Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows
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| Formato: | Preprint |
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2022
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| author | Bensouilah, A. Duong, G. K. Ghoul, T. E. |
| author_facet | Bensouilah, A. Duong, G. K. Ghoul, T. E. |
| contents | In this paper, we consider the Yang-Mills heat flow on $\mathbb R^d \times SO(d)$ with $d \ge 11$. Under a certain symmetry preserved by the flow, the Yang-Mills equation can be reduced to: $$ \partial_t u =\partial_r^2 u +\frac{d+1}{r} \partial_r u -3(d-2) u^2 - (d-2) r^2 u^3, \text{ and } (r,t) \in \mathbb R_+ \times \mathbb R_+. $$ We are interested in describing the singularity formation of this parabolic equation. We construct non-self-similar blowup solutions for $d \ge 11$ and prove that the asymptotic of the solution is of the form $$ u(r,t) \sim \frac{1}{λ_\ell(t)} \mathcal{Q} \left( \frac{r}{\sqrt{λ_\ell (t)}} \right), \text{ as } t \to T ,$$ where $\mathcal{Q}$ is the ground state with boundary conditions $\mathcal{Q}(0)=-1, \mathcal{Q}'(0)=0$ and the blowup speed $λ_\ell$ verifies $$λ_\ell (t) = \left( C(u_0) +o_{t\to T}(1) \right) (T-t)^{\frac{2\ell }α} \text{ as } t \to T,~~ \ell \in \mathbb{N}^*_+, ~~α>1.$$ In particular, when $\ell = 1$, this asymptotic is stable whereas for $ \ell \ge 2$ it becomes stable on a space of codimension $\ell-1$. Our approach here is not based on energy estimates but on a careful construction of time dependent eigenvectors and eigenvalues combined with maximum principle and semigroup pointwise estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_02297 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows Bensouilah, A. Duong, G. K. Ghoul, T. E. Analysis of PDEs Mathematical Physics In this paper, we consider the Yang-Mills heat flow on $\mathbb R^d \times SO(d)$ with $d \ge 11$. Under a certain symmetry preserved by the flow, the Yang-Mills equation can be reduced to: $$ \partial_t u =\partial_r^2 u +\frac{d+1}{r} \partial_r u -3(d-2) u^2 - (d-2) r^2 u^3, \text{ and } (r,t) \in \mathbb R_+ \times \mathbb R_+. $$ We are interested in describing the singularity formation of this parabolic equation. We construct non-self-similar blowup solutions for $d \ge 11$ and prove that the asymptotic of the solution is of the form $$ u(r,t) \sim \frac{1}{λ_\ell(t)} \mathcal{Q} \left( \frac{r}{\sqrt{λ_\ell (t)}} \right), \text{ as } t \to T ,$$ where $\mathcal{Q}$ is the ground state with boundary conditions $\mathcal{Q}(0)=-1, \mathcal{Q}'(0)=0$ and the blowup speed $λ_\ell$ verifies $$λ_\ell (t) = \left( C(u_0) +o_{t\to T}(1) \right) (T-t)^{\frac{2\ell }α} \text{ as } t \to T,~~ \ell \in \mathbb{N}^*_+, ~~α>1.$$ In particular, when $\ell = 1$, this asymptotic is stable whereas for $ \ell \ge 2$ it becomes stable on a space of codimension $\ell-1$. Our approach here is not based on energy estimates but on a careful construction of time dependent eigenvectors and eigenvalues combined with maximum principle and semigroup pointwise estimates. |
| title | Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2204.02297 |