Sharp regularity of gradient blow-up solutions in the Camassa-Holm equation
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_ | 1866909410513649664 |
|---|---|
| author | Kim, Yunjoo Kwon, Bongsuk Yoon, Jeongsik |
| author_facet | Kim, Yunjoo Kwon, Bongsuk Yoon, Jeongsik |
| contents | We study the formation of singularities in the Camassa-Holm (CH) equation, providing a detailed description of the blow-up dynamics and identifying the precise Hölder regularity of the gradient blow-up solutions. To this end, we first construct self-similar blow-up profiles and examine their properties, including the asymptotic behavior at infinity, which determines the type of singularity. Using these profiles as a reference and employing modulation theory, we establish global pointwise estimates for the blow-up solutions in self-similar variables, thereby demonstrating the stability of the self-similar profiles we construct. Our results indicate that the solutions, evolving from smooth initial data within a fairly general open set, form $C^{3/5}$ cusps as the first singularity in finite time. These singularities are analogous to \emph{pre-shocks} emerging in the Burgers equation, exhibiting unbounded gradients while the solutions themselves remain bounded. However, the nature of the singularity differs from that of the Burgers equation, which is a cubic-root singularity, i.e., $C^{1/3}$. Our work provides the first proof that generic pre-shocks of the CH equation exhibit $C^{3/5}$ Hölder regularity. It is our construction of new self-similar profiles, incorporating the precise leading-order correction to the CH equation in the blow-up regime, that enables us to identify the sharp Hölder regularity of the singularities and to capture the detailed spatial and temporal dynamics of the blow-up. We also show that the generic singularities developed in the Hunter-Saxton (HS) equation are of the same type as those in the CH equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00558 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp regularity of gradient blow-up solutions in the Camassa-Holm equation Kim, Yunjoo Kwon, Bongsuk Yoon, Jeongsik Analysis of PDEs 35Q35, 35A21, 76B15, 35C06, 76B25 We study the formation of singularities in the Camassa-Holm (CH) equation, providing a detailed description of the blow-up dynamics and identifying the precise Hölder regularity of the gradient blow-up solutions. To this end, we first construct self-similar blow-up profiles and examine their properties, including the asymptotic behavior at infinity, which determines the type of singularity. Using these profiles as a reference and employing modulation theory, we establish global pointwise estimates for the blow-up solutions in self-similar variables, thereby demonstrating the stability of the self-similar profiles we construct. Our results indicate that the solutions, evolving from smooth initial data within a fairly general open set, form $C^{3/5}$ cusps as the first singularity in finite time. These singularities are analogous to \emph{pre-shocks} emerging in the Burgers equation, exhibiting unbounded gradients while the solutions themselves remain bounded. However, the nature of the singularity differs from that of the Burgers equation, which is a cubic-root singularity, i.e., $C^{1/3}$. Our work provides the first proof that generic pre-shocks of the CH equation exhibit $C^{3/5}$ Hölder regularity. It is our construction of new self-similar profiles, incorporating the precise leading-order correction to the CH equation in the blow-up regime, that enables us to identify the sharp Hölder regularity of the singularities and to capture the detailed spatial and temporal dynamics of the blow-up. We also show that the generic singularities developed in the Hunter-Saxton (HS) equation are of the same type as those in the CH equation. |
| title | Sharp regularity of gradient blow-up solutions in the Camassa-Holm equation |
| topic | Analysis of PDEs 35Q35, 35A21, 76B15, 35C06, 76B25 |
| url | https://arxiv.org/abs/2412.00558 |