Singular, finite-time $L^2$ attractors for odd, smooth solutions of Burgers equation on the torus
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909899594661888 |
|---|---|
| author | Miller, Evan |
| author_facet | Miller, Evan |
| contents | In this paper, we show that the positive multiples of a particular function $F$ -- which is singular with a jump discontinuity at the origin -- are finite-time global attractors in $L^2$ for generic odd, smooth solutions of the one dimensional inviscid Burgers equation. Furthermore, the identity that leads to this result provides to an alternative proof of finite-time blowup for the fractal Burgers equation in the supercritical range $0<α<\frac{1}{2}$. This proof is based on lower bounds on a Lyapunov functional given by the inner product of the solution with the global attractor $F$. We will also show that this property holds for a broader class of odd functions that are strictly increasing on $(0,π)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09528 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singular, finite-time $L^2$ attractors for odd, smooth solutions of Burgers equation on the torus Miller, Evan Analysis of PDEs 35B44, 35Q53 In this paper, we show that the positive multiples of a particular function $F$ -- which is singular with a jump discontinuity at the origin -- are finite-time global attractors in $L^2$ for generic odd, smooth solutions of the one dimensional inviscid Burgers equation. Furthermore, the identity that leads to this result provides to an alternative proof of finite-time blowup for the fractal Burgers equation in the supercritical range $0<α<\frac{1}{2}$. This proof is based on lower bounds on a Lyapunov functional given by the inner product of the solution with the global attractor $F$. We will also show that this property holds for a broader class of odd functions that are strictly increasing on $(0,π)$. |
| title | Singular, finite-time $L^2$ attractors for odd, smooth solutions of Burgers equation on the torus |
| topic | Analysis of PDEs 35B44, 35Q53 |
| url | https://arxiv.org/abs/2511.09528 |