The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium

Abstract

Abstract. We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the 3-dimensional torus. The ultimate aim of this work is to obtain existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmannis a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an L 2 − L ∞ theoryà la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (e.g. Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the L 1 v L ∞ x framework is dealt with for any k > k 0 , recovering the optimal physical threshold of finite energy k 0 = 2 in the particular case of a multi-species hard spheres mixture with same masses.

Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions

“…Instead of trying to recover a full spectral gap property, our idea is to establish an explicit estimate for an upper bound of the entropy production functional associated to L ε . More precisely, we prove that the estimate known to exist for the Dirichlet form of L [16,10] remains valid for the Dirichlet form of L ε , up to a correction term of order ε. The strategy exploits the fact that the non-equilibrium state M ε is close to the global equilibrium µ µ µ up to a factor of order ε, and the same property holds for L ε and L. Specifically, we analyze the structure of the operator L ε − L, showing that it can be seen as the sum of a multiplicative operator and an integral one.…”

“…As already mentioned in the introduction, in order to study the Cauchy problem (5) and the convergence of the solution to equilibrium, it is common [17,6,16,10] to consider a perturbative regime where each distribution function F i is close to the global equilibrium given by (9). More precisely, when writing (5) can be rewritten in the form of the following perturbed multi-species Boltzmann system, set in…”

“…To mention a few of them, we can refer to [34,23,27] for the first kinetic models for mixtures, to [19,1,12] for some BGK-type models, and to [32,34,8,33,17] for multi-species kinetic models where also chemical reactions are taken into account. Moreover, we recall the recent works [6,16,10,9], where some of these models are mathematically investigated.…”

“…The study of explicit coercivity estimates for the linearized Boltzmann multi-species operator L(f ) = Q(µ µ µ, f ) + Q(f , µ µ µ) has been very recently tackled, and the existence of a spectral gap for this operator has been shown in [16] for mixtures where the atomic masses of the different species are the same (m i = m j for any 1 ≤ i, j ≤ N ), and in [10], in the general case of species with different atomic masses. Both results are valid for Maxwellian, hard potentials and hard spheres collision kernels with cutoff.…”

“…This feature induces additional difficulties if one wants to analyze the spectrum of the linearized operator L ε (f ) = Q(M ε , f ) + Q(f , M ε ), because L ε does not satisfy any self-adjointness property and it appears problematic to determine the space of equilibrium states Ker L ε . Consequently, we did not manage to give a detailed description of the spectrum of L ε and show explicit spectral gap estimates, using the same methods developed in [16,10].…”