Polynomial tail solutions of the non-cutoff Boltzmann equation near local Maxwellians
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929417678225408 |
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| author | Duan, Renjun Li, Zongguang |
| author_facet | Duan, Renjun Li, Zongguang |
| contents | This paper aims to incorporate the Caflisch's decomposition into the macro-micro decomposition in Boltzmann theory for allowing the microscopic component to exhibit only the polynomial tail in large velocities. In particular, we treat the Cauchy problem on the non-cutoff Boltzmann equation under the compressible Euler scaling in case of three-dimensional whole space. Up to a finite time we construct the Boltzmann solution around a local Maxwellian corresponding to small-amplitude classical solutions of the full compressible Euler system around constant states. We design a new energy functional which can capture the convergence rate in the small Knudsen number $\varepsilon$ and allow the microscopic part of solutions to decay polynomially in large velocities. Moreover, the energy norm of perturbations can be of the order $\varepsilon^{1/2}$ which the usual method of Hilbert expansion fails to obtain. As a byproduct of the proof, our estimates immediately yield a global-in-time existence result when the Euler solutions are taken to be constant states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial tail solutions of the non-cutoff Boltzmann equation near local Maxwellians Duan, Renjun Li, Zongguang Analysis of PDEs This paper aims to incorporate the Caflisch's decomposition into the macro-micro decomposition in Boltzmann theory for allowing the microscopic component to exhibit only the polynomial tail in large velocities. In particular, we treat the Cauchy problem on the non-cutoff Boltzmann equation under the compressible Euler scaling in case of three-dimensional whole space. Up to a finite time we construct the Boltzmann solution around a local Maxwellian corresponding to small-amplitude classical solutions of the full compressible Euler system around constant states. We design a new energy functional which can capture the convergence rate in the small Knudsen number $\varepsilon$ and allow the microscopic part of solutions to decay polynomially in large velocities. Moreover, the energy norm of perturbations can be of the order $\varepsilon^{1/2}$ which the usual method of Hilbert expansion fails to obtain. As a byproduct of the proof, our estimates immediately yield a global-in-time existence result when the Euler solutions are taken to be constant states. |
| title | Polynomial tail solutions of the non-cutoff Boltzmann equation near local Maxwellians |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.08346 |