Concentric bubbles concentrating in finite time for the energy critical wave maps equation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866929676476219392 |
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| author | Jendrej, Jacek Krieger, Joachim |
| author_facet | Jendrej, Jacek Krieger, Joachim |
| contents | We show that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ to $\mathbb{S}^2$ and restricted to the co-rotational setting with co-rotation index $k = 2$ admits finite time blow up solutions of finite energy on $(0, t_0]\times \mathbb{R}^2$, $t_0>0$, and concentrating two concentric bubble profiles at the frequency scales $λ_1(t) = e^{α(t)},\,α(t)\sim \big|\log t\big|^{β+1}$, as well as $λ_2(t) = t^{-1}\cdot \big|\log t\big|^β$. The parameter $β>\frac32$ can be chosen arbitrarily. This shows that soliton resolution scenarios with finite time blow up and $N = 2$ collapsing profiles, i. e. bubble trees, do occur for this equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Concentric bubbles concentrating in finite time for the energy critical wave maps equation Jendrej, Jacek Krieger, Joachim Analysis of PDEs 35L05, 35B40 We show that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ to $\mathbb{S}^2$ and restricted to the co-rotational setting with co-rotation index $k = 2$ admits finite time blow up solutions of finite energy on $(0, t_0]\times \mathbb{R}^2$, $t_0>0$, and concentrating two concentric bubble profiles at the frequency scales $λ_1(t) = e^{α(t)},\,α(t)\sim \big|\log t\big|^{β+1}$, as well as $λ_2(t) = t^{-1}\cdot \big|\log t\big|^β$. The parameter $β>\frac32$ can be chosen arbitrarily. This shows that soliton resolution scenarios with finite time blow up and $N = 2$ collapsing profiles, i. e. bubble trees, do occur for this equation. |
| title | Concentric bubbles concentrating in finite time for the energy critical wave maps equation |
| topic | Analysis of PDEs 35L05, 35B40 |
| url | https://arxiv.org/abs/2501.08396 |