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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.22825 |
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| _version_ | 1866908876944703488 |
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| author | Krieger, Joachim Palacios, José M. |
| author_facet | Krieger, Joachim Palacios, José M. |
| contents | We show that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ into $\mathbb{S}^2$, restricted to the $k=2$ co-rotational setting, admits arbitrarily large numbers of concentrating concentric $n$ bubble profiles. For any $n\in\mathbb{N}$, we construct an $n$-bubble solution concentrating at scales $λ_1(t)\gg λ_2(t)\gg \ldots\gg λ_n(t)$, where $λ_n(t)=t^{-1}\vert \log t\vert^β$, and $λ_j(t)\gtrsim \exp( \int_t^{t_0} λ_{j+1}(s)ds)$, for any $j<n$. Here $β>\tfrac32$ is a parameter that can be chosen arbitrarily. This shows that, as far as finite time blow-up case is concerned, the entirety of cases postulated in the soliton resolution theorem indeed occur, provided the concentric collapsing bubbles have alternating signs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22825 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Long finite time bubble trees for two co-rotational wave maps Krieger, Joachim Palacios, José M. Analysis of PDEs 35L05, 35B40, 35B44 We show that the energy critical Wave Maps equation from $\mathbb{R}^{2+1}$ into $\mathbb{S}^2$, restricted to the $k=2$ co-rotational setting, admits arbitrarily large numbers of concentrating concentric $n$ bubble profiles. For any $n\in\mathbb{N}$, we construct an $n$-bubble solution concentrating at scales $λ_1(t)\gg λ_2(t)\gg \ldots\gg λ_n(t)$, where $λ_n(t)=t^{-1}\vert \log t\vert^β$, and $λ_j(t)\gtrsim \exp( \int_t^{t_0} λ_{j+1}(s)ds)$, for any $j<n$. Here $β>\tfrac32$ is a parameter that can be chosen arbitrarily. This shows that, as far as finite time blow-up case is concerned, the entirety of cases postulated in the soliton resolution theorem indeed occur, provided the concentric collapsing bubbles have alternating signs. |
| title | Long finite time bubble trees for two co-rotational wave maps |
| topic | Analysis of PDEs 35L05, 35B40, 35B44 |
| url | https://arxiv.org/abs/2602.22825 |