The zero-dispersion limit for the Benjamin--Ono equation on the circle
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918365117808640 |
|---|---|
| author | Mæhlen, Ola |
| author_facet | Mæhlen, Ola |
| contents | Using the explicit formula of P. Gérard, we characterize the zero-dispersion limit for solutions of the Benjamin--Ono equation on the circle $\mathbb{T}= \mathbb{R}/2π\mathbb{Z}$ with bounded initial data $u_0\in L^\infty(\mathbb{T},\mathbb{R})$. The result generalizes the work of L. Gassot, who focused on periodic bell-shaped data, and complements the work of Gérard and X. Chen who identified the zero-dispersion limit on the line with $u_0\in L^2\cap L^\infty(\mathbb{R})$. Here, as well as in the mentioned cases, the characterization agrees with the one first obtained by Miller--Xu for bell-shaped data on the line: The zero-dispersion limit is given as an alternating sum of the branches of the multivalued solution of Burgers' equation. From this characterization, we compute regularity properties of the zero-dispersion limit, including maximum principles and an Oleinik estimate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14134 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The zero-dispersion limit for the Benjamin--Ono equation on the circle Mæhlen, Ola Analysis of PDEs 35B40, 35Q53, 37K15, 47B35 Using the explicit formula of P. Gérard, we characterize the zero-dispersion limit for solutions of the Benjamin--Ono equation on the circle $\mathbb{T}= \mathbb{R}/2π\mathbb{Z}$ with bounded initial data $u_0\in L^\infty(\mathbb{T},\mathbb{R})$. The result generalizes the work of L. Gassot, who focused on periodic bell-shaped data, and complements the work of Gérard and X. Chen who identified the zero-dispersion limit on the line with $u_0\in L^2\cap L^\infty(\mathbb{R})$. Here, as well as in the mentioned cases, the characterization agrees with the one first obtained by Miller--Xu for bell-shaped data on the line: The zero-dispersion limit is given as an alternating sum of the branches of the multivalued solution of Burgers' equation. From this characterization, we compute regularity properties of the zero-dispersion limit, including maximum principles and an Oleinik estimate. |
| title | The zero-dispersion limit for the Benjamin--Ono equation on the circle |
| topic | Analysis of PDEs 35B40, 35Q53, 37K15, 47B35 |
| url | https://arxiv.org/abs/2509.14134 |