English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the Cauchy theory for a multi-species mixture, where the differentspecies 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 themulti-species Boltzmann equation in $L^1_vL^\infty_x(m)$, where $m\sim(1+|v|^k)$ is a polynomial weight. We prove the existence of a spectral gap forthe linear multi-species Boltzmann operator allowing different masses, and thenwe establish a semigroup property thanks to a new explicit coercive estimatefor the Boltzmann operator. Then we develop an $L^2-L^\infty$ theory\textit{\`a la Guo} for the linear perturbed equation. Finally, we combine thelatter results with a decomposition of the multi-species Boltzmann equation inorder to deal with the full equation. We emphasize that dealing with differentmasses induces a loss of symmetry in the Boltzmann operator which prevents thedirect adaptation of standard mono-species methods (\textit{e.g.} Carlemanrepresentation, Povzner inequality). Of important note is the fact that allmethods used and developed in this work are constructive. Moreover, they do notrequire any Sobolev regularity and the $L^1_vL^\infty_x$ framework is dealtwith for any $k>k_0$, recovering the optimal physical threshold of finiteenergy $k_0=2$ in the particular case of a multi-species hard spheres mixturewith same masses.

The Boltzmann equation for a multi-species mixture close to global equilibrium