Cutoff Boltzmann equation with polynomial perturbation near Maxwellian

Chuqi Cao Yau Mathematical Science Center, Tsinghua University

Analysis of PDEs mathscidoc:2211.03002

Journal of Functional Analysis, 2022.11
In this paper, we consider the cutoff Boltzmann equation near Maxwellian, we proved the global existence and uniqueness for the cutoff Boltzmann equation in polynomial weighted space for all $\gamma \in (-3, 1]$. We also proved initially polynomial decay for the large velocity in $L^2$ space will induce polynomial decay rate, while initially exponential decay will induce exponential rate for the convergence. Our proof is based on newly established inequalities for the cutoff Boltzmann equation and semigroup techniques. Moreover, by generalizing the $L_x^\infty L^1_v \cap L^\infty_{x, v}$ approach, we prove the global existence and uniqueness of a mild solution to the Boltzmann equation with bounded polynomial weighted $L^\infty_{x, v}$ norm under some small condition on the initial $L^1_x L^\infty_v$ norm and entropy so that this initial data allows large amplitude oscillations.
Boltzmann equation; Global existence; Polynomial weighted space; Convergence to equilibrium.
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@inproceedings{chuqi2022cutoff,
  title={Cutoff Boltzmann equation with polynomial perturbation near Maxwellian},
  author={Chuqi Cao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20221108222612012120737},
  booktitle={Journal of Functional Analysis},
  year={2022},
}
Chuqi Cao. Cutoff Boltzmann equation with polynomial perturbation near Maxwellian. 2022. In Journal of Functional Analysis. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20221108222612012120737.
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