Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows

Abstract

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 $$u(r,t)\sim \frac{1}{{\lambda }_{\ell }(t)}\mathcal{Q}\left(\frac{r}{\sqrt{{\lambda }_{\ell }(t)}}\right),\text{ as }t\to T,$$ where $\mathcal{Q}$ is the ground state with boundary conditions $\mathcal{Q}(0)=-1,{\mathcal{Q}}^{\prime }(0)=0$ and the blowup speed ${\lambda }_{\ell }$ verifies $${\lambda }_{\ell }(t)=\left(C(u_0)+o_{t\to T}(1)\right)(T-t{)}^{\frac{2\ell }{\alpha }}\text{ as }t\to T,\ell \in {\mathbb{N}}_{+}^{\ast },\alpha >1.$$ 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.

arXiv comment88 pages