On the axially symmetric solutions to the spatially homogeneous Landau equation
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912202047356928 |
|---|---|
| author | Jang, Jin Woo Kim, Junha |
| author_facet | Jang, Jin Woo Kim, Junha |
| contents | In this paper, we consider the spatially homogeneous Landau equation, which is a variation of the Boltzmann equation in the grazing collision limit. For the Landau equation for hard potentials in the style of Desvillettes-Villani (Comm. Partial Differential Equations, 2000), we provide the proof of the existence of axisymmetric measure-valued solution for any axisymmetric $\mathcal{P}_p(\mathbb{R}^3)$ initial profile for any $p\ge 2$. Moreover, we prove that if the initial data is not a single Dirac mass, then the solution instantaneously becomes analytic for any time $t>0$ in the hard potential case. In the soft potential and the Maxwellian molecule cases, we show that there are no solutions whose support is contained in a fixed line even for any given line-concentrated data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07393 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the axially symmetric solutions to the spatially homogeneous Landau equation Jang, Jin Woo Kim, Junha Analysis of PDEs 35Q20 In this paper, we consider the spatially homogeneous Landau equation, which is a variation of the Boltzmann equation in the grazing collision limit. For the Landau equation for hard potentials in the style of Desvillettes-Villani (Comm. Partial Differential Equations, 2000), we provide the proof of the existence of axisymmetric measure-valued solution for any axisymmetric $\mathcal{P}_p(\mathbb{R}^3)$ initial profile for any $p\ge 2$. Moreover, we prove that if the initial data is not a single Dirac mass, then the solution instantaneously becomes analytic for any time $t>0$ in the hard potential case. In the soft potential and the Maxwellian molecule cases, we show that there are no solutions whose support is contained in a fixed line even for any given line-concentrated data. |
| title | On the axially symmetric solutions to the spatially homogeneous Landau equation |
| topic | Analysis of PDEs 35Q20 |
| url | https://arxiv.org/abs/2501.07393 |