Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2507.13535 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918161113153536 |
|---|---|
| author | Holmes, John Massey, Katie Thompson, Ryan C. |
| author_facet | Holmes, John Massey, Katie Thompson, Ryan C. |
| contents | In this paper we study a new integrable fifth-order Camassa-Holm (CH)-type equation derived by Reyes, Zhu, and Qiao, which we call the RZQ equation. The m-form of this equation possesses a striking similarity to the m-form of the CH equation. However, unlike the CH equation, the nonlocal form of this equation cannot be interpreted as a nonlocal perturbation of Burgers' equation. We prove that the initial value problem corresponding to the RZQ equation is well-posed in the sense of Hadamard, in Sobolev spaces $H^s$, $s>7/2$. We further show that the data-to-solution map is not uniformly continuous in the $H^s$ topology, though it is Hölder continuous in a weaker topology. The initial value problem corresponding to the RZQ equation is ill-posed in $H^s$ for $s<7/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13535 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Cauchy problem for the integrable RZQ equation Holmes, John Massey, Katie Thompson, Ryan C. Analysis of PDEs 35Q35 In this paper we study a new integrable fifth-order Camassa-Holm (CH)-type equation derived by Reyes, Zhu, and Qiao, which we call the RZQ equation. The m-form of this equation possesses a striking similarity to the m-form of the CH equation. However, unlike the CH equation, the nonlocal form of this equation cannot be interpreted as a nonlocal perturbation of Burgers' equation. We prove that the initial value problem corresponding to the RZQ equation is well-posed in the sense of Hadamard, in Sobolev spaces $H^s$, $s>7/2$. We further show that the data-to-solution map is not uniformly continuous in the $H^s$ topology, though it is Hölder continuous in a weaker topology. The initial value problem corresponding to the RZQ equation is ill-posed in $H^s$ for $s<7/2$. |
| title | The Cauchy problem for the integrable RZQ equation |
| topic | Analysis of PDEs 35Q35 |
| url | https://arxiv.org/abs/2507.13535 |