thermodynamic properties of mixtures (see the multispecies H-theorem in [16,20]) provided the main difficulty in the analysis for the multi-species deterministic problem.…”

On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

“…We consider the multi-species Boltzmann equation describing the evolution of a multi-species mono-atomic nonreactive gaseous mixture with additional uncertainty coming from the initial data and collision kernel, which was studied analytically in the deterministic setting in [1,3,5,7,8,10,14]. Compared to the single-species deterministic analysis of the Boltzmann equation, dealing with different conserved quantities due to different thermodynamic properties of mixtures (see the multispecies H-theorem in [16,20]) provided the main difficulty in the analysis for the multi-species deterministic problem.…”