Sharp regularity of gradient blow-up solutions in the Camassa-Holm equation
| Autor: | Kim, Yunjoo, Kwon, Bongsuk, Yoon, Jeongsik |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 H\"older 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 $C^{3/5}$ H\"older 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 H\"older 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. Comment: 40 pages, 2 figures |
| Databáze: | arXiv |
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