The Boltzmann equation in the homogeneous critical regularity framework
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909687461445632 |
|---|---|
| author | Liu, Jing Shou, Ling-Yun Xu, Jiang |
| author_facet | Liu, Jing Shou, Ling-Yun Xu, Jiang |
| contents | We construct a unique global solution to the Cauchy problem of the 3D Boltzmann equation for initial data around the Maxwellian in the spatially critical homogeneous Besov space $\widetilde{L}^2_ξ(\dot{B}_{2,1}^{1/2}\cap\dot{B}_{2,1}^{3/2})$. In addition, under the condition that the low-frequency part of initial perturbation is bounded in $\widetilde{L}^2_ξ(\dot{B}_{2,\infty}^{σ_{0}})$ with $-3/2\leqσ_{0}<1/2$, it is shown that the solution converges to its equilibrium in large times with the optimal rate of $\mathcal{O}(t^{-(σ-σ_{0})/2})$ in $\widetilde{L}^2_ξ(\dot{B}_{2,1}^σ)$ with some $σ>σ_0$, and the microscopic part decays at an enhanced rate of $\mathcal{O}(t^{-(σ-σ_{0})/2-1/2})$. In contrast to [19], the usual $L^2$ estimates are not necessary in our approach, which provides a new understanding of hypocoercivity theory for the Boltzmann equation allowing to construct the Lyapunov functional with different dissipation rates at low and high frequencies. Furthermore, a time-weighted Lyapunov energy argument can be developed to deduce the optimal time-decay estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_13610 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Boltzmann equation in the homogeneous critical regularity framework Liu, Jing Shou, Ling-Yun Xu, Jiang Analysis of PDEs 35Q20, 76N10 We construct a unique global solution to the Cauchy problem of the 3D Boltzmann equation for initial data around the Maxwellian in the spatially critical homogeneous Besov space $\widetilde{L}^2_ξ(\dot{B}_{2,1}^{1/2}\cap\dot{B}_{2,1}^{3/2})$. In addition, under the condition that the low-frequency part of initial perturbation is bounded in $\widetilde{L}^2_ξ(\dot{B}_{2,\infty}^{σ_{0}})$ with $-3/2\leqσ_{0}<1/2$, it is shown that the solution converges to its equilibrium in large times with the optimal rate of $\mathcal{O}(t^{-(σ-σ_{0})/2})$ in $\widetilde{L}^2_ξ(\dot{B}_{2,1}^σ)$ with some $σ>σ_0$, and the microscopic part decays at an enhanced rate of $\mathcal{O}(t^{-(σ-σ_{0})/2-1/2})$. In contrast to [19], the usual $L^2$ estimates are not necessary in our approach, which provides a new understanding of hypocoercivity theory for the Boltzmann equation allowing to construct the Lyapunov functional with different dissipation rates at low and high frequencies. Furthermore, a time-weighted Lyapunov energy argument can be developed to deduce the optimal time-decay estimates. |
| title | The Boltzmann equation in the homogeneous critical regularity framework |
| topic | Analysis of PDEs 35Q20, 76N10 |
| url | https://arxiv.org/abs/2408.13610 |