Stable blow-up solutions for the $SO(d)$-equivariant supercritical Yang-Mills heat flow
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866917935975497728 |
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| author | Yi, Yezhou |
| author_facet | Yi, Yezhou |
| contents | We consider the $SO(d)$-equivariant Yang-Mills heat flow
\begin{equation*}
\partial_t u-\partial_r^2 u-\frac{(d-3)}{r}\partial_r u+\frac{(d-2)}{r^2}u(1-u)(2-u)=0
\end{equation*} in dimensions $d>10.$ We construct a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration of a universal profile \begin{equation*}
u(t,r)\sim Q\left(\frac{r}{λ(t)}\right), \end{equation*}where $Q$ is a stationary state of the equation and the blow-up rates are quantized by \begin{equation*} λ(t)\sim c_{u}(T-t)^{\frac{l}γ},\,\,\,l\,\,\,\text{is any positive integer},\,\,\,γ=γ(d)=\frac{d-4-\sqrt{(d-6)^2-12}}{2}. \end{equation*} Moreover, such solutions are in fact $(l-1)$-codimension stable under pertubation of the initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_13325 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Stable blow-up solutions for the $SO(d)$-equivariant supercritical Yang-Mills heat flow Yi, Yezhou Analysis of PDEs 35B40, 35K15, 35K55 We consider the $SO(d)$-equivariant Yang-Mills heat flow \begin{equation*} \partial_t u-\partial_r^2 u-\frac{(d-3)}{r}\partial_r u+\frac{(d-2)}{r^2}u(1-u)(2-u)=0 \end{equation*} in dimensions $d>10.$ We construct a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration of a universal profile \begin{equation*} u(t,r)\sim Q\left(\frac{r}{λ(t)}\right), \end{equation*}where $Q$ is a stationary state of the equation and the blow-up rates are quantized by \begin{equation*} λ(t)\sim c_{u}(T-t)^{\frac{l}γ},\,\,\,l\,\,\,\text{is any positive integer},\,\,\,γ=γ(d)=\frac{d-4-\sqrt{(d-6)^2-12}}{2}. \end{equation*} Moreover, such solutions are in fact $(l-1)$-codimension stable under pertubation of the initial data. |
| title | Stable blow-up solutions for the $SO(d)$-equivariant supercritical Yang-Mills heat flow |
| topic | Analysis of PDEs 35B40, 35K15, 35K55 |
| url | https://arxiv.org/abs/2112.13325 |